# Maths MCQs for Class 12 with Answers Chapter 5 Continuity and Differentiability

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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

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. Explaination: 3. for x = 2, then value of k for which f is continuous is
(a) -2
(b) -1
(c) 0
(d) 1

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

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 = $$\frac{3}{2}$$

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

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°

Explaination: 8.  Explaination: 9. If f(x) = $$\log _{x^{2}}(\log x)$$, then f(e) is
(a) 0
(b) 1
(c) $$\frac{1}{e}$$
(d) $$\frac{1}{2e}$$

Explaination: 10. If f(x) = ex and g(x) = loge x, then (gof)’ (x) is
(a) 0
(b) 1
(c) e
(d) 1 + e

Explaination: 11.  Explaination: 12. If y = xx-∞, , then x(l -y log x)$$\frac{d y}{d x}$$ is equal to
(a) x²
(b) y²
(c) xy²
(d) x²y

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

Explaination: 14. Ify = Ae5x,+ Be-5x x then $$\frac{d^{2} y}{d x^{2}}$$ is equal to
(a) 25y
(b) 5y
(c) -25y
(d) 10y

Explaination: 15. Given the function the function is continuous at x = 0, state true or false.

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 _________ .

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

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.

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.

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 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.

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.

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’ Explaination: 24. Determine the value of ‘k’ for which the following function is continuous at x = 3: Explaination: 25. For what value of ‘k’ is the function continuous at x = 0?

Explaination:  26. Find the value of k, so that the function is continuous at x = 1

Explaination: 27. Determine the value of the constant ‘k’ so that the function is continuous at x = 0. [Delhi]

Explaination: 28. For what value of ‘k’ is the function continuous at x = 0? [Foreign]

Explaination: 29. The derivative of State true or false.

Explaination: 30. Find $$\frac{d y}{d x}$$, if x² + y² = 5

Explaination: 31. Differentiate sin-1x², with resepct to x.

Explaination: 32. Find $$\frac{d y}{d x}$$, if sin y + x = log x

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

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

Explaination: 35. If y = 2√x, then $$\frac{d y}{d x}$$ is _______ .

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

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

Explaination: 38. Differentiate e-2x with respect to x.

Explaination: 39. Differentiate 5sin x, with respect to x.

Explaination: 40. Differentiate loge(sin x) with respect to x.

Explaination: 41. Differentiate log x² w.r.t x.

Explaination: 42. If y = e-3 log x then find $$\frac{d y}{d x}$$.

Explaination: 43. 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]

Explaination: 45. Find $$\frac{d^{2} y}{d x^{2}}$$, if y = log x

Explaination: 46. If y = sin 3x, find y2

Explaination:
y = sin 3x
y1 = 3 cos 3x
y2 = -9 sin 3x.

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

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

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]

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]. 