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## Continuity and Differentiability Class 12 Maths MCQs Pdf

1. Given functions f(x) = \(\frac{x^{2}-4}{x-2}\) and g(x) = x + 2, x <= R. Then which of the following is correct?

(a) f is continuous at x = 2,

g is continuous at x = 2

(b) f is continuous at x = 2,

g is not continuous at x = 2

(c) f is not continuous at x = 2,

g is continuous at x = 2

(d) f is not continuous at x = 2,

g is not continuous at x = 2

**Answer/Explanation**

Answer: c

Explaination: (c), as f(2) is not defined so / is not continuous at x = 2 ‘g’ is a polynomial function, so continuous at x = 2.

2.

**Answer/Explanation**

Answer: d

Explaination:

3.

for x = 2, then value of k for which f is continuous is

(a) -2

(b) -1

(c) 0

(d) 1

**Answer/Explanation**

Answer: d

Explaination:

4. A function /is said to be continuous for x ∈ R, if

(a) it is continuous at x = 0

(b) differentiable at x = 0

(c) continuous at two points

(d) differentiable for x ∈ R

**Answer/Explanation**

Answer: d

Explaination: (d), as differentiable functions is continuous also.

5. Afunction

is continuous at x = 0 for

(a) k = 1

(b) k = 2

(c) k = \(\frac{1}{2}\)

(d) k = \(\frac{3}{2}\)

**Answer/Explanation**

Answer: a

Explaination:

6. Write the number of points where f(x) = |x + 2| + |x – 3| is not differentiable.

(a) 2

(b) 3

(c) 0

(d) 1

**Answer/Explanation**

Answer: a

Explaination: (a), as f(x) = |x – a| is continuous at x = a but not differentiable thereat.

7. Derivative of cot x° with respect to x is

(a) cosec x°

(b) cosec x° cot x°

(c) -1° cosec2 x°

(d) -1° cosec x° cot x°

**Answer/Explanation**

Answer: c

Explaination:

8.

**Answer/Explanation**

Answer: a

Explaination:

9. If f(x) = \(\log _{x^{2}}(\log x)\), then f(e) is

(a) 0

(b) 1

(c) \(\frac{1}{e}\)

(d) \(\frac{1}{2e}\)

**Answer/Explanation**

Answer: d

Explaination:

10. If f(x) = e^{x} and g(x) = log_{e} x, then (gof)’ (x) is

(a) 0

(b) 1

(c) e

(d) 1 + e

**Answer/Explanation**

Answer: b

Explaination:

11.

**Answer/Explanation**

Answer:

Explaination:

12. If y = x^{x-∞}, , then x(l -y log x)\(\frac{d y}{d x}\) is equal to

(a) x²

(b) y²

(c) xy²

(d) x²y

**Answer/Explanation**

Answer: b

Explaination:

13. The derivative of sin x with respect to log x is

(a) cos x

(b) x cos x

(c) \(\frac{\cos x}{\log x}\)

(d) \(\frac{1}{x}\) cos x

**Answer/Explanation**

Answer: b

Explaination:

14. Ify = Ae^{5x},+ Be^{-5x} x then \(\frac{d^{2} y}{d x^{2}}\) is equal to

(a) 25y

(b) 5y

(c) -25y

(d) 10y

**Answer/Explanation**

Answer:

Explaination:

15. Given the function

the function is continuous at x = 0, state true or false.

**Answer/Explanation**

Answer:

Explaination: False, as ‘f’ is not defined at x = 0. i.e.f (0) does not exist.

46. The function f(x) = \(\frac{x+1}{1+\sqrt{1+x}}\) continuous at x = 0 if/(0) is _________ .

**Answer/Explanation**

Answer:

Explaination:

17. A function f(x) = \(\frac{x}{x-5}\) x ∈ R, is a continuous function. State true or false.

**Answer/Explanation**

Answer:

Explaination: False, as for x = 5, f(5) is not defined.

18. A function f(x) = sin x + cos x is continuous function. State true or false.

**Answer/Explanation**

Answer:

Explaination: True, as sum of two continuous functions is a continuous function.

19. Discuss the continuity of the function fix)= \(\frac{1}{x-5}\) for x ∈ R.

**Answer/Explanation**

Answer:

Explaination: f(x) = \(\frac{1}{x-5}\), as f(5) is not defined, therefore function is not continuous at x = 5.

20. Discuss the continuity of the function

**Answer/Explanation**

Answer:

Explaination:

f(x)= \(\frac{x^{2}-25}{x-5}\), x ≠ 5. As x ≠ 5, therefore, value of function exists for all x(≠5) ∈ R.

Also \(\lim _{x \rightarrow a}\) f(x) = f(a) = a + 5, (a ≠ 5). Hence, continuous.

