Question 1 :
Let $f(x)=[x]$ and $g(x)=x-[x]$, then which of the following is false?
Question 3 :
If $X \in R,$ then sgn $\left( {{X^2} + 1} \right)$ is equal to
Question 4 :
$A=\left \{ x\in R: x\neq 0,-4\leq x\leq 4 \right \} f:A\rightarrow R$ is defined as $f(x)=\dfrac{\left | x \right |}{x}$ then the range of $f$ is <br>
Question 5 :
Evaluate $\displaystyle \left ( 4a + 3b \right )^{2} - \left ( 4a - 3b \right )^{2} + 48ab$
Question 6 :
The population of a town increases by $5\%$ every year. If the present population is $5,40,000$ find the population after 2 years.
Question 8 :
$A$ and $B$ are two sets having $3$ and $4$ elements respectively and having $2$ elements in common. The number of relations which can be defined from $A$ to $B$ is:
Question 9 :
If function $y=f(x)$ satisfies $(x+1)$ $f(x)-2({x}^{2}+x)f(x)= \dfrac { { e }^{ { x }^{ 2 } } }{ \left( x+1 \right) } ,\forall x>-1$
Question 10 :
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a function such that for any irrational number $r,$ and any real number $x$ we have $f(x)=f(x+r)$. Then, $f$ is
Question 11 :
If $\displaystyle \left[ x \right] $ is the greatest integer less than or equal to $x$, what is the value of $\displaystyle \left[ -1.6 \right] +\left[ 3.4 \right] +\left[ 2.7 \right] $?
Question 12 :
Let R be the relation in the set N given by = {(a, b): a = b - 2, b > 6}. Choose the correct answer
Question 13 :
If $\displaystyle f\left( { x }_{ 1 } \right) -f\left( { x }_{ 2 } \right) =f\left( \frac { { x }_{ 1 }-{ x }_{ 2 } }{ 1-{ x }_{ 1 }{ x }_{ 2 } } \right) $ for $\displaystyle { x }_{ 1 },{ x }_{ 2 }\in \left[ -1,1 \right] $ then $f\left( x \right) $ is
Question 14 :
If $2\leq a < 3$, then the value of $\cos^{-1} \cos [a] + \text{cosec}^{-1} \text{cosec }[a] + \cot^{-1} \cot [a]$. where [.] denotes greater integer less then equal to $x$) is equal to:
Question 15 :
The relation $R$ defined on the set $A=\left\{ 1,2,3,4,5 \right\} $ by $R=\left\{ \left( a,b \right) :\left| { a }^{ 2 }-{ b }^{ 2 } \right| <16 \right\} $, is not given by
Question 17 :
Let $R$ be a relation on the set $N$ given by $R=\left\{ \left( a,b \right) :a=b-2,b>6 \right\}$. Then
Question 18 :
Let [ ] represents the greatest integer function and $[x^3+x^2+1+x]=[x^3+x^2+1]+x.$ The number of solution(s) of the equation in $\left| [x] \right| =2-\left| [x] \right| $ is
Question 22 : A relation $R$ is defined from $\left\{ 2,3,4,5 \right\} $ to $\left\{ 3,6,7,10 \right\} $ by:<div>$xRy\Leftrightarrow x$ is relatively prime to $y$. Then, domain of $R$ is</div>
Question 23 :
Let $R$ be a relation on $N$ defined by $x+2y=8$. The domain of $R$ is
Question 25 :
If $f(x) = {x^2} - {x^{ - 2}}$ then $f\left(\cfrac{1}{x}\right)$ is equal to
Question 26 :
If $f: R^+ \rightarrow R$ such that $f(x)=\log_3 x,$ find $f^{-1}(x)$.<br/>
Question 27 :
$f:\left( 0,\infty \right) \rightarrow R$ is continuous. If $F\left(x\right)$ is a differentiable function such that $F\left(x\right)= f\left(x\right), \forall x>0$ and $ f\left( { x }^{ 2 } \right) ={ x }^{ 2 }+{ x }^{ 3 }$, then $f\left(4\right)$ equals
Question 28 :
The minimum value of $f\left( x \right) ={ x }^{ 2 }+2x+3 ,x\in R$ is equal to
Question 31 :
If $\displaystyle a - \frac{1}{a} = 8$ and $\displaystyle a \neq 0$; find :$\displaystyle a + \frac{1}{a}$.
