# Class 11

## NCERT class 9 English Snapshot Chapter 6 summary The Ghat of the Only World (notes) explained (translation) in hindi

The Ghat of the Only World in English In this write-up, Amitav Ghosh pays glowing tribute to Agha Shahid Ali, a teacher and poet. Shahid was an expatriate from Kashmir. He moved to Pennysylvania in 1975 and after that he lived mainly in America. His brother was already there and they were later joined by …

## NCERT class 9 English Snapshot Chapter 8 summary The Tale of Melon City (notes) explained (translation) in hindi

The Tale of Melon City Summary In English Once there was a just and gentle king who ruled over a country. The king announced publicly that a curved structure should be built which should extend across the major public road in a victorious manner to benefit the beholders spiritually. Obeying the king’s orders, the workmen …

## NCERT class 9 English Snapshot Chapter 7 summary Birth (notes) explained (translation) in hindi

Birth Summary In English It was nearly midnight when Andrew Manson, the young doctor reached Bryngower. He found driller Joe Morgan waiting anxiously for him. Joe told Andrew that his wife, Susan, wanted his help and that too before time. Andrew went into his house, took his bag and left with Joe for number 12 …

## NCERT class 9 English Snapshot Chapter 5 summary Mother’s Day (notes) explained (translation) in hindi

Mother’s Day Summary In English ‘Mother’s Day’is a hilarious drawing room comedy by J.B. Priestley. It raises a serious issue and deals with it in a humorous manner. The comic undertone, however, does not belittle the importance of the issues raised in the play. The play centres round Mrs Annie Pearson, a devoted wife and …

## NCERT class 9 English Snapshot Chapter 4 summary Albert Einstein at School (notes) explained (translation) in hindi

Albert Einstein at School Summary In English Albert’s class was on. The History teacher, Mr. Braun, asked Albert when the Prussians defeated the French at Waterloo. Albert told him that he didn’t know. The teacher said that he had often told them. Albert said that he must have forgotten. This irritated the teacher. He asked …

## NCERT class 9 English Snapshot Chapter 3 summary Ranga’s Marriage (notes) explained (translation) in hindi

Ranga’s Marriage Summary In English The story ‘Ranga’s Marriage’ is located in Hosahalli, a village in the former Mysore state, now a part of Karnataka. Ten years ago, the village didn’t have many people who knew English. The village accountant’s son, Ranga was the first one to be sent to Bangalore, to study. A decade …

## NCERT class 9 English Snapshot Chapter 2 summary The Address (notes) explained (translation) in hindi

The Address Summary In English This is a moving story of a daughter who goes in search of her mother’s belongings after the War, in Holland. The narrator is the daughter of Mrs S., who died during the war. The narrator went to number 46, Marconi Street to see Mrs Dorling who was an old …

## NCERT class 9 English Snapshot Chapter 1 summary The Summer of the Beautiful White Horse (notes) explained (translation) in hindi

The Summer of the Beautiful White Horse Summary In English This story revolves around two poor Armenian boys Mourad and Aram. They are members of the Garoghlanian family. The hallmarks of their tribe are trust and honesty. The story begins in a mood of nostalgia. Aram, the narrator was then a boy of nine and …

## NCERT class 9 English Hornbill Chapter 5 summary Father to Son (notes) explained (translation) in hindi

Father to Son Summary In English This poem highlights a universal problem—the generation gap and the lack of communication between father and son. The poem begins with a father’s lament that he does not understand his child though they have been living together in the same house for so many years. He confesses that he …

## NCERT class 8 English Hornbill Chapter 5 summary Childhood (notes) explained (translation) in hindi

Childwood Summary In English The poet seems puzzled about the loss of childhood. It is natural in the process of growing up. Still, the poet tries to find an answer to his two queries: When did my childhood go?’ and Where did my childhood go?” The first possibility of the time of departure of his …

## NCERT class 11 English Hornbill Chapter 5 summary The Voice of the Rain (notes) explained (translation) in hindi

The Voice of the Rain Summary In English The poet asked the soft falling shower who it was. Strangely enough, the shower gave him an answer. The voice of the rain told him that it was the poem of earth. It is everlasting and perpetual. It is something that cannot be touched. It is born …

## NCERT class 11 English Hornbill Chapter 2 summary A Photograph (notes) explained (translation) in hindi

A Photograph Summary In English The photograph pasted on the cardboard shows two girl cousins Betty and Dolly who went paddling in the sea with the poetess mother. Each of them was holding one of the hands of the poet’s mother, who was a big girl of some twelve year or so at that time. …

## NCERT class 11 English Hornbill Chapter 7 summary The Browning Version (notes) explained (translation) in hindi

The Browning Version Summary In English This is an excerpt from Terence Rattigan’s play’i te Browning Version’. The scene is set in a good school. Taplow, a boy of sixteen has come in to do extra work for his master Mr. CrockerHarris. He has not yet arrived. Another master, Frank, younger in years than Mr. …

## NCERT class 11 English Hornbill Chapter 6 summary The Ailing Planet: the Green Movement’s Role (notes) explained (translation) in hindi

The Ailing Planet: the Green Movement’s Role Summary In English The Green Movement started nearly twenty-five years ago. The world’s first nationwide Green party was founded in New Zealand in 1972. Since then, the movement has not looked back. In fact, no other movement in world history has excited human race so much as the …

## NCERT class 11 English Hornbill Chapter 4 summary Discovering Tut: the Saga Continues (notes) explained (translation) in hindi

Discovering Tut: the Saga Continues Summary In English It was 6 p.m. on January 5, 2005 when the mummy of Tutankhamun moved smoothly and quietly into CT scanner which had been carried to Tut’s resting place. The aim was to probe the persisting medical mysteries of this young ruler who died more than 3,300 years …

## NCERT class 11 English Hornbill Chapter 3 summary We’re Not Afraid to Die…if We Can All Be Together (notes) explained (translation) in hindi

We’re Not Afraid to Die… if We Can All Be Together Summary In English In July 1976, the narrator, a 37 year-old businessman, his wife Mary, six year old son, Jonathan and seven year old daughter, Suzanne set sail from Plymouth, England. They wished to go round the world on a long sea journey as …

## NCERT class 11 English Hornbill Chapter 1 summary The Portrait of a Lady (notes) explained (translation) in hindi

The Portrait of a Lady Summary In English In ‘The Portrait of a Lady’, Khushwant Singh has given an account of his grandmother. He draws a life-like portrait. She was very old. Her face was wrinkled. Her hair was white. It was hard to believe that once she had been young and pretty. His grandfather’s …

## RD Sharma Solution CLass 12 Mathematics Chapter 6 Determinants

Chapter 6 – Determinants Exercise Ex. 6.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 Solution 3 Question 4 Solution 4 Question 5 Solution 5 Question 6 Solution 6 Question 7 Solution 7 Question 8 Solution 8 Question 9 Solution 9 Question 10 Solution 10 Question 11 Solution 11 Question 12 Solution 12 …

## RD Sharma Solution CLass 12 Mathematics Chapter 4 Inverse Trigonometric Functions

Chapter 4 – Inverse Trigonometric Functions Exercise Ex. 4.1 Question 1 Find the value of each of the following : Solution 1 Question 2 Find the value of each of the following : Solution 2 Question 3 Find the value of each of the following : Solution 3 Question 4 Find the value of each …

## RD Sharma Solution CLass 12 Mathematics Chapter 3 Binary Operations

Chapter 3 – Binary Operations Exercise Ex. 3.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 Solution 3 Question 4 Solution 4 Question 5 Solution 5 Question 6 Determine whether the following operation defines a binary operation on the Solution 6 We have, A ⊙ b = ab + ba for all a, …

## RD Sharma Solution CLass 12 Mathematics Chapter 2 Functions

Chapter 2 – Functions Exercise Ex. 2.1 Question 1 Give an example of function which is one-one but not onto. Solution 1 Question 2 Give an example of a function which is not one-one but onto. Solution 2 Question 3 Given an example of a function which is neither one-one nor onto. Solution 3 Question …

## RD Sharma Solution CLass 11 Mathematics Chapter 33 Probability

Chapter 33 – Probability Exercise Ex. 33.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 Solution 3 Question 4 Solution 4 Question 5 Solution 5 Question 6 Solution 6 Question 7 Solution 7 Question 8 Solution 8 Question 9 Solution 9 Question 10 An experiment consists of tossing a coin and then tossing …

## RD Sharma Solution CLass 11 Mathematics Chapter 32 Statistics

Chapter 32 – Statistics Exercise Ex. 32.1 Question 1 Calculate the mean deviation on the following income groups of five and seven members from their medians: I Income in Rs. II income in Rs. 4000 3800 4200 4000 4400 4200 4600 4400 4800 4600 4800 5800 Solution 1 Note: Answer given in the book is …

## RD Sharma Solution CLass 11 Mathematics Chapter 31 Mathematical Reasoning

Chapter 31 – Mathematical Reasoning Exercise Ex. 31.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 Solution 3 Question 4 Solution 4 Question 5 Solution 5 Question 6 Solution 6 Question 7 Solution 7 Question 8 Solution 8 Question 9 Solution 9 It is not a statement. The sentence “This sentence is a …

## RD Sharma Solution CLass 11 Mathematics Chapter 30 Derivatives

Chapter 30 – Derivatives Exercise Ex. 30.2 Question 1 Differentiate -x using first principles. Solution 1 Question 2 Differentiate (-x)-1 using first principles. Solution 2 Question 3 Differentiate sin(x + 1) using first principles. Solution 3 Question 4 Differentiate cos using first principles. Solution 4 Question 5 Solution 5 Question 6 Solution 6 Question 7 …

## RD Sharma Solution CLass 11 Mathematics Chapter 29 Limits

Chapter 29 – Limits Exercise Ex. 29.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 Solution 3   Question 4 Solution 4 Question 5 Solution 5 Question 6   Solution 6 Question 7 Solution 7 Question 8 Solution 8 Question 9 Solution 9 Question 10 Solution 10 Question 11 Solution 11 Question 12 …

## RD Sharma Solution CLass 11 Mathematics Chapter 28 Introduction to 3D Coordinate Geometry

Chapter 28 – Introduction to 3-D coordinate geometry Exercise Ex. 28.1 Question 1 Name the octants in which the following points lie:  (i) (5, 2, 3)           Solution 1 All are positive, so octant is XOYZ   Question 2 Name the octants in which the following points lie: (ii) (-5, 4, …

## RD Sharma Solution CLass 11 Mathematics Chapter 27 Hyperbola

Chapter 27 – Hyperbola Exercise Ex. 27.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 Solution 3 Question 4 Solution 4 Question 5 Solution 5 Question 6 Solution 6 Question 7 Solution 7 Question 8 Solution 8 Question 9 Solution 9 Question 10 Solution 10 Question 11 Solution 11 Question 12 Solution 12 …

## RD Sharma Solution CLass 11 Mathematics Chapter 26 Ellipse

Chapter 26 – Ellipse Exercise Ex. 26.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 Solution 3 Question 4 Solution 4 Question 5 Solution 5 Question 6 Solution 6 Question 7 Solution 7 Question 8 Solution 8 Question 9 Solution 9 Question 10 Find the eccentricity, coordinates of foci, length of the latus-rectum …

## RD Sharma Solution CLass 11 Mathematics Chapter 24 The Circle

Chapter 24 – The Circle Exercise Ex. 24.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 Solution 3 Question 4 Solution 4 Question 5 Solution 5 Question 6 Solution 6 Question 7 Solution 7 Question 8 Solution 8 Question 9 Solution 9 Question 10 Solution 10 Question 11 Solution 11 Question 12 Solution …

## RD Sharma Solution CLass 11 Mathematics Chapter 23 The Straight Lines

Chapter 23 – The Straight Lines Exercise Ex. 23.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 Solution 3 Question 4 Solution 4 Question 5 Solution 5 Question 6 Solution 6 Question 7 Solution 7 Question 8 Solution 8 Question 9 Solution 9 Question 10 Solution 10 Question 11 Solution 11 Question 12 …

## RD Sharma Solution CLass 11 Mathematics Chapter 22 Brief Review of Cartesian System of Rectangular Coordinates

Chapter 22 – Brief Review of Cartesian System of Rectangular Coordinates Exercise Ex. 22.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 Solution 3 Question 4 Solution 4 Question 5 Solution 5 Question 6 Solution 6 Question 7 Solution 7 Question 8 Solution 8 Chapter 22 – Brief Review of Cartesian System of …

## RD Sharma Solution Class 11 Mathematics Chapter 21 Some Special Series

Chapter 21 – Some Special Series Exercise Ex. 21.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 Solution 3 Question 4 Solution 4 Question 5 Solution 5 Question 6 Solution 6 Question 7 Solution 7 Question 8 Solution 8 Question 9 Solution 9 Question 10 Solution 10 Question 11 Solution 11 Question 12 …

## RD Sharma Solution Class 11 Mathematics Chapter 20 Geometric Progressions

Chapter 20 – Geometric Progressions Exercise Ex. 20.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 Solution 3 Question 4 Solution 4 Question 5 Solution 5 Question 6 Solution 6 Question 7 Solution 7 Question 8 Solution 8 Question 9 Solution 9 Question 10 Solution 10 Question 11 Solution 11 Question 12 Solution …

## RD Sharma Solution Class 11 Mathematics Chapter 19 Arithmetic Progressions

Chapter 19 – Arithmetic Progressions Exercise Ex. 19.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 Find the first four terns of the sequence defined by a1 = 3 and, an = 3an– 1 + 2, for all n > 1 Solution 3 Question 4 Solution 4 Question 5 Solution 5 Question 6 …

## RD Sharma Solution Class 11 Mathematics Chapter 18 Binomial Theorem

Chapter 18 – Binomial Theorem Exercise Ex. 18.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 Solution 3 Question 4 Solution 4 Question 5 Solution 5 Question 6 Solution 6 Question 7 Solution 7 Question 8 Solution 8 Question 9 Solution 9 Question 10 Solution 10 Question 11 Solution 11 Question 12 Solution …

## RD Sharma Solution Class 11 Mathematics Chapter 17 Combinations

Chapter 17 – Combinations Exercise Ex. 17.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 Solution 3 Question 4 Solution 4 Question 5 Solution 5 Question 6 Solution 6 Question 7 Solution 7 Question 8 Solution 8 Question 9 Solution 9 Question 10 Solution 10 Question 11 Solution 11 Question 12 Solution 12 …

## RD Sharma Solution Class 11 Mathematics Chapter 16 Permutations

Chapter 16 – Permutations Exercise Ex. 16.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 Solution 3 Question 4 Solution 4 Question 5 Solution 5 Question 6 Solution 6 Question 7 Solution 7 Question 8 Solution 8 Question 9 Solution 9 Question 10 Solution 10 Question 11 Solution 11 Question 12 Solution 12 …

## RD Sharma Solution Class 11 Mathematics Chapter 15 Linear Inequations

Chapter 15 – Linear Inequations Exercise Ex. 15.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 Solution 3 Question 4 Solution 4 Question 5 Solution 5 Question 6 Solution 6 Question 7 Solution 7 Question 8 Solution 8 Question 9 Solution 9 Question 10 Solution 10 Question 11 Solution 11 Question 12 Solution …

## RD Sharma Solution Class 11 Mathematics Chapter 14 Quadratic Equations

Chapter 14 – Quadratic Equations Exercise Ex. 14.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 Solution 3 Question 4 Solution 4 Question 5 Solution 5 Question 6 Solution 6 Question 7 Solution 7 Question 8 Solution 8 Question 9 Solution 9 Question 10 Solution 10 Question 11 Solution 11 Question 12 Solution …

## RD Sharma Solution Class 11 Mathematics Chapter 13 Complex Numbers

Chapter 13 – Complex Numbers Exercise Ex. 13.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 Solution 3 Question 4 Solution 4 Question 5 Solution 5 Question 6 Solution 6 Question 7 Solution 7 Question 8 Solution 8 Question 9 Solution 9 Question 10 Solution 10 Question 11 Solution 11 Question 12 Solution …

