## MP Board Class 12th Maths Important Questions Chapter 5A Continuity and Differentiability

### Continuity and Differentiability Important Questions

**Continuity And Differentiability Objective Type Questions:**

Question 1.

Choose the correct answer:

Question 1.

If x = at^{2}, y = 2at, then \(\frac{dy}{dx}\) will be:

(a) t

(b) t^{2}

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

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

Answer:

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

Question 2.

If y = 2\(\sqrt { cot(x^{ 2 }) } \) ,then \(\frac{dy}{dx}\) will be:

Answer:

(a)

Question 3.

The value of \(\frac{d}{dx}\) (x^{3} + sin x^{2}) is to be:

(a) 3x^{2} + cos x^{2}

(b) 3x^{2} + x sin x^{2}

(c) 3x^{2} + 2x cos x^{2}

(d) 3x^{2} + x cos x^{2}

Answer:

(c) 3x^{2} + 2x cos x^{2}

Question 4.

The value of \(\frac{d}{dx}\) a^{x} is to be:

(a) a^{x}

(b) a^{x} log_{a} e

(c) a^{x} log_{e} a

(d) \(\frac { a^{ x } }{ log_{ e }a } \)

Answer:

(c) a^{x} log_{e} a

Question 5.

If y = 500e^{7x} + 600e^{-7x}, then the value of \(\frac { d^{ 2 }y }{ dx^{ 2 } } \) will be:

(a) 45 y

(b) 47 y

(c) 49 y

(d) 50 y

Answer:

(c) 49 y

Question 2.

Fill in the blanks:

- Differential coefficient of cos x
^{0}w.r.t x is …………………………….. - Differential coefficient of e
^{logea}w.r.t x is ………………………………. - Differential coefficient of log
_{e }a w.r.t x is ……………………………… - Differential coefficient of a
^{x}w.r.t x is ………………………………. - Differential coefficient of sin 3x w.r.t x is ……………………………….
- If y = sin
^{-1}(2x \(\sqrt { (1-x^{ 2 }) } \)), then \(\frac{dy}{dx}\) = ……………………………… - Differential coefficient of sin x w.r.t. cos x is …………………………………..
- The value of \(\frac{d}{dx}\) (log tan x) is ………………………………………
- Differential coefficient of log (log sin x) ………………………………
- If x = y\(\sqrt { (1-y^{ 2 }) } \), then \(\frac{dy}{dx}\) will be …………………………………
- n
^{th}differentiation of sin x will be ……………………………………… - If y = \(\sqrt { x+\sqrt { x+……..\infty } } \), then \(\frac{dy}{dx}\) will be ……………………………….
- If x = r cos θ, y = r sin θ, then \(\frac{dy}{dx}\) will be ………………………..
- Differential coefficient of e
^{x}w.r.t \(\sqrt { x } \) will be ………………………….

Answer:

- – \(\frac { \pi }{ 180 } \) sin x
^{0} - 0
- 0, 4
- log
_{e}a.a^{x} - cos 3x
- \(\frac { 2 }{ \sqrt { 1-x^{ 2 } } } \)
- – cot x
- 2 cosec 2x
- \(\frac { cotx }{ logsinx } \)
- \(\frac { \sqrt { 1-y^{ 2 } } }{ 1-2y^{ 2 } } \)
- sin (\(\frac { n\pi }{ 2 } \) + x)
- \(\frac { 1 }{ 2y-1 } \)
- – cot θ
- 2\(\sqrt { x } \)e
^{x}.

Question 3.

Write True/False:

- Differential coefficient of e
^{logex}is \(\frac{1}{x}\)? - If f(x) = \(\sqrt { x } \); x>0, then value of f'(2) is \(\frac { 1 }{ 2\sqrt { 2 } } \)?
- Any function f(x) is said to be differentiatiable at any point x = a when Lf'(a) ≠ Rf'(a)?
- Differential coefficient of sec
^{-1}a w.r.t x is 0? - If y = Ae
^{mx}+ Be^{-mx}, then \(\frac { d^{ 2 }y }{ dx^{ 2 } } \) = – m^{2}y? - If y = sin
^{-1}( \(\frac { x-1 }{ x+1 } \) ) + cos^{-1}( \(\frac { x-1 }{ x+1 } \) ), then \(\frac{dy}{dx}\) = 0? - Every differentiatiable function is continous?
- Differential coefficient of a
^{2x}is a^{2x}log_{e}a?

