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

Continuity and Differentiability Class 12 Maths MCQs PDF

Continuity and differentiability are fundamental concepts in calculus that form the backbone of many mathematical principles. Continuity refers to the smoothness of a function, where there are no sudden jumps or breaks in its graph. Differentiability, on the other hand, deals with the existence of a derivative at a given point, indicating the rate at which the function changes. Understanding these concepts is crucial for solving complex mathematical problems and real-life applications.

MCQs (Multiple Choice Questions) are an effective tool for students preparing for the Class 12 Maths board exam. They help students assess their understanding of continuity and differentiability concepts by offering a range of possible answers. By practicing MCQs, students can identify areas of weakness and focus on improving them.

Additionally, MCQs simulate the exam environment, helping students manage time effectively and develop a strategy for answering questions efficiently. Overall, MCQs play a significant role in helping students ace the Class 12 Maths board exam by providing targeted practice and enhancing conceptual understanding.

Latest: CBSE Class 12 Maths Answer Key 2024 | CBSE Class 12 Maths Paper Analysis 2024

Question 1. The derivative of f(tan x) w.r.t. g(sec x) at x = \(\frac{\pi}{4}\), where f'(1) = 2 and g'(√2) = 4, is

(a) \(\frac{1}{\sqrt{2}}\)

(b) √2

(c) 1

(d) 0

Answer:

(a) \(\frac{1}{\sqrt{2}}\)

Question 2.

Answer:

(c) \(\frac{2}{3}\)

Question 3.

Answer:

(b) 1

Question 5.

Answer:

(a) n²y

Question 6.

Answer:

(d) \(-\frac{b}{a^{2}} \sec ^{3} \theta\)

Question 7.

Answer:

(c) y. (log ab²)²

Question 8.

Answer:

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

Also Check: CBSE Class 12 Maths Notes on Continuity & Differentiability

Question 9.

Answer:

(a) \(\frac{\sec ^{3} \theta}{a \theta}\)

Question 10.

Answer:

(d) 0

Question 11.

Answer:

(b) \(-\sqrt{\frac{\pi}{6}}\)

Question 12.

Answer:

(a) \(\frac{\sqrt{(x+y)}-\sqrt{y-x}}{\sqrt{y-x}+\sqrt{x+y}}\)

Question 13.

Answer:

(b) \(\frac{2 a x+b y-y^{2}}{2 x y-b x-2 y}\)

Question 14.

Answer:

(d) 1

Question 15.

Answer:

(c) \(\frac{1}{2 \sqrt{1-x^{2}}}\)

Question 16.

Answer:

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

Question 17.

Answer:

(c) \(\frac{2\left(1-x^{2}\right)}{\left(1+x^{2}\right)\left|1-x^{2}\right|}, x \neq\pm 1,0\)

Question 18.

Answer:

(b) 0

Question 19.

Answer:

(c) sec x tan x

Question 20.

Answer:

(d) 3e⁷

Question 21. If x² + y² = 1, then

(a) yy” – (2y’)² + 1 = 0

(b) yy” + (y’)² + 1 = 0

(c) yy” – (y’)² – 1 = 0

(d) yy” + (2y’)² + 1 = 0

Answer:

(b) yy” + (y’)² + 1 = 0

Question 22.

Answer:

(c) -9y

Question 23. The value of c in Rolle’s theorem for the function, f(x) = sin 2x in [0, \(\frac{\pi}{2}\)] is

(a) \(\frac{\pi}{2}\)

(b) \(\frac{\pi}{4}\)

(c) \(\frac{\pi}{3}\)

(d) \(\frac{\pi}{6}\)

Answer:

(b) \(\frac{\pi}{4}\)

Question 24. The value of c in Rolle’s Theorem for the function f(x) = e^x sin x, x ∈ [0, π] is

(a) \(\frac{\pi}{6}\)

(b) \(\frac{\pi}{4}\)

(c) \(\frac{\pi}{2}\)

