Continuity and Differentiability - Exemplar Solutions

 CBSE Class–12 Mathematics

NCERT Exemplar
Chapter - 5 
CONTINUITY AND DIFFERENTIABILITY - Short Answer Questions

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1. Find the value of the constant k so that the function defined below is continuous at 

Sol. It is given that the function is continuous at x = 0. Therefore, 

 (1−cos2x=2sin2x)$(1-\cos 2x=2{\sin }^{2}x)$

 (limx→0sinxx=1)$(\underset{x\to 0}{lim}\frac{\sin x}{x}=1)$

Thus, is continuous at x = 0 if k = 1.

2. Discuss the continuity of the function .

Sol. Since sin x and cos x are continuous functions and product of two continuous

function is a continuous function, therefore is a continuous function.

3. If is continuous at x = 2, find the value of k.

Sol. Given = k.

Now, 

As is continuous at x= 2, we have

4. Show that the function defined by

is continuous at x = 0.

Sol. Left hand limit at x = 0 is given by

Similarly, 

Thus, 

5. Given . Find the points of discontinuity of the composite function 

Sol. We know that is discontinuous at x = 1

Now, for x1,

Which is discontinuous at x = 2.

Hence, the points of discontinuity are x = 1 and x = 2.

6. Let for allDiscuss the derivability of at x = 0

Sol. We may rewrite as 

Now, 

Now, 

Since the left-hand derivative and right hand derivative both are equal, hence is differentiable at x = 0.

7. Differentiate w.r.t. x

Sol. Let . Using chain rule, we get

8. If 

Sol. Given y = tan (x + y). differentiating both sides w.r.t. x, we get

Therefore, 

9. If prove that dydx=−ex−y(1−ey)1−ex$\frac{dy}{dx}=\frac{-e^{x-y}(1-e^y)}{1-e^x}$

Sol. Given that Differentiating both sides w.r.t. x, we have

Which implies that dydx=ex+y−exey−ex+y=ex.ey−exey−ex.ey=ex(ey−1)ey(1−ex)=−ex−y(1−ey)1−ex$\frac{dy}{dx}=\frac{e^{x+y}-e^x}{e^y-e^{x+y}}=\frac{e^x.e^y-e^x}{e^y-e^x.e^y}=\frac{e^x(e^y-1)}{e^y(1-e^x)}=\frac{-e^{x-y}(1-e^y)}{1-e^x}$

10. Find 

Sol. Put x = tan, where 

Therefore, 

Hence, dydx=31+x2$\frac{dy}{dx}=\frac{3}{1+x^2}$

11. If 

Sol. We have 

Put 

Therefore, 

Thus 

Differentiating w.r.t. x, we get

12. If and , find 

Sol. We have x = a sec3and y = a tan3.

Differentiating w.r.t., we get

and 

Thus 

Hence 

13. If prove that 

Sol. We have Taking logarithm on both sides, we get

i.e. 

Differentiating both sides w.r.t. x, we get

14. If , prove that 

Sol. We have . Differentiating w.r.t. x, we get

Thus

Now, differentiating again w.r.t. x, we get

15. If find.

Sol. When < x <, cos x < 0 so that = - cos x, i.e., f (x) = - cos x f ‘ (x) = sin x.

Hence, 

16. If find 

Sol. When cos x >sin x, so that cos x – sin x > 0, i.e.,

Hence, 

17. Verify Rolle’s theorem for the function, 

Sol. Given Note that:

(i) The function is continuous in as is a sine function, which is always continuous.

(ii) exists in , hence is derivable in 

(iii) and = sin= 0 .

Conditions of Rolle’s theorem are satisfied. Hence there exists at least one such that . Thus

18. Verify mean value theorem for the function in [3, 5].

Sol. (i) Function is continuous in [3, 5] as product of polynomial functions is a polynomial, which is continuous.

(ii) exists in (3, 5) and hence derivable in (3, 5).

Thus conditions of mean value theorem are satisfied. Hence, there exists at least one such that

Hence (since other value is not permissible).

 

Long Answer Questions

 

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19. If find the value of so that becomes continuous at 

Sol. Given, 

Therefore, 

Thus, 

If we define then will become continuous at Hence for to be continuous at 

20. Show that the function given by is discontinuous at x = 0.

Sol. The left hand limit of at x = 0 is given by

Similarly, 

Thus therefore, does not exist. Hence is discontinuous at x = 0.

