JEE Main Mathematics - Functions
Exam Duration: 60 Mins Total Questions : 30
If f(x) =ax+b, where a and b are integers, f(-1)=-5 and f(3)=3, then a and b are
- (a)
a=-3, b=-1
- (b)
a=2, b=-3
- (c)
a=0, b=2
- (d)
a=2, b=3
Let A={1,2,3}, B={4,5,6,7} and f={(1,4),(2,5),(3,6)} be a function from A to B. Then, f is
- (a)
one-one
- (b)
onto
- (c)
neither one-one nor onto
- (d)
many -one onto
If \(f(x)=\frac { sinx }{ sin\left( \frac { x }{ n } \right) } for\quad n\epsilon Z\) , then value of n, if its period is equal to \(4\pi \) , is
- (a)
2
- (b)
13
- (c)
4
- (d)
5
Let \(f:(4,6)\rightarrow (6,8)\) be a function defined by \(f(x)=x+\left[ \frac { x }{ 2 } \right] \) where [] denotes the greatest integer function, then f-1(x) is equal to
- (a)
\(x-\left\{ \frac { x }{ 2 } \right\} \)
- (b)
-x-2
- (c)
x-2
- (d)
\(\frac { 1 }{ x+\left\{ \frac { \pi }{ 2 } \right\} } \)
If f(x) and g(x) are periodic functions with periods 7 and 11, respectively. Then, the period of \(F(x)=f(x).g\left( \frac { x }{ 5 } \right) +g(x).f\left( \frac { x }{ 3 } \right) \) is
- (a)
177
- (b)
222
- (c)
433
- (d)
1155
The domain of the function \(f(x)=\frac { 1 }{ \sqrt { |x|-x } } \) is
- (a)
\(\left( 0,\infty \right) \)
- (b)
\(\left( -\infty ,0 \right) \)
- (c)
\(\left( -\infty ,\infty \right) -\{ 0\} \)
- (d)
\(\left( -\infty ,\infty \right) \)
Let f(x)=ax2+bx+c,
g(x)=a1x2+b1x+c1 and
p(x)=f(x)-g(x),\(\quad a\neq { a }_{ 1 }\neq 0\)
If p(x)=0 only for x=-1 and p(-2)=2, then the value of P(2) is
- (a)
18
- (b)
3
- (c)
9
- (d)
6
For real x, let \(f(x)={ x }^{ 3 }+5x+1,\) then
- (a)
f is one-one but not onto R
- (b)
f is onto R but not one-one
- (c)
f is one-one and onto R
- (d)
f is neither one-one nor onto R
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,\(\sqrt2\)}
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]. Which statement is correct
- (a)
the domain of G(x) and H(x) are same
- (b)
the range of G(x) and H(x) are same
- (c)
the union of domain of G(x) and H(x) are all real numbers
- (d)
the union of domain of G(x) and H(x) are rational numbers
Total number of solutions of 2x+3x+4x-5x=0 is
- (a)
0
- (b)
1
- (c)
2
- (d)
Infinitely many
Let f: R\(\rightarrow\)R defined by f(x)=\(\frac { { e }^{ { x }^{ 2 } }-{ e }^{ { -x }^{ 2 } } }{ { e }^{ { x }^{ 2 } }+{ e }^{ { -x }^{ 2 } } } ,\)then
- (a)
f(x) is one-one but not onto
- (b)
f(x) is neither one-one nor onto
- (c)
f(x) is many one but onto
- (d)
f(x) is one-one and onto
If g(x)=[x2]-[x]2, where [.] denotes the greatest integer function and x∈[0,2], then the set of values of g(x) is
- (a)
{-1,0}
- (b)
{-1,0,1}
- (c)
{0}
- (d)
{0,1,2}
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
If f(x) is a polynomial satisfying f(x). f(1/x)=f(x)+f(1/x) and f(3)=28, then f(4) is equal to
- (a)
63
- (b)
65
- (c)
17
- (d)
none of these
