Test: Continuity And Differentiability (CBSE Level) - 2


25 Questions MCQ Test Mathematics (Maths) Class 12 | Test: Continuity And Differentiability (CBSE Level) - 2


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QUESTION: 1

Solution:

L'Hopital's rules says that the  
lim x→a f(x)/g(x)
⇒ f'(a)/g'(a)
Using this, we get  
lim x→0 (1−cosx)/x2
⇒ − sin0/2(0)
Yet as the denominator is 0, this is impossible. So we do a second limit:
lim(x→0) sinx/2x
⇒ cos0/2 = 1/2 = 0.5
So, in total lim x→0 (1−cosx)/x2
⇒ lim x→0 sinx/2x
⇒ cosx/2
⇒ cos0/2= 1/2

QUESTION: 2

If f (x) is a polynomial of degree m (⩾1) , then which of the following is not true ?

Solution:

As all the three remaining statements are true for the given function.

QUESTION: 3

Let f and g be differentiable functions such that fog = I, the identity function. If g’ (a) = 2 and g (a) = b, then f ‘ (b) =

Solution:

f(g(x)) = x 
f'(g(x)) g'(x) = 1 
put x = a
f'(b) g'(a) = 1
2 f'(b) = 1
f'(b) = 1/2

QUESTION: 4

Solution:


QUESTION: 5

If f (x) =x2g(x) and g (x) is twice differentiable then f’’’ (x) is equal to

Solution:



QUESTION: 6

Solution:

 

QUESTION: 7

If f(x) = | x | ∀ x ∈ R, then

Solution:


The graph of f(x) = |x|
As observed from the graph, f(x) = |x| is continuous at x = 0.
As this curve is pointed at x = 0, f(x) is not derivable at x = 0.

QUESTION: 8

Differential coefficient of a function f (g (x)) w.r.t. the function g (x) is

Solution:

QUESTION: 9

Solution:

Given xpyq = (x+y)p+qTaking log on both sides we get:
plogx+qlogy = (p+q) log (x+y).  Differentiating both sides w.r.t. x we get , 

QUESTION: 10

If y = aemx + be−mx, then y2 is equal to

Solution:

y = aemx + be-mx ⇒ y1 = amemx + (-m)be-mx ⇒y2

= am2emx + (m2)be-mx ⇒y2 = m2 (aemx + be-mx) ⇒ y2 = m2y

QUESTION: 11

Solution:



QUESTION: 12

 
then   is equal to 

Solution:
QUESTION: 13

Solution:

 

QUESTION: 14

If f(x) be any function which assumes only positive values and f’ (x) exists then f’ (x) is equal to

Solution:

 

QUESTION: 15

If y = a sin mx + b cos m x, then  is equal to

Solution:

y  = a sin mx+b cos mx ⇒ y1 = am cos mx − bm sin mx
⇒ y2 = −am2sin mx−bm2cosmx ⇒ y2 = −m2(a sin mx+b cos mx)

QUESTION: 16

Solution:

 

QUESTION: 17

Solution:

 

QUESTION: 18

If y = tan−1x and z = cot−1x then  is equal to

Solution:

QUESTION: 19

Solution:


QUESTION: 20

Solution:

 

QUESTION: 21

Solution:

 

QUESTION: 22

If both f and g are defined in a nhd of 0 ; f(0) = 0 = g(0) and f ‘ (0) = 8 = g’ (0), then  equal to

Solution:

 (by using L’Hospital Rule)

QUESTION: 23

Solution:

 

QUESTION: 24

The differential coefficient of log (| log x |) w.r.t. log x is

Solution:

 

QUESTION: 25

If f is derivable at x = a , then    is equal to

Solution: