## MCQ (Single Correct Answer)

The sum of the absolute maximum and minimum values of the function $$f(x)=\left|x^{2}-5 x+6\right|-3 x+2$$ in the interval $$[-1,3]$$ is equal to :...

Let $$f:\mathbb{R}-{0,1}\to \mathbb{R}$$ be a function such that $$f(x)+f\left(\frac{1}{1-x}\right)=1+x$$. Then $$f(2)$$ is equal to

Let $$f(x) = \left| {\matrix{
{1 + {{\sin }^2}x} & {{{\cos }^2}x} & {\sin 2x} \cr
{{{\sin }^2}x} & {1 + {{\cos }^2}x} & {\sin 2x} \cr
{{{...

Let $f: \mathbb{R}-\{2,6\} \rightarrow \mathbb{R}$ be real valued function defined as $f(x)=\frac{x^2+2 x+1}{x^2-8 x+12}$.
Then range of $f$ is...

The absolute minimum value, of the function
$f(x)=\left|x^{2}-x+1\right|+\left[x^{2}-x+1\right]$,
where $[t]$ denotes the greatest integer function, ...

If the domain of the function $$f(x)=\frac{[x]}{1+x^{2}}$$, where $$[x]$$ is greatest integer $$\leq x$$, is $$[2,6)$$, then its range is

Consider a function $$f:\mathbb{N}\to\mathbb{R}$$, satisfying $$f(1)+2f(2)+3f(3)+....+xf(x)=x(x+1)f(x);x\ge2$$ with $$f(1)=1$$. Then $$\frac{1}{f(2022...

The domain of $$f(x) = {{{{\log }_{(x + 1)}}(x - 2)} \over {{e^{2{{\log }_e}x}} - (2x + 3)}},x \in \mathbb{R}$$ is

Let $$f:R \to R$$ be a function such that $$f(x) = {{{x^2} + 2x + 1} \over {{x^2} + 1}}$$. Then

The number of functions
$$f:\{ 1,2,3,4\} \to \{ a \in :Z|a| \le 8\} $$
satisfying $$f(n) + {1 \over n}f(n + 1) = 1,\forall n \in \{ 1,2,3\} $$ is...

Let $$f:\mathbb{R}\to\mathbb{R}$$ be a function defined by $$f(x) = {\log _{\sqrt m }}\{ \sqrt 2 (\sin x - \cos x) + m - 2\} $$, for some $$m$$, such ...

Let $$f(x) = 2{x^n} + \lambda ,\lambda \in R,n \in N$$, and $$f(4) = 133,f(5) = 255$$. Then the sum of all the positive integer divisors of $$(f(3) -...

Let $$f:(0,1)\to\mathbb{R}$$ be a function defined $$f(x) = {1 \over {1 - {e^{ - x}}}}$$, and $$g(x) = \left( {f( - x) - f(x)} \right)$$. Consider two...

Let $$f(x)$$ be a function such that $$f(x+y)=f(x).f(y)$$ for all $$x,y\in \mathbb{N}$$. If $$f(1)=3$$ and $$\sum\limits_{k = 1}^n {f(k) = 3279} $$, t...

If $$f(x) = {{{2^{2x}}} \over {{2^{2x}} + 2}},x \in \mathbb{R}$$, then $$f\left( {{1 \over {2023}}} \right) + f\left( {{2 \over {2023}}} \right)\, + \...

The domain of the function $$f(x)=\sin ^{-1}\left(\frac{x^{2}-3 x+2}{x^{2}+2 x+7}\right)$$ is:

$$
\text { Let } f(x)=a x^{2}+b x+c \text { be such that } f(1)=3, f(-2)=\lambda \text { and } $$ $$f(3)=4$$. If $$f(0)+f(1)+f(-2)+f(3)=14$$, then $$\...

The function $$f(x)=x \mathrm{e}^{x(1-x)}, x \in \mathbb{R}$$, is :

Considering only the principal values of the inverse trigonometric functions, the domain of the function $$f(x)=\cos ^{-1}\left(\frac{x^{2}-4 x+2}{x^{...

Let $$\alpha, \beta$$ and $$\gamma$$ be three positive real numbers. Let $$f(x)=\alpha x^{5}+\beta x^{3}+\gamma x, x \in \mathbf{R}$$ and $$g: \mathbf...

The domain of the function $$f(x)=\sin ^{-1}\left[2 x^{2}-3\right]+\log _{2}\left(\log _{\frac{1}{2}}\left(x^{2}-5 x+5\right)\right)$$, where [t] is t...

