Mathematics - Differential Calculus

Question - 1

The function \(f\left( x \right) =\log { \frac { 1+x }{ 1-x } } \),satisfies the equation

  • A \(f\left( x+2 \right) -2f(x+1)+f(x)=0\)
  • B f(x)+f(x+1)=f(\({ x }^ {2 }\)+x)
  • C \(f({ x }_{ 1 })f({ x }_{ 2 })=f({ x }_{ 1 }+{ x }_{ 2 })\)
  • D \(f({ x }_{ 1 })+f({ x }_{ 2 })=f(\frac { { x }_{ 1 }+{ x }_{ 2 } }{ 1+{ x }_{ 1 }{ x }_{ 2 } } )\)

Question - 2

A polynomial function f(x) satisfies the equation \(f(x)f(\frac { 1 }{ x } )=f(x)+f(\frac { 1 }{ x } )andf(3)=28\);then value of f(4) is:

  • A 67
  • B 65
  • C 63
  • D 29

Question - 3

The graph of the function \(cosx\quad cos(x+2)-{ cos }^{ 2 }(x+1)\)is

  • A a straight line passing through the point(0,\({ sin }^{ 2 }\)1)with slope 2.
  • B a straight line passing through(0,0)
  • C a parabola with vertex (1,\({ -sin }^{ 2 }\)1)
  • D a straight line passing through the point \((\frac { \Pi }{ 2 } ,{ -sin }^{ 2 }1)\)and parallel to the -axis

Question - 4

If \(f(x)=cos[{ \Pi }^{ `2 }]x+cos[{ -\Pi }^{ `2 }]x,\)where [x] stands for the greatest integer function,then

  • A \(f(\frac { \Pi }{ 2 } )=-1\)
  • B \(f(\pi )=1\)
  • C \(f(-\pi )=1\)
  • D \(f(\frac { \Pi }{ 4 } )=2\)

Question - 5

Let g(x) be a function defined on [-1,1].If the area of the equilateral triangle with two of its vertices at (0,0) and [x,g(x)]is \(\frac { \sqrt { 3 } }{ 4 } \),then function g(x), is

  • A \(\pm \sqrt { 1-{ x }^{ 2 } } \)
  • B either \(\sqrt { 1-{ x }^{ 2 } } \)or -\(\sqrt { 1-{ x }^{ 2 } } \)
  • C \(\sqrt {1+{x} ^ {2 }}\)
  • D None of these

Question - 6

If \(f(x)=cos(log\quad x),\)then \(f(x)f(y)-\frac { 1 }{ 2 } [f(x/y)+f(xy)]\)equals

  • A 0
  • B 1
  • C -1
  • D 2

Question - 7

Let {x} and [x] denote the fractional and integral parts of a real number x-respectively, and 4{x}=x+[x],then x=

  • A 0 or \(\frac { 5 } { 3 }\)
  • B 1 or \(\frac { 4 } { 3 }\)
  • C \(\frac { 3 } { 2}\)
  • D \(\frac { 5 } { 4}\)

Question - 8

If \(f(x)=log(\frac { 1+x }{ 1-x } )andg(x)=\frac { 3x+{ x }^{ 3 } }{ 1+3{ x }^{ 2 } } \)then f(g(x))is equal to

  • A -f(x)
  • B 3f(x)
  • C \({ (f(x)) }^{ 3 }\)
  • D f(3x)

Question - 9

If f(1)=1 and f(n+1)=2f(n)+1,if \(n\ge 1\),then f(x) is defined as

  • A \({ 2 } ^ { n+1 }\)
  • B \({ 2 } ^ { n }\)
  • C \({ 2 } ^ { n }-1\)
  • D \({ 2 } ^ { n-1 }-1\)

Question - 10

If \(f(x+\frac { 1 }{ x } )={ x }^{ 2 }+\frac { 1 }{ { x }^{ 2 } } \),then equals

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