Mathematics - Functions
Exam Duration: 45 Mins Total Questions : 30
If \(f(x)=\frac { x-1 }{ x+1 } \) , then \(f\left( -\frac { 1 }{ x } \right) \) is equal to
- (a)
\(-\frac { 1 }{ f(x) } \)
- (b)
\(\frac { 1 }{ f(x) } \)
- (c)
f(x)
- (d)
-f(x)
The range of f(x)=1+3cos2x is equal to
- (a)
[3,2]
- (b)
[4,2]
- (c)
[-2,4]
- (d)
None of these
If \(f:R\rightarrow R\) is given by \(f(x)=\begin{cases} -1,\quad when\quad x\quad is\quad rational \\ 1,\quad when\quad x\quad is\quad irrational \end{cases}\) \(\left( fof \right) \left( 1-\sqrt { 3 } \right) \) is equal to
- (a)
1
- (b)
-1
- (c)
\(\sqrt { 3 } \)
- (d)
0
The domain of the function f denoted by \(f(x)=\sqrt { 4-x } +\frac { 1 }{ \sqrt { { x }^{ 2 }-1 } } \) is equal to
- (a)
\((-\infty, -1) \cup[1, 4)\)
- (b)
\((-\infty ,-1)\cup (1,4]\)
- (c)
\((-\infty ,-1]\cup [1,4]\)
- (d)
\((-\infty ,-1)\cup (1,4]\)
Let E={1,2,3,4} and F=(1,2}, then the number of onto functions from E to F is
- (a)
14
- (b)
16
- (c)
12
- (d)
8
Let function \(f:R\rightarrow R\) be defined by \(f(x)=2x+sinx\quad for\quad X\epsilon R\) then, f is
- (a)
one-to-one and onto
- (b)
one-one but not onto
- (c)
onto but not one-one
- (d)
neither one-to-one nor onto
If \(f(x)=3sin\sqrt { \left( \frac { { \pi }^{ 2 } }{ 16 } -{ x }^{ 2 } \right) } \) then its range is
- (a)
\(\left[ \frac { -3 }{ \sqrt { 2 } } ,\frac { 3 }{ \sqrt { 2 } } \right] \)
- (b)
\(\left[ 0,\frac { 3 }{ \sqrt { 2 } } \right] \)
- (c)
\(\left[ -\frac { 3 }{ \sqrt { 2 } } ,0 \right] \)
- (d)
None of these
Let f be a function defined by \(f(x)={ \left( x-1 \right) }^{ 2 }+1,(x\ge 1)\)
Statement I The set {x : f(x) = f-1(x)}={1,2}
Statement II f is bijection and \(f'(x)=1+\sqrt { x-1 } ,x\ge 1\) .
- (a)
Statement I is true, Statement II is true; Statement II is the correct explanation for Statement I
- (b)
Statement I is true, Statement II is true; Statement II is not the correct explanation for Statement I
- (c)
Statement I is true, Statement II is false
- (d)
Statement I is false, Statement II is true
If \(f:R\rightarrow R\) satisfies f(x+y)=f(x)+f(y), for all \(x,y\epsilon R\) and f(1)=7, then \(\sum _{ r=1 }^{ n }{ f(r) } \) is
- (a)
\(\frac { 7n }{ 2 } \)
- (b)
\(\frac { 7(n+1) }{ 2 } \)
- (c)
\(7n(n+1)\)
- (d)
\(\frac { 7n(n+1) }{ 2 } \)
If f(x)=\(\sqrt { (3|x|-x-2) } \) and g(x)=sin x, then domain of definition of (fog) x is
- (a)
\({ \left\{ 2n\pi +\frac { \pi }{ 2 } \right\} }_{ n\epsilon I }\)
- (b)
\(\underset { n\epsilon I }{ U } \left[ 2n\pi +\frac { 7\pi }{ 6 } ,2n\pi +\frac { 11\pi }{ 6 } \right] \)
- (c)
\({ \left\{ 2n\pi +\frac { 7\pi }{ 6 } \right\} }_{ n\epsilon I }\)
- (d)
\(\left\{ x\epsilon \{ (4m+1)\} \frac { \pi }{ 2 } ,m\epsilon I\} \right\} \underset { n\epsilon I }{ U } \left[ 2n\pi +\frac { 7\pi }{ 6 } ,2n\pi +\frac { 11\pi }{ 6 } \right] \)
f(x)=(sin x7)\({ e }^{ { x }^{ 5sgn{ x }^{ 9 } } }\) is
- (a)
an even function
- (b)
an odd function
- (c)
neither even nor odd
- (d)
none of these
Let f(x)=\(\sqrt { ([sin\quad 2x]-[cos\quad 2x]) } \)(where [.] denotes the greatest integer function), then range of f(x) will be
- (a)
{0}
- (b)
{1}
- (c)
{0,1}
- (d)
{0,1,\(\sqrt2\)}
If 2f(x - 1) - f \(\left( \frac { 1-x }{ x } \right) \) = x, then f(x) is
- (a)
\(\frac { 1 }{ 3 } \left\{ 2(1+x)+\frac { 1 }{ (1+x) } \right\} \)
- (b)
2 (x-1) - \(\frac { (1-x) }{ x } \)
- (c)
\({ x }^{ 2 }+\frac { 1 }{ { x }^{ 2 } } +4\)
- (d)
\(\frac { 1 }{ 4 } \left\{ (x+2)+\frac { 1 }{ (x+2) } \right\} \)
The range of the functior f(x) = \({ sin }^{ -1 }\left[ { x }^{ 2 }+\frac { 1 }{ 2 } \right] +{ cos }^{ -1 }\left[ { x }^{ 2 }-\frac { 1 }{ 2 } \right] \), where [.] denotes the greatest integer function) is
- (a)
\(\left\{ \frac { \pi }{ 2 } \right\} \)
- (b)
\(\left\{ \pi \right\} \)
- (c)
\(\left\{ -\frac { 1 }{ 2 } ,0 \right\} \)
- (d)
\(\left( 0,\frac { \pi }{ 2 } \right) \)
Let F(x)=f(x)+g(x), G(x)=f(x)-g(x) and H(x)=\(\frac { f(x) }{ g(x) } \), where f(x)=1-2sin2x and g(x)=cos 2x, ∀ f: R⟶ [-1,1] and g : R⟶ [-1,1]. If the solutions of F(x)-G(x)=0 are x1,x2,x3,...,xn where x∈[0,5π], then
- (a)
x1,x2,x3,...,xn are in AP with common difference π/4.
