Continuity and Differentiability Class 12 Mathematics Exam Questions

Please refer to Continuity and Differentiability Class 12 Mathematics Exam Questions provided below. These questions and answers for Class 12 Mathematics have been designed based on the past trend of questions and important topics in your Class 12 Mathematics books. You should go through all Class 12 Mathematics Important Questions provided by our teachers which will help you to get more marks in upcoming exams.

Class 12 Mathematics Exam Questions Continuity and Differentiability

Class 12 Mathematics students should read and understand the important questions and answers provided below for Continuity and Differentiability which will help them to understand all important and difficult topics.

Very Short Answer Type Questions

Question. Write the derivative of sin x w.r.t. cos x.
Answer. Let y = sin x ; Let z = cos x

Continuity and Differentiability Class 12 Mathematics Exam Questions

Question. If the following function f (x) is continuous at x = 0, then write the value of k

Answer.

Short Answer Type Questions-I

Question.

Answer. sin² y + cos xy = K
Differentiating both sides w.r.t. x, we have

Question. Find the value of k for which the function f(x)

Answer.

Question. Find the value of p for which the function f (x)

Answer.

Question.

Answer. f (x) = sin 2x – cos 2x
Differentiating both sides w.r.t. x, we get
f«(x) = 2 cos 2x + 2 sin 2x

Short Answer Type Questions-II

Question. Differentiate the following function w.r.t. x ; x^sin x + (sin x)^cos x
Answer.

Question.

Answer. (x² + y²)² = xy
Differentiating both sides w.r.t. x, we get

Question. Find the value of ‘a’ for which the function f defined as

Continuity and Differentiability Class 12 Mathematics Important Questions

Answer.

Question. Find the values of a and b such that the following function f (x) is a continuous function :

Answer.

Since f(x) is continuous.
∴ L.H.L. = R.H.L. = f (10)
10a + b = 21 = 21
10a + b = 21
10a + 5 – 2a = 21 [From (i)
8a = 21 – 5 = 16
∴ a = 2
Putting the value of a in (i), we get
b = 5 – 2(2) = 5 – 4 = 1
∴ a = 2, b = 1

Question.

Answer. sin y = x sin (a + y)

Question. If (cos x)^y= (sin y)^x, find dy/dx
Answer. (cos x)^y = (sin y)^x
Taking log on both sides, we get
∴ log (cos x) = x log (sin y)
Differentiating both sides w.r.t. x, we get

Question.

Answer.

Question.

Answer. y = e^x(sin x + cos x) …(i)
Differentiating both sides w.r.t. x, we get

Question.

Answer.

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

Answer.

Question. If y = (log x)^x + (x)^{cos x}, find dy/dx
Answer. y = (log x)^x + (x)^{cos x}
Let y = A + B

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

Answer. Case I : At x = – 3

Question.

Answer.

Question.

Answer.

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

Answer.

Question. If y = (sin x)^x + sin^–1 √x , then find.
Answer. Let A = (sin x)^x
Taking log on both sides,
log A = x log sin x
Differentiating both sides w.r.t. x, we have

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.

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

Answer.

Question. If x = cos θ and y = sin³ θ, then prove that

Answer.

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

Answer.

Question. Find the value of k, for which the function f defined below is continuous :

Answer.

Question. If the function f (x), defined as

Answer.

f (x) is continuous at x = 1
∴ L.H.L. = R.H.L. = f (1)
5a – 2b = 3a + b = 11
3a + b = 11 ⇒ b = 11 – 3a …(i)
5a – 2b = 11
5a – 2(11 – 3a) = 11 [From (i)]
5a – 22 + 6a = 11
11a = 33 ⇒ a = 3
Putting the value of a in (i), we get
b = 11 – 3(3) = 11 – 9 = 2 ∴ a = 3, b = 2

Question. If the function

Answer.

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

Answer. (i) For continuity at x = 2

Question.

Answer.

Question.

Answer. y = x^x (Given)
Taking log on both sides
log y = log x^x
log y = x log x
Differentiating both sides w.r.t. x, we get

Question. Find dy/dx , if y = (cos ^x)x + (sin x)^1/x
Answer. y = (cos ^x)x + (sin x)^1/x

A= (cos x)x
Taking log on both sides
log A = x (log cos x)
Differentiating both sides w.r.t. x, we get

Question.

Answer.

Question.

Answer. y = e^a sin−1 …(i)
Differentiating both sides w.r.t. x, we get

Question.

Answer.

Question.

Answer. y = (cot–1 x)²
Differentiating both sides w.r.t. x, we get

Question.

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

CASE STUDY :

P(x) = -5x² +125x + 37500 is the total profit function of a company, where x is the production of the company.

Question. What will be the production when the profit is maximum?
a. 37500
b. 12.5
c. -12.5
d. –37500

Answer

Question. What will be the maximum profit?
a. Rs 38,28,125
b. Rs 38281.25
c. Rs 39,000
d. None

Answer

Question. Check in which interval the profit is strictly increasing .
a. (12.5,∞ )
b. for all real numbers
c. for all positive real numbers
d. (0, 12.5)

Answer

Question. When the production is 2units what will be the profit of the company?
a. 37500
b. 37,730
c. 37,770
d. None

Answer

Question. What will be production of the company when the profit is Rs 38250?
a. 15
b. 30
c. 2
d. data is not sufficient to find

Answer

CASE STUDY :

The bridge connects two hills 100 feet apart. The arch on the bridge is in a parabolic form.
The highest point on the bridge is 10 feet above the road at the middle of the bridge as seen in the figure.
Based on the information given above, answer the following questions:

Question. The equation of the parabola designed on the bridge is
a. 𝑥² = 250𝑦
b. 𝑥² = −250𝑦
c. 𝑦² = 250𝑥
d. 𝑦² = 250𝑦

Answer

Question. The value of the integral