## RD Sharma Solution Class 11 Mathematics Chapter 12 Mathematical Induction

Chapter 12 – Mathematical Induction Exercise Ex. 12.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 Solution 3 Question 4 Solution 4 Question 5 Solution 5 Question 6 Solution 6 Chapter 12 – Mathematical Induction Exercise Ex. 12.2 Question 1 Solution 1 Question 2 Solution 2 Question 3 Solution 3 Question 4 Solution …

## RD Sharma Solution Class 11 Mathematics Chapter 11 Trigonometric Equations

Chapter 11 – Trigonometric Equations Exercise Ex. 11.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 Solution 3 Question 4 Solution 4 Question 5 Solution 5 Question 6 Solution 6 Question 7 Solution 7 Question 8 Solution 8 Question 9 Solution 9 Question 10 Solution 10 Question 11 Solution 11 Question 12 Solution …

## RD Sharma Solution Class 11 Mathematics Chapter 10 Sine and Cosine Formulae and Their Applications

Chapter 10 – Sine and Cosine Formulae and Their Applications Exercise Ex. 10.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 Solution 3 Question 4 Solution 4 Question 5 Solution 5 Question 6 Solution 6 Question 7 Solution 7 Question 8 Solution 8 Question 9 Solution 9 Question 10 Solution 10 Question 11 …

## RD Sharma Solution CLass 11 Mathematics Chapter 9 Trigonometric Ratios of Multiple and Submultiple Angles

Chapter 9 – Trigonometric Ratios of Multiple and Submultiple Angles Exercise Ex. 9.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 Solution 3 Question 4 Solution 4 Question 5 Solution 5 Question 6 Solution 6 Question 7 Solution 7 Question 8 Solution 8 Question 9 Solution 9 Question 10 Solution 10 Question 11 …

## RD Sharma Solution CLass 11 Mathematics Chapter 8 Transformation Formulae

Chapter 8 – Transformation Formulae Exercise Ex. 8.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 Solution 3 Question 4 Solution 4 Question 5 Solution 5 Question 6 Solution 6 Question 7 Solution 7 Question 8 Solution 8 Question 9 Solution 9 Question 10 Solution 10 Question 11 prove that ten 20o tan …

## RD Sharma Solution CLass 11 Mathematics Chapter 7 Trigonometric Ratios of Compound Product

Chapter 7 – Trigonometric Ratios of Compound Angles Exercise Ex.7.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 Solution 3 Question 4 Solution 4 Question 5 Solution 5 Question 6 Solution 6 Question 7 Solution 7 Question 8 Solution 8 Question 9 Solution 9 Question 10 Solution 10 Question 11 Solution 11 Question …

## RD Sharma Solution CLass 11 Mathematics Chapter 6 Graphs of Trigonometric Functions

Chapter 6 – Graphs of Trigonometric Functions Exercise Ex. 6.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 Solution 3 Question 4 Solution 4 Question 5 Solution 5 Question 6 Solution 6 Question 7 Sketch the graph of the following pairs of functions on the same axes: y = sin x, y = …

## RD Sharma Solution CLass 11 Mathematics Chapter 5 Trigonometric Functions

Chapter 5 – Trigonometric Functions Exercise Ex. 5.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 Solution 3 Question 4 Solution 4 Question 5 Solution 5 Question 6 Solution 6 Question 7 Solution 7 Question 8 Solution 8 Question 9 Solution 9 Question 10 Solution 10 Question 11 Solution 11 Question 12 Solution …

## RD Sharma Solution CLass 11 Mathematics Chapter 4 Measurement of Angle

Chapter 4 – Measurement of Angles Exercise Ex. 4.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 Solution 3 Question 4 Solution 4 Question 5 Solution 5 Question 6 Solution 6 Question 7 Solution 7 Question 8 Solution 8 Question 9 Solution 9 Question 10 Solution 10 Question 11 Solution 11 Question 12 …

## RD Sharma Solution CLass 11 Mathematics Chapter 3 Functions

Chapter 3 – Functions Exercise Ex. 3.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 Solution 3 Question 4 Solution 4 Question 5 Solution 5 Question 6 Solution 6 Question 7 Solution 7 Question 8 Solution 8 Question 9 Solution 9 Question 10 Solution 10 Question 11 Solution 11 Question 12 Solution 12 …

## RD Sharma Solution CLass 11 Mathematics Chapter 2 Relations

Chapter 2 – Relations Exercise Ex. 2.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 Solution 3 Question 4 Solution 4 Question 5 Solution 5 Question 6 Solution 6 Question 7 Solution 7 Question 8 Solution 8 Question 9 Solution 9 Question 10 Solution 10 Question 11 Solution 11 Question 12 Solution 12 …

## RD Sharma Solution Class 11 Mathematics Chapter 1 Sets

Chapter 1 – Sets Exercise Ex. 1.1 Question 1 Solution 1 Question 2 Solution 2 Question 3 If A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then insert the appropriate symbol or in each of the following blank spaces: 4…A -4 …A 12 ….A 9 …A 0 …..A -12 ….A …

## NCERT solution class 11 English chapter 6 The Ghat of the Only World

Page No: 64 Reading with Insight 1. What impressions of Shahid do you gather from the piece? Answer Shahid Ali was a multi faceted personalityand appears to be sensitive soul. He was born in Srinagar and had studied in Delhi. Later, he migrated to America and served in various colleges and universities. Shahid was a …

## NCERT solution class 11 English chapter 8 The Tale of Melon City

Page No: 76 Reading with Insight 1. Narrate ‘The Tale of Melon City’ in your own words. Answer ‘The Tale of Melon City’ runs like a folk tale. The city is called Melon City because its ruler is a melon. There is a curious tale about it. Once a fair and easygoing king ruled over a …

## NCERT solution class 11 English chapter 7 Birth

Page No: 70 1. “I have done something; oh, God! I’ve done something real at last.” Why does Andrew say this? What does it mean? Answer   Andrew, the protagonist of the story Birth, utters these words as he is able to bring a still born child back to life which seemed impossible in the beginning. …

## NCERT solution class 11 English chapter 5 Mother’s Day

Page No: 52 Reading with Insight 1.This play, written in the 1950s, is a humorous and satirical depiction of the status of the mother in the family. (i) What are the issues it raises? (ii) Do you think it caricatures these issues or do you think that the problems it raises are genuine? How does the …

## NCERT solution class 11 English chapter 4 Albert Einstein at School

Page No: 31 Reading with Insight 1. What do you understand of Einstein’s nature from his conversations with his history teacher, his mathematics teacher and the head teacher? Answer   Einstein’s behavior seemed to be extremely unruly. He didn’t believe in the then prevailing system of education. His nature was a spontaneous one. He found memorising …

## NCERT solution class 11 English chapter 3 Ranga’s Marriage

Page No: 24 Reading with Insight 1. Comment on the influence of English – the language and the way of life – on Indian life as reflected in the story. What is the narrator’s attitude to English? Answer The story ‘Ranga’s Marriage’ is set in a village Hosahalli, which was in the erstwhile Mysore state. In …

## NCERT solution class 11 English chapter 2 The Address

Page No: 15 Reading with Insight 1. ‘Have you come back?’ said the woman.’I thought that no one had come back.’ Does this statement give some clue about the story? If yes, what is it? Answer Yes, these words by Mrs Dorling to the narrator shows that she least expected such a visit. She had …

## NCERT solution class 11 English chapter 1 The Summer of the Beautiful White Horse

Page No: 8 Reading with Insight 1. You will probably agree that this story does not have breathless adventure and exciting action. Then what in your opinion makes it interesting? Answer It is true that though the story “The Summer of the Beautiful White Horse” has neither any breathless adventure nor any exciting action, yet it …

## NCERT solution class 11 English chapter 27 Bridges

Page No: 196 Understanding the Text 1. How did the author feel about her mother’s passion to make her a dancer? Answer The author did not feel comfortable about her mother’s passion to make her a dancer. The author tells us that her mother discovered in her the innate ability to dance. This motivated the author’s …

## NCERT solution class 11 English chapter 26 The Story

Page No: 186 Understanding the Text 1. What do you understand of the three voices in response to the question ‘What does a novel do’ ? Answer The three voices stand for three different types of readers of novels. The first voice represents a good tempered person who is engaged in some other activity but is …

## NCERT solution class 11 English chapter 25 What is a Good Book?

Page No: 180 Understanding the Text 1. What, according to Ruskin, are the limitations of the good book of the hour? Answer According to Ruskin, the limitations of the good book of the hour are that these are not true books but merely letters or newspapers in good print. The good books of the hour …

## NCERT solution class 11 English chapter 24 Tribal Verse

Page No: 172 Understanding the Text 1. Identify the common characteristics shared by tribal communities all over the world. Answer The essayist identifies some common characteristics shared by tribal communities all over the world. The tribals live in groups that are cohesive and organically unified. They show very little interest in accumulating wealth or in using …

## NCERT solution class 11 English chapter 23 Patterns of Creativity

Page No: 158 Understanding the Text 1. How does Shelley’s attitude to science differ from that of Wordsworth and Keats? Answer Wordsworth in his A Poet’s Epitaph looks at science with a critical mind. Even in Tables turned he praises nature and appreciates the beauty it bequeaths to the humanity and is critical of how humans …

## NCERT solution class 11 English chapter 22 My Three Passions

Page No: 151 Understanding the Text 1. Why does Russell call the three passions ‘simple’? Answer The essay actually is the preface to Bertrand Russell’s autobiography. Every human is driven by a force, a passion all her/his life. It keeps her/him going. Some desire money, other, fame. There are some who desire simple satisfaction. Bertie’s desires …

## NCERT solution class 11 English chapter 21 My Watch

Page No: 148 Understanding the Text 1. What was the importance of the watch to the author? Answer The watch was important to the author as it showed him the correct time thus keeping him punctual. He had it working properly for 18 months until he let it run down. He had staunch faith on its …

## NCERT solution class 11 English chapter 20 Ajamil and the Tigers

Page No: 142 Understanding the Poem 1. The poem has a literal level and a figurative level. Why has the poet chosen ‘tigers’ and ‘sheep’ to convey his message? Answer Though literally the poem talks of a short story in which a shepherd made a concord with a king of tigers to maintain peace among the …

## NCERT solution class 11 English chapter 19 Ode to a Nightingale

Page No: 138 Understanding the Poem 1. How does the nightingale’s song plunge the poet into a state of ecstasy? Answer When Keats was sitting under a plum tree in the garden of his house, in Hampstead, he composed this poem. He was inspired by Nightingale’s song and completed the poem within one day. The poet …

## NCERT solution class 11 English chapter 18 Felling of the Banyan Tree

Page No: 133 Understanding the Poem 1. Identify the lines that reveal the critical tone of the poet towards the felling of the tree. Answer   There are many expressions in the poem that reveal the critical tone of the poet towards the felling of the tree: “Its scraggy aerial roots fell to the ground” …

## NCERT solution class 11 English chapter 17 Refugee Blues

Page No: 131 Understanding the Poem 1. The title, ‘Refugee Blues’ encapsulates the theme of the poem. Comment. Answer   Blues are African-American form of ballads that originated by the end of 19th century. They primarily are melancholic and sad and can be called as “the sad black slave’s songs”. In this Blue Ballad, Auden …

## NCERT solution class 11 English chapter 16 For Elkana

Page No: 126 Understanding the Poem 1. Comment on the subtlety with which the poet captures the general pattern of communication within a family. Answer Nissim in his poem For Elkana has portrayed a common scene of any Indian home. On an April evening, strolling in the porch, the wife tries to discuss certain issues that …

## NCERT solution class 11 English chapter 15 Hawk Roosting

Page No: 122 Understanding the Poem 1. Comment on the physical features of the hawk highlighted in the poem and their significance. Answer Hawk Roosting signifies self-esteem or self-assertion of a Hawk that is so alienated from the human world. The poem is a dramatic monologue in a non-human voice; i.e., of the Hawk, who carries …

## NCERT solution class 11 English chapter 14 Mother Tongue

Page No: 120 Understanding the Poem 1. The quill is the central element of the poem – what does it symbolise? Answer The quill symbolises the Sharade script. A script is central to propagate and preserve any language. In this case the poet is eager to make a point for her mother tongue Dogri which was …

## NCERT solution class 11 English chapter 12 Telephone Conversation

Page No: 115 Word Meanings Notice these expressions in the poem and guess their meaning from the context: rancid breath squelching tar spectroscopic flight of fancy rearing on the thunderclap brunette peroxide blonde clinical assent raven black Answer rancid breath: Rancid means a matter which is offensive or disagreeable. Thus, the voice in which the …

## NCERT solution class 11 English chapter 11 Coming

Page No: 112 Understanding the Poem   1. What does the bird in the poem announce? How is this related to the title, ‘Coming’? Answer The poem ‘Coming’ by Philip Larkin is a celebration ofthe advent of the spring. To express the happiness the poet sets the plot of house fronts clothedin chilly and yellow …

## NCERT solution class 11 English chapter 10 Let Me Not to the Marriage of True Minds

Page No: 110 Word Meanings The following two common words are used in a different sense in the poem. Guess what they mean bark compass Answer Bark: Ship Compass: Time range Understanding the Poem 1. ‘Constancy’ is the theme of the poem. Indicate the words, phrases and images that suggest the theme. Answer Constancy is …

## NCERT solution class 11 English chapter 9 The Peacock

Page No: 108 Word Meanings   Notice these words in the poem and guess their meaning from the context turquoise                darts Answer Turquoise: A greenish-blue colour Dart: To move suddenly and quickly in a particular direction Understanding the Poem 1. Comment on the lines that make you visualise the colourful image of the peacock. Answer …

## NCERT solution class 11 English chapter 8 The Luncheon

Page No 104: Question 1: Although the author was not a vindictive man he was very happy to see the twenty one stone lady who had impoverished him twenty years ago, and says he had finally had his revenge. What makes him say this? Answer: The story Luncheon relates incidents, replete with humour and irony, …

## NCERT solution class 11 English chapter 7 Glory at Twilight

Page No 85: Question 1: Look for these expressions in the story and guess the meaning from the context brusquely attuned himself queer rhythmic frenzy wrenching flush of prosperity daze of bewilderment wide-eyed wonder and eager homage talking animatedly tremulous deliberation on terms of a perpetual feud   Answer: Brusquely: It is an adverb which means …

## NCERT solution class 11 English chapter 6 The Third and Final Continent

Page No: 83 1. Indicate the details that tell us that the narrator was not very financially comfortable during his stay in London. Answer The following details depict a very uncomfortable picture of life, faced by financial crisis: The author travelled in a third class cabin. The author talks about ‘struggling’ to earn a livelihood and …

## NCERT solution class 11 English chapter 5 Pappachi’s Moth

Page No 60: Question 1: Comment on the relationship shared by Mammachi and Pappachi. Answer: Pappachi and Mammachi had a gap of 17 years. A retired high ranked officer, Pappachi was always jealous of Mammach’s talent and of the attention she received. Whether it was pickle making or her violin classes in Vienna, Pappachi was …

## NCERT solution class 11 English chapter 4 The Adventure of the Three Garridebs

Page No 54: Question 1: What clues did Sherlock Holmes work upon to get at the fact that the story of the three Garridebs was a ruse? Answer: The first time Holmes felt wry of the story when the American Garrideb was angry at Nathan to have involved a detective. When Sherlock noticed the Garrideb …

## NCERT solution class 11 English chapter 3 The Rocking-horse Winner

Page No 35: Question 1: What was the reason for young Paul’s restlessness at the beginning of the story? How did it find expression? Answer: Paul, the first born of Hester, desired to be lucky for his mother’s sake. He desired her affection and wanted her not to worry. The mother however, considered her husband …