Question 32 :
$f(x)=\begin{cases} 2-|{ x }^{ 2 }+5x+6|,\quad \quad \quad x\neq -2 \\ a^{ 2 }+1,\quad \quad \quad \quad \quad \quad \quad \quad \quad x=-2 \end{cases}$. then the range of $a$, so that $f(x)$ has maxima at $x=-2$, is
Question 33 :
Find the value of $\displaystyle f\left( -3 \right) $ if the function $\displaystyle f(x)$ is defined as $\displaystyle f\left( x \right) =-8{ x }^{ 2 }$
Question 34 : <div>State the whether given statement is true or false</div>If $f\left( x \right) = \dfrac{{x + 1}}{{x - 1}},$ then $f\left( x \right) + f\left( {\dfrac{1}{x}} \right) = 0$
Question 36 :
We want to find a polynomial f(x) of degree n such that f(1) = $\sqrt2$ and f(3) =$\pi$. Which of the following is true?
Question 37 :
A relation $\phi$ from $C$ to $R$ is defined by $x\phi y\Leftrightarrow \left| x \right| =y$. Which one is correct?
Question 39 :
Let the number of elements of the sets $A$ and $B$ be $p$ and $q$ respectively. Then, the number of relations from the set $A$ to the set $B$ is
Question 40 :
If A = (a, b, c, d), B= (p, q, r, s). then which of the following are relations from A to B? Give reasons for your answer:
Question 41 :
Let $f$ be a linear function for which $f (6) - f (2) = 12$. The value of $f (12) - f(2)$ is equal the
Question 42 :
If $A=\left\{ 1,2,3 \right\} , B=\left\{ 1,4,6,9 \right\} $ and $R$ is a relation from $A$ to $B$ defined by $x$ is greater than $y$. The range of $R$ is
Question 43 :
If $f: R \rightarrow R$ and $g: R \rightarrow R$ are defined by $f(x) =2x +3, g(x)=x^2 + 7$, what are the values of $x$ for which $f[g(x)]=25$?<br/>
Question 44 :
If $f(x) = 3x -7$, then what is the value of $3f(3)$
Question 46 :
If $f(x) = \dfrac{x + 1}{x - 1}$ show that f(f(x)) is an identify function.
Question 48 : <div>Let $x$ be a real number $\left [ x \right ]$ denotes the greatest integer function, and $\left \{ x \right \}$ denotes the fractional part and $(x)$ denotes the least integer function,then solve the following.<br/></div>$\left [ 2x \right ]-2x=\left [ x+1 \right ]$<br/>
Question 49 :
In $[0, 1]$ Lagranges Mean Value theorem is NOT applicable to<br>
Question 51 : <p>The set of real values of x satisfying $|x - 1| \leq 3$ and $|x - 1| \geq 1$ is</p>
Question 52 :
If $f(x) = 3x + 10, g(x) = x^2 - 1$, then $(fog)^{-1}$ is equal to
Question 53 :
If $f(x) \displaystyle = \left ( \frac{x-1}{x+1} \right )$, then which of the following statements is/are correct?
Question 54 :
$f:R\rightarrow R$ is defined as $f(x)=2x+\left | x \right |$ then $f(2x)-f(-x)-4x=$
Question 55 :
If $f:R\rightarrow (-1,1)$ is defined by $f(x)=\cfrac { -x\left| x \right| }{ 1+{ x }^{ 2 } } $ then ${f}^{-1}(x)$ equals
Question 56 :
Let $f : R\rightarrow R$ be such that $f(2x - 1) = f(x)$ for all $x\epsilon R$. If $f$ is continuous at $x = 1$ and $f(1) = 1$, then
Question 57 :
The value of $b$ and $c$ for which the identify $f (x + 1) - f (x) = 8x + 3$ is satisfied, where $f (x)\, =\, bx^{2}\, +\, cx\, +\, d$, are -
Question 59 :
If $|z|\geq 3$, then the least value of $\left |z + \dfrac {1}{4}\right |$ is
Question 60 :
Let $A=\{9,10,11,12,13\}$ and $f:A\rightarrow N$ be defined by $f(n)=$ highest prime factor of $n$, then its range is<br>
Question 62 :
If $f(x) + f( 1- x) = 10$, then the value of $f\left ( \frac{1}{10} \right )\, +\, f\left ( \frac{2}{10} \right )\, +\, ..........\, +\, f\left ( \frac{9}{10} \right )$ is
Question 63 :
If $e^{x} = y + \sqrt {1 + y^{2}}$, then the value of $y$ is
Question 64 :
If f is a function such that $f(0) = 2, f(1) = 3, f(x + 2) = 2f(x) - f(x +1)$ for every real x, then $f(5)$ is
Question 65 :
The curve $ay^2 = x^2 (3a - x)$ cuts the $y$-axis at
Question 67 :
If is a mapping from P to Q The set of all images of elements of P is called _________
Question 69 :
If $f\left( x \right) = {x^2} + \cfrac{1}{{{x^2}}}$, then find the value of $f(5)$.
