# Exam Question for Class 12 Mathematics Chapter 5 Continuity and Differentiability

Please refer to below Exam Question for Class 12 Mathematics Chapter 5 Continuity and Differentiability. These questions and answers have been prepared by expert Class 12 Mathematics teachers based on the latest NCERT Book for Class 12 Mathematics and examination guidelines issued by CBSE, NCERT, and KVS. We have provided Class 12 Mathematics exam questions for all chapters in your textbooks. You will be able to easily learn problems and solutions which are expected to come in the upcoming class tests and exams for standard 10th.

## Chapter 5 Continuity and Differentiability Class 12 Mathematics Exam Question

All questions and answers provided below for Exam Question Class 12 Mathematics Chapter 5 Continuity and Differentiability are very important and should be revised daily.

**Exam Question Class 12 Mathematics Chapter 5 Continuity and Differentiability**

**Very Short Answer Type Questions**

**Question.****Find the value of k so that the following function** **is continuous at x = 2.**

**Answer.** **∵** f(x) is continuous at x = 2

**Question.****Find the relationship between a and b so that** **the function ‘f ’ defined by**

**Answer.** **∵** f(x) is continuous at x = 3

**Question.** Discuss the continuity of the function f(x) at**x = 1/2, when f(x) is defined as follows:**

**Answer.**

**Question.**

**Answer.**

**Question.** If x = cos t(3 – 2 cos^{2}t) and y = sint (3 – 2 sin^{2}t),**find the value of**

**Answer.** Here, x = cos t(3 – 2cos2t), y = sin t(3 – 2 sin2t)

**Short Answer Type Questions**

**Question.** Find the values of p and q, for which

**Answer.** ∴ f(x) is continuous at π/2.

**Question.** Find the value of the constant k so that the**function f, defined below, is continuous at x= 0,** **where**

**Answer.** **∵** f(x) is continuous at x = 0.

∴ f(0) = k

**Question.****Find the value of k, for which**

**Answer.** **∵** f(x) is continuous at x = 0

**Question.**

**Answer.** **∵** f(x) is continuous at x = 0.

**Question.**

**Find the values of a and b so that f(x) is a continuous function.****Answer.** Continuity at x = 3**∵** f (x) is continuous at x = 3

**Question.** For what value of a is the function f defined by

**Answer.** **∵** f(x) is continuous at x = 0,

**Question.** Show that the function f(x) defined by

**Answer.**

**Question.** If f(x) defined by the following is continuous at**x = 0, find the value of a, b and c.**

**Answer.** For f(x) to be continuous at x = 0, we must have

**Question.** Find the value of k if the function

**Answer.** **∵** f(x) is continuous at x = 1

**Question.**

**Answer.** **∵** f(x) is continuous at x = 5

**Question.** Find the values of a and b, if the function f

**Answer.** Given that f(x) is differentiable at x = 1.

Therefore, f(x) is continuous at x = 1

**Question.**

**Answer.** We have,

**Question.**

**Answer.**

**Question.** Show that the function f(x) = |x – 1| + |x + 1|, for all x ∈ R, is not differentiable at the points x = –1 and x = 1.**Answer.** The given function is f(x) = |x – 1| + |x + 1|

**Question.** Find whether the following function is differentiable at x = 1 and x = 2 or not.

**Answer.** At x = 1:

**Question.****If cosy = xcos(a + y), where cos a ≠ ±1, prove that**

**Answer.** We have cosy = xcos(a + y)

**Question.**

**Answer.**

**Question.****Show that the function f(x) = |x – 3|, x ∈ R, is** **continuous but not differentiable at x = 3.****Answer.**

**Question.**

**Answer.**

**Question.**

**Answer.**

**Question.** If y = a sin x + b cos x, prove that

**Answer.** Here, y = a sinx + b cos x

**Question.** Show that the function defined as follows, is continuous at x = 2, but not differentiable.

**Answer.**

**Question.**

**Answer.**

**Question.**

**Answer.** We have, (x^{2} + y^{2})^{2} = xy

Differentiating w.r.t. x, we get

**Question.**

**Answer.** We have,

**Question.**

**Answer.**

**Question.** **If xy + y ^{2} = tan x + y, find dy/dx**

**Answer.**Given, xy + y

^{2}= tan x + y

Differentiating w.r.t. x, we get

**Question.**

**Answer.**

**Question.** If e^{x} + e^{y} = e^{x + y}, prove that dy/dx + e ^{y − x} = 0**Answer.** Given e^{x} + e^{y} = e^{x} + y ⇒ 1 + e^{y–x} = e^{y} …(1)

