# maths

## NCERT solution class 9 chapter Surface area and volume

Exercise 13.1 Question 1 A plastic box 1.5 m long, 1.25 m wide and 65 cm deep, is to be made. It is to be open at the top. Ignoring the thickness of the plastic sheet, determine: (i) The area of the sheet required for making the box. (ii) The cost of sheet for it, if …

## Rd sharma solution class 12 chapter 2 functions

Exercise 3.1 Question 1 Give an example of a function (i) which is one-one but not onto Sol : Injectivity : Let x and y be any two elements in the domain (Z) , such that  So , is one-one   Surjectivity : Let y be any element in the co-domain (Z) , such that for some …

## NCERT solution class 10 chapter 2 polynomials

Exercise 2.2 Question 1 Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (i) Sol : or or Therefore , the zeroes of are 4 and -2 . Sum of zeroes Product of zeroes   (ii) Sol : Therefore , the zeroes of are and Sum …

## NCERT solution class 10 chapter 2 polynomials

Exercise 2.1 Question 1 The graphs of y = p(x) are given in following figure, for some polynomials p(x). Find the number of zeroes of p(x), in each case. (i)   Sol : (i) The number of zeroes is 0 as the graph does not cut the x-axis at any point.   (ii)   Sol …

## NCERT solution class 10 chapter 1 Real numbers

Exercise 1.1    Exercise 1.2   Exercise 1.3   Exercise 1.4 Exercise 1.4 Question 1 Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion: (i) Sol : The denominator is in the form of Hence, the decimal expansion of is terminating. …

## NCERT solution class 10 chapter 1 Real numbers

Exercise 1.1    Exercise 1.2   Exercise 1.3   Exercise 1.4 Exercise 1.3 Question 1 Prove that is irrational . Sol : Let is a rational number. Therefore, we can find two integers a , b such that Let a and b have a common factor other than 1 . Then we can divide them by the …

## NCERT solution class 10 chapter 1 Real numbers

Exercise 1.1    Exercise 1.2   Exercise 1.3   Exercise 1.4 Exercise 1.2 Question 1 Express each number as product of its prime factors: (i) 140 Sol :   (ii) 156 Sol :   (iii) 3825 Sol :   (iv) 5005 Sol :   (v) 7429 Sol :   Question 2 Find the LCM and HCF of …

## Rd sharma solution class 12 chapter Relations

Exercise 1.1 Question 1 Let A be the set of all human beings in a town at a particular time. Determine whether each of the following relations are reflexive, symmetric and transitive: (i) R = {(x , y) : x and y work at the same place } Sol : (Reflexivity) Let x be an …

## Rd sharma solution class 12 chapter Differentiation

Exercise 11.2 Question 1 Differentiate  Sol : Let  Differentiating y with respect to x we get, So,     Question 2 Differentiate Sol : Let  which can be written as Differentiating y with respect to x we get, So,     Question 3 Differentiate Sol : Let  Differentiating y with respect to x we get, …

## Rd sharma solution class 12 chapter Inverse trigonometric functions

Exercise 4.1 Question 1  Find the principal value of each of the following: (i) Sol :   (ii) Sol :   (iii)  Sol :       (iv)   Sol :             (v) Sol :   (vi) Sol : Let Therefore We know that principal value of and Therefore principle …

## Rd sharma solution class 12 chapter Continuity

Exercise 9.1 Question 1 Test the continuity of the function on at the origin: Sol : Given We observe (LHL at x= 0) =-1   (RHL at x = 0) =1 Hence, is discontinuous at the origin.   Question 2 A function is defined as  Show that is continuous that Sol : Given We observe …

## NCERT solution class 10 chapter 1 Real numbers

Exercise 1.1    Exercise 1.2   Exercise 1.3   Exercise 1.4 Exercise 1.1 Question 1 Use Euclid’s division algorithm to find the HCF of: (i) 135 and 225 Sol : (i) 135 and 225 Since 225 > 135, we apply the division lemma to 225 and 135 to obtain 225 = 135 × 1 + 90 …