Question 70 :
When $ x \ge 2, $ then function $ f(x) = 2 |x-2| - |x+1| + x $ is reduced to :
Question 72 :
Sum of all integral values of $a \in [1,500]$ for which the equation $[x]^{3}+x-a=0$ has a solution ($[.]$ denotes the greatest integer function) is
Question 73 :
An equation that defines $y$ as a function of $x$ is given. Solve for $y$ in terms of $x$, and replace $y$ with the function notation $f\left( x \right)$.<br>$x - 2y = 18$
Question 74 :
Let $f:\left\{ x,\cfrac { { x }^{ 2 } }{ 1+{ x }^{ 2 } } :x\in R \right\} $ be a function from $R\rightarrow R$, the range of $f$ is
Question 77 :
If $A = \left\{ {x\left| {{x^2} - x = 0} \right.,\,x\, \in \,R} \right\},\,f\,:\,N\, \to \,A$ and $f\left( x \right) = \left\{ \begin{array}{l}0,\,when\,x\,is\,even\\1,\,when\,x\,is\,odd\end{array} \right.$, then range of $f$ is
Question 79 :
If a $f(x)+bf\left( \cfrac { 1 }{ x } \right) =\dfrac{1}{x}-5,x\neq 0$ and $a\neq b$, then $f(2) $ is equal to
Question 81 :
If $y=2\tan ^{ -1 }{ x } +\sin ^{ -1 }{ \cfrac { 2x }{ 1+{ x }^{ 2 } } } \quad $ then
Question 82 :
Let the function f be defined by $f(x)=5x$ for all numbers x. Which of the following is equivalent to $f(p+r)$?
Question 83 :
The range of the function $f(x) = \dfrac{{1 - \tan x}}{{1 + \tan x}}$ is
Question 84 :
If $h(x) = x^{2} - 3x + 1$, what is $[h(-2)^{2}]$?
Question 86 :
Let $\circ$ be defined as $a\circ b = a^{2} + ba - 16b \div a$. Calculate the value of $8\circ 3$.
Question 87 :
If $a = b^3$ and $b = \dfrac{\sqrt{5}}{c}$, calculate the value of $ a$ when $c = \dfrac{1}{3}$.
Question 89 :
Give the domain and range of the relation<br>$\left\{ \left( 2,2 \right) ,\left( -3,-3 \right) ,\left( -6,-6 \right) ,\left( 6,6 \right) \right\} $
Question 90 :
Let f and g are two real functions defined as <br>$f(x) = 2x -3$<br>$g(x) =\frac{3+x}{2}$. Find gof and fog.
