# NCERT Solutions for Class 12 Maths Maths Part-1 Chapter 5

## Continuity and Differentiability Class 12

### Exercise 5.1 : Solutions of Questions on Page Number : 159

Q1 :

Prove that the functionis continuous at

Therefore, f is continuous at x= 0

Therefore, f is continuous at x= - 3

Therefore, f is continuous at x= 5

Q2 :

Examine the continuity of the function.

Thus, f is continuous at x= 3

Q3 :

Examine the following functions for continuity.

(a) (b)

(c) (d)

Q4 :

Prove that the function is continuous at x= n, where n is a positive integer.

Q5 :

Is the function fdefined by

continuous at x= 0? At x= 1? At x= 2?

Q6 :

Find all points of discontinuity of f, where f is defined by

Q7 :

Find all points of discontinuity of f, where f is defined by

Q8 :

Find all points of discontinuity of f, where f is defined by

Q9 :

Find all points of discontinuity of f, where f is defined by

Q10 :

Find all points of discontinuity of f, where f is defined by

Q11 :

Find all points of discontinuity of f, where f is defined by

Q12 :

Find all points of discontinuity of f, where f is defined by

Q13 :

Is the function defined by

a continuous function?

Q14 :

Discuss the continuity of the function f, where f is defined by

Q15 :

Discuss the continuity of the function f, where f is defined by

Q16 :

Discuss the continuity of the function f, where f is defined by

Q17 :

Find the relationship between aand b so that the function fdefined by

is continuous at x = 3.

Q18 :

For what value of is the function defined by

continuous at x = 0? What about continuity at x= 1?

Q19 :

Show that the function defined by is discontinuous at all integral point. Here denotes the greatest integer less than or equal to x.

Q20 :

Is the function defined by continuous at x = π?

Q21 :

Discuss the continuity of the following functions.

(a) f(x) = sin x + cos x

(b) f(x) = sin x - cos x

(c) f(x) = sin x x cos x

Q22 :

Discuss the continuity of the cosine, cosecant, secant and cotangent functions,

Q23 :

Find the points of discontinuity of f, where

Q24 :

Determine if fdefined by

is a continuous function?

Q25 :

Examine the continuity of f, where f is defined by

Q26 :

Find the values of k so that the function fis continuous at the indicated point.

Q27 :

Find the values of k so that the function fis continuous at the indicated point.

Q28 :

Find the values of k so that the function fis continuous at the indicated point.

Q29 :

Find the values of k so that the function fis continuous at the indicated point.

Q30 :

Find the values of aand b such that the function defined by

is a continuous function.

Q31 :

Show that the function defined by f (x) = cos (x2) is a continuous function.

Q32 :

Show that the function defined by is a continuous function.

Q33 :

Examine that is a continuous function.

Q34 :

Find all the points of discontinuity of f defined by.

### Exercise 5.2 : Solutions of Questions on Page Number : 166

Q1 :

Differentiate the functions with respect to x.

Q2 :

Differentiate the functions with respect to x.

Q3 :

Differentiate the functions with respect to x.

Q4 :

Differentiate the functions with respect to x.

Q5 :

Differentiate the functions with respect to x.

Q6 :

Differentiate the functions with respect to x.

Q7 :

Differentiate the functions with respect to x.

Q8 :

Differentiate the functions with respect to x.

Q9 :

Prove that the function f given by

is notdifferentiable at x = 1.

Q10 :

Prove that the greatest integer function defined byis not

differentiable at x = 1 and x = 2.

Q1 :

Find :

Q2 :

Find :

Q3 :

Find :

Q4 :

Find :

Q5 :

Find :

Q6 :

Find :

Q7 :

Find :

Q8 :

Find :

Q9 :

Find :

Q10 :

Find :

Q11 :

Find :

Q12 :

Find :

Q13 :

Find :

Q14 :

Find :

Q15 :

Find :

### Exercise 5.4 : Solutions of Questions on Page Number : 174

Q1 :

Differentiate the following w.r.t. x:

Q2 :

Differentiate the following w.r.t. x:

Q3 :

Differentiate the following w.r.t. x:

Q4 :

Differentiate the following w.r.t. x:

Q5 :

Differentiate the following w.r.t. x:

Q6 :

Differentiate the following w.r.t. x:

Q7 :

Differentiate the following w.r.t. x:

Q8 :

Differentiate the following w.r.t. x:

Q9 :

Differentiate the following w.r.t. x:

Q10 :

Differentiate the following w.r.t. x:

### Exercise 5.5 : Solutions of Questions on Page Number : 178

Q1 :

Differentiate the function with respect to x.

