### 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 $\frac { 1 } {{ x } ^ { 2 }}$