## NCERT solution class 11 English chapter 2 A Pair of Mustachios

Page No 17: Question 1: What do you understand of the natures of Ramanand and Azam Khan from the episode described? Answer: Seth Ramanand was a man whose every move was a calculated one. He was not one from a rich background, though he built up his business on the maxim that the customer is …

## NCERT solution class 11 English chapter 1 The Lament

Page No 8: Question 1: Comment on the indifference that meets Iona’s attempts to share his grief with his fellow human beings. Answer: Iona Potapov had lost his son, who died a week before. He wished to share his suffering and his emotions and grieve at his loss. However,  the people he came across, whether …

## NCERT solution class 11 English chapter 15 Creative Writting

Page No 118: Question 1: Activity I Put down the images that come to your mind immediately when you see the words in the box.       Cat          cupboard           wall           pond            bird Answer: Cat: rat, milk, fur Cupboard: Clothes, books Wall: Clock, paint, colours Pond: ducks, water, lotus Bird: tree, nest, sky Question 2: …

## NCERT solution class 11 English chapter 14 Letter Writing

Page No 114: Question 1: You have not received your Roll number card for the Class XII examination. Write a letter to the Registrar Examination Branch CBSE asking for it. Answer: Rakesh Sharma Class: XII, Sec. B St. Thomas School Delhi February 20, 2012 The Registrar Examination Branch CBSE Delhi Dear Sir/Madam, Subject: Inquiry about …

## NCERT solution class 11 English chapter 13 Essay Writing

Page No 106: Question 1: Himalayan Quake 2005 Answer: The mesmerising beauty of the Himalayan mountain range was put to question on 8th of October 2005, when a massive earthquake rocked some parts of the Himalayan region. The earthquake, measuring 7.6 on the Richter scale, was devastating. According to the seismologists, the epicentre of the …

## NCERT solution class 11 English chapter 12 Sub-titling

Page No 101: Question 1: Notice the italicized sentence placed at the top of the article which tells us at a glance what the article is about. Answer: (For this, students just need to read the italicized sentence at the top of the article and notice how it puts forth the main idea of the …

## NCERT solution class 11 English chapter 11 Summarising

Page No 96: Question 1: Read the text below and summarise it. Green Sahara The Great Desert Where Hippos Once Wallowed The Sahara sets a standard for dry land. It’s the world’s largest desert. Relative humidity can drop into the low single digits. There are places where it rains only about once a century. There …

## NCERT solution class 11 English chapter 10 Note-making

Page No 93: Question 1: Underline the important words and phrases. Answer: The energy stored in coal and petroleum originally came to the earth from the sun. The bulk of the present-day supplies was laid down some 200 to 600 million years ago, when tropical conditions were widespread. Lush, swampy forests produced huge trees; warm coastal seas swarmed with …

## NCERT solution class 11 English chapter 9 Father to Son

Page No 86: Question 1: Does the poem talk of an exclusively personal experience or is it fairly universal? Answer: Though the poem seems to talk exclusively of the personal experiences of a father, it is fairly universal in the problems that it addresses from the father’s perspective. At the surface, the poem communicates the …

## NCERT solution class 11 English chapter 8 Childhood

Page No 59: Question 1: Identify the stanzas that talks of each of the following.    Individuality                    rationalism                                      hypocrisy   Answer: Individuality- Third stanza Rationalism- First stanza Hypocrisy- Second stanza Question 2: What according to the poem is involved in the process of growing up? Answer: According to the poem, the process of growing up involves …

## NCERT solution class 11 English chapter 7 The Browning Version

Page No 50: Question 1: Notice these expressions in the text. Infer their meaning from the context. Remove slackers muck kept in got carried away cut sadist shrivelled up Answer: emove: a division in a school slackers: unmotivated and lazy students muck: useless, of no practical good kept in: grounded, detained, work after the official …

## NCERT solution class 11 English chapter 6 The Ailing Planet: the Green Movement’s Role

Page No 43: Question 1: Notice these expressions in the text. Infer their meaning from the context. a holistic and ecological view  inter alia sustainable development decimated languish  catastrophic depletion ignominious darkness transcending concern Answer: a holistic and ecological view – It refers to the view that calls for the preservation of the planet. The holistic …

## NCERT solution class 11 English chapter 5 The Voice of the Rain

Page No 42: Question 1: 1. There are two voices in the poem. Who do they belong to? Which lines indicate this? 2. What does the phrase “strange to tell” mean? 3.  There is a parallel drawn between rain and music. Which words indicate this? Explain the similarity between the two. 4. How is the cyclic movement of rain …

## NCERT solution class 11 English chapter 4 Discovering Tut: the Saga Continues

Page No 22: Question 1: Notice these expressions in the text. Infer their meaning from the context. forensic reconstruction scudded across casket grey Resurrection funerary treasures Circumvented computed tomography eerie detail Answer: Forensic reconstruction– It refers to the process of creating a face on the skull and see how the owner of the skull looked …

## NCERT solution class 11 English chapter 3 We’re Not Afraid to Die…if We Can All Be Together

Page No 13: Question 1: Notice these expressions in the text. Infer their meaning from the context. honing our seafaring skills ominous silence Mayday calls pinpricks in the vast ocean a tousled head Answer: honing our seafaring skills: this refers to the efforts made by the author and his wife, to perfect or sharpen their seafaring …

## NCERT solution class 11 English chapter 2 A Photograph

Page No 11: Question 1: Infer the meaning of the following words from the context.    Padding                                         transient Now look up the dictionary to see if your inference is right. Answer: Paddling: To move a boat by means of paddles. Transient: Something that stays at a place for a short time-period. Page No 12: Question …

## NCERT solution class 11 English chapter 1 The Portrait of a Lady

Page No 3: Question 1: Notice these expressions in the text. Infer their meaning from the context. the thought was almost revolting an expanse of pure white serenity a turning-point accepted her seclusion with resignation a veritable bedlam of chirrupings frivolous rebukes the sagging skins of the dilapidated drum Answer: the thought was almost revolting –  The …

## NCERT solution class 12 science chemistry part 1 chapter 3 Electrochemistry

Page No 68: Question 3.1: How would you determine the standard electrode potential of the systemMg2+ | Mg? Answer: The standard electrode potential of Mg2+ | Mg can be measured with respect to the standard hydrogen electrode, represented by Pt(s), H2(g) (1 atm) | H+(aq)(1 M). A cell, consisting of Mg | MgSO4 (aq 1 M) as the anode and …

## NCERT solution class 11 science biology chapter 22 Chemical Coordination and Integration

Page No 341: Question 1: Define the following: (a) Exocrine gland (b) Endocrine gland (c) Hormone Answer: (a) Exocrine glands: Glands that discharge secretions into ducts are known as exocrine glands. Sebaceous gland in the skin, salivary gland in the buccal cavity, etc. are examples of exocrine glands. (b) Endocrine glands: Glands that do not discharge their secretions into ducts …

## NCERT solution class 11 science biology chapter 21 Neural Control and Coordination

Page No 328: Question 1: Briefly describe the structure of the following: Brain (b) Eye (c) Ear Answer: (A) Brain: Brain is the main coordinating centre of the body. It is a part of nervous system that controls and monitors every organ of the body. It is well protected by cranial meninges that are made up of …

## NCERT solution class 11 science biology chapter 20 Locomotion and Movement

Page No 313: Question 1: Draw the diagram of a sarcomere of skeletal muscle showing different regions. Answer: The diagrammatic representation of a sarcomere is as follows: Question 2: Define sliding filament theory of muscle contraction. Answer: The sliding filament theory explains the process of muscle contraction during which the thin filaments slide over the …

## NCERT solution class 11 science biology chapter 19 Excretory Products and their Elimination

Page No 300: Question 1: Define Glomerular Filtration Rate (GFR) Answer: Glomerular filtration rate is the amount of glomerular filtrate formed in all the nephrons of both the kidneys per minute. In a healthy individual, it is about 125 mL/minute. Glomerular filtrate contains glucose, amino acids, sodium, potassium, urea, uric acid, ketone bodies, and large …

## NCERT solution class 11 science biology chapter 18 Body Fluids and Circulation

Page No 289: Question 1: Name the components of the formed elements in the blood and mention one major function of each of them. Answer: The component elements in the blood are: (1) Erythrocytes: They are the most abundant cells and contain the red pigment called haemoglobin. They carry oxygen to all parts of the …

## NCERT solution class 11 science biology chapter 17 Breathing and Exchange of Gases

Page No 277: Question 1: Define vital capacity. What is its significance? Answer: Vital capacity is the maximum volume of air that can be exhaled after a maximum inspiration. It is about 3.5 – 4.5 litres in the human body. It promotes the act of supplying fresh air and getting rid of foul air, thereby …

## NCERT solution class 11 science biology chapter 16 Digestion and Absorption

Page No 267: Question 1: Choose the correct answer among the following: (a) Gastric juice contains (i) pepsin, lipase and rennin (ii) trypsin lipase and rennin (iii) trypsin, pepsin and lipase (iv) trypsin, pepsin and renin (b) Succus entericus is the name given to (i) a junction between ileum and large intestine (ii) intestinal juice …

## NCERT solution class 11 science biology chapter 15 Plant Growth and Development

Page No 253: Question 1: Define growth, differentiation, development, dedifferentiation, redifferentiation, determinate growth, meristem and growth rate. Answer: (a) Growth It is an irreversible and permanent process, accomplished by an increase in the size of an organ or organ parts or even of an individual cell. (b) Differentiation It is a process in which the …

## NCERT solution class 11 science biology chapter 14 Respiration in Plants

Page No 238: Question 1: Differentiate between (a) Respiration and Combustion (b) Glycolysis and Krebs’ cycle (c) Aerobic respiration and Fermentation Answer: (a) Respiration and combustion Respiration Combustion 1. It is a biochemical process. 1. It is a physiochemical process. 2. It occurs in the living cells. 2. It does not occur in the living cells. …

## NCERT solution class 11 science biology chapter 13 Photosynthesis in Higher Plants

Page No 224: Question 1: By looking at a plant externally can you tell whether a plant is C3 or C4? Why and how? Answer: One cannot distinguish whether a plant is C3 or C4 by observing its leaves and other morphological features externally. Unlike C3 plants, the leaves of C4 plants have a special anatomy called Kranz anatomy and …

## NCERT solution class 11 science biology chapter 12 Mineral Nutrition

Page No 205: Question 1: ‘All elements that are present in a plant need not be essential to its survival’. Comment. Answer: Plants tend to absorb different kinds of nutrients from soil. However, a nutrient is inessential for a plant if it is not involved in the plant’s physiology and metabolism. For example, plants growing …

## NCERT solution class 11 science biology chapter 11 Transport in Plants

Page No 193: Question 1: What are the factors affecting the rate of diffusion? Answer: Diffusion is the passive movement of substances from a region of higher concentration to a region of lower concentration. Diffusion of substances plays an important role in cellular transport in plants. Rate of diffusion is affected by concentration gradient, membrane …

## NCERT solution class 11 science biology chapter 10 Cell Cycle and Cell Division

Page No 171: Question 1: What is the average cell cycle span for a mammalian cell? Answer: The average cell cycle span for a mammalian cell is approximately 24 hours. Question 2: Distinguish cytokinesis from karyokinesis. Answer: Cytokinesis  Karyokinesis  (i) Cytokinesis is the biological process involving the division of a cellâ€™s cytoplasm during mitosis or …

## NCERT solution class 11 science biology chapter 9 Biomolecules

Page No 160: Question 1: What are macromolecules? Give examples. Answer: Macromolecules are large complex molecules that occur in colloidal state in intercellular fluid. They are formed by the polymerization of low molecular weight micromolecules. Polysaccharides, proteins, and nucleic acids are common examples of macromolecules. Question 2: Illustrate a glycosidic, peptide and a phospho-diester bond. …

## NCERT solution class 11 science biology chapter 8 Cell: The Unit of Life

Page No 141: Question 1: Which of the following is not correct? (a) Robert Brown discovered the cell. (b) Schleiden and Schwann formulated the cell theory. (c) Virchow explained that cells are formed from pre-existing cells. (d) A unicellular organism carries out its life activities within a single cell. Answer: (a) Robert Brown did not discover …

## NCERT solution class 11 science biology chapter 7 Structural Organisation in Animals

Page No 122: Question 2: Answer the following: (i) What is the function of nephridia? (ii) How many types of nephridia are found in earthworm based on their location? Answer: (i) Nephridia are segmentally arranged excretory organs present in earthworms. (ii) On the basis of their location, three types of nephridia are found in earthworms. They are: (a) Septal nephridia: These …

## NCERT solution class 11 science biology chapter 6 Anatomy of Flowering Plants

Page No 99: Question 1: State the location and function of different types of meristem. Answer: Meristems are specialised regions of plant growth. The meristems mark the regions where active cell division and rapid division of cells take place. Meristems are of three types depending on their location. Apical meristem It is present at the …

## NCERT solution class 11 science biology chapter 5 Morphology of Flowering Plants

Page No 82: Question 1: What is meant by modification of root? What type of modification of root is found in the (a) Banyan tree (b) Turnip (c) Mangrove trees Answer: Primarily, there are two types of root systems found in plants, namely the tap root system and fibrous root system. The main function of the roots is …

## NCERT solution class 11 science biology chapter 4 Animal Kingdom

Page No 62: Question 1: What are the difficulties that you would face in classification of animals, if common fundamental features are not taken into account? Answer: For the classification of living organisms, common fundamental characteristics are considered. If we consider specific characteristics, then each organism will be placed in a separate group and the …

## NCERT solution class 11 science biology chapter 3 Plant Kingdom

Page No 44: Question 1: What is the basis of classification of algae? Answer: Algae are classified into three main classes – Chlorophyceae, Phaeophyceae, and Rhodophyceae. These divisions are based on the following factors: (a) Major photosynthetic pigments present (b) Form of stored food (c) Cell wall composition (d) Number of flagella and position of …

## NCERT solution class 11 science biology chapter 2 Biological Classification

Page No 28: Question 1: Discuss how classification systems have undergone several changes over a period of time? Answer: The classification systems have undergone several changes with time. The first attempt of classification was made by Aristotle. He classified plants as herbs, shrubs, and trees. Animals, on the other hand, were classified on the basis …

## NCERT solution class 11 science biology chapter 1 The Living World

Page No 15: Question 1: Why are living organisms classified? Answer: A large variety of plants, animals, and microbes are found on earth. All these living organisms differ in size, shape, colour, habitat, and many other characteristics. As there are millions of living organisms on earth, studying each of them is impossible. Therefore, scientists have …

## NCERT solution class 11 science chemistry part 2 chapter 7 Environmental Chemistry

Page No 413: Question 14.1: Define environmental chemistry. Answer: Environmental chemistry is the study of chemical and biochemical processes occurring in nature. It deals with the study of origin, transport, reaction, effects, and fates of various chemical species in the environment. Question 14.2: Explain tropospheric pollution in 100 words. Answer: Tropospheric pollution arises due to …

## NCERT solution class 11 science chemistry part 2 chapter 6 Hydrocarbons

Page No 396: Question 13.1: How do you account for the formation of ethane during chlorination of methane? Answer: Chlorination of methane proceeds via a free radical chain mechanism. The whole reaction takes place in the given three steps. Step 1: Initiation: The reaction begins with the homolytic cleavage of Cl – Cl bond as: Step …

## NCERT solution class 11 science chemistry part 2 chapter 5 Organic Chemistry: Some Basic Principles and Techniques

Page No 361: Question 12.1: What are hybridisation states of each carbon atom in the following compounds? CH2=C=O, CH3CH=CH2, (CH3)2CO, CH2=CHCN, C6H6 Answer: (i)  C–1 is sp2 hybridised. C–2 is sp hybridised. (ii)  C–1 is sp3 hybridised. C–2 is sp2 hybridised. C–3 is sp2 hybridised. (iii) C–1 and C–3 are sp3 hybridised. C–2 is sp2 hybridised. (iv)  C–1 is sp2 hybridised. C–2 is sp2 hybridised. C–3 is sp hybridised. (v) C6H6 All the 6 carbon atoms …