Question 91 :
If $P$ and $Q$ are the sum and product respectively of all integral values of $x$ satisfying the equation $|3[x]-4x|=4$, then<br>(where [.] denotes represents greatest integer function )
Question 92 :
The domain of $\displaystyle f(x)=\frac{\tan \pi [x]}{1+ \sin \pi [x] + [x^{2}]}$ is (where [.] denotes greatest integer function)<br>
Question 94 :
Let $f(x) = \dfrac {ax + b}{cx + d}$, then $fof(x) = x$, provided that
Question 95 :
Let $R$ be the set of all real numbers and let f be a function $R$ to $R$ such that $f(x) + (x + \dfrac{1}{2} ) f(1 - x) = 1$, for all $x \in R$. Then $2f(0) + 3f(1)$ is equal to
Question 96 :
The range of the function $f\left( x \right) = \sqrt { 4-{ x }^{ 2 } } + \sqrt { { x }^{ 2 }-1 } $ is
Question 97 :
If $ f(x) = ax^2 + bx + c $ satisfies the identity $ f(x+1) - f(x) = 8x + 3 $ for all $ x \epsilon R $ . Then $ (a,b) =$
Question 98 :
If $f(x) = log_e \left(\dfrac{1 - x}{1 + x} \right) $ then $f \left(\dfrac{2x}{1 + x^2} \right)$ is equal to
Question 99 :
Let $f_k(x)=\dfrac{1}{k}(sin^kx+cos^kx)$ for $k=1,2,3,....$ Then for all $x\in R$, the value of $f_4(x)-f_6(x)$ is equal to:-
Question 100 :
$\displaystyle \cos x \left ( \cos x + \sin x \right ) = \left [ x \right ] $ where $\displaystyle \left [ \: . \: \right ]$ represents the greatest integer function. The interval of values of $x$ which satisfy the equation is
Question 101 :
The abscissae of a point of intersection of the curves $\displaystyle y=\sin\frac{\pi x}{2}$ and $\displaystyle y=\left[\sin\frac{\pi}{12}+\cos\frac{\pi}{12}\right]$, where $[.]$ stands for the greatest integer function can be<br>
Question 102 :
The value of $[\sin x]+[1+ \sin x]+2+[\sin x]+3+ [\sin x]$ in $\left(\pi, \displaystyle \frac{3\pi}{2}\right]$ is<br/>
Question 103 :
If $\left | 2-\left | [x]-1 \right | \right |\leq 2$, then $x$ belongs to the solution set (where [.] represents greatest integer function)
Question 105 :
The solution set of $(x)^{2}+(x+1)^{2}=25,$ where $(x)$ denotes the nearest integer greater than or equal to $x$ is<br/>
Question 106 :
Let $[a]$ denotes the greatest integer less than or equal to $a$. Given that the quadratic equation $ { x }^{ 2 }+\left[ { a }^{ 2 }-5a+b+4 \right] x+b=0$ has roots $-5$ and $1$, which of the following is true <br>
Question 107 :
If $S$ is the set containing values of $x$ satisfying $[x]^2-5[x]+ 6 \le 0$, where $[x]$ denotes greatest integer function then $S$ contains :
Question 108 :
Complete solution set of $2[x]^{2}-11[x]+12\leq 0$ where $[ . ]$ represents greatest integer function, is<br/>
Question 109 : Match the statements of Column $ I $ with values of Column $II$<br><table class="wysiwyg-table"><tbody><tr><td></td><td><b>Column I</b></td><td></td><td><b>Column II</b></td></tr><tr><td>A.</td><td>If $f(x)=x+1$, when $x<0$,<br>$f(x)=x^{2}-1$,for $x\geq 0,$<br>then $f(f(x))$, for $-1\leq x\leq 0$ is</td><td>1.</td><td>$\displaystyle \frac{x-3}{2}$</td></tr><tr><td>B.</td><td>If $\displaystyle f \left(\frac{2\tan x}{1+\tan^{2}x}\right)=$ <br> $\dfrac{(\cos 2x+1)(\sec^{2}x+2\tan x)}{2}$ then $f(x)$ is</td><td>2.</td><td>$x^{2}+2x$</td></tr><tr><td>C.</td><td>If $f(x+y+1)=(\sqrt{f(x)}+\sqrt{f(y)})^{2}$ for all $x,y \in R$ and $f(0)=1,$ then $f(x)$ is</td><td>3.</td><td>$1+x$</td></tr><tr><td>D.</td><td>If $4 < x < 5 $ and $\displaystyle f(x)=\left[\frac{x}{4}\right] +2x+2$, where $[y]$ is the greatest integer $\leq y,$ then $f^{-1}(x)$ is</td><td>4.</td><td>$(x+1)^{2}$</td></tr></tbody></table>
Question 111 :
Given : $\displaystyle ax + by = d$ and $\displaystyle y= mx + c$. <br/>Find $x$ in terms of $b, c, d$ and $m$.<br/>
Question 114 :
$\displaystyle s=u+\frac{1}{2}a\left ( 2t-1 \right )$ ; make 't' the subject of formula.<br/>
Question 115 :
$\displaystyle V=\pi \left ( R^2-r^2 \right )h$ ; make 'r' the subject of formula.<br>
Question 118 :
$f(x)$ is a function defined on the set of Real Numbers $f(1) = 1$.<br>Also, $f(x + 5) \geq f(x) + 5$ for all $x\epsilon R$ and $f(x + 1) \leq f(x) + 1$<br>If, $g(x) = (f(x))^{2} - f(x)$
Question 119 :
Let f(x) = x and g(x) = |x| for all $x \in R$. The function $\phi (x) \ given \ that \ \phi (x)\geq 0$ satisfying $(\phi (x) f(x))^2 + (\phi (x) g(x))^2 = x^4$ is.