Answer:

- Flase
- True
- Flase
- False
- True
- True
- True
- Flase

Question 4.

Match the Column:

Answer:

- (d)
- (e)
- (a)
- (f)
- (h)
- (g)
- (c)
- (b)

Question 5.

Write the answer in one word/sentence:

- Find differential coefficient of \(\frac { 6^{ x } }{ x^{ 6 } } \) w.r.t. x?
- Find differential coefficient of y =
^{logetanx}w.r.t x? - Find n
^{th}derivative of a^{x}? - If y = sin(ax + b), then find the value of \(\frac { d^{ 2 }y }{ dx^{ 2 } } \)?
- If x
^{2}+ y^{2}= sin xy, then find the value of \(\frac{dy}{dx}\)? - Find differential coefficient of log tan \(\frac{x}{2}\) w.r.t. x?
- Find differential coefficient of sin
^{-1}\(\frac { 2x }{ 1+x^{ 2 } } \) w.r.t. x? - Find differential coefficient of e
^{-logex}w.r.t x?

Answer:

- \(\frac { 6^{ x } }{ x^{ 6 } } \) [log 6 – \(\frac{6}{x}\) ]
- sec
^{2}x - a
^{x}(log_{e}a)^{n} - -a
^{2}y - \(\frac { ycosxy-2x }{ 2y-xcosxy } \)
- cosec x,
- \(\frac { 2 }{ 1+x^{ 2 } } \)
- – \(\frac { 1 }{ x^{ 2 } } \)

**Continuity And Differentiability Short Answer Type Questions**

Question 1.

Find all the points of discontinuity of f, when f is defined as:

f(x) = \(\left\{\begin{array}{lll}

{2 x+3,} & {\text { if }} & {x \leq 2} \\

{2 x-3,} & {\text { if }} & {x>2}

\end{array}\right.\) (NCERT)

Solution:

For x< 2, f(x) = 2x + 3 is polynomial function.

Hence, for x < 2, f(x) is continuous function. For x > 2, f(x) = 2x – 3 is polynomial function.

Hence, x > 2, f(x) is continous.

Now, we shall examine the continuty of f(x) at x = 2 only.

Put x = 2 + h,

When x → 2, then h → 0

= 2(2 + 0) – 3 = 4 – 3 = 1.

Put x = 2 – h,

When x → 2, then h → 0

= 2(2 – 0) + 3 = 7

f(2) = 2(2) + 3 = 7

Hence, f(x) is discontinous at x = 2 only.

Question 2.

Find all the points of discontunity of f, when f is defined as follow:

f(x) = \(\left\{\begin{array}{ccc}

{\frac{|x|}{x},} & {\text { if }} & {x \neq 0} \\

{0} & {\text { if }} & {x=0}

\end{array}\right.\). (NCERT)

Solution:

Hence, we shall examine the continuty of f(x) at x = 0 only,

Put x = 0 + h,

When x → 0, then h → 0

Put x = 0 – h,

When x → 0, then h → 0

= -1

Given: f(0) = 0

Hence, the given function f(x) is discontinous at x = 0.

Question 3.

Examine the continuty of function f(x) at point x = 0?

f(x) = \(\left\{\begin{array}{cc}

{\frac{1-\cos x}{x^{2}},} & {x \neq 0} \\

{\frac{1}{2},} & {x=0}

\end{array}\right.\)

Solution:

f(x) = \(\frac { 1-cosx }{ x^{ 2 } } \), when x ≠ 0.

Put x = 0 + h, when x → 0, then h → 0

Again, put x = 0 – h, when x → 0, then h → 0

Hence, f(x) is continous at x = 0.

Question 4.

Function f is defined as:

f(x) = \(\left\{\begin{aligned}

\frac{|x-4|}{x-4} ; & x \neq 4 \\

0 ; & x = 4

\end{aligned}\right.\)

Then prove that function f is continous function for all points except x = 4?

Solution:

Hence, for x = 4 the function f(x) is dicontinous

When x < 4 then f(x) = -1 which is constant function Hence, it is continous function when x > 4 then f(x) = 1 which is constant function.