Let $$f, g: \mathbb{N}-\{1\} \rightarrow \mathbb{N}$$ be functions defined by $$f(a)=\alpha$$, where $$\alpha$$ is the maximum of the powers of those ...

If the maximum value of $$a$$, for which the function $$f_{a}(x)=\tan ^{-1} 2 x-3 a x+7$$ is non-decreasing in $$\left(-\frac{\pi}{6}, \frac{\pi}{6}\r...

Let f : R $$\to$$ R be a continuous function such that $$f(3x) - f(x) = x$$. If $$f(8) = 7$$, then $$f(14)$$ is equal to :

The number of bijective functions $$f:\{1,3,5,7, \ldots, 99\} \rightarrow\{2,4,6,8, \ldots .100\}$$, such that $$f(3) \geq f(9) \geq f(15) \geq f(21)...

The total number of functions,
$$
f:\{1,2,3,4\} \rightarrow\{1,2,3,4,5,6\}
$$
such that $$f(1)+f(2)=f(3)$$, is equal to :

If the absolute maximum value of the function $$f(x)=\left(x^{2}-2 x+7\right) \mathrm{e}^{\left(4 x^{3}-12 x^{2}-180 x+31\right)}$$ in the interval $$...

Let $${S_1} = \left\{ {x \in R - \{ 1,2\} :{{(x + 2)({x^2} + 3x + 5)} \over { - 2 + 3x - {x^2}}} \ge 0} \right\}$$ and $${S_2} = \left\{ {x \in R:{3^{...

The domain of the function $${\cos ^{ - 1}}\left( {{{2{{\sin }^{ - 1}}\left( {{1 \over {4{x^2} - 1}}} \right)} \over \pi }} \right)$$ is :

Let a function f : N $$\to$$ N be defined by
$$f(n) = \left[ {\matrix{
{2n,} & {n = 2,4,6,8,......} \cr
{n - 1,} & {n = 3,7,11,15,......} \cr...

Let f : R $$\to$$ R be defined as f (x) = x $$-$$ 1 and g : R $$-$$ {1, $$-$$1} $$\to$$ R be defined as $$g(x) = {{{x^2}} \over {{x^2} - 1}}$$.
Then t...

Let $$f(x) = {{x - 1} \over {x + 1}},\,x \in R - \{ 0, - 1,1\} $$. If $${f^{n + 1}}(x) = f({f^n}(x))$$ for all n $$\in$$ N, then $${f^6}(6) + {f^7}(7)...

Let $$f(x) = 2{\cos ^{ - 1}}x + 4{\cot ^{ - 1}}x - 3{x^2} - 2x + 10$$, $$x \in [ - 1,1]$$. If [a, b] is the range of the function f, then 4a $$-$$ b i...

Let f : N $$\to$$ R be a function such that $$f(x + y) = 2f(x)f(y)$$ for natural numbers x and y. If f(1) = 2, then the value of $$\alpha$$ for which
...

Let f : R $$\to$$ R be defined as $$f(x) = {x^3} + x - 5$$. If g(x) is a function such that $$f(g(x)) = x,\forall 'x' \in R$$, then g'(63) is equal to...

Let f(x) be a polynomial function such that $$f(x) + f'(x) + f''(x) = {x^5} + 64$$. Then, the value of $$\mathop {\lim }\limits_{x \to 1} {{f(x)} \ove...

Let $$f:R \to R$$ and $$g:R \to R$$ be two functions defined by $$f(x) = {\log _e}({x^2} + 1) - {e^{ - x}} + 1$$ and $$g(x) = {{1 - 2{e^{2x}}} \over {...

For the function $$f(x) = 4{\log _e}(x - 1) - 2{x^2} + 4x + 5,\,x > 1$$, which one of the following is NOT correct?

The sum of absolute maximum and absolute minimum values of the function $$f(x) = |2{x^2} + 3x - 2| + \sin x\cos x$$ in the interval [0, 1] is :

The domain of the function $$f(x) = {{{{\cos }^{ - 1}}\left( {{{{x^2} - 5x + 6} \over {{x^2} - 9}}} \right)} \over {{{\log }_e}({x^2} - 3x + 2)}}$$ is...

The range of the function, $$f(x) = {\log _{\sqrt 5 }}\left( {3 + \cos \left( {{{3\pi } \over 4} + x} \right) + \cos \left( {{\pi \over 4} + x} \righ...

The domain of the function$$f(x) = {\sin ^{ - 1}}\left( {{{3{x^2} + x - 1} \over {{{(x - 1)}^2}}}} \right) + {\cos ^{ - 1}}\left( {{{x - 1} \over {x +...

Let f : N $$\to$$ N be a function such that f(m + n) = f(m) + f(n) for every m, n$$\in$$N. If f(6) = 18, then f(2) . f(3) is equal to :

Which of the following is not correct for relation R on the set of real numbers ?

Let [t] denote the greatest integer less than or equal to t. Let f(x) = x $$-$$ [x], g(x) = 1 $$-$$ x + [x], and h(x) = min{f(x), g(x)}, x $$\in$$ [$$...

The domain of the function $${{\mathop{\rm cosec}\nolimits} ^{ - 1}}\left( {{{1 + x} \over x}} \right)$$ is :

Out of all patients in a hospital 89% are found to be suffering from heart ailment and 98% are suffering from lungs infection. If K% of them are suffe...

Let f : R $$\to$$ R be defined as $$f(x + y) + f(x - y) = 2f(x)f(y),f\left( {{1 \over 2}} \right) = - 1$$. Then, the value of $$\sum\limits_{k = 1}^{...

Let N be the set of natural numbers and a relation R on N be defined by $$R = \{ (x,y) \in N \times N:{x^3} - 3{x^2}y - x{y^2} + 3{y^3} = 0\} $$. Then...

If [x] be the greatest integer less than or equal to x, then $$\sum\limits_{n = 8}^{100} {\left[ {{{{{( - 1)}^n}n} \over 2}} \right]} $$ is equal to :...

Consider function f : A $$\to$$ B and g : B $$\to$$ C (A, B, C $$ \subseteq $$ R) such that (gof)$$-$$1 exists, then :

Let g : N $$\to$$ N be defined asg(3n + 1) = 3n + 2,g(3n + 2) = 3n + 3,g(3n + 3) = 3n + 1, for all n $$\ge$$ 0. Then which of the following statements...

Let [x] denote the greatest integer less than or equal to x. Then, the values of x$$\in$$R satisfying the equation $${[{e^x}]^2} + [{e^x} + 1] - 3 = 0...

The number of solutions of sin7x + cos7x = 1, x$$\in$$ [0, 4$$\pi$$] is equal to

If the domain of the function $$f(x) = {{{{\cos }^{ - 1}}\sqrt {{x^2} - x + 1} } \over {\sqrt {{{\sin }^{ - 1}}\left( {{{2x - 1} \over 2}} \right)} }}...

Let $$f:R - \left\{ {{\alpha \over 6}} \right\} \to R$$ be defined by $$f(x) = {{5x + 3} \over {6x - \alpha }}$$. Then the value of $$\alpha$$ for wh...

Let [ x ] denote the greatest integer $$\le$$ x, where x $$\in$$ R. If the domain of the real valued function $$f(x) = \sqrt {{{\left| {[x]} \right| -...

Let f : R $$-$$ {3} $$ \to $$ R $$-$$ {1} be defined by f(x) = $${{x - 2} \over {x - 3}}$$.Let g : R $$ \to $$ R be given as g(x) = 2x $$-$$ 3. Then, ...

The real valued function $$f(x) = {{\cos e{c^{ - 1}}x} \over {\sqrt {x - [x]} }}$$, where [x] denotes the greatest integer less than or equal to x, is...

If the functions are defined as $$f(x) = \sqrt x $$ and $$g(x) = \sqrt {1 - x} $$, then what is the common domain of the following functions :f + g, f...

Consider the function f : R $$ \to $$ R defined by
$$f(x) = \left\{ \matrix{
\left( {2 - \sin \left( {{1 \over x}} \right)} \right)|x|,x \ne 0 \hfi...

The inverse of $$y = {5^{\log x}}$$ is :

In a school, there are three types of games to be played. Some of the students play two types of games, but none play all the three games. Which Venn ...

Let f be a real valued function, defined on R $$-$$ {$$-$$1, 1} and given by f(x) = 3 loge $$\left| {{{x - 1} \over {x + 1}}} \right| - {2 \over {x - ...

Let A = {2, 3, 4, 5, ....., 30} and '$$ \simeq $$' be an equivalence relation on A $$\times$$ A, defined by (a, b) $$ \simeq $$ (c, d), if and only if...

The number of elements in the set {x $$\in$$ R : (|x| $$-$$ 3) |x + 4| = 6} is equal to :

Let [ x ] denote greatest integer less than or equal to x. If for n$$\in$$N, $${(1 - x + {x^3})^n} = \sum\limits_{j = 0}^{3n} {{a_j}{x^j}} $$, then $$...

The range of a$$\in$$R for which the function f(x) = (4a $$-$$ 3)(x + loge 5) + 2(a $$-$$ 7) cot$$\left( {{x \over 2}} \right)$$ sin2$$\left( {{x \ove...

Let $$A = \{ 1,2,3,....,10\} $$ and $$f:A \to A$$ be defined as$$f(k) = \left\{ {\matrix{
{k + 1} & {if\,k\,is\,odd} \cr
k & {if\,k\,i...

Let R = {(P, Q) | P and Q are at the same distance from the origin} be a relation, then the equivalence class of (1, $$-$$1) is the set :

A function f(x) is given by $$f(x) = {{{5^x}} \over {{5^x} + 5}}$$, then the sum of the series $$f\left( {{1 \over {20}}} \right) + f\left( {{2 \over ...

Let x denote the total number of one-one functions from a set A with 3 elements to a set B with 5 elements and y denote the total number of one-one fu...

Let f, g : N $$ \to $$ N such that f(n + 1) = f(n) + f(1) $$\forall $$ n$$\in$$N and g be any arbitrary function. Which of the following statements is...

Let f : R → R be defined as f (x) = 2x – 1 and g : R - {1} → R be defined as g(x) =
$${{x - {1 \over 2}} \over {x - 1}}$$.
Then the composition functi...

For a suitably chosen real constant a, let a
function, $$f:R - \left\{ { - a} \right\} \to R$$ be defined by
$$f(x) = {{a - x} \over {a + x}}$$. Furt...

If f(x + y) = f(x)f(y) and $$\sum\limits_{x = 1}^\infty {f\left( x \right)} = 2$$ , x, y $$ \in $$ N, where N is the set of all natural number, then...

A survey shows that 73% of the persons working in an office like coffee, whereas 65% like tea. If x denotes the percentage of them, who like both coff...

If the minimum and the maximum values of the function $$f:\left[ {{\pi \over 4},{\pi \over 2}} \right] \to R$$, defined by
$$f\left( \theta \right...

Let $$\mathop \cup \limits_{i = 1}^{50} {X_i} = \mathop \cup \limits_{i = 1}^n {Y_i} = T$$ where each Xi contains 10 elements and each Yi contains ...

A survey shows that 63% of the people in a city read newspaper A whereas 76% read
newspaper B. If x% of the people read both the newspapers, then a po...

Let R1
and R2
be two relation defined as
follows :
R1
= {(a, b) $$ \in $$ R2
: a2
+ b2 $$ \in $$ Q} and
R2
= {(a, b) $$ \in $$ R2
: a2
+ b2 $$...

Consider the two sets :
A = {m $$ \in $$ R : both the roots of x2
– (m + 1)x + m + 4 = 0 are real}
and B = [–3, 5).
Which of the following is not tr...

Let f : R $$ \to $$ R be a function which satisfies
f(x + y) = f(x) + f(y) $$\forall $$ x, y $$ \in $$ R. If f(1) = 2 and
g(n) = $$\sum\limits_{k = 1}...

If R = {(x, y) : x, y
$$ \in $$ Z, x2 + 3y2
$$ \le $$ 8} is a relation
on the set of integers Z, then the domain of R–1 is :...

The domain of the function
f(x) = $${\sin ^{ - 1}}\left( {{{\left| x \right| + 5} \over {{x^2} + 1}}} \right)$$ is (–
$$\infty $$, -a]$$ \cup $$[a, $$...

If A = {x $$ \in $$ R : |x| < 2} and B = {x $$ \in $$ R : |x – 2| $$ \ge $$ 3};
then :

Let a – 2b + c = 1.
If $$f(x)=\left| {\matrix{
{x + a} & {x + 2} & {x + 1} \cr
{x + b} & {x + 3} & {x + 2} \cr
{x + c} &a...

Let ƒ : (1, 3) $$ \to $$ R be a function defined by
$$f(x) = {{x\left[ x \right]} \over {1 + {x^2}}}$$ , where [x] denotes the greatest
integer $$ \le...

