### Mathematics - Differential Coefficient

#### Question - 1

If y=tan-1$\sqrt { \left( \frac { 1+sin\quad x }{ 1-sin\quad x } \right) } ,\frac { \pi }{ 2 } <x<\pi ,then\frac { dy }{ dx }$equals

• A -1/2
• B -1
• C 1/2
• D 1

#### Question - 2

If f(x) = |x|, then f' (x), where x$\neq$0 is equal to

• A -1
• B 0
• C 1
• D $\frac { \left| x \right| }{ x }$

#### Question - 3

If x=2 cos t - cos 2t, y=2 sin t - sin 2t, then $\frac { { d }^{ 2 }y }{ { dx }^{ 2 } }$ at t = $\frac { \pi }{ 2 }$ is

• A -5/2
• B -3/2
• C 3/2
• D 5/2

#### Question - 4

Let f(x) =$\left| \begin{matrix} sin\quad 3x & 1 & 2\left( cos\left( \frac { 3x }{ 2 } \right) +sin\left( \frac { 3x }{ 2 } \right) \right) ^{ 2 } \\ cos\quad 3x & -1 & 2\left( cos^{ 2 }\left( \frac { 3x }{ 2 } \right) -sin^{ 2 }\left( \frac { 3x }{ 2 } \right) \right) \\ tan\quad 3x & 4 & 1+2\quad tan\quad 3x \end{matrix} \right|$
then the value of f' (x) at x = (2n + 1) $\pi$, n $\in$ I (the set of integers) is equal to

• A (-1)n
• B 3
• C (-1)n+1
• D 9

#### Question - 5

The set of values of a for which the function f(x) = (4a - 3) (x + In 5) + 2 (a -7) cot $\left( \frac { x }{ 2 } \right) { sin }^{ 2 }\left( \frac { x }{ 2 } \right)$ does not possess critical points is

• A $(1,\infty )$
• B $[1,\infty )$
• C $(-\infty ,2)$
• D $(-8,-4/3)\cup (2,\infty )$

#### Question - 6

If f(x) = x + tan x and f is inverse of g' then g' (x) is equal to

• A $\frac { 1 }{ 1+(g(x)-x)^{ 2 } }$
• B $\frac { 1 }{ 1-(g(x)-x)^{ 2 } }$
• C $\frac { 1 }{ 2+(g(x)-x)^{ 2 } }$
• D $\frac { 1 }{ 2-(g(x)-x)^{ 2 } }$

#### Question - 7

If $\sqrt { (1-{ x }^{ 2n }) } +\sqrt { (1-y^{ 2n }) } =a({ x }^{ n }-{ y }^{ n }),then\quad \sqrt { \left( \frac { 1-{ x }^{ 2n } }{ 1-{ y }^{ 2n } } \right) } \frac { dy }{ dx }$ is equal to

• A $\frac { { x }^{ n-1 } }{ { y }^{ n-1 } }$
• B $\frac { { y }^{ n-1 } }{ { x }^{ n-1 } }$
• C $\frac { x }{ y }$
• D 1

#### Question - 8

If  y2=ax2+bx+c, then y3.$\frac { { d }^{ 2 }y }{ { dx }^{ 2 } }$ is

• A a constant
• B a function of x only
• C a function of y only
• D a function of x and y

#### Question - 9

If $x=sec\theta -cos\theta ,y={ sec }^{ 10 }\theta -{ cos }^{ 10 }\theta \quad and\quad \left( { x }^{ 2 }+4 \right) \left( \frac { dy }{ dx } \right) ^{ 2 }=k({ y }^{ 2 }+4),$then k is equal to

• A $\frac { 1 }{ 100 }$
• B 1
• C 10
• D 100

#### Question - 10

If x=ex-y, then $\frac { dy }{ dx }$ is equal to

• A (1+1n x)-1
• B (1+1n x)-2
• C 1n x (1+1n x)-2
• D none of these