Question 120 :
$Let\quad g(x)=\int _{ 0 }^{ x }{ f(t)dt,\quad f\quad is\quad such\quad that\quad \quad \frac { 1 }{ 2 } } \le f(t)\le 1\\ for\quad t\epsilon [0,1]\quad and\quad 0\le f(t)\le \frac { 1 }{ 2 } \quad for\quad t\epsilon [1,2].\quad Then\\ g(2)\quad satisfies\quad the\quad inequality$
Question 121 :
If $D_{30}$ is the set of all divisors of $30, x, y\epsilon D_{30}$, we define $x+y=LCM(x, y), x\cdot y=GCD(x, y), x'=\displaystyle\frac{30}{x}$ and $f(x, y, z)=(x+y)\cdot (y'+z)$, then $f(2, 5, 15)$ is equal to
Question 122 :
If $f\left(\dfrac {x+y}{2}\right)=\dfrac {f(x)+f(y)}{2}\forall,\ x y \epsilon R$ and $f'(0)=-1,f(0)=1$, then $f(2)=$
Question 124 :
If $f(x) = 3x - 2$ and $g(x) = 7, f[g(x)] =$
Question 125 :
The number of different solutions to the equation $f (20x -11/ x) = 0$ cannot be<br/><br/>
Question 126 :
If $f(x)$ is a polynomial function of degree $n$ satisfying eqation $f(x) f(2x) = xf(3x)$ then $f(x)$ is
Question 127 :
Let $f(x) = \dfrac {2025^{x}}{45 + 2025^{x}}$, then the value of $f\left (\dfrac {1}{2025}\right ) + f\left (\dfrac {2}{2025}\right ) + f\left (\dfrac {3}{2025}\right ) + ... + f\left (\dfrac {2024}{2025}\right )$ is equal to
Question 128 :
If f(x) is real valued continuous and differentiable satisfying $(f(x))^{2}=\int_{0}^{x}\left ( f^{2}(t)+(f'(t))^{2} \right )dt+2013$ then which of following are possible.<br/><br/>
Question 129 :
Let $f(x)=\frac{\left ( x-a \right )\left ( x-b \right )}{\left ( x-c \right )}$ Which of the following are <b>TRUE</b> ?
Question 130 :
What positive value(s) of x, less than $360^o$, will give a minimum value for 4 - 2 sin x cos x?