## Rd sharma solution class 1 chapter Indefinite integrals

Exercise 19.2 Question 1

## Rd sharma solution class 1 chapter Indefinite integrals

Exercise 19.1 Question 1 Evaluate each of the following integrals: (i) Sol :   (ii) Sol :   (iii) Sol :   (iv)  Sol :   (v)  Sol :   (vi)  Sol :   (vii) Sol :   (viii) Sol :     Question 2 Evaluate : (i) Sol :   (ii)  Sol :   …

## RD Sharma solution class 12 chapter 28 Straight line in space

Exercise 28.1 Question 1 Find the vector and cartesian equations of the line through the point and which is parallel to the vector   Sol : We know that the vector equation of a line passing through a point with position vector and parallel to is   Here, Vector equation of the required line is given …

## NCERT solution class 8 chapter 16 Playing with numbers

Exercise 16.1   Exercise 16.2 Exercise 16.2 Question 1 If 21y5 is a multiple of 9, where y is a digit, what is the value of y? Sol : If a number is a multiple of 9, then the sum of its digits will be divisible by 9. Sum of digits of 21y5 = 2 + …

## NCERT solution class 8 chapter 16 Playing with numbers

Exercise 16.1   Exercise 16.2 Exercise 16.1 Question 1 Find the values of the letters in the following and give reasons for the steps involved. Sol : The addition of A and 5 is giving 2 i.e., a number whose ones digit is 2. This is possible only when digit A is 7. In that case, …

## NCERT solution class 8 chapter 15 Introduction to Graphs

Exercise 15.1   Exercise 15.2   Exercise 15.3 Exercise 15.3 Question 1  Draw the graphs for the following tables of values, with suitable scales on the axes. (a) Cost of apples Number of apples 1 2 3 4 5 Cost (in Rs) 5 10 15 20 25 (b) Distance travelled by a car Time (in hours) 6 …

## NCERT solution class 8 chapter 15 Introduction to Graphs

Exercise 15.1   Exercise 15.2   Exercise 15.3 Exercise 15.2 Question 1  Plot the following points on a graph sheet. Verify if they lie on a line (a) A(4, 0), B(4, 2), C(4, 6), D(4, 2.5) (b) P(1, 1), Q(2, 2), R(3, 3), S(4, 4) (c) K(2, 3), L(5, 3), M(5, 5), N(2, 5) Sol : (a) …

## NCERT solution class 8 chapter 15 Introduction to Graphs

Exercise 15.1   Exercise 15.2   Exercise 15.3 Exercise 15.1 Question 1  The following graph shows the temperature of a patient in a hospital, recorded every hour. (a) What was the patient’s temperature at 1 p.m.? (b) When was the patient’s temperature 38.5°C? (c) The patient’s temperature was the same two times during the period given. What …

## NCERT solution class 8 chapter 14 Factorisation

Exercise 14.1   Exercise 14.2   Exercise 14.3   Exercise 14.4 Exercise 14.4 Question 1  Find and correct the errors in the statement: 4(x − 5) = 4x − 5 Sol : L.H.S. = 4(x − 5) = 4 × x − 4 × 5 = 4x − 20 ≠ R.H.S. The correct statement is 4(x − 5) …

## NCERT solution class 8 chapter 14 Factorisation

Exercise 14.1   Exercise 14.2   Exercise 14.3   Exercise 14.4 Exercise 14.3 Question 1  Carry out the following divisions. (i) 28×4 ÷ 56x (ii) −36y3 ÷ 9y2 (iii) 66pq2r3 ÷ 11qr2 (iv) 34x3y3z3 ÷ 51xy2z3 (v) 12a8b8 ÷ (−6a6b4) Sol : (i) 28×4 = 2 × 2 × 7 × x × x × x × x …

## NCERT solution class 8 chapter 14 Factorisation

Exercise 14.1   Exercise 14.2   Exercise 14.3   Exercise 14.4 Exercise 14.2 Question 1  Factorise the following expressions. (i) a2 + 8a + 16 (ii) p2 − 10p + 25 (iii) 25m2 + 30m + 9 (iv) 49y2 + 84yz + 36z2 (v) 4×2 − 8x + 4 (vi) 121b2 − 88bc + 16c2 (vii) (l + …