## NCERT solution class 11 science chemistry part 2 chapter 4 The p-Block Elements

Page No 323: Question 11.1: Discuss the pattern of variation in the oxidation states of (i) B to Tl and (ii) C to Pb. Answer: (i) B to Tl The electric configuration of group 13 elements is ns2 np1. Therefore, the most common oxidation state exhibited by them should be +3. However, it is only boron and aluminium …

## NCERT solution class 11 science chemistry part 2 chapter 3 The s-Block Elements

Page No 305: Question 10.1: What are the common physical and chemical features of alkali metals? Answer: Physical properties of alkali metals are as follows. (1) They are quite soft and can be cut easily. Sodium metal can be easily cut using a knife. (2) They are light coloured and are mostly silvery white in appearance. (3) They …

## NCERT solution class 11 science chemistry part 2 chapter 2 Hydrogen

Page No 288: Question 9.1: Justify the position of hydrogen in the periodic table on the basis of its electronic configuration. Answer: Hydrogen is the first element of the periodic table. Its electronic configuration is [1s1]. Due to the presence of only one electron in its 1s shell, hydrogen exhibits a dual behaviour, i.e., it resembles …

## NCERT solution class 11 science chemistry part 2 chapter 1 Redox Reactions

Page No 272: Question 8.1: Assign oxidation numbers to the underlined elements in each of the following species: (a) NaH2PO4 (b) NaHSO4 (c) H4P2O7 (d) K2MnO4 (e) CaO2 (f) NaBH4 (g) H2S2O7 (h) KAl(SO4)2.12 H2O Answer: (a)  Let the oxidation number of P be x. We know that, Oxidation number of Na = +1 Oxidation number of H = +1 Oxidation number …

## NCERT solution class 11 science chemistry part 1 chapter 7 Equilibrium

Page No 224: Question 7.1: A liquid is in equilibrium with its vapour in a sealed container at a fixed temperature. The volume of the container is suddenly increased. a) What is the initial effect of the change on vapour pressure? b) How do rates of evaporation and condensation change initially? c) What happens when …

## NCERT solution class 11 science chemistry part 1 chapter 6 Chemical Thermodynamics

Page No 182: Question 6.1: Choose the correct answer. A thermodynamic state function is a quantity (i) used to determine heat changes (ii) whose value is independent of path (iii) used to determine pressure volume work (iv) whose value depends on temperature only. Answer: A thermodynamic state function is a quantity whose value is independent …

## NCERT solution class 11 science chemistry part 1 chapter 5 States of Matter

Page No 152: Question 5.1: What will be the minimum pressure required to compress 500 dm3 of air at 1 bar to 200 dm3 at 30°C? Answer: Given, Initial pressure, p1 = 1 bar Initial volume, V1 = 500 dm3 Final volume, V2 = 200 dm3 Since the temperature remains constant, the final pressure (p2) can be calculated using Boyle’s law. According to …

## NCERT solution class 11 science chemistry part 1 chapter 4 Chemical Bonding and Molecular Structure

Page No 129: Question 4.1: Explain the formation of a chemical bond. Answer: A chemical bond is defined as an attractive force that holds the constituents (atoms, ions etc.) together in a chemical species. Various theories have been suggested for the formation of chemical bonds such as the electronic theory, valence shell electron pair repulsion …

## NCERT solution class 11 science chemistry part 1 chapter 3 Classification of Elements and Periodicity in Properties

Page No 92: Question 3.1: What is the basic theme of organisation in the periodic table? Answer: The basic theme of organisation of elements in the periodic table is to classify the elements in periods and groups according to their properties. This arrangement makes the study of elements and their compounds simple and systematic. In …

## NCERT solution class 11 science chemistry part 1 chapter 2 Structure of Atom

Page No 65: Question 2.1: (i) Calculate the number of electrons which will together weigh one gram. (ii) Calculate the mass and charge of one mole of electrons. Answer: (i) Mass of one electron = 9.10939 × 10–31 kg Number of electrons that weigh 9.10939 × 10–31 kg = 1 Number of electrons that will weigh 1 …

## NCERT solution class 11 science chemistry part 1 chapter 1 Some Basic Concepts of Chemistry

Page No 22: Question 1.1: Calculate the molecular mass of the following: (i) H2O (ii) CO2 (iii) CH4 Answer: (i) H2O: The molecular mass of water, H2O = (2 × Atomic mass of hydrogen) + (1 × Atomic mass of oxygen) = [2(1.0084) + 1(16.00 u)] = 2.016 u + 16.00 u = 18.016 = 18.02 u …

## NCERT solution class 11 science physics part 2 chapter 7 Waves

Page No 388: Question 15.1: A string of mass 2.50 kg is under a tension of 200 N. The length of the stretched string is 20.0 m. If the transverse jerk is struck at one end of the string, how long does the disturbance take to reach the other end? Answer: Mass of the string, M = …

## NCERT solution class 11 science physics part 2 chapter 6 Oscillations

Page No 358: Question 14.1: Which of the following examples represent periodic motion? (a) A swimmer completing one (return) trip from one bank of a river to the other and back. (b) A freely suspended bar magnet displaced from its N-S direction and released. (c) A hydrogen molecule rotating about its center of mass. (d) …

## NCERT solution class 11 science physics part 2 chapter 5 Kinetic Theory

Page No 333: Question 13.1: Estimate the fraction of molecular volume to the actual volume occupied by oxygen gas at STP. Take the diameter of an oxygen molecule to be 3Å. Answer: Diameter of an oxygen molecule, d = 3Å Radius, r  = 1.5 Å = 1.5 × 10–8 cm Actual volume occupied by 1 mole of oxygen gas …

## NCERT solution class 11 science physics part 2 chapter 4 Thermodynamics

Page No 316: Question 12.1: A geyser heats water flowing at the rate of 3.0 litres per minute from 27 °C to 77 °C. If the geyser operates on a gas burner, what is the rate of consumption of the fuel if its heat of combustion is 4.0 × 104 J/g? Answer: Water is flowing at …

## NCERT solution class 11 science physics part 2 chapter 3 Thermal Properties Of Matter

Page No 294: Question 11.1: The triple points of neon and carbon dioxide are 24.57 K and 216.55 K respectively. Express these temperatures on the Celsius and Fahrenheit scales. Answer: Kelvin and Celsius scales are related as: TC = TK – 273.15 … (i) Celsius and Fahrenheit scales are related as:  … (ii) For neon: TK = 24.57 K …

## NCERT solution class 11 science physics part 2 chapter 2 Mechanical Properties Of Fluids

Page No 268: Question 10.1: Explain why (a) The blood pressure in humans is greater at the feet than at the brain (b) Atmospheric pressure at a height of about 6 km decreases to nearly half of its value at the sea level, though the height of the atmosphere is more than 100 km (c) Hydrostatic pressure is …

## NCERT solution class 11 science physics part 2 chapter 1 Mechanical Properties Of Solids

Page No 242: Question 9.1: A steel wire of length 4.7 m and cross-sectional area 3.0 × 10–5 m2 stretches by the same amount as a copper wire of length 3.5 m and cross-sectional area of 4.0 × 10–5 m2 under a given load. What is the ratio of the Young’s modulus of steel to that of copper? Answer: …

## NCERT solution class 11 science physics part 1 chapter 8 Gravitation

Page No 201: Question 8.1: Answer the following: (a) You can shield a charge from electrical forces by putting it inside a hollow conductor. Can you shield a body from the gravitational influence of nearby matter by putting it inside a hollow sphere or by some other means? (b) An astronaut inside a small space ship orbiting …

## NCERT solution class 11 science physics part 1 chapter 7 System Of Particles And Rotational Motion

Page No 178: Question 7.1: Give the location of the centre of mass of a (i) sphere, (ii) cylinder, (iii) ring, and (iv) cube, each of uniform mass density. Does the centre of mass of a body necessarily lie inside the body? Answer: Geometric centre; No The centre of mass (C.M.) is a point where …

## NCERT solution class 11 science physics part 1 chapter 6 Work, Energy And Power

Page No 134: Question 6.1: The sign of work done by a force on a body is important to understand. State carefully if the following quantities are positive or negative: (a) work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket. (b) work …

## NCERT solution class 11 science physics part 1 chapter 5 Laws of Motion

Page No 109: Question 5.1: Give the magnitude and direction of the net force acting on (a) a drop of rain falling down with a constant speed, (b) a cork of mass 10 g floating on water, (c) a kite skillfully held stationary in the sky, (d) a car moving with a constant velocity of …

## NCERT solution class 11 science physics part 1 chapter 4 Motion in a Plane

Page No 85: Question 4.1: State, for each of the following physical quantities, if it is a scalar or a vector: volume, mass, speed, acceleration, density, number of moles, velocity, angular frequency, displacement, angular velocity. Answer: Scalar: Volume, mass, speed, density, number of moles, angular frequency Vector: Acceleration, velocity, displacement, angular velocity A scalar quantity is specified …

## NCERT solution class 11 science physics part 1 chapter 3 Motion in a Straight Line

Page No 55: Question 3.1: In which of the following examples of motion, can the body be considered approximately a point object: (a) a railway carriage moving without jerks between two stations. (b) a monkey sitting on top of a man cycling smoothly on a circular track. (c) a spinning cricket ball that turns sharply …

## NCERT solution class 11 science physics part 1 chapter 2 Units And Measurements

Page No 35: Question 2.1: Fill in the blanks (a) The volume of a cube of side 1 cm is equal to…..m3 (b) The surface area of a solid cylinder of radius 2.0 cm and height 10.0 cm is equal to … (mm)2 (c) A vehicle moving with a speed of 18 km h–1covers….m in …

## NCERT solution class 11 science physics part 1 chapter 1 Physical World Physics

Page No: 13 Exercises 1.1. Some of the most profound statements on the nature of science have come from Albert Einstein, one of the greatest scientists of all time. What do you think did Einstein mean when he said : “The most incomprehensible thing about the world is that it is comprehensible”? Answer The Physical world around us …

## NCERT solution class 8 SCIENCE

S.no Chapters Links 1 Crop Production and Management Exercise 1.1 Exercise 1.2 Exercise 1.3 Exercise 1.4 Exercise 1.5 2 Microorganisms: Friend and Foe Exercise 2.1 Exercise 2.2 Exercise 2.3 3 Synthetic Fibres and Plastics Exercise 3.1 Exercise 3.2 Exercise 3.3 Exercise 3.4 Exercise 3.5 4 Materials: Metals and Non-Metals Exercise 4.1 Exercise 4.2 Exercise …

## NCERT solution class 12 chapter 7 Probability exercise 7.6 mathematics part 2

EXERCISE 7.6 Page No 582: Question 1: A and B are two events such that P (A) ≠ 0. Find P (B|A), if (i) A is a subset of B (ii) A ∩ B = Φ Answer: It is given that, P (A) ≠ 0 (i) A is a subset of B. (ii)  Question 2: …

## NCERT solution class 12 chapter 7 Probability exercise 7.5 mathematics part 2

EXERCISE 7.5 Page No 576: Question 1: A die is thrown 6 times. If ‘getting an odd number’ is a success, what is the probability of (i) 5 successes? (ii) at least 5 successes? (iii) at most 5 successes? Answer: The repeated tosses of a die are Bernoulli trials. Let X denote the number of …

## NCERT solution class 12 chapter 7 Probability exercise 7.4 mathematics part 2

EXERCISE 7.4 Page No 569: Question 1: State which of the following are not the probability distributions of a random variable. Give reasons for your answer. (i) X 0 1 2 P (X) 0.4 0.4 0.2 (ii) X 0 1 2 3 4 P (X) 0.1 0.5 0.2 − 0.1 0.3 (iii) Y −1 0 1 P …

## NCERT solution class 12 chapter 7 Probability exercise 7.3 mathematics part 2

EXERCISE 7.3 Page No 555: Question 1: An urn contains 5 red and 5 black balls. A ball is drawn at random, its colour is noted and is returned to the urn. Moreover, 2 additional balls of the colour drawn are put in the urn and then a ball is drawn at random. What is …

## NCERT solution class 12 chapter 7 Probability exercise 7.2 mathematics part 2

EXERCISE 7.2 Page No 546: Question 1: If, find P (A ∩ B) if A and B are independent events. Answer: It is given that A and B are independent events. Therefore, Question 2: Two cards are drawn at random and without replacement from a pack of 52 playing cards. Find the probability that both …

## NCERT solution class 12 chapter 7 Probability exercise 7.1 mathematics part 2

EXERCISE 7.1 Page No 538: Question 1: Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E ∩ F) = 0.2, find P (E|F) and P(F|E). Answer: It is given that P(E) = 0.6, P(F) = 0.3, and P(E ∩ F) = 0.2 Question 2: Compute P(A|B), if …

## NCERT solution class 12 chapter 6 Linear Programming exercise 6.3 mathematics part 2

EXERCISE 6.3 Page No 525: Question 1: Refer to Example 9. How many packets of each food should be used to maximize the amount of vitamin A in the diet? What is the maximum amount of vitamin A in the diet? Answer: Let the diet contain x and y packets of foods P and Q respectively. Therefore, x ≥ 0 …

## NCERT solution class 12 chapter 6 Linear Programming exercise 6.2 mathematics part 2

EXERCISE 6.2 Page No 519: Question 1: Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 11 units of vitamin B. Food P costs Rs 60/kg and Food Q costs Rs 80/kg. Food …

## NCERT solution class 12 chapter 6 Linear Programming exercise 6.1 mathematics part 2

EXERCISE 6.1 Page No 513: Question 1: Maximise Z = 3x + 4y Subject to the constraints: Answer: The feasible region determined by the constraints, x + y ≤ 4, x ≥ 0, y ≥ 0, is as follows. The corner points of the feasible region are O (0, 0), A (4, 0), and B (0, 4). The values of Z at these points …

## NCERT solution class 12 chapter 5 Three Dimensional Geometry exercise 5.4 mathematics part 2

EXERCISE 5.4 Page No 497: Question 1: Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, −1), (4, 3, −1). Answer: Let OA be the line joining the origin, O (0, 0, 0), and the point, A (2, 1, 1). …

## NCERT solution class 12 chapter 5 Three Dimensional Geometry exercise 5.3 mathematics part 2

EXERCISE 5.3 Page No 493: Question 1: In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin. (a)z = 2 (b)  (c)  (d)5y + 8 = 0 Answer: (a) The equation of the plane is z = 2 or 0x + 0y + z = 2 … (1) The direction ratios of …

## NCERT solution class 12 chapter 5 Three Dimensional Geometry exercise 5.2 mathematics part 2

EXERCISE 5.2 Page No 477: Question 1: Show that the three lines with direction cosines  are mutually perpendicular. Answer: Two lines with direction cosines, l1, m1, n1 and l2, m2, n2, are perpendicular to each other, if l1l2 + m1m2 + n1n2 = 0 (i) For the lines with direction cosines,  and , we obtain Therefore, the lines are perpendicular. (ii) For the lines with direction cosines,  and , we obtain Therefore, the lines …

## NCERT solution class 12 chapter 5 Three Dimensional Geometry exercise 5.1 mathematics part 2

EXERCISE 5.1 Page No 467: Question 1: If a line makes angles 90°, 135°, 45° with x, y and z-axes respectively, find its direction cosines. Answer: Let direction cosines of the line be l, m, and n. Therefore, the direction cosines of the line are Question 2: Find the direction cosines of a line which makes equal angles with the coordinate axes. …

## NCERT solution class 12 chapter 4 Differential Equations exercise 4.5 mathematics part 2

EXERCISE 4.5 Page No 458: Question 1: Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis. Answer: If is a unit vector in the XY-plane, then  Here, θ is the angle made by the unit vector with the positive direction of the x-axis. Therefore, for θ = 30°: Hence, the required unit vector …

