Please refer to Continuity and Differentiability Class 12 Mathematics Important Questions with solutions provided below. These questions and answers have been provided for Class 12 Mathematics based on the latest syllabus and examination guidelines issued by CBSE, NCERT, and KVS. Students should learn these problem solutions as it will help them to gain more marks in examinations. We have provided Important Questions for Class 12 Mathematics for all chapters in your book. These Board exam questions have been designed by expert teachers of Standard 12.

## Class 12 Mathematics Important Questions Continuity and Differentiability

**Very Short Answer Type Questions**

**Question.****If x = 2 cos θ – cos 2θ and y = 2sin θ – sin 2θ,then prove that**

**Answer.** Here, x = 2 cos θ – cos^{2}θ, y = 2 sin θ – sin^{2}θ

**Question.** Find the value of ‘a’ if the function f(x) defined**by**

**Answer.** For f to be continuous at x = 2, we must have

**Question.** Find all points of discontinuity of f, where f is**defined as follows :**

**Answer.** Continuity at x = – 3 :

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

**continuous at x = 0 ? Also, write whether the function is continuous at x = 1.****Answer.**

**∴** f(x) is continuous at x = 0, if k = 1/2.

Also, f(x) is continuous at x = 1 as f(x) = 3x + 1 is a polynomial function

**Question.****Find the values of a and b such that the function** **defined as follows is continuous :**

**Answer.** **∵** f(x) is continuous at x = 2 and x = 5

**Question.** **Find dy/dx if x = a (θ + sinθ, y = a (1 – cosθ)Answer.** Given that x = a (θ + sinθ, y = a (1 – cosθ)

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

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

**Question.****If the function f(x) given by**

**is continuous at x = 1, find the values of a and b.****Answer.** **∵** f(x) is continuous at x = 1

**Short Answer Type Questions**

**Question. Find all points of discontinuity of the function ƒ, where ƒ is defined by**

**Answer.**

**Question. Prove that the function ƒ given by ƒ (x) = |x – 1|, for all x∈R Is not differentiable at .Answer.**

**Question.**

**Answer.**

**Question. Find the values of k so that the function ƒ**,

**defined by**

**Answer.**

**Question.**

**Answer.**

**Question.**

**Answer.**

**Question. Differentiate sin (tan ^{-1} e^{-x}) with respect to**

**Answer.**Let y = sin (tan

^{-1}e

^{-x})

Differentiating both sides w.r.t.x, we get

**Question. Differentiate the function f (x) = (sin x) ^{x} + sin √x With respect to x.Answer.**

**Question.** If x = ae^{t}(sin t + cos t) and y = ae^{t}(sin t – cos t),

**Answer.** We have x = ae^{t}(sin t + cos t)

**Question.**

**Answer.**

**Question.** If x = a(θ – sin θ) and y = a (1 + cos θ), find dy/dx at θ = π/3**Answer.**

**Question.**

**Answer.** We have, y = (cosx)^{x} + (sinx)^{1/x}

**Question. If y = (sin x – cos x) ^{sin x – cos x,}**

**Answer.** We have, y = (sin x – cos x)^{(sin x – cos x)}

Taking log on both sides, we get

log y = (sin x – cos x) log (sin x – cos x)

Differentiating w.r.t. x, we get

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

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

^{cos x }+ (sin x)

^{tan x}

⇒ y = e

^{cos x log x}+ e

^{tan x log sin x}

Differentiating w.r.t. x, we get

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

⇒ y = e^{xlog(logx)} + ec^{osx ·logx}

Differentiating w.r.t. x, we get

**Question.****If y = x ^{x} – (sin x)^{x}, find dy/dx.**

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

^{x}– (sinx)

^{x}

⇒ y = e

^{xlogx}– e

^{xlog sinx}

Differentiating w.r.t. x, we get

**Question.**

**Answer.**

**Question.** Differentiate (sin x)^{tan x} + (cos x)^{sec x} w.r.t. x.**Answer.** Let y = (sin x)^{tan x} + (cos x)^{sec x}

⇒ y = e^{tan x · log sin x} + e^{sec x · log cos x}

Differentiating w.r.t. x, we get

**Question.** If x = a sin 2t(1 + cos 2t) and**y = b cos 2t(1 – cos 2t), find the values of**

**Answer.** x = a sin2t (1 + cos2t), y = b cos2t (1 – cos2t)

**Question.** If x = a (cos t + log tan t/2) and y = a sin t, find dy/dx**Answer.** Here, x = a (cos t + log tan t/2)

**Question.**

**Answer.** We have, y = x^{x}

⇒ y = e^{x log x}

Differentiating w.r.t. x, we get

**Question.** If x = a cos θ + b sin θ, y = a sin θ – b cosθ, show**that**

**Answer.** Given, x = a cosθ + b sinθ, y = a sinθ – b cosθ

⇒ x^{2} = a^{2} cos^{2}θ + b^{2} sin^{2}θ + 2ab cosθ sinθ

and y^{2} = a^{2} sin^{2}θ + b^{2} cos^{2}θ – 2ab sinθ cosθ

Adding (1) and (2), we get

x^{2} + y^{2} = a^{2} + b^{2}

Differentiating w.r.t x, we get

**Question.**

**Answer.**

**Question. If x = cosθ and y = sin ^{3}θ, then prove that**

**Answer.** Given x = cosθ and y = sin^{3}θ

= sin^{2}θ(3cos^{2}θ –3sin^{2}θ) + 9sin^{2}θ cos^{2}θ

= 3 sin^{2}θ(cos^{2}θ – sin^{2}θ + 3cos^{2}θ)

= 3 sin^{2}θ(4cos^{2}θ – sin^{2}θ)

= 3 sin^{2}θ(4cos^{2}θ – 1 + cos^{2}θ)

= 3 sin^{2}θ(5cos^{2}θ –1) = R.H.S.