## NCERT solution class 8 chapter 13 Direct and Inverse proportions

Exercise 13.1   Exercise 13.2 Exercise 13.2 Question 1  Which of the following are in inverse proportion? (i) The number of workers on a job and the time to complete the job. (ii) The time taken for a journey and the distance travelled in a uniform speed. (iii) Area of cultivated land and the crop harvested. …

## NCERT solution class 8 chapter 13 Direct and Inverse proportions

Exercise 13.1   Exercise 13.2 Exercise 13.1 Question 1  Following are the car parking charges near a railway station up to 4 hours Rs 60 8 hours Rs 100 12 hours Rs 140 24 hours Rs 180 Check if the parking charges are in direct proportion to the parking time. Sol : A table of the …

## NCERT solution class 8 chapter 12 Exponents and Power

Exercise 12.1   Exercise 12.2 Exercise 12.2 Question 1 Express the following numbers in standard form. (i) 0.0000000000085 (ii) 0.00000000000942 (iii) 6020000000000000 (iv) 0.00000000837 (v) 31860000000 Sol : (i) 0.0000000000085 = 8.5 × 10−12 (ii) 0.00000000000942 = 9.42 × 10−12 (iii) 6020000000000000 = 6.02 × 1015 (iv) 0.00000000837 = 8.37 × 10−9 (v) 31860000000 = 3.186 …

## NCERT solution class 8 chapter 12 Exponents and Power

Exercise 12.1   Exercise 12.2 Exercise 12.1 Question 1 Evaluate (i) 3−2 (ii) (−4)−2 (iii) Sol : (i) (ii) (iii)     Question 2 Simplify and express the result in power notation with positive exponent. (i) (ii) (iii) (iv) (v) Sol : (i) (−4)5 ÷ (−4)8 = (−4)5 − 8 (am ÷ an = am − …

## NCERT solution class 8 chapter 11 Mensuration

Exercise 11.1   Exercise 11.2   Exercise 11.3   Exercise 11.4 Exercise 11.4 Question 1 Given a cylindrical tank, in which situation will you find surface area and in which situation volume. (a) To find how much it can hold (b) Number of cement bags required to plaster it (c) To find the number of smaller tanks that …

## NCERT solution class 8 chapter 11 Mensuration

Exercise 11.1   Exercise 11.2   Exercise 11.3   Exercise 11.4 Exercise 11.3 Question 1 There are two cuboidal boxes as shown in the adjoining figure. Which box requires the lesser amount of material to make? Sol : We know that, Total surface area of the cuboid = 2 (lh + bh + lb) Total surface area of …

## NCERT solution class 8 chapter 11 Mensuration

Exercise 11.1   Exercise 11.2   Exercise 11.3   Exercise 11.4 Exercise 11.2 Question 1 The shape of the top surface of a table is a trapezium. Find its area if its parallel sides are 1 m and 1.2 m and perpendicular distance between them is 0.8 m. E Sol : Area of trapezium = (Sum of parallel …

## NCERT solution class 8 chapter 11 Mensuration

Exercise 11.1   Exercise 11.2   Exercise 11.3   Exercise 11.4 Exercise 11.1 Question 1 A square and a rectangular field with measurements as given in the figure have the same perimeter. Which field has a larger area? Sol : Perimeter of square = 4 (Side of the square) = 4 (60 m) = 240 m Perimeter of …

## NCERT solution class 8 chapter 10 Visualising Solid Shapes

Exercise 10.1  Exercise 10.2  Exercise 10.3 Exercise 10.3 Question 1 Can a polyhedron have for its faces (i) 3 triangles? (ii) 4 triangles? (iii) a square and four triangles? Sol : (i) No, such a polyhedron is not possible. A polyhedron has minimum 4 faces. (ii) Yes, a triangular pyramid has 4 triangular faces. (iii) Yes, a …