The inverse function of
f(x) = $${{{8^{2x}} - {8^{ - 2x}}} \over {{8^{2x}} + {8^{ - 2x}}}}$$, x $$ \in $$ (-1, 1), is :

Let ƒ(x) = xcos–1(–sin|x|), $$x \in \left[ { - {\pi \over 2},{\pi \over 2}} \right]$$, then
which of the following is true?

If g(x) = x2 + x - 1 and (goƒ) (x) = 4x2 - 10x + 5, then ƒ$$\left( {{5 \over 4}} \right)$$ is equal to:

For x $$ \in $$ (0, 3/2), let f(x) = $$\sqrt x $$ , g(x) = tan x and h(x) = $${{1 - {x^2}} \over {1 + {x^2}}}$$. If $$\phi $$ (x) = ((hof)og)(x), th...

Let f(x) = loge(sin x), (0 < x < $$\pi $$) and g(x) = sin–1
(e–x
), (x $$ \ge $$ 0). If $$\alpha $$ is a positive real number such that
a = (fog...

Let f(x) = x2
, x $$ \in $$ R. For any A $$ \subseteq $$ R, define g (A) = { x $$ \in $$ R : f(x) $$ \in $$ A}. If S = [0,4], then which one of the
fo...

Let f(x) = ex – x and g(x) = x2 – x, $$\forall $$ x $$ \in $$ R. Then the set of all x $$ \in $$ R, where the function h(x) = (fog) (x) is increasing,...

The domain of the definition of the function
$$f(x) = {1 \over {4 - {x^2}}} + {\log _{10}}({x^3} - x)$$ is

Let $$\sum\limits_{k = 1}^{10} {f(a + k) = 16\left( {{2^{10}} - 1} \right)} $$ where the function
ƒ satisfies
ƒ(x + y) = ƒ(x)ƒ(y) for all natural
num...

If the function ƒ : R – {1, –1} $$ \to $$ A defined by
ƒ(x) = $${{{x^2}} \over {1 - {x^2}}}$$ , is surjective, then A is equal to

Let ƒ(x) = ax
(a > 0) be written as
ƒ(x) = ƒ1
(x) + ƒ2
(x), where ƒ1
(x) is an even
function of ƒ2
(x) is an odd function. Then
ƒ1
(x + y) + ƒ1
(x...

Let $$f(x) = \int\limits_0^x {g(t)dt} $$ where g is a non-zero even
function. If ƒ(x + 5) = g(x), then $$ \int\limits_0^x {f(t)dt} $$ equals-

If $$f(x) = {\log _e}\left( {{{1 - x} \over {1 + x}}} \right)$$, $$\left| x \right| < 1$$ then $$f\left( {{{2x} \over {1 + {x^2}}}} \right)$$ is eq...

If the function f given by f(x) = x3 – 3(a – 2)x2 + 3ax + 7, for some a$$ \in $$R is increasing in (0, 1] and decreasing in [1, 5), then a root of th...

Let a function f : (0, $$\infty $$) $$ \to $$ (0, $$\infty $$) be defined by f(x) = $$\left| {1 - {1 \over x}} \right|$$. Then f is :

Let f : R $$ \to $$ R be defined by f(x) = $${x \over {1 + {x^2}}},x \in R$$. Then the range of f is :

Let fk(x) = $${1 \over k}\left( {{{\sin }^k}x + {{\cos }^k}x} \right)$$ for k = 1, 2, 3, ... Then for all x $$ \in $$ R, the value of f4(x) $$-$$ f6(x...

Let N be the set of natural numbers and two functions f and g be defined as f, g : N $$ \to $$ N such that
f(n) = $$\left\{ {\matrix{
{{{n + 1} \ov...

Let A = {x $$ \in $$ R : x is not a positive integer}.
Define a function $$f$$ : A $$ \to $$ R as $$f(x)$$ = $${{2x}...

For $$x \in R - \left\{ {0,1} \right\}$$, Let f1(x) = $$1\over x$$, f2 (x) = 1 – x and f3 (x) = $$1 \over {1 - x}$$
be three given
functions. If a ...

Let N denote the set of all natural numbers. Define two binary relations on N as R = {(x, y) $$ \in $$ N $$ \times $$ N : 2x + y = 10} and R2 = {(x, y...

Let f : A $$ \to $$ B be a function defined as f(x) = $${{x - 1} \over {x - 2}},$$ Where A = R $$-$$ {2} and B = R $$-$$ {1}. Then f i...

Consider the following two binary relations on the set A = {a, b, c} :
R1 = {(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)} and
R2 = {(a, b), (b, a...

