Mathematics - Functions
Exam Duration: 45 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
The period of \(f(x)=tan3x+cos\frac { 5x }{ 2 } \) is equal to
- (a)
\(2\pi \)
- (b)
\(7\pi \)
- (c)
\(4\pi \)
- (d)
\(9\pi \)
If 2
- (a)
0
- (b)
1
- (c)
3
- (d)
2
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)
\(\left( -\infty ,\infty \right) \)
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)
f(x)=-f(-x)
The domain of \({ sin }^{ -1 }\left[ { log }_{ 3 }\left( \frac { x }{ 3 } \right) \right] \) is
- (a)
[1,9]
- (b)
[-1,9]
- (c)
[-9,1]
- (d)
[-9,-1]
Range of values of f(x)=1+sinx+sin3x+sin5x+......;\(x\epsilon \left( -\frac { \pi }{ 2 } ,\frac { \pi }{ 2 } \right) \) is
- (a)
(0,1)
- (b)
(0,2)
- (c)
(-2,2)
- (d)
\((-\infty ,\infty )\)
The graph of f(x)=\(\left| \left| \left( \frac { 1 }{ \left| x \right| } -n \right) \right| -n \right| \) is lie in the (n>0)
- (a)
I and II quadrant
- (b)
I and III quadrant
- (c)
I and IV quadrant
- (d)
II and III quadrant
Let f(x)=sin-1sin(tan x) and g(x)=cos-1sin\(\sqrt { (1-\tan ^{ 2 }{ x } ) } \) are same functions, then x∊
- (a)
\(\left[ 0,\tan ^{ -1 }{ \frac { \pi }{ 2 } } \right] \)
- (b)
[0,1]
- (c)
[0,\(\infty\)]
- (d)
none of these
Let f(x)=min {x-[x], -x-[-x]}, -2\(\le\)x\(\le\)2; g9x)=|2-|x-2||, -2\(\le\)x\(\le\)2 and h(x)=, -2\(\le\)x\(\le\)2 and x\(\ne\)0 (where [x] denotes the greatest integer function\(\le\)x). The range fo f(x) is
- (a)
\(\left[ 0,\frac { 1 }{ 2 } \right] \)
- (b)
[0,1]
- (c)
[0,2]
- (d)
none of these
Let f be a function satisfying f(x)=\(\frac { { a }^{ x } }{ { a }^{ x }+\sqrt { a } } \)=ga(x)(a>0). Let f(x)=g9(x), then the value of \(\left[ \sum _{ r=1 }^{ 1995 }{ f\left( \frac { r }{ 1996 } \right) } \right] \) is (where [.] denotes the greatest integer function)
- (a)
995
- (b)
996
- (c)
997
- (d)
998
Let f be a function satisfying f(x)=\(\frac { { a }^{ x } }{ { a }^{ x }+\sqrt { a } } \)=ga(x)(a>0). If the value of \(\sum _{ r=0 }^{ 2n }{ f\left( \frac { r }{ 2n+1 } \right) } =\frac { 1 }{ 1+\sqrt { a } } \)+987, then the value of n is
- (a)
493
- (b)
494
- (c)
987
- (d)
988
Let f be a function satisfying f(x)=\(\frac { { a }^{ x } }{ { a }^{ x }+\sqrt { a } } \)=ga(x)(a>0). The value of \(\sum _{ r=1 }^{ 2n-1 }{ 2f\left( \frac { r }{ 2n } \right) } \) is
- (a)
0
- (b)
2n-1
- (c)
2n
- (d)
none of these
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]. Domain and range of H(x) are respectively.
- (a)
R and {1}
- (b)
R and {0,1}
- (c)
R~{(2n+1)\(\frac {\pi}{4}\), and {1}, n∈I
- (d)
R~{(2n+1) and {0,1}, n∈I
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 R⟶[-2,2], then
- (a)
F(x) is one-one function
- (b)
F(x) is onto function
- (c)
F(x) is into function
- (d)
none of the above
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 range of the function f(x)=3|sin x|-|-2|cos x| is
- (a)
[-2,\(\sqrt{13}\)]
- (b)
[-2,3]
- (c)
[3,\(\sqrt{13}\)]
- (d)
[-3,2]
The value of b and c for which the identity f(x+1)-f(X)=8x+3 is satisfied, where f(x)=bx2+cx+d are
- (a)
b=2, c=1
- (b)
b=4, c=-1
- (c)
b=-1, c=4
- (d)
b=-1, c=1
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]
Let f: R⇢Q be a continuous function such that f(2)=3, then
- (a)
f(x) is always an even function
- (b)
f(x) is always an odd function
- (c)
nothing can be said about f(x) being even or odd
- (d)
f(x) is an increasing function
Let f: R\(\rightarrow\)R be a function defined by \(f(x)=\frac { { x }^{ 2 }+2x+5 }{ { x }^{ 2 }+x+1 } \) is
- (a)
one-one and into
- (b)
one-one and onto
- (c)
many one and onto
- (d)
many one and into
The domain of f(x)=\(\sqrt { \{ x-4-2\sqrt { (x-5 } \} } -\sqrt { \{ x-4+2\sqrt { (x-5 } \} } \) is
- (a)
\((-5,\infty)\)
- (b)
\((-\infty,2)\)
- (c)
\((5,\infty)\)
- (d)
none of these
If 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)) \(\sqrt { 3 } /4\) is , then
- (a)
\(g(x)=\pm \sqrt { (1-{ x }^{ 2 }) } \)
- (b)
\(g(x)=- \sqrt { (1-{ x }^{ 2 }) } \)
- (c)
\(g(x)=\sqrt { (1-{ x }^{ 2 }) } \)
- (d)
\(g(x)=\sqrt { (1+{ x }^{ 2 }) } \)
Let f be a real valued function defined by \(f(x)=\frac { { e }^{ x }-{ e }^{ -|x| } }{ { e }^{ x }+{ e }^{ |x| } } \), then range of f is
- (a)
R
- (b)
[0,1]
- (c)
[0,1)
- (d)
[0,1/2)
if \(f(x)=\left( \frac { x-1 }{ x+1 } \right) ,\) then which of the following statement(s) is/are correct
- (a)
\(f\left( \frac { 1 }{ x } \right) =f(x)\)
- (b)
\(f\left( \frac { 1 }{ x } \right) =-f(x)\)
- (c)
\(f\left( -\frac { 1 }{ x } \right) =\frac { 1 }{ f(x) } \)
- (d)
\(f\left( -\frac { 1 }{ x } \right) =-\frac { 1 }{ f(x) } \)
Which of the following functions are periodic? where [.] denotes the greatest integer fucntion.
- (a)
f(x)=sin x+|sin x|
- (b)
\(g(x)=\frac { (1+sin\quad x)(1+sec\quad x) }{ (1+cos\quad x)(1+cosec\quad x) } \)
- (c)
h(x)=max (sin x, cos x)
- (d)
\(p(x)=[x]+\left[ x+\frac { 1 }{ 3 } \right] +\left[ x+\frac { 2 }{ 3 } \right] -3x+10\)