## NCERT solution class 8 chapter 10 Visualising Solid Shapes

Exercise 10.1  Exercise 10.2  Exercise 10.3 Exercise 10.2 Question 1 Look at the given map of a city. Answer the following. (a) Colour the map as follows: Blue − water plant, red − fire station, orange − library, yellow − schools, green − park, pink − college, purple − hospital, brown − cemetery. (b) Mark a green …

## NCERT solution class 8 chapter 10 Visualising Solid Shapes

Exercise 10.1  Exercise 10.2  Exercise 10.3 Exercise 10.1 Question 1 For each of the given solid, the two views are given. Match for each solid the corresponding top and front views. Sol : The given solids, matched to their respective side view and top view, are as follows. Object Side view Top view   Question 2 For …

## NCERT solution class 8 chapter 9 Algebraic Expression and Identities

Exercise 9.1  Exercise 9.2  Exercise 9.3  Exercise 9.4  Exercise 9.5 Exercise 9.4 Question 1 Multiply the binomials. (i) (2x + 5) and (4x − 3) (ii) (y − 8) and (3y − 4) (iii) (2.5l − 0.5m) and (2.5l + 0.5m) (iv) (a + 3b) and (x + 5) (v) (2pq + 3q2) and (3pq − 2q2) (vi) …

## NCERT solution class 8 chapter 9 Algebraic Expression and Identities

Exercise 9.1  Exercise 9.2  Exercise 9.3  Exercise 9.4  Exercise 9.5 Exercise 9.5 Question 1 Use a suitable identity to get each of the following products. (i) (x + 3) (x + 3) (ii) (2y + 5) (2y + 5) (iii) (2a ­− 7) (2a − 7) (iv) (v) (1.1m − 0.4) (1.1 m + 0.4) (vi) (a2 + …

## NCERT solution class 8 chapter 9 Algebraic Expression and Identities

Exercise 9.1  Exercise 9.2  Exercise 9.3  Exercise 9.4  Exercise 9.5 Exercise 9.3 Question 1 Carry out the multiplication of the expressions in each of the following pairs. (i) 4p, q + r (ii) ab, a − b (iii) a + b, 7a2b2 (iv) a2 − 9, 4a (v) pq + qr + rp, 0   Sol : (i) …

## NCERT solution class 8 chapter 9 Algebraic Expression and Identities

Exercise 9.1  Exercise 9.2  Exercise 9.3  Exercise 9.4  Exercise 9.5 Exercise 9.2 Question 1 Find the product of the following pairs of monomials. (i) 4, 7p (ii) − 4p, 7p (iii) − 4p, 7pq (iv) 4p3, − 3p (v) 4p, 0 Sol : The product will be as follows. (i) = 4 × 7p = 4 × 7 …

## NCERT solution class 8 chapter 9 Algebraic Expression and Identities

Exercise 9.1  Exercise 9.2  Exercise 9.3  Exercise 9.4  Exercise 9.5 Exercise 9.1 Question 1 Identify the terms, their coefficients for each of the following expressions. (i) 5xyz2 − 3zy (ii) 1 + x + x2 (iii) 4x2y2 − 4x2y2z2 + z2 (iv) 3 − pq + qr − rp (v) (vi) 0.3a − 0.6ab + 0.5b   Sol …

## NCERT solution class 8 chapter 8 Comparing quantities

Exercise 8.1  Exercise 8.2  Exercise 8.3 Exercise 8.3 Question 1 Calculate the amount and compound interest on (a) Rs 10800 for 3 years at per annum compounded annually. Sol : (a) Principal (P) = Rs 10, 800 Rate (R) = = % (annual) Number of years (n) = 3 Amount, A = C.I. = A …

## NCERT solution class 8 chapter 8 Comparing quantities

Exercise 8.1  Exercise 8.2  Exercise 8.3 Exercise 8.2 Question 1 A man got a 10% increase in his salary. If his new salary is Rs 1,54,000, find his original salary. Sol : Let the original salary be x. It is given that the new salary is Rs 1,54,000. Original salary + Increment = New salary …

## NCERT solution class 8 chapter 8 Comparing quantities

Exercise 8.1  Exercise 8.2  Exercise 8.3 Exercise 8.1 Question 1 Find the ratio of the following: (a) Speed of a cycle 15 km per hour to the speed of scooter 30 km per hour. (b) 5 m to 10 km (c) 50 paise to Rs 5   Sol : (a) Ratio of the speed of …