**Question.** If y = sin^{–1} x, show that

**Answer.** Given y = sin^{–1} x

Differentiating w.r.t. x, we get

**Question.** If y = (tan^{–1}x)^{2}, show that

**Answer.** y = (tan^{–1}x)^{2}

Differentiating w.r.t. x, we get

**Question.** If y = cosec^{–1}x, x > 1, then show that

**Answer.** We have, y = cosec^{–1} x

**Question.** If y = (cot^{–1} x)^{2}, then show that

**Answer.** We have, y = (cot–1x)^{2}

**Question.**

**Answer.**

**Question.** If y = e^{x} (sin x + cos x), then prove that

**Answer.** y = e^{x} (sin x + cos x)

**Question.****If y = e ^{x} sin x, then prove that**

**Answer.** We have, y = e^{x} sin x

**Question.** If y = sin(log x), then prove that

**Answer.** We have, y = sin (log x)

Differentiating w.r.t. x, we get

**Question.** If y = e^{m sin-1} x, – 1 ≤ x ≤ 1, then show that

**Answer.** We have, y = e^{m sin−1} ^{x}

Differentiating w.r.t. x, we get

**Question.**

**Answer.**

**Question.**

**Answer.** Here x = a sec^{3}θ

**Question.** If y = Ae^{mx} + Be^{nx}, show that

**Answer.** Given y = Ae^{mx} + Be^{nx}

Differentiating w.r.t. x, we get

**Question.** If x = a(cos t + t sin t) and y = a(sin t – t cos t),**then find the value of**

**Answer.** Here, x = a(cos t + t sin t)

**Question. Find dy/dx if y _{x} + x^{y} + x^{x} = a^{b}** y

Answer.

_{x}+ x

^{y}+ x

^{x}= a

^{b}

**Question. if y = xx – 2 ^{sinx}, find dy/dxAnswer.** y = xx – 2

^{sinx}

**Question. if y = (tan ^{-1} x)^{2}, then show that (x^{2} + 1)^{2}y_{2} + 2x(x^{2} + 1)y^{1 }= 2** Given that y = (tan

Answer.

^{-1}x)

^{2}

**Question. Differentiate sin ^{2} x with respect to e^{cosx}**

Answer.

**Question.**

**Answer.**

**Question.**

**Answer.**

**Question. if y = 3 cos (log x + 4 sin (log x), then show that x ^{2}y2 + xy_{1} + y = 0**

Answer.

**Long Answer Type Questions**

**Question.**

**Answer.**

**Question.**

**Answer.**

**Question.** Differentiate

**Answer.**

**CASE STUDY: ****The Relation between the height of the plant (y in cm) with respect to exposure to sunlight is governed by the following equation y = 4x -1/2 x ^{2} where x is the number of days exposed to sunlight.**

**Question. The rate of growth of the plant with respect to sunlight is ______ .**a. 4x-1/2 x

^{2}

b. 4 – x

c. x – 4

d. x – 12 x

^{2}

## Answer

B

** Question. What is the number of days it will take for the plant to grow to the maximum height?**a. 4

b. 6

c. 7

d. 10

## Answer

A

** Question. What is the maximum height of the plant?**a. 12 cm

b. 10 cm

c. 8 cm

d. 6 cm

## Answer

C

** Question. What will be the height of the plant after 2 days?**a. 4cm

b. 6 cm

c. 8cm

d. 10cm

## Answer

B

** Question. If the height of the plant is 7/2 cm, the number of days it has been exposed to the sunlight is ______ .**a. 2

b. 3

c. 4

d. 1

## Answer

D

**CASE STUDY :**

**The shape of a toy is given as f(x) = 6(2x**

^{4}– x^{2}). To make the toy beautiful 2 sticks which are perpendicular to each other were placed at a point (2,3), above the toy.**Question. Which value from the following may be abscissa of critical point?**a. ±14

b. ±12

c. ± 1

d. None

## Answer

B

** Question. Find the slope of the normal based on the position of the stick.**a. 360

b. –360

c. 1/360

d. −1/360

## Answer

D

** Question. What will be the equation of the tangent at the critical point if it passes through (2,3)?**a. x + 360y = 1082

b. y = 360x – 717

c. x = 717y + 360

d. none

## Answer

B

** Question. Find the second order derivative of the function at x = 5.**a. 598

b. 1176

c. 3588

d. 3312

## Answer

C

** Question. At which of the following intervals will f(x) be increasing?**a. (-∞, -1/2) ꓴ (1/2, ∞)

b. (-1/2, 0) ꓴ (1/2, ∞)

c. (0, ½) ꓴ (1/2, ∞)

d. (-∞, -1/2) ꓴ (0, ½)

## Answer

B