- (b)
the number of solutions of F(x)-G(x)=0 is 10, ∀ x∈[0,5π]
- (c)
the sum of all solution of F(x)-G(x)=0, ∀ x∈[0,5π] is 25π
- (d)
(b) and (c) correct
The domain of the function f(x)=\(\sqrt { \left( \frac { 1 }{ sin\quad x } -1 \right) } \)is
- (a)
\(\left( 2n\pi ,2n\pi +\frac { \pi }{ 2 } \right) ,\quad \forall n\epsilon I\)
- (b)
\(\left( 2n\pi ,(2n+1)\pi \right) ,\quad \forall n\epsilon I\)
- (c)
\(\left( 2n-1)\pi ,2n\pi \right) ,\quad \forall n\epsilon I\)
- (d)
none of the above
The domain of the function y=\(\underbrace { { log }_{ 10 }{ log }_{ 10 }{ log }_{ 10 }....{ log }_{ 10 }x }_{ n\quad times } \) is
- (a)
[10n,\(\infty\))
- (b)
(10n-1,\(\infty\))
- (c)
[10n-2,\(\infty\))
- (d)
none of these
The graph of the function y=f(x) is symmetrical about the line x=2, then
- (a)
f(x+2)=f(x-2)
- (b)
f(2+x)=f(2-x)
- (c)
f(x)=f(-x)
- (d)
none of these
The domain of the function f(x)=\(\sqrt { \sin ^{ -1 }{ \left( \log _{ 2 }{ x } \right) } } +\sqrt { \cos { (sinx) } } +\sin ^{ -1 }{ \left( \frac { 1+{ x }^{ 2 } }{ 2x } \right) } \)
- (a)
{x:1x2}
- (b)
{x}
- (c)
not defined for any value of x
- (d)
{-1,1}
If [.] denotes the greatest integer function, then the domain of the real valued function log[x+1/2]|x2-x-2| is
- (a)
\([ \frac { 3 }{ 2 } , \infty ) \)
- (b)
\([\frac { 3 }{ 2 } ,2 ,\ ) \cup (2,\infty )\)
- (c)
\((\frac { 1}{ 2 } ,2 ,\ ) \cup (2,\infty )\)
- (d)
none of these
The period of the function f(x)=asin2x+sin2(x+\(\pi\)/3)+cos x cos (x+\(\pi\)/3) is (where a is constant)
- (a)
1
- (b)
\(\pi\)/2
- (c)
\(\pi\)
- (d)
cannot be determined
Let f(x)=cos\(\sqrt {k}\)x, where K=[m]=the greatest integer\(\le\)m, if the period of f(x) is \(\pi\), then
- (a)
\(m\epsilon [4,5)\)
- (b)
m=4,5
- (c)
\(m\epsilon [4,5]\)
- (d)
none of these
The function f(x)=\(\int _{ 0 }^{ x }{ \log _{ e }{ \left( \frac { 1-x }{ 1+x } \right) } } dx\) is
- (a)
an even function
- (b)
an odd function
- (c)
a periodic function
- (d)
none of these
If f: R\(\rightarrow\)R, g: R\(\rightarrow\)R be two given functions, then f(x)=2 min(f(x)-g(x),0) equals
- (a)
f(x)+g(x)-|g(x)-f(x)|
- (b)
f(x)+g(x)+|g(x)-f(x)|
- (c)
f(x)-g(x)+|g(x)-f(x)|
- (d)
f(x)-g(x)-|g(x)-f(x)|
If g(x) is a polynomial satisfying g(x)g(y)=g(x)+g(y)+g(xy)-2 for all real x and y g(2)=5, then g(3) is equal to
- (a)
10
- (b)
24
- (c)
21
- (d)
none of these
If f(x)=\(\frac { x }{ 1+{ x }^{ 2 } } \) and f(A)=\(\left\{ y:-\frac { 1 }{ 2 } \le y<0 \right\} \), then set A is
- (a)
[-1,0)
- (b)
(-\(\infty\),-1)
- (c)
(-\(\infty\),0)
- (d)
(-\(\infty\),\(\infty\))
If y=f(x)=\(\frac { x+2 }{ x-1 } \), then
- (a)
x=f(y)
- (b)
f(1)=3
- (c)
y increases with for x<1
- (d)
f is rational function of x
If f(x)=cos([π2]x)+cos([-π2]x), where [x] stands for the greatest integer function, then
- (a)
\(f\left( \frac { \pi }{ 2 } \right) =-1\)
- (b)
\(f\left( \pi \right) =1\)
- (c)
\(f\left( -\pi \right) =0\)
- (d)
\(f\left( \frac { \pi }{ 4 } \right) =1\)
The possible values of 'a' for which the function f(x)=ex-[x]+cos ax (where [.] denotes the greatest integer function) is periodic with finite funcdamental period is
- (a)
\(\pi\)
- (b)
2\(\pi\)
- (c)
3\(\pi\)
- (d)
1