## NCERT solution class 12 chapter 4 Differential Equations exercise 4.4 mathematics part 2

EXERCISE 4.4 Page No 454: Question 1: Find, if and. Answer: We have, and Question 2: Find a unit vector perpendicular to each of the vector and, where and. Answer: We have, and Hence, the unit vector perpendicular to each of the vectors and is given by the relation, Question 3: If a unit vector  makes an angleswith with and an acute angle θ with, …

## NCERT solution class 12 chapter 4 Differential Equations exercise 4.3 mathematics part 2

EXERCISE 4.3 Page No 447: Question 1: Find the angle between two vectorsandwith magnitudesand 2, respectively having. Answer: It is given that, Now, we know that. Hence, the angle between the given vectors andis. Question 2: Find the angle between the vectors Answer: The given vectors are. Also, we know that. Question 3: Find the projection …

## NCERT solution class 12 chapter 4 Differential Equations exercise 4.2 mathematics part 2

EXERCISE 4.2 Page No 440: Question 1: Compute the magnitude of the following vectors: Answer: The given vectors are: Question 2: Write two different vectors having same magnitude. Answer: Hence, are two different vectors having the same magnitude. The vectors are different because they have different directions. Question 3: Write two different vectors having same direction. …

## NCERT solution class 12 chapter 4 Differential Equations exercise 4.1 mathematics part 2

EXERCISE 4.1 Page No 428: Question 1: Represent graphically a displacement of 40 km, 30° east of north. Answer: Here, vector represents the displacement of 40 km, 30° East of North. Question 2: Classify the following measures as scalars and vectors. (i) 10 kg (ii) 2 metres north-west (iii) 40° (iv) 40 watt (v) 10–19 coulomb (vi) …

## NCERT solution class 12 chapter 3 Differential Equations exercise 3.7 mathematics part 2

EXERCISE 3.7 Page No 419: Question 1: For each of the differential equations given below, indicate its order and degree (if defined). (i)  (ii)  (iii)  Answer: (i) The differential equation is given as: The highest order derivative present in the differential equation is. Thus, its order is two. The highest power raised to is one. Hence, its …

## NCERT solution class 12 chapter 3 Differential Equations exercise 3.6 mathematics part 2

EXERCISE 3.6 Page No 413: Question 1: Answer: The given differential equation is  This is in the form of  The solution of the given differential equation is given by the relation, Therefore, equation (1) becomes: This is the required general solution of the given differential equation. Question 2: Answer: The given differential equation is  The …

## NCERT solution class 12 chapter 3 Differential Equations exercise 3.5 mathematics part 2

EXERCISE 3.5 Page No 406: Question 1: Answer: The given differential equation i.e., (x2 + xy) dy = (x2 + y2) dx can be written as: This shows that equation (1) is a homogeneous equation. To solve it, we make the substitution as: y = vx Differentiating both sides with respect to x, we get: Substituting the values of v and in equation (1), we get: Integrating both sides, …

## NCERT solution class 12 chapter 3 Differential Equations exercise 3.4 mathematics part 2

EXERCISE 3.4 Page No 395: Question 1: Answer: The given differential equation is: Now, integrating both sides of this equation, we get: This is the required general solution of the given differential equation. Question 2: Answer: The given differential equation is: Now, integrating both sides of this equation, we get: This is the required general …

## NCERT solution class 12 chapter 3 Differential Equations exercise 3.3 mathematics part 2

EXERCISE 3.3 Page No 391: Question 1: Answer: Differentiating both sides of the given equation with respect to x, we get: Again, differentiating both sides with respect to x, we get: Hence, the required differential equation of the given curve is Question 2: Answer: Differentiating both sides with respect to x, we get: Again, differentiating both sides with …

## NCERT solution class 12 chapter 3 Differential Equations exercise 3.2 mathematics part 2

EXERCISE 3.2 Page No 385: Question 1: Answer: Differentiating both sides of this equation with respect to x, we get: Now, differentiating equation (1) with respect to x, we get: Substituting the values ofin the given differential equation, we get the L.H.S. as: Thus, the given function is the solution of the corresponding differential equation. Question 2: …

## NCERT solution class 12 chapter 3 Differential Equations exercise 3.1 mathematics part 2

EXERCISE 3.1 Page No 382: Question 1: Determine order and degree(if defined) of differential equation  Answer: The highest order derivative present in the differential equation is. Therefore, its order is four. The given differential equation is not a polynomial equation in its derivatives. Hence, its degree is not defined. Question 2: Determine order and degree(if …

## NCERT solution class 12 chapter 2 Application of Integrals exercise 2.3 mathematics part 2

EXERCISE 2.3 Page No 375: Question 1: Find the area under the given curves and given lines: (i) y = x2, x = 1, x = 2 and x-axis (ii) y = x4, x = 1, x = 5 and x –axis Answer: The required area is represented by the shaded area ADCBA as The required area is represented by the shaded area ADCBA as Question 2: Find the area between the …

## NCERT solution class 12 chapter 2 Application of Integrals exercise 2.2 mathematics part 2

EXERCISE 2.2 Page No 371: Question 1: Find the area of the circle 4×2 + 4y2 = 9 which is interior to the parabola x2 = 4y Answer: The required area is represented by the shaded area OBCDO. Solving the given equation of circle, 4×2 + 4y2 = 9, and parabola, x2 = 4y, we obtain the point of intersection as. It can …

## NCERT solution class 12 chapter 2 Application of Integrals exercise 2.1 mathematics part 2

EXERCISE 2.1 Page No 365: Question 1: Find the area of the region bounded by the curve y2 = x and the lines x = 1, x = 4 and the x-axis. Answer: The area of the region bounded by the curve, y2 = x, the lines, x = 1 and x = 4, and the x-axis is the area ABCD. Question 2: Find the area of the region bounded by y2 = 9x, x = …

## NCERT solution class 12 chapter 1 Integrals exercise 1.12 mathematics part 2

EXERCISE 1.12 Page No 352: Question 1: Answer: Equating the coefficients of x2, x, and constant term, we obtain −A + B − C = 0 B + C = 0 A = 1 On solving these equations, we obtain From equation (1), we obtain Question 2: Answer: Question 3:  [Hint: Put] Answer: Question 4: Answer: Question 5:   Answer: On dividing, we obtain Question 6: …

## NCERT solution class 12 chapter 1 Integrals exercise 1.11 mathematics part 2

EXERCISE 1.11 Page No 347: Question 1: Answer: Adding (1) and (2), we obtain Question 2: Answer: Adding (1) and (2), we obtain Question 3: Answer: Adding (1) and (2), we obtain Question 4: Answer: Adding (1) and (2), we obtain Question 5: Answer: It can be seen that (x + 2) ≤ 0 on [−5, …

## NCERT solution class 12 chapter 1 Integrals exercise 1.10 mathematics part 2

EXERCISE 1.10 Page No 340: Question 1: Answer: When x = 0, t = 1 and when x = 1, t = 2 Question 2: Answer: Also, let  Question 3: Answer: Also, let x = tanθ ⇒ dx = sec2θ dθ When x = 0, θ = 0 and when x = 1,  Takingθas first function and sec2θ as second function and integrating by parts, we obtain Question 4: Answer: Let x + 2 = t2 ⇒ dx = 2tdt When x = 0,  and …

## NCERT solution class 12 chapter 1 Integrals exercise 1.9 mathematics part 2

EXERCISE 1.9 Page No 338: Question 1: Answer: By second fundamental theorem of calculus, we obtain Question 2: Answer: By second fundamental theorem of calculus, we obtain Question 3: Answer: By second fundamental theorem of calculus, we obtain Question 4: Answer: By second fundamental theorem of calculus, we obtain Question 5: Answer: By second fundamental …

## NCERT solution class 12 chapter 1 Integrals exercise 1.8 mathematics part 2

EXERCISE 1.8 Page No 334: Question 1: Answer: It is known that, Question 2: Answer: It is known that, Question 3: Answer: It is known that, Question 4: Answer: It is known that, From equations (2) and (3), we obtain Question 5: Answer: It is known that, Question 6: Answer: It is known that,

## NCERT solution class 12 chapter 1 Integrals exercise 1.6 mathematics part 2

EXERCISE 1.6 Page No 327: Question 1: x sin x Answer: Let I =  Taking x as first function and sin x as second function and integrating by parts, we obtain Question 2: Answer: Let I =  Taking x as first function and sin 3x as second function and integrating by parts, we obtain Question 3: Answer: Let  Taking x2 as first function and ex as second function and integrating by parts, …

## NCERT solution class 12 chapter 1 Integrals exercise 1.5 mathematics part 2

EXERCISE 1.5 Page No 322: Question 1: Answer: Let  Equating the coefficients of x and constant term, we obtain A + B = 1 2A + B = 0 On solving, we obtain A = −1 and B = 2 Question 2: Answer: Let  Equating the coefficients of x and constant term, we obtain A + B = 0 −3A + 3B = 1 On solving, we obtain Question 3: Answer: Let  Substituting x = …

## NCERT solution class 12 chapter 1 Integrals exercise 1.4 mathematics part 2

EXERCISE 1.4 Page No 315: Question 1: Answer: Let x3 = t ∴ 3×2 dx = dt Question 2: Answer: Let 2x = t ∴ 2dx = dt Question 3: Answer: Let 2 − x = t ⇒ −dx = dt Question 4: Answer: Let 5x = t ∴ 5dx = dt Question 5: Answer: Question 6: Answer: Let x3 = t ∴ 3×2 dx = dt Question 7: Answer: From (1), we obtain Question 8: Answer: Let x3 = t ⇒ 3×2 dx = dt …

## NCERT solution class 12 chapter 1 Integrals exercise 1.3 mathematics part 2

EXERCISE 1.3 Page No 307: Question 1: Answer: Question 2: Answer: It is known that,  Question 3: cos 2x cos 4x cos 6x Answer: It is known that, Question 4: sin3 (2x + 1) Answer: Let  Question 5: sin3 x cos3 x Answer: Question 6: sin x sin 2x sin 3x Answer: It is known that,  Question 7: sin 4x sin 8x Answer: It is known …

## NCERT solution class 12 chapter 1 Integrals exercise 1.2 mathematics part 2

EXERCISE 1.2 Page No 304: Question 1: Answer: Let = t ∴2x dx = dt Question 2: Answer: Let log |x| = t ∴  Question 3: Answer: Let 1 + log x = t ∴  Question 4: sin x ⋅ sin (cos x) Answer: sin x ⋅ sin (cos x) Let cos x = t ∴ −sin x dx = dt Question 5: Answer: Let  ∴ 2adx = dt Question 6: Answer: Let ax …

## NCERT solution class 12 chapter 1 Integrals exercise 1.1 mathematics part 2

EXERCISE 1.1 Page No 299: Question 1: sin 2x Answer: The anti derivative of sin 2x is a function of x whose derivative is sin 2x. It is known that, Therefore, the anti derivative of Question 2: Cos 3x Answer: The anti derivative of cos 3x is a function of x whose derivative is cos 3x. It is known that, Therefore, …

## NCERT solution class 12 chapter 6 Application of Derivatives exercise 6.6 mathematics part 1

EXERCISE 6.6 Page No 242: Question 1: Using differentials, find the approximate value of each of the following. (a)  (b)  Answer: (a) Consider Then, Now, dy is approximately equal to Δy and is given by, Hence, the approximate value of = 0.667 + 0.010 = 0.677. (b) Consider. Let x = 32 and Δx = 1. Then, Now, dy is approximately equal to Δy and is …

## NCERT solution class 12 chapter 6 Application of Derivatives exercise 6.5 mathematics part 1

EXERCISE 6.5 Page No 231: Question 1: Find the maximum and minimum values, if any, of the following functions given by (i) f(x) = (2x − 1)2 + 3        (ii) f(x) = 9×2 + 12x + 2 (iii) f(x) = −(x − 1)2 + 10    (iv) g(x) = x3 + 1 Answer: (i) The given function is f(x) = (2x − 1)2 + 3. It can be observed that (2x − …

## NCERT solution class 12 chapter 6 Application of Derivatives exercise 6.4 mathematics part 1

EXERCISE 6.4 Page No 216: Question 1: 1. Using differentials, find the approximate value of each of the following up to 3 places of decimal (i)  (ii)  (iii)  (iv)  (v)  (vi)  (vii)  (viii)  (ix)  (x)  (xi)  (xii)  (xiii)  (xiv)  (xv)  Answer: (i)  Consider. Let x = 25 and Δx = 0.3. Then, Now, dy is approximately equal to Δy and is given by, Hence, the approximate value ofis 0.03 + 5 = …

## NCERT solution class 12 chapter 6 Application of Derivatives exercise 6.3 mathematics part 1

EXERCISE 6.3 Page No 211: Question 1: Find the slope of the tangent to the curve y = 3×4 − 4x at x = 4. Answer: The given curve is y = 3×4 − 4x. Then, the slope of the tangent to the given curve at x = 4 is given by, Question 2: Find the slope of the tangent to the curve, x ≠ 2 at x = 10. …

## NCERT solution class 12 chapter 6 Application of Derivatives exercise 6.2 mathematics part 1

EXERCISE 6.2 Page No 205: Question 1: Show that the function given by f(x) = 3x + 17 is strictly increasing on R. Answer: Letbe any two numbers in R. Then, we have: Hence, f is strictly increasing on R. Question 2: Show that the function given by f(x) = e2x is strictly increasing on R. Answer: Letbe any two numbers in R. Then, we have: Hence, f is …

## NCERT solution class 12 chapter 6 Application of Derivatives exercise 6.1 mathematics part 1

EXERCISE 6.1 Page No 197: Question 1: Find the rate of change of the area of a circle with respect to its radius r when (a) r = 3 cm (b) r = 4 cm Answer: The area of a circle (A)with radius (r) is given by, Now, the rate of change of the area with respect to its radius is …

## NCERT solution class 12 chapter 5 Continuity and Differentiability exercise 5.9 mathematics part 1

EXERCISE 5.9 Page No 191: Question 1: Answer: Using chain rule, we obtain Question 2: Answer: Question 3: Answer: Taking logarithm on both the sides, we obtain Differentiating both sides with respect to x, we obtain Question 4: Answer: Using chain rule, we obtain Question 5: Answer: Question 6: Answer: Therefore, equation (1) becomes Question 7: …

## NCERT solution class 12 chapter 5 Continuity and Differentiability exercise 5.8 mathematics part 1

EXERCISE 5.8 Page No 186: Question 1: Verify Rolle’s Theorem for the function Answer: The given function,, being a polynomial function, is continuous in [−4, 2] and is differentiable in (−4, 2). ∴ f (−4) = f (2) = 0 ⇒ The value of f (x) at −4 and 2 coincides. Rolle’s Theorem states that there is a point c ∈ (−4, 2) …

## NCERT solution class 12 chapter 5 Continuity and Differentiability exercise 5.7 mathematics part 1

EXERCISE 5.7 Page No 183: Question 1: Find the second order derivatives of the function. Answer: Let Then, Question 2: Find the second order derivatives of the function. Answer: Let Then, Question 3: Find the second order derivatives of the function. Answer: Let Then, Question 4: Find the second order derivatives of the function. Answer: …

## NCERT solution class 12 chapter 5 Continuity and Differentiability exercise 5.6 mathematics part 1

EXERCISE 5.6 Page No 181: Question 1: If x and y are connected parametrically by the equation, without eliminating the parameter, find. Answer: The given equations are Question 2: If x and y are connected parametrically by the equation, without eliminating the parameter, find. x = a cos θ, y = b cos θ Answer: The given equations are x = a cos θ and y = b cos θ Question 3: If x and y are connected parametrically by the equation, without eliminating the parameter, find. …

