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
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