Differentiating (1) w.r.t. x, we get

**Question.**

**Answer.**

**Question.**

**Answer.**

**Question.**

**Answer.**

**Question.**

**Answer.**

**Question.****Differentiate x ^{sin x }+ (sin x)^{cos x} with respect to x.**

**Answer.**

**Question.****If y = (sinx) ^{x} + sin^{−1} √x, then find dy/dx**

**Answer.**

**Question.**

**Answer.**

Given x^{m}y^{n} = (x + y)^{m + n}

Taking log on both the sides, we get

logx^{m }+ log y^{n} = (m + n)log (x + y)

⇒ m logx + n log y = (m + n) log (x + y)

Differentiating w.r.t. x, we get

**Question.**

**Answer.**

**Question.****Differentiate the following function with respect to x : (log x) ^{x} + x^{logx.}**

**Answer.**Let y = (log x)

^{x}+ x

^{log x}

**∴**y = e

^{x log (log x)}+ e(log x)

^{2}

Differentiating w.r.t. x, we get

**Question.**

**Answer.** Here y^{x} = e^{y} ^{– x}

Taking log on both sides, we get

xlog y = (y – x)log e = y – x

⇒ x(1 + logy) = y

**Question.****Differentiate the following with respect to x :**

**Answer.**

**Question.**

**Answer.** We have, x^{y} = e^{x–y}

Taking log on both sides, we get

y log x =(x – y)log e = x – y

**Question.** If (cos x)^{y} = (cos y)^{x}, find dy/dx.**Answer.** We have, (cos x)^{y} = (cos y)^{x}

Taking log on both sides, we get

y log (cos x) = x log (cos y)

Differentiating w.r.t. x, we get

**Question.**

**Answer.**

**Question.** If y = 2cos(log x) + 3sin(log x), prove that

**Answer.** We have, y = 2 cos (log x) + 3 sin (log x)

Differentiating w.r.t. x, we get

**Question.** If x = sin t and y = sin pt. Prove that

**Answer.** We have, x = sin t and y = sin pt

**Question.**

**Answer.**

**Question.** If x = a cos^{3 }θ and y = a sin^{3} θ, then find the value of

**Answer.** Here x = a cos^{3} θy = a sin^{3} θ

**Question.**

**Answer.**

**Question.** If y = x + tan x, then prove that

**Answer.** We have, y = x + tan x

**Question.****Verify Rolle’s theorem for the function** **f(x) = x ^{2} – 4x + 3 on [1, 3].**

**Answer.**We have, f(x) = x

^{2}– 4x + 3

(i) f(x) being a polynomial function is continuous in [1,3]

(ii) f(x) being a polynomial function is differentiable in (1, 3)

(iii) f(3) = 3

^{2}– 4(3) + 3 = 0 and f(1) = 12 – 4(1) + 3 = 0. Thus f(1) = f(3)

Thus, all the conditions of Rolle’s theorem are satisfied, so there exists atleast one point c ∈(1, 3) such that f'(c) = 0

f ‘(x) = 2x – 4 ⇒ f ‘(c) = 2c – 4

∴ f ‘(c) = 2c – 4 = 0 ⇒ c = 2 ∈(1, 3)

Hence, the Rolle’s theorem is verified.

**Long Answer Type Questions**

**Question.** For what value λ of the function defined by

**is continuous at x = 0? Hence check the differentiability of f(x) at x = 0.****Answer.**

**Question.** If (tan–1x)^{y} + y^{cotx} = 1, then find dy/dx.**Answer.** Here, (tan^{–1}x)^{y }+ y^{cotx }= 1

⇒ u + v = 1 where u = (tan^{–1}x)^{y} and v = y^{cotx}

Differentiating w.r.t. x, we get