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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)/x^{2}

⇒ − 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)/x^{2}

⇒ 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) =x^{2}g(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 x^{p}y^{q} = (x+y)^{p+q}Taking log on both sides we get:

_{p}logx+_{q}logy = (p+q) log (x+y). Differentiating both sides w.r.t. x we get ,

QUESTION: 10

If y = ae^{mx} + be^{−mx}, then y_{2} is equal to

Solution:

y = ae^{mx} + be^{-mx} ⇒ y1 = ame^{mx} + (-m)be^{-mx} ⇒y2

= am^{2}e^{mx}_{ + (m}^{2})be^{-mx} ⇒y2 = m^{2} (ae^{mx} + be^{-mx}) ⇒ y2 = m^{2}y

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 ⇒ y_{1} = am cos mx − bm sin mx

⇒ y_{2} = −am^{2}sin mx−bm^{2}cosmx ⇒ y_{2} = −m^{2}(a sin mx+b cos mx)

QUESTION: 16

Solution:

QUESTION: 17

Solution:

QUESTION: 18

If y = tan^{−1}x and z = cot^{−1}x 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:

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