## NCERT solution class 8 chapter 7 Cube and Cube roots

Exercise 7.1  Exercise 7.2 Exercise 7.2 Question 1 Find the cube root of each of the following numbers by prime factorization method. (i) 64 Sol : (i) Prime factorization of ∴   (ii) 512 Sol : (ii) Prime factorization of ∴   (iii) 10648 Sol : (iii) Prime factorization of ∴   (iv) 27000 Sol …

## NCERT Solutions for class 8 Maths chapter 6 Square and square roots

Exercise 6.1  Exercise 6.2  Exercise 6.3  Exercise 6.4 Exercise 6.3 Question 1 What could be the possible ‘one’s’ digits of the square root of each of the following numbers? (i) 9801 (ii) 99856 (iii) 998001 (iv) 657666025 Sol : (i) If the number ends with 1, then the one’s digit of the square root of …

## NCERT Solutions for class 8 Maths chapter 6 Square and square roots

Exercise 6.1  Exercise 6.2  Exercise 6.3  Exercise 6.4 Exercise 6.4 Question 1 Find the square root of each of the following numbers by division method. (i) 2304 Sol : (i) The square root of 2304 can be calculated as follows. 48 4 88 704 704 0 ∴   (ii) 4489 Sol : (ii) The square …

## NCERT Solutions for class 8 Maths chapter 6 Square and square roots

Exercise 6.1  Exercise 6.2  Exercise 6.3  Exercise 6.4 Exercise 6.2 Question 1 Find the square of the following numbers (i) 32 (ii) 35 (iii) 86 (iv) 93 (v) 71 (vi) 46 Sol : (i) 322 = (30 + 2)2 = 30 (30 + 2) + 2 (30 + 2) = 302 + 30 × 2 …

## NCERT Solutions for class 8 Maths chapter 5 Data handling

Exercise 5.1    Exercise 5.2   Exercise 5.3 Exercise 5.2   Question 1 A survey was made to find the type of music that a certain group of young people liked in a city. Adjoining pie chart shows the findings of this survey. From this pie chart answer the following − (i) If 20 people liked …

## NCERT Solutions for class 8 Maths chapter 5 Data handling

Exercise 5.1    Exercise 5.2   Exercise 5.3 Exercise 5.1 Question 1 For which of these would you use a histogram to show the data? (a) The number of letters for different areas in a postman’s bag. (b) The height of competitors in an athletics meet. (c) The number of cassettes produced by 5 companies. (d) …

## RD sharma solution class 12 chapter determinants

Determinants Exercise 6.1 Question 1 Write the minors and cofactors of each element of the first column of the following matrices and hence evaluate the determinant in each case: (i) A = Sol : In a matrix, the minor is obtained for a particular element , by deleting that row and column where the element …

## NCERT Solutions for class 8 Maths chapter 4 Practical geometry

Exercise 4.1  Exercise 4.2  Exercise 4.3  Exercise 4.4  Exercise 4.5 Exercise 4.5 Question 1 Draw the following: The square READ with RE = 5.1 cm Sol : All the sides of a square are of the same measure and also all the interior angles of a square are of 90º measure. Therefore, the given square …

## NCERT Solutions for class 8 Maths chapter 4 Practical geometry

Exercise 4.1  Exercise 4.2  Exercise 4.3  Exercise 4.4  Exercise 4.5 Exercise 4.4 Question 1 Construct the following quadrilaterals, (i) Quadrilateral DEAR DE = 4 cm EA = 5 cm AR = 4.5 cm ∠E = 60° ∠A = 90° Sol : (Step-1)A rough sketch of this quadrilateral can be drawn as follows. (Step-2) Draw a …