## NCERT solution class 12 chapter 5 Continuity and Differentiability exercise 5.5 mathematics part 1

EXERCISE 5.5 Page No 178: Question 1: Differentiate the function with respect to x. Answer: Taking logarithm on both the sides, we obtain Differentiating both sides with respect to x, we obtain Question 2: Differentiate the function with respect to x. Answer: Taking logarithm on both the sides, we obtain Differentiating both sides with respect to x, we obtain …

## NCERT solution class 12 chapter 5 Continuity and Differentiability exercise 5.4 mathematics part 1

EXERCISE 5.4 Page No 174: Question 1: Differentiate the following w.r.t. x: Answer: Let By using the quotient rule, we obtain Question 2: Differentiate the following w.r.t. x: Answer: Let By using the chain rule, we obtain Question 3: Differentiate the following w.r.t. x: Answer: Let  By using the chain rule, we obtain Question 4: Differentiate the following …

## NCERT solution class 12 chapter 5 Continuity and Differentiability exercise 5.3 mathematics part 1

EXERCISE 5.3 Page No 169: Question 1: Find : Answer: The given relationship is Differentiating this relationship with respect to x, we obtain Question 2: Find  : Answer: The given relationship is Differentiating this relationship with respect to x, we obtain Question 3: Find  : Answer: The given relationship is Differentiating this relationship with respect to x, we …

## NCERT solution class 12 chapter 5 Continuity and Differentiability exercise 5.2 mathematics part 1

EXERCISE 5.2 Page No 166: Question 1: Differentiate the functions with respect to x. Answer: Let f(x)=sinx2+5, ux=x2+5, and v(t)=sint   Then, vou=vux=vx2+5=tanx2+5=f(x) Thus, f is a composite of two functions. Alternate method Question 2: Differentiate the functions with respect to x. Answer: Thus, f is a composite function of two functions. Put t = u (x) = sin x By chain rule, Alternate method Question 3: Differentiate the functions with …

## NCERT solution class 12 chapter 5 Continuity and Differentiability exercise 5.1 mathematics part 1

EXERCISE 5.1 Page No 159: Question 1: Prove that the functionis continuous at Answer: Therefore, f is continuous at x = 0 Therefore, f is continuous at x = −3 Therefore, f is continuous at x = 5 Question 2: Examine the continuity of the function. Answer: Thus, f is continuous at x = 3 Question 3: Examine the following functions for continuity. (a)  (b) (c)  (d)  Answer: (a) The given function …

## NCERT solution class 12 chapter 4 Determinants exercise 4.7 mathematics part 1

EXERCISE 4.7 Page No 141: Question 1: Prove that the determinant is independent of θ. Answer: Hence, Î” is independent of Î¸. Question 2: Without expanding the determinant, prove that Answer: Hence, the given result is proved. Question 3: Evaluate  Answer: Expanding along C3, we have: Question 4: If a, b and c are real numbers, and, Show that either a + b + c = 0 or a = b = c. Answer: …

## NCERT solution class 12 chapter 4 Determinants exercise 4.6 mathematics part 1

EXERCISE 4.6 Page No 136: Question 1: Examine the consistency of the system of equations. x + 2y = 2 2x + 3y = 3 Answer: The given system of equations is: x + 2y = 2 2x + 3y = 3 The given system of equations can be written in the form of AX = B, where ∴ A is non-singular. Therefore, A−1 exists. Hence, the given system of …

## NCERT solution class 12 chapter 4 Determinants exercise 4.5 mathematics part 1

EXERCISE 4.5 Page No 131: Question 1: Find adjoint of each of the matrices. Answer: Question 2: Find adjoint of each of the matrices. Answer: Question 3: Verify A (adj A) = (adj A) A = I . Answer: Question 4: Verify A (adj A) = (adj A) A = I . Answer: Page No 132: Question 5: Find the inverse of each of the matrices (if …

## NCERT solution class 12 chapter 4 Determinants exercise 4.4 mathematics part 1

EXERCISE 4.4 Page No 126: Question 1: Write Minors and Cofactors of the elements of following determinants: (i)  (ii)  Answer: (i) The given determinant is. Minor of element aij is Mij. ∴M11 = minor of element a11 = 3 M12 = minor of element a12 = 0 M21 = minor of element a21 = −4 M22 = minor of element a22 = 2 Cofactor of aij is Aij = (−1)i + j Mij. ∴A11 = …

## NCERT solution class 12 chapter 4 Determinants exercise 4.3 mathematics part 1

EXERCISE 4.3 Page No 122: Question 1: Find area of the triangle with vertices at the point given in each of the following: (i) (1, 0), (6, 0), (4, 3) (ii) (2, 7), (1, 1), (10, 8) (iii) (−2, −3), (3, 2), (−1, −8) Answer: (i) The area of the triangle with vertices (1, 0), …

## NCERT solution class 12 chapter 4 Determinants exercise 4.2 mathematics part 1

EXERCISE 4.2 Page No 119: Question 1: Using the property of determinants and without expanding, prove that: Answer: Question 2: Using the property of determinants and without expanding, prove that: Answer: Here, the two rows R1 and R3 are identical. Δ = 0. Question 3: Using the property of determinants and without expanding, prove that: Answer: Question …

## NCERT solution class 12 chapter 4 Determinants exercise 4.1 mathematics part 1

EXERCISE 4.1 Page No 108: Question 1: Evaluate the determinants in Exercises 1 and 2. Answer:  = 2(−1) − 4(−5) = − 2 + 20 = 18 Question 2: Evaluate the determinants in Exercises 1 and 2. (i)  (ii)  Answer: (i)  = (cos θ)(cos θ) − (−sin θ)(sin θ) = cos2 θ+ sin2 θ = 1 (ii)  = (x2 − x + 1)(x + 1) − (x − 1)(x + …

## NCERT solution class 12 chapter 3 Matrices exercise 3.5 mathematics part 1

EXERCISE 3.5 Page No 100: Question 1: Let, show that, where I is the identity matrix of order 2 and n ∈ N Answer: It is given that  We shall prove the result by using the principle of mathematical induction. For n = 1, we have: Therefore, the result is true for n = 1. Let the result be true for n = k. That is, Now, …

## NCERT solution class 12 chapter 3 Matrices exercise 3.4 mathematics part 1

EXERCISE 3.4 Page No 97: Question 1: Find the inverse of each of the matrices, if it exists. Answer: We know that A = IA Question 2: Find the inverse of each of the matrices, if it exists. Answer: We know that A = IA Question 3: Find the inverse of each of the matrices, if it exists. Answer: We know …

## NCERT solution class 12 chapter 3 Matrices exercise 3.3 mathematics part 1

EXERCISE 3.3 Page No 88: Question 1: Find the transpose of each of the following matrices: (i)  (ii)  (iii)  Answer: (i)  (ii)  (iii)  Question 2: If and, then verify that (i)  (ii)  Answer: We have: (i) (ii) Question 3: If and, then verify that (i)  (ii)  Answer: (i) It is known that Therefore, we have: (ii) Question 4: …

## NCERT solution class 12 chapter 3 Matrices exercise 3.2 mathematics part 1

EXERCISE 3.2 Page No 80: Question 1: Let Find each of the following (i)  (ii)  (iii)  (iv)  (v)  Answer: (i) (ii) (iii) (iv) Matrix A has 2 columns. This number is equal to the number of rows in matrix B. Therefore, AB is defined as: (v) Matrix B has 2 columns. This number is equal to the number of rows in matrix A. Therefore, BA is defined as: Question 2: …

## NCERT solution class 12 chapter 3 Matrices exercise 3.1 mathematics part 1

EXERCISE 3.1 Page No 64: Question 1: In the matrix, write: (i) The order of the matrix (ii) The number of elements, (iii) Write the elements a13, a21, a33, a24, a23 Answer: (i) In the given matrix, the number of rows is 3 and the number of columns is 4. Therefore, the order of the matrix is 3 × 4. (ii) Since …

## NCERT solution class 12 chapter 2 Inverse Trigonometric Functions exercise 2.3 mathematics part 1

EXERCISE 2.3 Page No 51: Question 1: Find the value of  Answer: We know that cos−1 (cos x) = x if, which is the principal value branch of cos −1x. Here, Now, can be written as: Question 2: Find the value of  Answer: We know that tan−1 (tan x) = x if, which is the principal value branch of tan −1x. Here, Now, can be written …

## NCERT solution class 12 chapter 2 Inverse Trigonometric Functions exercise 2.2 mathematics part 1

EXERCISE 2.2 Page No 47: Question 1: Prove  Answer: To prove:  Let x = sinθ. Then,  We have, R.H.S. = = 3θ = L.H.S. Question 2: Prove  Answer: To prove: Let x = cosθ. Then, cos−1 x =θ. We have, Question 3: Prove  Answer: To prove: Question 4: Prove  Answer: To prove:  Question 5: Write the function in the simplest form: …

## NCERT solution class 12 chapter 2 Inverse Trigonometric Functions exercise 2.1 mathematics part 1

EXERCISE 2.1 Page No 41: Question 1: Find the principal value of  Answer: Let sin-1  Then sin y =  We know that the range of the principal value branch of sin−1 is  and sin Therefore, the principal value of  Question 2: Find the principal value of  Answer: We know that the range of the principal value branch of cos−1 is …

## NCERT solution class 12 chapter 1 Relations and Functions exercise 1.5 mathematics part 1

EXERCISE 1.5 Page No 29: Question 1: Let f: R → R be defined as f(x) = 10x + 7. Find the function g: R → R such that g o f = f o g = 1R. Answer: It is given that f: R → R is defined as f(x) = 10x + 7. One-one: Let f(x) = f(y), where x, y ∈R. ⇒ 10x + 7 = 10y + 7 ⇒ x = y ∴ f is a one-one function. Onto: For y ∈ R, let y = 10x + 7. Therefore, for any y ∈ R, there exists such that  ∴ f is …

## NCERT solution class 12 chapter 1 Relations and Functions exercise 1.4 mathematics part 1

EXERCISE 1.4 Page No 24: Question 1: Determine whether or not each of the definition of given below gives a binary operation. In the event that * is not a binary operation, give justification for this. (i) On Z+, define * by a * b = a − b (ii) On Z+, define * by a * b = ab (iii) On R, define * by a * b = ab2 (iv) On Z+, define * …

## NCERT solution class 12 chapter 1 Relations and Functions exercise 1.3 mathematics part 1

EXERCISE 1.3 Page No 18: Question 1: Let f: {1, 3, 4} → {1, 2, 5} and g: {1, 2, 5} → {1, 3} be given by f = {(1, 2), (3, 5), (4, 1)} and g = {(1, 3), (2, 3), (5, 1)}. Write down gof. Answer: The functions f: {1, 3, 4} → {1, 2, 5} and g: {1, 2, 5} → …

## NCERT solution class 12 chapter 1 Relations and Functions exercise 1.2 mathematics part 1

EXERCISE 1.2 Page No 10: Question 1: Show that the function f: R* → R* defined byis one-one and onto, where R* is the set of all non-zero real numbers. Is the result true, if the domain R* is replaced by N with co-domain being same as R*? Answer: It is given that f: R* → R* is defined by One-one: ∴f is one-one. Onto: It is clear that for y∈ R*, there existssuch that …

## NCERT solution class 12 chapter 1 Relations and Functions exercise 1.1 mathematics part 1

EXERCISE 1.1 Page No 5: Question 1: Determine whether each of the following relations are reflexive, symmetric and transitive: (i)Relation R in the set A = {1, 2, 3…13, 14} defined as R = {(x, y): 3x − y = 0} (ii) Relation R in the set N of natural numbers defined as R = {(x, y): y = x + 5 and x < 4} (iii) Relation R in …

## NCERT solution class 11 chapter 16 Probability exercise 16.4 mathematics

EXERCISE 16.4 Page No 408: Question 1: A box contains 10 red marbles, 20 blue marbles and 30 green marbles. 5 marbles are drawn from the box, what is the probability that (i) all will be blue? (ii) atleast one will be green? Answer: Total number of marbles = 10 + 20 + 30 = …

## NCERT solution class 11 chapter 16 Probability exercise 16.3 mathematics

EXERCISE 16.3 Page No 403: Question 1: Which of the following can not be valid assignment of probabilities for outcomes of sample space S =  Assignment ω1 ω2 ω3 ω4 ω5 ω6 ω7 (a) 0.1 0.01 0.05 0.03 0.01 0.2 0.6 (b) (c) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 (d) –0.1 0.2 0.3 0.4 …

## NCERT solution class 11 chapter 16 Probability exercise 16.2 mathematics

EXERCISE 16.2 Page No 393: Question 1: A die is rolled. Let E be the event “die shows 4” and F be the event “die shows even number”. Are E and F mutually exclusive? Answer: When a die is rolled, the sample space is given by S = {1, 2, 3, 4, 5, 6} Accordingly, …

## NCERT solution class 11 chapter 16 Probability exercise 16.1 mathematics

EXERCISE 16.1 Page No 386: Question 1: Describe the sample space for the indicated experiment: A coin is tossed three times. Answer: A coin has two faces: head (H) and tail (T). When a coin is tossed three times, the total number of possible outcomes is 23 = 8 Thus, when a coin is tossed three …

## NCERT solution class 11 chapter 15 Statistics exercise 15.4 mathematics

EXERCISE 15.4 Page No 380: Question 1: The mean and variance of eight observations are 9 and 9.25, respectively. If six of the observations are 6, 7, 10, 12, 12 and 13, find the remaining two observations. Answer: Let the remaining two observations be x and y. Therefore, the observations are 6, 7, 10, 12, 12, 13, x, y. From …

## NCERT solution class 11 chapter 15 Statistics exercise 15.3 mathematics

EXERCISE 15.3 Page No 375: Question 1: From the data given below state which group is more variable, A or B? Marks 10-20 20-30 30-40 40-50 50-60 60-70 70-80 Group A 9 17 32 33 40 10 9 Group B 10 20 30 25 43 15 7 Answer: Firstly, the standard deviation of group A …

## NCERT solution class 11 chapter 15 Statistics exercise 15.2 mathematics

EXERCISE 15.2 Page No 371: Question 1: Find the mean and variance for the data 6, 7, 10, 12, 13, 4, 8, 12 Answer: 6, 7, 10, 12, 13, 4, 8, 12 Mean,  The following table is obtained. xi 6 –3 9 7 –2 4 10 –1 1 12 3 9 13 4 16 4 …

## NCERT solution class 11 chapter 15 Statistics exercise 15.1 mathematics

EXERCISE 15.1 Page No 360: Question 1: Find the mean deviation about the mean for the data 4, 7, 8, 9, 10, 12, 13, 17 Answer: The given data is 4, 7, 8, 9, 10, 12, 13, 17 Mean of the data,  The deviations of the respective observations from the mean are –6, – 3, –2, …

## NCERT solution class 11 chapter 14 Mathematical Reasoning exercise 14.6 mathematics

EXERCISE 14.6 Page No 345: Question 1: Write the negation of the following statements: (i) p: For every positive real number x, the number x – 1 is also positive. (ii) q: All cats scratch. (iii) r: For every real number x, either x > 1 or x < 1. (iv) s: There exists a number x such that 0 < x < 1. Answer: (i) The negation of statement p is as …

## NCERT solution class 11 chapter 14 Mathematical Reasoning exercise 14.5 mathematics

EXERCISE 14.5 Page No 342: Question 1: Show that the statement p: “If x is a real number such that x3 + 4x = 0, then x is 0” is true by (i) direct method (ii) method of contradiction (iii) method of contrapositive Answer: p: “If x is a real number such that x3 + 4x = 0, then x is 0”. Let q: x is a real number such that x3 + 4x = …

## NCERT solution class 11 chapter 14 Mathematical Reasoning exercise 14.4 mathematics

EXERCISE 14.4 Page No 338: Question 1: Rewrite the following statement with “if-then” in five different ways conveying the same meaning. If a natural number is odd, then its square is also odd. Answer: The given statement can be written in five different ways as follows. (i) A natural number is odd implies that its …