(d) \(\frac{3 \pi}{4}\)

Answer:

(d) \(\frac{3 \pi}{4}\)

Question 25. A value of c for which the Mean value theorem holds for the function f(x) = log_ex on the interval [1, 3] is

(a) 2log₃e

(b) \(\frac{1}{2} \log _{e} 3\)

(c) log₃e

(d) log_e3

Answer:

(a) 2log₃e

Also Check: CBSE Syllabus

Question 26. The value of c in mean value theorem for the function f(x) = (x – 3)(x – 6)(x – 9) in [3, 5] is

(a) 6 ± √(13/3)

(b) 6 + √(13/3)

(c) 6 – √(13/3)

(d) None of these

Answer:

(c) 6 – √(13/3)

Question 27. The value of c in Mean value theorem for the function f(x) = x(x – 2), x ∈ [1, 2] is

(a) \(\frac{3}{2}\)

(b) \(\frac{2}{3}\)

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

(d) \(\frac{5}{2}\)

Answer:

(a) \(\frac{3}{2}\)

Question 28.

Answer:

(b) ln a + ln b

Question 29.

Answer:

(c) 8

Question 30. The number of discontinuous functions y(x) on [-2, 2] satisfying x² + y² = 4 is

(a) 0

(b) 1

(c) 2

(d) >2

Answer:

(a) 0

Question 31.

Answer:

(c) \(-\frac{1}{2}\)

Question 32.

Answer:

(b) \(\frac{1}{4}\)

Question 33.

Answer:

(c) \(\frac{-1}{(1+x)^{2}}\)

Question 34. If y = (1 + x)(1 + x²)(1 + x⁴)…..(1 + x²ⁿ), then the value of \(\frac{d y}{d x}\) at x = 0 is

(a) 0

(b) -1

(c) 1

(d) None of these

Answer:

(c) 1

Question 35.

Answer:

(d) \(\frac{1}{\sqrt{24}}\)

Question 36. If y = ax² + b, then \(\frac{d y}{d x}\) at x = 2 is equal to

(a) 4a

(b) 3a

(c) 2a

(d) None of these

Answer:

(a) 4a

Question 37.

Answer:

(b) \(\frac{2 y \sqrt{y^{2}-1}\left(x^{2}+x-1\right)}{\left(x^{2}+1\right)^{2}}\)

Question 38.

Answer:

(a) \(\frac{1}{2}\)

Question 39.

Answer:

(c) \(\frac{\log _{10} e}{x}\left(\frac{y}{y-1}\right)\)

Question 40.

Answer:

(d) None of these

Question 41.

Answer:

(d) \(\[\frac{y}{x}\)\]

Question 42. If Rolle’s theorem holds for the function f(x) = x³ + bx² + ax + 5 on [1, 3] with c = (2 + \(\frac{1}{\sqrt{3}}\)), find the value of a and b.

(a) a = 11, b = -6

(b) a = 10, b = 6

(c) a = -11, b = 6

(d) a = 11, b = 6

Answer:

(a) a = 11, b = -6

Question 43. If y = (tan x)^{sin x}, then \(\frac{d y}{d x}\) is equal to

(a) sec x + cos x

(b) sec x + log tan x

(c) (tan x)^{sin x}

(d) None of these

Answer:

(d) None of these

Question 44.

Answer:

(d) \(\frac{\log x}{(1+\log x)^{2}}\)

Question 45. The derivative of y = (1 – x)(2 – x) ….. (n – x) at x = 1 is equal to

(a) 0

(b) (-1)(n – 1)!

(c) n! – 1

(d) (-1)^n-1(n – 1)!

Answer:

(b) (-1)(n – 1)!

Question 46. If x^y . y^x = 16, then the value of \(\frac{d y}{d x}\) at (2, 2) is

(a) -1

(b) 0

(c) 1

(d) none of these

Answer:

(a) -1

Question 47.

Answer:

(c) \(\frac{y}{1-y}\)