Maths MCQs for Class 12 with Answers Chapter 5 Continuity and Differentiability
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) = ex and g(x) = loge 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 = xx-∞, , 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 = Ae5x,+ 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 Answer:
is continuous at x = 0. [Delhi] Answer/Explanation
Explaination:
28. For what value of ‘k’ is the function Answer:
continuous at x = 0? [Foreign] Answer/Explanation
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-1x², 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:Answer/Explanation
Explaination:
34. Differentiate tan-1 \(\left(\frac{1+\cos x}{\sin x}\right)\) with respect to x. [CBSE 2018] Answer:Answer/Explanation
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 5sin x, with respect to x.
Answer/Explanation
Answer:
Explaination:
40. Differentiate loge(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:Answer/Explanation
Explaination:
45. Find \(\frac{d^{2} y}{d x^{2}}\), if y = log x
Answer/Explanation
Answer:
Explaination:
46. If y = sin 3x, find y2
Answer/Explanation
Answer:
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
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:Answer/Explanation
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