## NCERT solution class 11 chapter 14 Mathematical Reasoning exercise 14.3 mathematics

EXERCISE 14.3 Page No 334: Question 1: For each of the following compound statements first identify the connecting words and then break it into component statements. (i) All rational numbers are real and all real numbers are not complex. (ii) Square of an integer is positive or negative. (iii) The sand heats up quickly in …

## NCERT solution class 11 chapter 14 Mathematical Reasoning exercise 14.2 mathematics

EXERCISE 14.2 Page No 329: Question 1: Write the negation of the following statements: (i) Chennai is the capital of Tamil Nadu. (ii) is not a complex number. (iii) All triangles are not equilateral triangle. (iv) The number 2 is greater than 7. (v) Every natural number is an integer. Answer: (i) Chennai is not the …

## NCERT solution class 11 chapter 14 Mathematical Reasoning exercise 14.1 mathematics

EXERCISE 14.1 Page No 324: Question 1: Which of the following sentences are statements? Give reasons for your answer. (i) There are 35 days in a month. (ii) Mathematics is difficult. (iii) The sum of 5 and 7 is greater than 10. (iv) The square of a number is an even number. (v) The sides …

## NCERT solution class 11 chapter 13 Limits and Derivatives exercise 13.3 mathematics

EXERCISE 13.3 Page No 317: Question 1: Find the derivative of the following functions from first principle: (i) –x (ii) (–x)–1 (iii) sin (x + 1) (iv)  Answer: (i) Let f(x) = –x. Accordingly, By first principle, (ii) Let. Accordingly, By first principle, (iii) Let f(x) = sin (x + 1). Accordingly, By first principle, (iv) Let. Accordingly, By first principle, …

## NCERT solution class 11 chapter 13 Limits and Derivatives exercise 13.2 mathematics

EXERCISE 13.2 Page No 312: Question 1: Find the derivative of x2 – 2 at x = 10. Answer: Let f(x) = x2 – 2. Accordingly, Thus, the derivative of x2 – 2 at x = 10 is 20. Question 2: Find the derivative of 99x at x = 100. Answer: Let f(x) = 99x. Accordingly, Thus, the derivative of 99x at x = 100 is 99. Question 3: Find the derivative of x at x = …

## NCERT solution class 11 chapter 13 Limits and Derivatives exercise 13.1 mathematics

EXERCISE 13.1 Page No 301: Question 1: Evaluate the Given limit: Answer: Question 2: Evaluate the Given limit: Answer: Question 3: Evaluate the Given limit: Answer: Question 4: Evaluate the Given limit: Answer: Question 5: Evaluate the Given limit: Answer: Question 6: Evaluate the Given limit: Answer: Put x + 1 = y so that y → 1 as x → 0. Question …

## NCERT solution class 11 chapter 12 Introduction to Three Dimensional Geometry exercise 12.4 mathematics

EXERCISE 12.4 Page No 278: Question 1: Three vertices of a parallelogram ABCD are A (3, –1, 2), B (1, 2, –4) andC (–1, 1, 2). Find the coordinates of the fourth vertex. Answer: The three vertices of a parallelogram ABCD are given as A (3, –1, 2), B (1, 2, –4), and C (–1, …

## NCERT solution class 11 chapter 12 Introduction to Three Dimensional Geometry exercise 12.3 mathematics

EXERCISE 12.3 Page No 277: Question 1: Find the coordinates of the point which divides the line segment joining the points (–2, 3, 5) and (1, –4, 6) in the ratio (i) 2:3 internally, (ii) 2:3 externally. Answer: (i) The coordinates of point R that divides the line segment joining points P (x1, y1, z1) and Q …

## NCERT solution class 11 chapter 12 Introduction to Three Dimensional Geometry exercise 12.2 mathematics

EXERCISE 12.2 Page No 273: Question 1: Find the distance between the following pairs of points: (i) (2, 3, 5) and (4, 3, 1) (ii) (–3, 7, 2) and (2, 4, –1) (iii) (–1, 3, –4) and (1, –3, 4) (iv) (2, –1, 3) and (–2, 1, 3) Answer: The distance between points P(x1, y1, z1) and …

## NCERT solution class 11 chapter 12 Introduction to Three Dimensional Geometry exercise 12.1 mathematics

EXERCISE 12.1 Page No 271: Question 1: A point is on the x-axis. What are its y-coordinates and z-coordinates? Answer: If a point is on the x-axis, then its y-coordinates and z-coordinates are zero. Question 2: A point is in the XZ-plane. What can you say about its y-coordinate? Answer: If a point is in the XZ plane, then its y-coordinate is zero. …

## NCERT solution class 11 chapter 11 Conic Sections exercise 11.5 mathematics

EXERCISE 11.5 Question 1: If a parabolic reflector is 20 cm in diameter and 5 cm deep, find the focus. Answer: The origin of the coordinate plane is taken at the vertex of the parabolic reflector in such a way that the axis of the reflector is along the positive x-axis. This can be diagrammatically represented …

## NCERT solution class 11 chapter 11 Conic Sections exercise 11.4 mathematics

EXERCISE 11.4 Page No 262: Question 1: Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola Answer: The given equation is. On comparing this equation with the standard equation of hyperbola i.e.,, we obtain a = 4 and b = 3. We know that a2 + b2 = c2. Therefore, The coordinates …

## NCERT solution class 11 chapter 11 Conic Sections exercise 11.3 mathematics

EXERCISE 11.3 Page No 255: Question 1: Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse  Answer: The given equation is. Here, the denominator of is greater than the denominator of. Therefore, the major axis is along …

## NCERT solution class 11 chapter 11 Conic Sections exercise 11.2 mathematics

EXERCISE 11.2 Page No 246: Question 1: Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for y2 = 12x Answer: The given equation is y2 = 12x. Here, the coefficient of x is positive. Hence, the parabola opens towards the right. On comparing this equation with y2 = 4ax, we …

## NCERT solution class 11 chapter 11 Conic Sections exercise 11.1 mathematics

EXERCISE 11.1 Page No 241: Question 1: Find the equation of the circle with centre (0, 2) and radius 2 Answer: The equation of a circle with centre (h, k) and radius r is given as (x­ – h)2 + (y ­– k)2 = r2 It is given that centre (h, k) = (0, 2) and radius (r) = 2. Therefore, the equation of the circle …

## NCERT solution class 11 chapter 10 Straight Lines exercise 10.4 mathematics

EXERCISE 10.4 Page No 233: Question 1: Find the values of k for which the lineis (a) Parallel to the x-axis, (b) Parallel to the y-axis, (c) Passing through the origin. Answer: The given equation of line is (k – 3) x – (4 – k2) y + k2 – 7k + 6 = 0 … (1) (a) If the given line is parallel to the x-axis, then Slope …

## NCERT solution class 11 chapter 10 Straight Lines exercise 10.3 mathematics

EXERCISE 10.3 Page No 227: Question 1: Reduce the following equations into slope-intercept form and find their slopes and the y-intercepts. (i) x + 7y = 0 (ii) 6x + 3y – 5 = 0 (iii) y = 0 Answer: (i) The given equation is x + 7y = 0. It can be written as This equation is of the form y = mx + c, where. Therefore, equation (1) is …

## NCERT solution class 11 chapter 10 Straight Lines exercise 10.2 mathematics

EXERCISE 10.2 Page No 219: Question 1: Write the equations for the x and y-axes. Answer: The y-coordinate of every point on the x-axis is 0. Therefore, the equation of the x-axis is y = 0. The x-coordinate of every point on the y-axis is 0. Therefore, the equation of the y-axis is x = 0. Question 2: Find the equation of the line which passes through the …

## NCERT solution class 11 chapter 10 Straight Lines exercise 10.1 mathematics

EXERCISE 10.1 Page No 211: Question 1: Draw a quadrilateral in the Cartesian plane, whose vertices are (–4, 5), (0, 7), (5, –5) and (–4, –2). Also, find its area. Answer: Let ABCD be the given quadrilateral with vertices A (–4, 5), B (0, 7), C (5, –5), and D (–4, –2). Then, by plotting …

## NCERT solution class 11 chapter 9 Sequences and Series exercise 9.5 mathematics

EXERCISE 9.5 Page No 199: Question 1: Show that the sum of (m + n)th and (m – n)th terms of an A.P. is equal to twice the mth term. Answer: Let a and d be the first term and the common difference of the A.P. respectively. It is known that the kth term of an A. P. is given by ak = a + (k –1) d ∴ am + n = a + (m + n –1) d am – n = a + (m – n –1) d am = a + (m –1) d …

## NCERT solution class 11 chapter 9 Sequences and Series exercise 9.4 mathematics

EXERCISE 9.4 Page No 196: Question 1: Find the sum to n terms of the series 1 × 2 + 2 × 3 + 3 × 4 + 4 × 5 + … Answer: The given series is 1 × 2 + 2 × 3 + 3 × 4 + 4 × 5 + … nth term, an = n ( n + 1) …

## NCERT solution class 11 chapter 9 Sequences and Series exercise 9.3 mathematics

EXERCISE 9.3 Page No 192: Question 1: Find the 20th and nthterms of the G.P. Answer: The given G.P. is  Here, a = First term =  r = Common ratio =  Question 2: Find the 12th term of a G.P. whose 8th term is 192 and the common ratio is 2. Answer: Common ratio, r = 2 Let a be the first term of the G.P. …

## NCERT solution class 11 chapter 9 Sequences and Series exercise 9.2 mathematics

EXERCISE 9.2 Page No 185: Question 1: Find the sum of odd integers from 1 to 2001. Answer: The odd integers from 1 to 2001 are 1, 3, 5, …1999, 2001. This sequence forms an A.P. Here, first term, a = 1 Common difference, d = 2 Thus, the sum of odd numbers from 1 to 2001 is 1002001. …

## NCERT solution class 11 chapter 9 Sequences and Series exercise 9.1 mathematics

EXERCISE 9.1 Page No 180: Question 1: Write the first five terms of the sequences whose nth term is  Answer: Substituting n = 1, 2, 3, 4, and 5, we obtain Therefore, the required terms are 3, 8, 15, 24, and 35. Question 2: Write the first five terms of the sequences whose nth term is  Answer: Substituting n = 1, …

## NCERT solution class 11 chapter 8 Binomial Theorem exercise 8.3 mathematics

EXERCISE 8.3 Page No 175: Question 1: Find a, b and n in the expansion of (a + b)n if the first three terms of the expansion are 729, 7290 and 30375, respectively. Answer: It is known that (r + 1)th term, (Tr+1), in the binomial expansion of (a + b)n is given by . The first three terms of the expansion are given as 729, 7290, and 30375 …

## NCERT solution class 11 chapter 8 Binomial Theorem exercise 8.2 mathematics

EXERCISE 8.2 Page No 171: Question 1: Find the coefficient of x5 in (x + 3)8 Answer: It is known that (r + 1)th term, (Tr+1), in the binomial expansion of (a + b)n is given by . Assuming that x5 occurs in the (r + 1)th term of the expansion (x + 3)8, we obtain Comparing the indices of x in x5 and in Tr +1, we obtain r = 3 Thus, the coefficient of x5 is …

## NCERT solution class 11 chapter 8 Binomial Theorem exercise 8.1 mathematics

EXERCISE 8.1 Page No 166: Question 1: Expand the expression (1– 2x)5 Answer: By using Binomial Theorem, the expression (1– 2x)5 can be expanded as Question 2: Expand the expression Answer: By using Binomial Theorem, the expression  can be expanded as Question 3: Expand the expression (2x – 3)6 Answer: By using Binomial Theorem, the expression (2x – 3)6 can …

## NCERT solution class 11 chapter 7 Permutations and Combinations exercise 7.5 mathematics

EXERCISE 7.5 Page No 156: Question 1: How many words, with or without meaning, each of 2 vowels and 3 consonants can be formed from the letters of the word DAUGHTER? Answer: In the word DAUGHTER, there are 3 vowels namely, A, U, and E, and 5 consonants namely, D, G, H, T, and R. …

## NCERT solution class 11 chapter 7 Permutations and Combinations exercise 7.4 mathematics

EXERCISE 7.4 Page No 153: Question 1: If, find. Answer: It is known that, Therefore, Question 2: Determine n if (i)  (ii)  Answer: (i) (ii) Question 3: How many chords can be drawn through 21 points on a circle? Answer: For drawing one chord on a circle, only 2 points are required. To know the number of chords …

## NCERT solution class 11 chapter 7 Permutations and Combinations exercise 7.3 mathematics

EXERCISE 7.3 Page No 148: Question 1: How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit is repeated? Answer: 3-digit numbers have to be formed using the digits 1 to 9. Here, the order of the digits matters. Therefore, there will be as many 3-digit numbers as …

## NCERT solution class 11 chapter 7 Permutations and Combinations exercise 7.2 mathematics

EXERCISE 7.2 Page No 140: Question 1: Evaluate (i) 8! (ii) 4! – 3! Answer: (i) 8! = 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 = 40320 (ii) 4! = 1 × 2 × 3 × 4 = 24 3! = 1 × 2 × 3 = …

## NCERT solution class 11 chapter 7 Permutations and Combinations exercise 7.1 mathematics

EXERCISE 7.1 Page No 138: Question 1: How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that (i) repetition of the digits is allowed? (ii) repetition of the digits is not allowed? Answer: (i) There will be as many ways as there are ways of filling 3 …

## NCERT solution class 11 chapter 6 Linear Inequalities exercise 6.4 mathematics

EXERCISE 6.4 Page No 132: Question 1: Solve the inequality 2 ≤ 3x – 4 ≤ 5 Answer: 2 ≤ 3x – 4 ≤ 5 ⇒ 2 + 4 ≤ 3x – 4 + 4 ≤ 5 + 4 ⇒ 6 ≤ 3x ≤ 9 ⇒ 2 ≤ x ≤ 3 Thus, all the real numbers, x, which are greater than or …

## NCERT solution class 11 chapter 6 Linear Inequalities exercise 6.3 mathematics

EXERCISE 6.3 Page No 129: Question 1: Solve the following system of inequalities graphically: x ≥ 3, y ≥ 2 Answer: x ≥ 3 … (1) y ≥ 2 … (2) The graph of the lines, x = 3 and y = 2, are drawn in the figure below. Inequality (1) represents the region on the right hand side of the line, x = 3 (including the …

## NCERT solution class 11 chapter 6 Linear Inequalities exercise 6.2 mathematics

EXERCISE 6.2 Page No 127: Question 1: Solve the given inequality graphically in two-dimensional plane: x + y < 5 Answer: The graphical representation of x + y = 5 is given as dotted line in the figure below. This line divides the xy-plane in two half planes, I and II. Select a point (not on the line), which lies in one of the half planes, to …

## NCERT solution class 11 chapter 6 Linear Inequalities exercise 6.1 mathematics

EXERCISE 6.1 Page No 122: Question 1: Solve 24x < 100, when (i) x is a natural number (ii) x is an integer Answer: The given inequality is 24x < 100. (i) It is evident that 1, 2, 3, and 4 are the only natural numbers less than. Thus, when x is a natural number, the solutions of the given inequality are 1, …

## NCERT solution class 11 chapter 5 Complex Numbers and Quadratic Equations exercise 5.4 mathematics

EXERCISE 5.4 Page No 112: Question 1: Evaluate:  Answer: Question 2: For any two complex numbers z1 and z2, prove that Re (z1z2) = Re z1 Re z2 – Im z1 Im z2 Answer: Question 3: Reduce to the standard form. Answer: Question 4: If x – iy =prove that. Answer: Question 5: Convert the following in the polar form: (i) , (ii)  Answer: (i) …

## NCERT solution class 11 chapter 5 Complex Numbers and Quadratic Equations exercise 5.3 mathematics

EXERCISE 5.3 Page No 109: Question 1: Solve the equation x2 + 3 = 0 Answer: The given quadratic equation is x2 + 3 = 0 On comparing the given equation with ax2 + bx + c = 0, we obtain a = 1, b = 0, and c = 3 Therefore, the discriminant of the given equation is D = b2 – 4ac = 02 – 4 × 1 × 3 = –12 …