21. Let 

For what value of a, is continuous at x = 0?

Sol. Here Left-hand limit of at 0 is

and right hand limit of at 0 is

Thus, Hence is continuous at x = 0 only if a = 8.

22. Examine the differentiability of the function defined by

f(x)=⎧⎩⎨⎪⎪2x+3,if−3≤x<−2x+1.if−2≤x<0x+2,if0≤x≤1$f(x)=\{\begin{array}{c}2x+3,if-3\le x<-2\hfill \\ x+1.if-2\le x<0\hfill \\ x+2,if0\le x\le 1\hfill \end{array}$

Sol. The only doubtful points for differentiability of are x = –2 and x = 0. Differentiability at x = - 2.

Now 

and 

Thus Therefore is not differentiable at x = -2.

Similarly, for differentiability at x = 0, we have

which does not exist. Hence is not differentiable at x = 0.

23. Differentiate with respect to , where 

Sol. Let and .

We want to find 

Now 

Then 

Hence 

Now 

Hence 

 

Objective Questions

 

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Choose the correct answer from the given four options in each of the Examples 24 to 35.

24. The function is continuous at x = 0, then the value of k is

(A) 3

(B) 2

(C) 1

(D) 1.5

Sol. (B) is the Correct answer.

25. The function where [x] denotes the greatest integer function, is continuous at

(A) 4

(B) - 2

(C) 1

(D) 1.5

Sol. (D) is the correct answer. The greatest integer function[x] is discontinuous at all integral values of x. Thus D is the correct answer.

26. The number of points at which the function is not continuous is

(A) 1

(B) 2

(C) 3

(D) None of these

Sol. (D) is the correct answer. As x – [x] = 0, when x is an integer so is discontinuous for all 

27. The function given by is discontinuous on the set

(A) 

(B) 

(C) 

(D) 

Sol. C is the correct answer.

28. Let Then

(A) is everywhere differentiable.

(B) is everywhere continuous but not differentiable at 

(C) is everywhere continuous but not differentiable at 

(D) None of these.

Sol. C is the correct answer.

29. The function is

(A) continuous at x = 0 as well as at x = 1.

(B) continuous at x = 1 but not at x = 0.
(C) discontinuous at x = 0 as well as at x = 1.
(D) continuous at x = 0 but not at x = 1.

Sol. Correct answer is A.

30. The value of k which makes the function defined by

(A) 8

(B) 1

(C) -1

(D) None of these

Sol. (D) is the correct answer. Indeed does not exist.

31. The set of points where the functions f given by differentiable is

(A) R

(B) 

(C) 

(D) None of these

Sol. B is the correct answer.

32. Differential coefficient of sec (tan-1 x) w.r.t. x is

(A) 

(B) 

(C) 

(D) 

Sol. (A) is the correct answer.

33. If and then is

(A) 

(B) x

(C) 

(D) 1

Sol. (D) is the correct answer.

34. The value of c in Rolle’s Theorem for the function is

(A) 

(B) 

(C) 

(D) 

Sol. (D) is the correct answer.

35. The value of c in Mean value theorem for the function is

(A) 

(B) 

(C) 

(D) 

Sol. (A) is the correct answer.

36. Match the following

COLUMN – I	COLUMN – II
(A) If a function is continuous at x = 0, then k is equal to	(A) |x|
(B) Every continuous function is differentiable	(B) True
(C) An example of a function which is continuous everywhere but not differentiable at exactly one point	(C) 6
(D) The identify function i.e. is a continuous function.	(D) False

Sol. 

Fill in the blanks in each of the Examples 37 to 41.

37. The number of points at which the function is discontinuous is _____________.

Sol. The given function is discontinuous at x = 0, 1 and hence the number of points of discontinuity is 3.

38. If is continuous, then a should be equal to ____________.

Sol. a = 2

39. The derivative of w.r.t. x is _____________.

Sol. 

40. If then is equal to ___________.

Sol. 0.

41. The derivative of sin x w.r.t. cos x is ___________.

Sol. –cot x

Sate whether the statements are True of False in each of the Exercises 42 to 46.

42. For continuity, at x = a, each of and is equal to .

Sol. True.

43. y = |x – 1| is a continuous function.

Sol. True.

44. A continuous function can have some points where limit does not exist.

Sol. False.

45. |sin x| is a differentiable function for every value of x.

Sol. False.

46. cos |x| is differentiable everywhere.

Sol. True.