class 12

Rd sharma solution class 12 chapter continuity

Exercise 9.2 Question 1 Prove that the function is everywhere continuous . Sol : When , we have We know that sin x as well as the identity function x are everywhere continuous . So , the quotient function is continuous at each point When , we have , which is a polynomial function . Therefore, is …

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 …

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 …

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 …

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 …

Derivative as a rate measure

Derivative as a rate measure Exercise 13.1 1. Find the rate of change of the total surface area of a cylinder of radius r and height h, when the radius varies. Sol :   2. Find the rate of change of the volume of a sphere with respect to its diameter. Sol :   3. …

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 :

Rd sharma solution Chapter 5 Algebra of matrices

Exercise 5.1 Exercise 5.2  Exercise 5.3 EXERCISE 5.1 Question 1 If a matrix has 8 elements, what are the possible orders it can have? What if it has 5 elements? Sol: Order of matrix = number of rows ​ number of column OR Total number of elements = number of rows number of column Thus, …

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