Q2 :

Differentiate the function with respect to x.

Q3 :

Differentiate the function with respect to x.

Q4 :

Differentiate the function with respect to x.

Q5 :

Differentiate the function with respect to x.

Q6 :

Differentiate the function with respect to x.

Q7 :

Differentiate the function with respect to x.

Q8 :

Differentiate the function with respect to x.

Q9 :

Differentiate the function with respect to x.

Q10 :

Differentiate the function with respect to x.

Q11 :

Differentiate the function with respect to x.

Q12 :

Find of function.

Q13 :

Find of function.

Q14 :

Find of function.

Q15 :

Find of function.

Q16 :

Find the derivative of the function given by and hence find.

Q17 :

Differentiate in three ways mentioned below

(i) By using product rule.

(ii) By expanding the product to obtain a single polynomial.

(iii By logarithmic differentiation.

Do they all give the same answer?

Q18 :

If u, v and w are functions of x, then show that

in two ways-first by repeated application of product rule, second by logarithmic differentiation.

### Exercise 5.6 : Solutions of Questions on Page Number : 181

Q1 :

If x and y are connected parametrically by the equation, without eliminating the parameter, find.

Q2 :

If x and y are connected parametrically by the equation, without eliminating the parameter, find.

x = a cos θ, y = b cos θ

Q3 :

If x and y are connected parametrically by the equation, without eliminating the parameter, find.

x = sin t, y = cos 2t

Q4 :

If x and y are connected parametrically by the equation, without eliminating the parameter, find.

Q5 :

If x and y are connected parametrically by the equation, without eliminating the parameter, find.

Q6 :

If x and y are connected parametrically by the equation, without eliminating the parameter, find.

Q7 :

If x and y are connected parametrically by the equation, without eliminating the parameter, find.

Q8 :

If x and y are connected parametrically by the equation, without eliminating the parameter, find.

Q9 :

If x and y are connected parametrically by the equation, without eliminating the parameter, find.

Q10 :

If x and y are connected parametrically by the equation, without eliminating the parameter, find.

Q11 :

If

### Exercise 5.7 : Solutions of Questions on Page Number : 183

Q1 :

Find the second order derivatives of the function.

Q2 :

Find the second order derivatives of the function.

Q3 :

Find the second order derivatives of the function.

Q4 :

Find the second order derivatives of the function.

Q5 :

Find the second order derivatives of the function.

Q6 :

Find the second order derivatives of the function.

Q7 :

Find the second order derivatives of the function.

Q8 :

Find the second order derivatives of the function.

Q9 :

Find the second order derivatives of the function.

Q10 :

Find the second order derivatives of the function.

Q11 :

If, prove that

Q12 :

If findin terms of y alone.

Q13 :

If, show that

Q14 :

Ifshow that

Q15 :

If, show that

Q16 :

If, show that

Q17 :

If, show that

### Exercise 5.8 : Solutions of Questions on Page Number : 186

Q1 :

Verify Rolle's Theorem for the function

Q2 :

Examine if Rolle's Theorem is applicable to any of the following functions. Can you say some thing about the converse of Rolle's Theorem from these examples?

(i)

(ii)

(iii)

Q3 :

If is a differentiable function and if does not vanish anywhere, then prove that.

Q4 :

Verify Mean Value Theorem, if in the interval, where and.

Q5 :

Verify Mean Value Theorem, if in the interval [a, b], where a = 1 and b = 3. Find all for which

Q6 :

Examine the applicability of Mean Value Theorem for all three functions given in the above exercise 2.

### Exercise Miscellaneous : Solutions of Questions on Page Number : 191

Q1 :

Q2 :

Q3 :

Q4 :

Q5 :

Q6 :

Q7 :

Q8 :

, for some constant aand b.

Q9 :

Q10 :

, for some fixed and

Q11 :

, for

Q12 :

Find, if

Q13 :

Find, if

Q14 :

If, for, - 1 < x<1, prove that

Q15 :

If, for some prove that

is a constant independent of aand b.

Q16 :

If with prove that

Q17 :

If and, find

Q18 :

If, show that exists for all real x, and find it.

Q19 :

Using mathematical induction prove that for all positive integers n.

Q20 :

Using the fact that sin (A + B) = sin Acos B + cos A sin B and the differentiation, obtain the sum formula for cosines.

Q21 :   Does there exist a function which is continuos everywhere but not differentiable at exactly two points? Justify your answer ?

Q22 :

If, prove that

Q23 :

If, show that