Question 131 :
Let $f$ be a function satisfying $\displaystyle f(x+y)=f(x)+f(y)$ for $x,y\:\epsilon\:R$. If $f(1)=k$ then $f(n)$,$n\:\in\:N$, is equal to
Question 132 :
Let $x,y$ be real numbers. Let $f(x,y)= |x+y| ;\ F [f(x,y)]= -f(x,y) $ and $\displaystyle G [f(x,y)]= -F [f(x,y)]$ then which of the following is true ?<br/>
Question 133 :
A real valued function $f(x)$ satisfies the functional equation $f(x-y)=f(x)f(y)-f(a-x)f(a+y)$, where $a$ is a given constant and $f(0)=1$, then
Question 134 :
For every pair of continuous functions $f, g : [0, 1] \rightarrow R$ such that $\mathrm{m}\mathrm{a}\mathrm{x} \{f(x) : x \in [0, 1]\} = \mathrm{m}\mathrm{a}\mathrm{x} \{g(x) : x \in [0, 1]\}$, the correct statement(s) is (are)
Question 135 : Assertion: <div>Consider two functions $f(x)=1+e^{\cot^{2}x}$ and $ \displaystyle g(x)=\sqrt{2\left| \sin x\right|-1}+\frac{1-\cos 2x}{1+\sin ^{4}x}$<br/></div><br/><div>Statement I: The solution of the equation $ f(x)=g(x)$ is given by $x=(2n+1)\dfrac{\pi}{2},\forall n\in I.$<br/></div>
Reason: <br/>Statement II: If $f(x)\leq k$ and $ g(x)\leq k$(where $k\in R),$ then solutions of the equation $ f(x)=g(x)$ is the solution corresponding to the equation $ f(x)=k$
Question 136 :
If the polynomial satisfies $f\left( x \right) =\dfrac { 1 }{ 2 } \begin{vmatrix} f\left( x \right) & f\left( \dfrac { 1 }{ x } \right) -f\left( x \right) \\ 1 & f\left( \dfrac { 1 }{ x } \right) \end{vmatrix}$ and $\displaystyle f \left ( 2 \right ) = 17$, then the value of $\displaystyle f\left ( 3 \right )$ is <br>
Question 137 : <div>Let a function $f(x)$ satisfies $f^{2}(x)-f^{2}(y)=4(x-y)$ and $f(0)= 2\left ( f\left ( x \right )\geq 0 \right )$whose domain is $\left [ a ,\infty \right )$ and it is differentiable on $\left (a ,\infty \right )$<br/></div>The value of $f(3)$ is<br/>
Question 138 :
A function $f : R \rightarrow R$ satisfies the equation $f(x)f(y) - f(xy) = x + y$ for all $x, y \in R $ and $f(1) > 0$ then
Question 139 :
The number of solution of the equation $\displaystyle a^{f\left ( x \right )}+g\left ( x \right )=0,$ where $\displaystyle a >0, g\left ( x \right )\neq 0$ and $\displaystyle g\left ( x \right )$ has minimum value $\dfrac14$, is<br/>
Question 140 :
Let $f:(0,1)\rightarrow \mathbb{R}$ be defined by$\displaystyle f(x)=\frac{b-x}{1-bx}$, where $b$ is a constant such that $0 < b < 1$, then:<br/>
Question 141 :
Find the correct statement pertaining to the functions $\displaystyle f\left( x \right) ={ \left( x-3 \right) }^{ 2 }+2$ and $\displaystyle g\left( x \right) =\frac { 1 }{ 2 } x+1$ graphed above
Question 142 :
Let $F_n (\theta) = \displaystyle \sum_{k = 0}^n \frac{1}{4^K} \sin^4 (2^{k} \theta)$, then which of the following is true
Question 143 :
If $f(x) = \cos( \log x)$ then $f(x^2)f(y^2)-\dfrac{1}{2} \left [ f(x^2y^2)+f\left ( \dfrac{x^2}{y^2} \right) \right]=$
Question 144 :
Let $f(x)=2-|x-3|, 1 \le x \le 5$ and for rest of the values $f(x)$ can be obtained by using the relation $f(5x)=\alpha\, f(x)\forall\, x \in R$.<br>The value of $f(2007)$ taking $\alpha = 5$, is:
Question 145 :
Let $a, b, c,$ $\epsilon\ R$. If $f(x)=ax^2+bx+c$ is such that $a+b+c=3$ and $f(x+y)=f(x)+f(y)+xy, \forall \, x, y\,\epsilon\, R,$ the $\displaystyle \sum_{n=1}^{10}f(n)$ is equal to.
Question 147 :
Find $m$, if $\displaystyle v = 3, g = 10, h = 5$ and $\displaystyle E = 109$.<br/>
Question 148 :
In the problem below, $f(x)={x}^{2}$ and $g(x)=4x-2$<br/>Find the following function: $(f\circ g)(x)$
Question 149 : <div>Given a function $f : A \rightarrow B$; where $A = \left \{1, 2, 3, 4, 5\right \}$ and $B = \left \{6, 7, 8\right \}$.<br/></div>The number of mappings of $g(x) : B\rightarrow A$ such that $g(i) \leq g(j)$ whenever $i < j$ is
Question 150 :
Find $r$, if $\displaystyle V = 22,\ R = 2,\ l = 4$ and $\displaystyle \pi =3\frac{1}{7}$.<br/>