## NCERT solution class 11 chapter 5 Complex Numbers and Quadratic Equations exercise 5.2 mathematics

EXERCISE 5.2 Page No 108: Question 1: Find the modulus and the argument of the complex number Answer: On squaring and adding, we obtain Since both the values of sin θ and cos θ are negative and sinθ and cosθ are negative in III quadrant, Thus, the modulus and argument of the complex number are 2 and respectively. Question 2: Find the modulus …

## NCERT solution class 11 chapter 5 Complex Numbers and Quadratic Equations exercise 5.1 mathematics

EXERCISE 5.1 Page No 103: Question 1: Express the given complex number in the form a + ib: Answer: Question 2: Express the given complex number in the form a + ib: i9 + i19 Answer: Question 3: Express the given complex number in the form a + ib: i–39 Answer: Page No 104: Question 4: Express the given complex number in the form a + ib: 3(7 + i7) + i(7 + i7) …

## NCERT solution class 11 chapter 4 Principle of Mathematical Induction exercise 4.1 mathematics

EXERCISE 4.1 Page No 94: Question 1: Prove the following by using the principle of mathematical induction for all n ∈ N:    Answer: Let the given statement be P(n), i.e., P(n): 1 + 3 + 32 + …+ 3n–1 = For n = 1, we have P(1): 1 =, which is true. Let P(k) be true for some positive integer k, i.e., …

## NCERT solution class 11 chapter 3 Trigonometric Functions exercise 3.5 mathematics

EXERCISE 3.5 Page No 81: Question 1: Prove that:  Answer: L.H.S. = 0 = R.H.S Question 2: Prove that: (sin 3x + sin x) sin x + (cos 3x – cos x) cos x = 0 Answer: L.H.S. = (sin 3x + sin x) sin x + (cos 3x – cos x) cos x = RH.S. Page No 82: Question 3: Prove that:  Answer: L.H.S. =  Question 4: Prove that:  …

## NCERT solution class 11 chapter 3 Trigonometric Functions exercise 3.4 mathematics

EXERCISE 3.4 Page No 78: Question 1: Find the principal and general solutions of the equation  Answer: Therefore, the principal solutions are x =and. Therefore, the general solution is Question 2: Find the principal and general solutions of the equation  Answer: Therefore, the principal solutions are x =and. Therefore, the general solution is, where n ∈ Z Question 3: Find the principal …

## NCERT solution class 11 chapter 3 Trigonometric Functions exercise 3.3 mathematics

EXERCISE 3.3 Page No 73: Question 1: Answer: L.H.S. =  Question 2: Prove that  Answer: L.H.S. =  Question 3: Prove that  Answer: L.H.S. = Question 4: Prove that  Answer: L.H.S = Question 5: Find the value of: (i) sin 75° (ii) tan 15° Answer: (i) sin 75° = sin (45° + 30°) = sin 45° …

## NCERT solution class 11 chapter 3 Trigonometric Functions exercise 3.2 mathematics

EXERCISE 3.2 Page No 63: Question 1: Find the values of other five trigonometric functions if , x lies in third quadrant. Answer: Since x lies in the 3rd quadrant, the value of sin x will be negative. Question 2: Find the values of other five trigonometric functions if , x lies in second quadrant. Answer: Since x lies in the 2nd quadrant, the value of cos x will be negative …

## NCERT solution class 11 chapter 3 Trigonometric Functions exercise 3.1 mathematics

EXERCISE 3.1 Page No 54: Question 1: Find the radian measures corresponding to the following degree measures: (i) 25° (ii) – 47° 30′ (iii) 240° (iv) 520° Answer: (i) 25° We know that 180° = π radian (ii) –47° 30′ –47° 30′ = degree [1° = 60′]  degree Since 180° = π radian (iii) 240° We …

## NCERT solution class 11 chapter 2 Sets exercise 2.4 mathematics

EXERCISE 2.4 Page No 46: Question 1: The relation f is defined by  The relation g is defined by  Show that f is a function and g is not a function. Answer: The relation f is defined as It is observed that for 0 ≤ x < 3, f(x) = x2 3 < x ≤ 10, f(x) = 3x Also, at x = 3, f(x) = 32 = 9 or f(x) = 3 × 3 = 9 …

## NCERT solution class 11 chapter 2 Sets exercise 2.3 mathematics

EXERCISE 2.3 Page No 44: Question 1: Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range. (i) {(2, 1), (5, 1), (8, 1), (11, 1), (14, 1), (17, 1)} (ii) {(2, 1), (4, 2), (6, 3), (8, 4), (10, 5), (12, 6), (14, 7)} (iii) …

## NCERT solution class 11 chapter 2 Sets exercise 2.2 mathematics

EXERCISE 2.2 Page No 35: Question 1: Let A = {1, 2, 3, … , 14}. Define a relation R from A to A by R = {(x, y): 3x – y = 0, where x, y ∈ A}. Write down its domain, codomain and range. Answer: The relation R from A to A is given as R = {(x, y): 3x – y = 0, …

## NCERT solution class 11 chapter 2 Sets exercise 2.1 mathematics

EXERCISE 2.1 Page No 33: Question 1: If, find the values of x and y. Answer: It is given that. Since the ordered pairs are equal, the corresponding elements will also be equal. Therefore,  and.   ∴ x = 2 and y = 1 Question 2: If the set A has 3 elements and the set B = {3, 4, 5}, then …

## NCERT solution class 11 chapter 1 Sets exercise 1.7 mathematics

EXERCISE 1.7 Page No 26: Question 1: Decide, among the following sets, which sets are subsets of one and another: A = {x: x ∈ R and x satisfy x2 – 8x + 12 = 0}, B = {2, 4, 6}, C = {2, 4, 6, 8…}, D = {6}. Answer: A = {x: x ∈ R and x satisfies x2 – 8x + 12 = 0} 2 and …

## NCERT solution class 11 chapter 1 Sets exercise 1.6 mathematics

EXERCISE 1.6 Page No 24: Question 1: If X and Y are two sets such that n(X) = 17, n(Y) = 23 and n(X ∪ Y) = 38, find n(X ∩Y). Answer: It is given that: n(X) = 17, n(Y) = 23, n(X ∪ Y) = 38 n(X ∩ Y) = ? We know that: Question 2: If X and Y …

## NCERT solution class 11 chapter 1 Sets exercise 1.5 mathematics

EXERCISE 1.5 Page No 20: Question 1: Let U ={1, 2, 3; 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}. Find (i)  (ii)  (iii)  (iv)  (v)  (vi)  Answer: U ={1, 2, 3, 4, 5, 6, 7, 8, 9} …

## NCERT solution class 11 chapter 1 Sets exercise 1.4 mathematics

EXERCISE 1.4 Page No 17: Question 1: Find the union of each of the following pairs of sets: (i) X = {1, 3, 5} Y = {1, 2, 3} (ii) A = {a, e, i, o, u} B = {a, b, c} (iii) A = {x: x is a natural number and multiple of 3} B = {x: x is a natural number less than 6} (iv) A = …

## NCERT solution class 11 chapter 1 Sets exercise 1.3 mathematics

EXERCISE 1.3 Page No 12: Question 1: Make correct statements by filling in the symbols ⊂ or ⊄ in the blank spaces: (i) {2, 3, 4} … {1, 2, 3, 4, 5} (ii) {a, b, c} … {b, c, d} (iii) {x: x is a student of Class XI of your school} … {x: x student of your school} (iv) {x: x is a circle in the plane} … …

## NCERT solution class 11 chapter 1 Sets exercise 1.2 mathematics

EXERCISE 1.2 Page No 8: Question 1: Which of the following are examples of the null set (i) Set of odd natural numbers divisible by 2 (ii) Set of even prime numbers (iii) {x:x is a natural numbers, x < 5 and x > 7 } (iv) {y:y is a point common to any two parallel lines} Answer: (i) A set of …

## NCERT solution class 11 chapter 1 Sets exercise 1.1 mathematics

EXERCISE 1.1 Page No 4: Question 1: Which of the following are sets? Justify our answer. (i) The collection of all months of a year beginning with the letter J. (ii) The collection of ten most talented writers of India. (iii) A team of eleven best-cricket batsmen of the world. (iv) The collection of all boys in your class. …

## RD sharma class solution 11 chapter 1 Set

Exercise 2.1 Question 1 (i) If find the values of and Sol : By the definition of equality of ordered pairs, we have: and and and b = 1   (ii) If find the values of x and y Sol : By the definition of equality of ordered pairs, we have: and  x = 2 …

## Geometric progression

EXERCISE 20.2 QUESTION 1 Find three numbers in G.P. whose sum is 65 and whose product is 3375 . Sol : Let the terms of the the given G.P. be , a and Then, product of the G.P. =3375 a = 15 Similarly, sum of the G.P. Substituting the value of a Hence, the G.P.for …

## Geometric Progressions

EXERCISE 20.1 QUESTION 1 Show that each one of the following progressions is a G.P. Also, find the common ratio in each case: (i) Sol : We have : ,, , , , Thus, , , and are in G.P., where a=4 and   (ii) Sol : We have : ,, , Thus, , and …

## Arithmetic progression

EXERCISE 19.6 QUESTION 1 Find the A.M. between: (i) 7 and 13 Sol : Let be the A.M. between 7 and 13 = 10   (ii) 12 and -8 Sol : Let be the A.M. between 12 and -8 = 2   (iii) (x – y) and (x + y) Sol : Let be the …

## Arithmetic progression

EXERCISE 19.4 QUESTION 1 Find the sum of the following arithmetic progressions: (i) to 10 terms Sol : We have : a = 50 d = (46 -50) = -4 n = 10 = 320   (ii) to 12 terms Sol : We have : a = 1 d = (3 – 1) = 2 …

## Arithmetic progression

EXERCISE 19.3 QUESTION 1 The sum of three terms of an A.P. is 21 and the product of the first and the third terms exceeds the second term by 6, find three terms. Sol : Let the three terms of the A.P. be  ,  a , Then , we have   Also ,     …

## Arithmetic progression

EXERCISE 19.2 QUESTION 1 Find : (i) 10 th term of the A.P. Sol : We have  a = 1 d = 4 – 1 = 3   = 28   (ii) 18 th term of the A.P. Sol : We have :   (iii) n th term of the A.P. Sol : We have …

## Permutation

Exercise 16.2 QUESTION 1 In a class there are 27 boys and 14 girls . The teacher wants to select 1 boy and 1 girl to represent the class in a function . In how many ways can the teacher make selection ? Sol : No. of boys in the class = 27 No. of …

## Rd sharma solution of class 11 chapter 16 permutation

EXERCISE 16.1 QUESTION 1 Compute : (i) Sol : = 870   (ii) Sol : = 100   (iii) L.C.M. Sol : Therefore, and can be rewritten as: LCM of and   QUESTION 2 Prove that Sol : Hence , proved   QUESTION 3 Find in each of the following: (i) Sol :   (ii) …

## complex number

Exercise 13.4 QUESTION 1 Find the modulus and argument of the following complex number and hence express each of them in the polar form : (i) Sol : Let Since point lies in the first quadrant , the argument of z is given by Polar form   (ii) Sol : Let Since point lies in the …

Exercise 14.1 QUESTION 1 Sol : Hence , the roots of the equation are i and -i    QUESTION 2 Sol : Hence, the roots of the equation are   QUESTION 3 Sol :   Hence, the roots of the equation are    QUESTION 4 Sol : Hence , the roots of the equation are …

## Rd sharma solution of class 11 chapter 13 complex number

Exercise 13.3 QUESTION 1 (i) Sol : , and   (ii) Sol : ,   , ṇṇ   (iii) Sol : , ,   (iv) Sol : , ,   (v) Sol : , ,   (vi) Sol : , ,   (vii) Sol : , ,   (viii) Sol : , ,   (ix) …

## Rd sharma solution of chapter 13 complex number class 11

Exercise 13.2 QUESTION 1 Express the following complex numbers in the standard form a+ib : (i) Sol :   (ii) Sol :   (iii) Sol :     (iv) Sol :     (v) Sol :     (vi) Sol :   (vii) Sol :       (viii) Sol :   (ix) Sol : …

## Rd sharma solution of chapter 13 Complex numbers

Complex numbers Exercise 13.1 QUESTION 1 Evaluate the following: (i) Sol :   (ii) Sol : =1   (iii) Sol : = -1   (iv) Sol :     (v) Sol :   = 0   (vi) Sol : = 0   (vii) Sol : = 1   (viii) Sol :     QUESTION 2 …

## Hydrocarbons

HYDROCARBONS Organic compound are composed of only carbons and hydrogen are called hydrocarbons. ON THE BASIS OF STRUCTURE HYDROCARBONS ARE OF TWO TYPES  1. Acyclic or open chain hydrocarbons 2. Cyclic or closed chain hydrocarbons   Acyclic or open chain hydrocarbons These hydrocarbons contains open chains of carbon atoms in their molecules. These hydrocarbons are …

## WORK, ENERGY AND POWER

WORK, ENERGY AND POWER In your daily life you have seen many things like person pushing heavy box , a teacher teaching students , mom cooking food , all are said to be working . And the person who have capacity to do heavy work is said to be having high stamina or energy. In …

## REDOX REACTIONS

REDOX REACTIONS Chemical reaction in which electron transfer from one chemical substance to another are termed as oxidation-reduction reactions or redox reactions. You can see many examples from your daily life like production of electricity in batteries , production of heat by burning chemical substances , electroplating , manufacturing of useful products caustic soda, potassium …

## Thermodynamics

Thermodynamics The word “thermodynamics” is derived from Greek words therme (heat) and dinamics (flow or motion) . Thermodynamics mainly deals with the transformation of heat into mechanical energy and vice versa .   LIMITATIONS OF THERMODYNAMICS (i) It does not give any direct information about the nature or structure of matter. (ii) It does not give …

## straight line

Straight line EXERCISE 23.1 1. Find the slopes of the lines which make the following angles with the positive direction of x-axis : (i) ​$$-\dfrac{\pi}{4}$$​ sol: m=tan​$$\theta$$​ =tan(​$$-\dfrac{\pi}{4}$$​)  = -1 (ii) ​$$\dfrac{2\pi}{3}$$​ sol: m=tan​$$\theta$$​ =tan(​$$\dfrac{2\pi}{3}$$​)  = tan(​$$\pi-\dfrac{\pi}{3}$$​)  = ​$$-\sqrt{3}$$​ (iii) ​$$\dfrac{3\pi}{4}$$​ …

## Chapter Some Basic Concepts of Chemistry

SOME BASIC CONCEPT OF CHEMISTRY Classification of matter on the basis of chemical properties: Matter classified into two groups pure substances and mixtures.Pure substances further divided into elements and compounds. 1. Elements : it is a simple individual which has a defined atomic number and has a definite position in periodic table. All the elements …

## Chapter classification of elements and periodicity in properties

CLASSIFICATION OF ELEMENTS AND PERIODICITY IN  PROPERTIES Brief Historical Development Of Periodic Table 1.Johann Dobereiner’s (Law of triads) In between 1815 and 1829 he gave law of triads.According to which group of three elements which posses similar chemical properties, then the central element mass is equal to the arithmetic mean of atomic masses of the …

## Chapter Kinematics

KINEMATICS The branch of physics which deals with the study of the objects at rest and in motion is called mechanics. It is further divided into two parts are: 1.Statics Statics is the branch of mechanics which deals with the study of objects at rest. An object can be at rest only when all the …

## Chapter Unit and measurement

UNIT AND MEASUREMENT Units of measurement are vital parts of any physical quantity. Just as the person is known by his or her name, the physical quantities are known by the units of measurement. System of unit There are four categories of system of unit. 1.C.G.S : In the CGS system of measurement length is …

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