Important Questions for Class 12 Maths Chapter 5 - Continuity and Differentiability

Important questions of Maths Chapter 5 continuity and differentiability for class 12 are provided here as per the CBSE syllabus. The important questions provided here will help the students to prepare and score well in the board examination. The questions given here covers the latest syllabus as per guidelines given by the board. Also, practice the important questions for all the chapters of 12th Maths subject here.

Here, we provided some practice questions so that students should get confidence to write the answer with efficiency. In class 12 chapter 5, continuity and differentiability covers the concepts such as continuity and differentiability, exponential function, logarithmic function, logarithmic differentiation, mean value theorem, second-order derivative, derivatives of functions in parametric forms.

Also, check:

• Important 4 Marks Questions for CBSE Class 12 Maths
• Important 6 marks Questions for CBSE Class 12 Maths

Class 12 Chapter 5 Continuity and Differentiability Important Questions with Solutions

A few important questions for class 12 continuity and differentiability are provided below with solutions. The solved problems include both short and long answer questions along with HOTS questions to let the students get completely familiarised with the in-depth knowledge of the concepts.

Question 1:

Explain the continuity of the function f(x) = sin x . cos x

Solution:

We know that sin x and cos x are continuous functions. It is known that the product of two continuous functions is also a continuous function.

Hence, the function f(x) = sin x . cos x is a continuous function.

Question 2:

Determine the points of discontinuity of the composite function y = f[f(x)], given that, f(x) = 1/x-1.

Solution:

Given that, f(x) = 1/x-1

We know that the function f(x) = 1/x-1 is discontinuous at x = 1

Now, for x ≠1,

f[f(x)]= f(1/x-1)

= 1/[(1/x-1)-1]

= x-1/ 2-x, which is discontinuous at the point x = 2.

Therefore, the points of discontinuity are x = 1 and x=2.

Question 3:

If f (x) = |cos x|, find f’(3π/4)

Solution:

Given that, f(x) = |cos x|

When π/2 <x< π, cos x < 0,

Thus, |cos x| = -cos x

It means that, f(x) = -cos x

Hence, f’(x) = sin x

Therefore, f’(3π/4) = sin (3π/4) = 1/√2

f’(3π/4) = 1/√2

Question 4:

Verify the mean value theorem for the following function f (x) = (x – 3) (x – 6) (x – 9) in [3, 5]

Solution:

f(x)=(x−3)(x−6)(x−9)

=(x−3)(x²−15x+54)

=x³−18x²+99x−162

fc∈(3,5)

f′(c)=f(5)−f(3)/5−3

f(5)=(5−3)(5−6)(5−9)

=2(−1)(−4)=−8

f(3)=(3−3)(3−6)(3−9)=0

f′(c)=8−0/2=4

∴f′(c)=3c²−36c+99

3c²−36c+99=4

3c²−36c+95=0

ax²+bx+c=0

a=3

b=−36

c=95

c=36±√(36)²−4(3)(95)/2(3)

=36±√1296−1140/6

=36±12.496

c=8.8&c=4.8

c∈(3,5)

f(x)=(x−3)(x−6)(x−9) on [3,5]

Question 5:

Explain the continuity of the function f = |x| at x = 0.

Solution:

From the given function, we define that,

f(x) = {-x, if x<0 and x, if x≥0

It is clearly mentioned that the function is defined at 0 and f(0) = 0. Then the left-hand limit of f at 0 is

Lim_x→0- f(x)= lim_x→0- (-x) = 0

Similarly for the right hand side,

Lim_x→0+ f(x)= lim_x→0+ (x) = 0

Therefore, for the both left hand and the right hand limit, the value of the function coincide at the point x = 0.

Therefore, the function f is continuous at the point x =0.

Question 6:

If y= tan x + sec x , then show that d^2.y / dx² = cos x / (1-sin x)²

Solution:

Given that, y= tan x + sec x

Now, the differentiate wih respect to x, we get

dy/dx = sec² x + sec x tan x

= (1/ cos² x) + (sin x/ cos² x)

= (1+sinx)/ (1+sinx)(1-sin x)

Thus, we get.

dy/dx = 1/(1-sin x)

Now, again differentiate with respect to x, we will get

d²y / dx² = -(-cosx )/(1- sin x)²

d²y / dx² = cos x / (1-sinx)².

Practice Problems for Class 12 Maths Chapter 5

Solve the below-given important questions for class 12 maths chapter 5 continuity and differentiability problems.

1. Prove that every polynomial function is continuous.
2. Discuss the continuity of the sine function and the cosine function.
3. Determine the derivative of the function given by f(x) = sin (x²).
4. Find dy/dx, if y + sin y = cos x.
5. Log (cos e^x) (b) cos x/ log x, x>0

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