Hence, it is continous function

The given functions f is continous at all points except x = 4. Proved.

Question 5.

Find the value of k for which the function

f(x) = \(\left\{\begin{array}{c}

{\frac{k \cos x}{\pi-2 x}, \text { if } x \neq \frac{\pi}{2}} \\

{3, \text { if } x=\frac{\pi}{2}}

\end{array}\right.\)

is contionuous at x = \(\frac { \pi }{ 2 } \). (NCERT)

Solution:

Put x = \(\frac { \pi }{ 2 } \) + h,

When x → \(\frac { \pi }{ 2 } \), then h → 0

\(\frac{k}{2}\) × 1 = \(\frac{k}{2}\)

Put x = \(\frac { \pi }{ 2 } \) – h

When x → \(\frac { \pi }{ 2 } \), then h → 0

\(\frac{k}{2}\) × 1 = \(\frac{k}{2}\)

Given that f ( \(\frac { \pi }{ 2 } \) ) = 3

The given function is continous,

\(\frac{k}{2}\) = \(\frac{k}{2}\) = 3

k = 6.

Question 6.

Find the value of k, if function

f(x) = \(\left\{\begin{array}{lll}

{k x+1,} & {\text { if }} & {x \leq \pi} \\

{\cos x,} & {\text { if }} & {x>\pi}

\end{array}\right.\) is continous at x = π? (NCERT)

Solution:

Put x = π + h,

When x → π, then h → 0

= cos (π + 0)

= -1.

Put x = π – h,

When x → π, then h → 0

= k(π – 0) + 1

= πk + 1

f(π) = kπ + 1

∴ The given function is continous at x = π

-1 = kπ + 1 = kπ + 1

kπ = -2

k =- \(\frac { 2 }{ \pi } \)

Question 7.

Function f is continous at x = 0:

f (x) = \(\left\{\begin{array}{c}

{\frac{1-\cos k x}{x \sin x} ; x \neq 0} \\

{\frac{1}{2} \quad ; x=0}

\end{array}\right.\) Find the value of k?

Solution:

Given:

f(x) = \(\frac { 1-coskx }{ xsinx } \), x ≠ 0

Given: f(0) = \(\frac{1}{2}\)

Question 8.

Find the relationship between a and b so the following function f defined by:

f(x) = \(\left\{\begin{array}{lll}

{a x+1,} & {\text { if }} & {x \leq 3} \\

{b x+3,} & {\text { if }} & {x>3}

\end{array}\right.\) is continous at x = 3. (NCERT; CBSE 2011)

Solution:

Put x = 3 + h,

When x → 3, then h → 0

= b(3 + 0) + 3

= 3b + 3

Put x = 3 – h,

When x → 3, then h → 0

= a(3 – 0) + 1

= 3a + 1

f(3) = 3a + 1

The given function is continous at x = 3.

3b + 3 = 3a + 1 = 3a + 1

3a + 1 = 3b + 3

3a = 3b + 2

a = b + \(\frac{2}{3}\).

Question 9.

Prove that the funcion f(x) = |x – 1|, x ∈ R is not differentiable at x = 1? (NCERT)

Solution:

Given:

f(1) = 1 – 1 = 0

Put x = 1 – h, when x → 1, then h → 0

Put x = 1 + h, when x → 1, then h → 0

Lf'(1) ≠ Rf'(1)

Question 10.

Show that the function:

f(x) = \(\left\{\begin{aligned}

x-1, & \text { if } x<2 \\

2 x-3, & \text { if } x \geq 2

\end{aligned}\right.\), is not differentaible at point x = 2?

Solution:

We know that:

RHD =

From the above, it is clear that,

LHD at x = 2 ≠ RHD at x = 2

∴ f(x) is not differentiable at x = 2. Proved.

Question 11.

Determine the function of defined by

f(x) = \(\left\{\begin{array}{cc}

{x^{2} \sin \frac{1}{x},} & {\text { when } x \neq 0} \\

{0,} & {\text { when } x=0}

\end{array}\right.\) in continous function? (NCERT)

Solution:

Here, f(0) = 0

= 0 × a finite quantiy, [∵ sin \(\frac{1}{h}\) is between -1 and 1]

= 0

Hence, the given function is continous at x = 0.