## NCERT Solutions for class 8 Maths chapter 4 Practical geometry

Exercise 4.1  Exercise 4.2  Exercise 4.3  Exercise 4.4  Exercise 4.5 Exercise 4.3 Question 1 Construct the following quadrilaterals. (i) Quadrilateral MORE MO = 6 cm OR = 4.5 cm ∠M = 60° ∠O = 105° ∠R = 105° Sol : (Step-1)A rough sketch of this quadrilateral can be drawn as follows. (Step-2) Draw a line …

## NCERT Solutions for class 8 Maths chapter 4 Practical geometry

Exercise 4.1  Exercise 4.2  Exercise 4.3  Exercise 4.4  Exercise 4.5 Exercise 4.2 Question 1 Construct the following quadrilaterals. (i) Quadrilateral LIFT LI = 4 cm IF = 3 cm TL = 2.5 cm LF = 4.5 cm IT = 4 cm Sol : A rough sketch of this quadrilateral can be drawn as follows. (Step-1) …

## NCERT Solutions for class 8 Maths chapter 4 Practical geometry

Exercise 4.1  Exercise 4.2  Exercise 4.3  Exercise 4.4  Exercise 4.5 Exercise 4.1 Question 1 Construct the following quadrilaterals. (i) Quadrilateral ABCD AB = 4.5 cm BC = 5.5 cm CD = 4 cm AD = 6 cm AC = 7 cm Sol : Firstly, a rough sketch of this quadrilateral can be drawn as follows. …

## NCERT Solutions for class 8 Maths chapter 3 Understanding Quadrilaterals

Exercise 3.1  Exercise 3.2  Exercise 3.3  Exercise 3.4 Exercise 3.4 Question 1 State whether True or False. (a) All rectangles are squares. (b) All rhombuses are parallelograms. (c) All squares are rhombuses and also rectangles. (d) All squares are not parallelograms. (e) All kites are rhombuses. (f) All rhombuses are kites. (g) All parallelograms are …

## NCERT Solutions for class 8 Maths chapter 3 Understanding Quadrilaterals

Exercise 3.1  Exercise 3.2  Exercise 3.3  Exercise 3.4 Exercise 3.3 Question 1 Given a parallelogram ABCD. Complete each statement along with the definition or property used. (i) AD = … (ii) ∠DCB = … (iii) OC = … (iv) m∠DAB + m∠CDA = …   Sol : (i) In a parallelogram, opposite sides are equal …

## NCERT Solutions for class 8 Maths chapter 2 Linear Equation In One Variable

Exercise 2.1  Exercise 2.2  Exercise 2.3  Exercise 2.4  Exercise 2.5  Exercise 2.6 Exercise 2.6 Question 1 : Solve : ​$$\dfrac{8x-3}{3x}=2$$​ Sol :  ​$$\dfrac{8x-3}{3x}=2$$​ On multiplying both sides by 3x , we obtain 8x -3 = 6x 8x – 6x = 3 2x = 3 ​$$x=\dfrac{3}{2}$$​   Question 2 : Solve : ​$$\dfrac{9x}{7-6x}=15$$​ Sol : …

## NCERT Solutions for class 8 Maths chapter 2 Linear Equation In One Variable

Exercise 2.1  Exercise 2.2  Exercise 2.3  Exercise 2.4  Exercise 2.5  Exercise 2.6 EXERCISE 2.1 Question 1 : Solve : x – 2=7 sol : x – 2=7 On transposing 2 to R.H.S , we obtain  x = 7 + 2 x = 9   Question 2 : Solve : y+3=10 sol :  y+3=10 On transposing 3 to R.H.S , we …

## NCERT Solution for class 8 chapter 1 RATIONAL NUMBER

Exercise 1.1 Exercise 1.2 EXERCISE 1.2 Question 1 :  Represent these numbers on the number line. (i)  Sol: can be represented on the number line as follows.   (ii) Sol : can be represented on the number line as follows.   Question 2 :  Represent on the number line. can be represented on the number …

## DIFFERENTIATION

Differentiation Exercise 11.1 Differentiate the following functions from first principles : QUESTION 1 Sol :   QUESTION 2 Sol : QUESTION 3 Sol :   QUESTION 4 Sol :   QUESTION 5 Sol :   Question 6 Sol : Question 7 Sol :   Sol :   Sol :   Sol :

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