IISER Mathematics - Functions
Exam Duration: 45 Mins Total Questions : 30
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 period of \(f(x)=tan3x+cos\frac { 5x }{ 2 } \) is equal to
- (a)
\(2\pi \)
- (b)
\(7\pi \)
- (c)
\(4\pi \)
- (d)
\(9\pi \)
Let A={-1,0,1,2}, B={-4,-2,0,2} and \(f,g:A\rightarrow B\) be the function defined by \(f(x)={ x }^{ 2 }-x,\quad x\epsilon A\) and \(g(x)=2\left| x-\frac { 1 }{ 2 } \right| -1,\quad x\epsilon A\) . Then,
- (a)
f>g
- (b)
g>f
- (c)
f=g
- (d)
2f=g
A function f from the set of natural numbers to integers defined by \(f(n)=\begin{cases} \frac { n-1 }{ 2 } ,when\quad n\quad is\quad odd \\ -\frac { n }{ 2 } ,when\quad n\quad is\quad even \end{cases}\) is
- (a)
one- one but not onto
- (b)
onto but not one-one
- (c)
both one-one and onto
- (d)
neither one-one nor onto
The period of \({ sin }^{ 2 }\theta \) is
- (a)
\({ \pi }^{ 2 }\)
- (b)
\({ \pi }\)
- (c)
\(2{ \pi }\)
- (d)
\(\frac { \pi }{ 2 } \)
If a f(x) + b\(f\left( \frac { 1 }{ x } \right) \) =x-1, x\(\neq\)0 and a\(\neq\)b, then f(2) is
- (a)
\(\frac { a }{ { a }^{ 2 }-{ b }^{ 2 } } \)
- (b)
\(\frac { (a+2b) }{2( { a }^{ 2 }-{ b }^{ 2 }) } \)
- (c)
\(\frac { (a-2b) }{ ({ a }^{ 2 }-{ b }^{ 2 }) } \)
- (d)
\(\frac { (2a+b) }{2( { a }^{ 2 }-{ b }^{ 2 }) } \)
Let f(x)=\(\frac { sin2nx }{ 1+\cos ^{ 2 }{ nx } } ,\quad n\epsilon N\) has\(\frac { \pi }{ 6 } \) as its fundamental period, then n is equal to
- (a)
2
- (b)
4
- (c)
6
- (d)
8
Let f(x)=[9x-3x+1]∀x∈(-∞,1), then range of f(x) is ([.] denotes the greatest integer function)
- (a)
{0,1,2,3,4,5,6}
- (b)
{0,1,2,3,4,5,6,7}
- (c)
{1,2,3,4,5,6}
- (d)
{1,2,3,4,5,6,7}
Period of the function f(x)=\(\frac { sin\{ sin(nx)\} }{ tan\left( \frac { x }{ n } \right) } ,\quad n\epsilon N,\) , is 6\(\pi\), then n is equal to
- (a)
1
- (b)
2
- (c)
3
- (d)
none of these
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)
\(\frac { 1 }{ 4 } \left\{ (x+2)+\frac { 1 }{ (x+2) } \right\} \)
Let f(x)=x2-5x+6, g(x)=f(|x|), h(x)=|g(x)| and \(\phi \)(x)=h(x)-(x) are four functios, where (x) is the least integral functions x\(\ge\)x. The number of solutions of the equation g(x)=0 is
- (a)
0
- (b)
2
- (c)
4
- (d)
6
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 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(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
If f(2x+3y,2x-7y)=20x, then f(x,y) equals
- (a)
7x-3y
- (b)
7x+3y
- (c)
3x-7y
- (d)
3x+7y
if f(x+y)=f(x)+f(y)-xy-1 for all x,y and f(1)=1, then the number of solutions of f(n)=n, n∈N is
- (a)
one
- (b)
two
- (c)
three
- (d)
none of these
Let f be a function satisfying 2f(xy)={f(x)}y+{f(y)}x and f(1)=k\(\ne\)1, then \(\sum _{ r=1 }^{ n }{ f(r) } \) is equal to
- (a)
kn-1
- (b)
kn
- (c)
kn +1
- (d)
none of the above
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
If f: X⇾Y defined by f(x)=\(\sqrt3\)sin x+cos x+r is one-one and onto, then Y is
- (a)
[1,4]
- (b)
[2,5]
- (c)
[1,5]
- (d)
[2,6]
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)
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) is a polynomial function ot the second degree such that f(-3) = 6, f(O);=: 6 and f(2) = 11, then the graph of the function f(x) cuts the ordinate x = 1 at the point
- (a)
(1,8)
- (b)
(1,-2)
- (c)
1,4
- (d)
none of these
If f(x + y, x - y) = x:v, then the arithrneric mean of f(x, y) and f(y, x) is
- (a)
x
- (b)
y
- (c)
0
- (d)
none of these
Under the condition ----------, the domain of f1+f2 is equal to
- (a)
dom f1\(\ne\)dom f2
- (b)
dom f1=dom f2
- (c)
dom f1>dom f2
- (d)
dom f1<dom f2
Let f: R\(\rightarrow\)[0, \(\pi\)/2] be a function defined by f(x)=tan-1(x2+x+a). If f is onto, then a equals
- (a)
0
- (b)
1
- (c)
1/2
- (d)
1/4
The interval into which the function \(y=\frac { (x-1) }{ ({ x }^{ 2 }-3x+3) } \) transforms the entire real line is
- (a)
\(\left[ \frac { 1 }{ 3 } ,2 \right] \)
- (b)
\(\left[ -\frac { 1 }{ 3 } ,1 \right] \)
- (c)
\(\left[ -\frac { 1 }{ 3 } ,2 \right] \)
- (d)
none of these
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)
\(f\left( \frac { \pi }{ 4 } \right) =1\)
Let \(f(x)=\frac { 5\sqrt { \sin { x } } }{ 1+\sqrt [ 3 ]{ \sin { x } } } .\) If Dis the domain of f, then D contains
- (a)
\((0, \pi )\)
- (b)
\((-2\pi, \pi)\)
- (c)
\((2\pi, 3\pi)\)
- (d)
\((4\pi,6\pi)\)