The value of the parameter \(\alpha\), for which the function f(x)=1+\(\alpha\)x, \(\alpha\)\(\ne\)0is the inverse of itself, is
- (a)
-2
- (b)
-1
- (c)
1
- (d)
2
If f(x)=sin2x+\(\sin ^{ 2 }{ \left( x+\frac { \pi }{ 3 } \right) } +cosx.\cos { \left( x+\frac { \pi }{ 3 } \right) } \) and g(5/4)=1, then (gof) x is
- (a)
a polynomial of the first degree in sin x, cos x
- (b)
a constant function
- (c)
a polynomial of the second degree in sin x, cos x
- (d)
none of the above
The range of function f: [0,1]⇢R, f(x)=x3-x2+4x+2sin-1x is
- (a)
[-π-2,0]
- (b)
[2,3]
- (c)
[0,4+π]
- (d)
(0,2+π]
If f: R⟶R is a function such that f(x)=x3+x2f'(1)+xf''(2)+f'''(3) for all x∈R, the f(2)-f(1)
- (a)
f(0)
- (b)
-f(0)
- (c)
f'(0)
- (d)
-f'(0)
The period of ecos4πx+x-[x]+cos2πx is [.] denotes the greatest integer function)
- (a)
2
- (b)
1
- (c)
0
- (d)
-1
Let f(x)=\(\begin{cases} 1+x & 0\le x\le 2 \\ 3-x & 2<x\le 3 \end{cases}\), then fof(x)
- (a)
\(=\begin{cases} 2+x & 0\le x\le 1 \\ 2-x & 1<x\le 2 \end{cases}\)
- (b)
\(=\begin{cases} 2+x & 0\le x\le 2 \\ 4-x & 2<x\le 3 \end{cases}\)
- (c)
\(=\begin{cases} 2+x & 0\le x\le 2 \\ 2-x & 2<x\le 3 \end{cases}\)
- (d)
none of these
The domain of f(x)=\(\frac { 1 }{ \sqrt { |cosx|+cosx } } \) is
- (a)
\(\left[ -2n\pi ,\quad 2n\pi \right] ;\quad \forall n\epsilon I\)
- (b)
\(\left[ 2n\pi ,\quad \bar { 2n+1\pi } \right] ;\quad \forall n\epsilon I\)
- (c)
\(\left( \frac { (4n+1)\pi }{ 2 } ,\frac { (4n+3)\pi }{ 2 } \right) ;\quad \forall n\epsilon I\)
- (d)
\(\left( \frac { (4n11)\pi }{ 2 } ,\frac { (4n+1)\pi }{ 2 } \right) ;\quad \forall n\epsilon I\)
If f: [-4,0]\(\rightarrow\)R is defined by ex+sin x, its even extension to [-4,4] is given by
- (a)
-e-|x|-sin|x|
- (b)
e-|x|-sin|x|
- (c)
e-|x|+sin|x|
- (d)
-e-|x|+sin|x|
If the function f : R \(\Rightarrow\) R be such that f(x) = x - [x], where [.] denotes the greatest integer function, then f-1(x) is
- (a)
\(\frac {1}{x-[x]}\)
- (b)
x-[x]
- (c)
not defined
- (d)
none of these
Domain of sin-1[secx] ([.] is greatest integer less then or equal to x) is
- (a)
\(\left\{ (2n+1)\pi ,(2n+9)\pi \right\} \cup \left\{ [(2m-1)\pi ,2m\pi +\pi /3,m\epsilon I \right\} \)
- (b)
\(\left\{ 2n\pi ,n\epsilon I \right\} \cup \left\{ [2m\pi ,(2m+1)\pi ,m\epsilon I \right\} \)
- (c)
\(\left\{ (2n+1)\pi ,n\epsilon I \right\} \cup \left\{ [2m\pi ,2m\pi +\pi /3],m\epsilon I \right\} \)
- (d)
none of these
sin ax+cos ax and |sin x|+|cos x| are periodic of same fundamental period, if a equals
- (a)
0
- (b)
1
- (c)
2
- (d)
4
Let f(x)=\(\begin{cases} 0, & for & x=0 \\ { x }^{ 2 }\sin { \left( \frac { \pi }{ x } \right) } , & for & -1<x<1,(x\neq 0), \\ x|x|, & for & x\ge 1\quad or\quad x\le -1 \end{cases}\) then
- (a)
f(x) is an odd function
- (b)
f(x) is an even function
- (c)
f(x) is neither odd not even
- (d)
f''(x) is an even function
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)
\(f\left( \frac { \pi }{ 4 } \right) =1\)