21. Check whether the function f(x) = 2x² + 1 is continuous at x = 0.

**Answer/Explanation**

Answer:

Explaination: f(x) = 2x² + 1, as \(\frac{x^{2}-25}{x-5}\) f(x) = f(0) = 1. Hence, continuous

22. Give an example of a function which is continuous but not differentiable at exactly two points.

**Answer/Explanation**

Answer:

Explaination:

We know function f(x)=|x – a| is continuous at x = a but not differentiable at x = a.

∴ functions |x| and |x – 1| are continuous but not differentiable at x = 0 and 1.

∴ function is h(x) = |x| + |x – 1|.

23. Determine the value of the constant ‘k’

**Answer/Explanation**

Answer:

Explaination:

24. Determine the value of ‘k’ for which the following function is continuous at x = 3:

**Answer/Explanation**

Answer:

Explaination:

25. For what value of ‘k’ is the function

continuous at x = 0?

**Answer/Explanation**

Answer:

Explaination:

26. Find the value of k, so that the function

is continuous at x = 1

**Answer/Explanation**

Answer:

Explaination:

27. Determine the value of the constant ‘k’ so that the function

is continuous at x = 0. [Delhi]

**Answer/Explanation**

Answer:

Explaination:

28. For what value of ‘k’ is the function

continuous at x = 0? [Foreign]

**Answer/Explanation**

Answer:

Explaination:

29. The derivative of

State true or false.

**Answer/Explanation**

Answer:

Explaination:

30. Find \(\frac{d y}{d x}\), if x² + y² = 5

**Answer/Explanation**

Answer:

Explaination:

31. Differentiate sin^{-1}x², with resepct to x.

**Answer/Explanation**

Answer:

Explaination:

32. Find \(\frac{d y}{d x}\), if sin y + x = log x

**Answer/Explanation**

Answer:

Explaination:

33. Find \(\frac{d y}{d x}\) at x = 1, y = \(\frac{\pi}{4}\) if sin²y + cos xy = K. [Delhi 2017]

**Answer/Explanation**

Answer:

Explaination:

34. Differentiate tan^{-1} \(\left(\frac{1+\cos x}{\sin x}\right)\) with respect to x. [CBSE 2018]

**Answer/Explanation**

Answer:

Explaination:

35. If y = 2^{√x}, then \(\frac{d y}{d x}\) is _______ .

**Answer/Explanation**

Answer:

Explaination:

36. If y = log (tan x) + log (cot x), then \(\frac{d y}{d x}\) is _______ .

**Answer/Explanation**

Answer:

Explaination:

37. If \(f(x)=9^{x^{2}+2 x}\), then f(-1) is _______ .

**Answer/Explanation**

Answer:

Explaination:

38. Differentiate e^{-2x} with respect to x.

**Answer/Explanation**

Answer:

Explaination:

39. Differentiate 5^{sin x}, with respect to x.

**Answer/Explanation**

Answer:

Explaination:

40. Differentiate log_{e}(sin x) with respect to x.

**Answer/Explanation**

Answer:

Explaination:

41. Differentiate log x² w.r.t x.

**Answer/Explanation**

Answer:

Explaination:

42. If y = e^{-3 log x} then find \(\frac{d y}{d x}\).

**Answer/Explanation**

Answer:

Explaination:

43.

**Answer/Explanation**

Answer:

Explaination:

44. Find \(\frac{d y}{d x}\) at t = \(\frac{2 \pi}{3}\) when x = 10(t – sin t) and y = 12(1 – cos t). [Foreign 2017]

**Answer/Explanation**

Answer:

Explaination:

45. Find \(\frac{d^{2} y}{d x^{2}}\), if y = log x

**Answer/Explanation**

Answer:

Explaination:

46. If y = sin 3x, find y_{2}

**Answer/Explanation**

Answer:

Explaination:

y = sin 3x

y_{1} = 3 cos 3x

y_{2} = -9 sin 3x.

47. Find \(\frac{d^{2} y}{d x^{2}}\) if y = e^{-3x}

**Answer/Explanation**

Answer:

Explaination:

48. Verify the Rolle’s Theorem for the function f(x) = x² in the inverval [-1, 1].

**Answer/Explanation**

Answer:

Explaination:

Function f(x) = x² is continuous in [-1,1 ], differentiable in ( -1, 1) and f(-1) = f(1). Hence, Rolle’s Theorem verified.

⇒ f'(c) = 0

⇒ 2c = 0

⇒ c = 0 for c ∈ (-1, 1)

49. Verify the Rolle’s Theorem for die functiony(x) = |x| in the inverval [-1, 1]. [HOTS]

**Answer/Explanation**

Answer:

Explaination: Not verified, as /(x) =|x| is not derivable at x = 0.

50. Verify the Rolle’s Theorem for the function f(x) = sin 2x in [0, π].

**Answer/Explanation**

Answer:

Explaination:

Function f(x) = sin 2x is continuous in [0, π], differentiable in (0, π) and (0)= f(π)

Hence, Rolle’s theorem verified.

⇒ f'(c) = 2 cos 2c

⇒ 2cos 2c = 0

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