The function f : N $$ \to $$ N defined by f (x) = x $$-$$ 5 $$\left[ {{x \over 5}} \right],$$ Where N is the set of natural numbers and [x] denotes th...

Let f(x) = 210.x + 1 and g(x)=310.x $$-$$ 1. If (fog) (x) = x, then x is equal to :

The function $$f:R \to \left[ { - {1 \over 2},{1 \over 2}} \right]$$ defined as
$$f\left( x \right) = {x \over {1 + {x^2}}}$$, is

Let P = {$$\theta $$ : sin$$\theta $$ $$-$$ cos$$\theta $$ = $$\sqrt 2 \,\cos \theta $$} and Q = {$$\theta $$ : sin$$\theta $$ + cos$$\theta $$ = $$\...

For x $$ \in $$ R, x $$ \ne $$ 0, Let f0(x) = $${1 \over {1 - x}}$$ and
fn+1 (x) = f0(fn(x)), n = 0, 1, 2, . . . .
Then the value of f100(3) + f1$$...

If $f(x)+2 f\left(\frac{1}{x}\right)=3 x, x \neq 0$, and $\mathrm{S}=\{x \in \mathbf{R}: f(x)=f(-x)\}$; then $\mathrm{S}:$

The domain of the function f(x) = $${1 \over {\sqrt {\left| x \right| - x} }}$$ is

Let $$f\left( x \right) = {\left( {x + 1} \right)^2} - 1,x \ge - 1$$
Statement - 1 : The set $$\left\{ {x:f\left( x \right) = {f^{ - 1}}\left( x \rig...

For real x, let f(x) = x3 + 5x + 1, then

Let $$f:N \to Y$$ be a function defined as f(x) = 4x + 3 where
Y = { y $$ \in $$ N, y = 4x + 3 for some x $$ \in $$ N }.
Show that f is invertible an...

The largest interval lying in $$\left( { - {\pi \over 2},{\pi \over 2}} \right)$$ for which the function
$$f\left( x \right) = {4^{ - {x^2}}} + {\co...

Let $$f:( - 1,1) \to B$$, be a function defined by
$$f\left( x \right) = {\tan ^{ - 1}}{{2x} \over {1 - {x^2}}}$$,
then $$f$$ is both one-one and ont...

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 given constant and f(0) = 1, f(2a - x)...

A function is matched below against an interval where it is supposed to be
increasing. Which of the following pairs is incorrectly matched?

The range of the function f(x) = $${}^{7 - x}{P_{x - 3}}$$ is

If $$f:R \to S$$, defined by
$$f\left( x \right) = \sin x - \sqrt 3 \cos x + 1$$,
is onto, then the interval of $$S$$ is

The graph of the function y = f(x) is symmetrical about the line x = 2, then

The domain of the function
$$f\left( x \right) = {{{{\sin }^{ - 1}}\left( {x - 3} \right)} \over {\sqrt {9 - {x^2}} }}$$

The function $$f\left( x \right)$$ $$ = \log \left( {x + \sqrt {{x^2} + 1} } \right)$$, is

A function $$f$$ from the set of natural numbers to integers defined by
$$$f\left( n \right) = \left\{ {\matrix{
{{{n - 1} \over 2},\,when\,n\,is\,...

If $$f:R \to R$$ satisfies $$f$$(x + y) = $$f$$(x) + $$f$$(y), for all x, y $$ \in $$ R and $$f$$(1) = 7, then $$\sum\limits_{r = 1}^n {f\left( r \rig...

Domain of definition of the function f(x) = $${3 \over {4 - {x^2}}}$$ + $${\log _{10}}\left( {{x^3} - x} \right)$$, is

The domain of $${\sin ^{ - 1}}\left[ {{{\log }_3}\left( {{x \over 3}} \right)} \right]$$ is

## Numerical

Let $A=\{1,2,3,5,8,9\}$. Then the number of possible functions $f: A \rightarrow A$ such that $f(m \cdot n)=f(m) \cdot f(n)$ for every $m, n \in A$ wi...

Let $$S=\{1,2,3,4,5,6\}$$. Then the number of one-one functions $$f: \mathrm{S} \rightarrow \mathrm{P}(\mathrm{S})$$, where $$\mathrm{P}(\mathrm{S})$$...

Suppose $$f$$ is a function satisfying $$f(x + y) = f(x) + f(y)$$ for all $$x,y \in N$$ and $$f(1) = {1 \over 5}$$. If $$\sum\limits_{n = 1}^m {{{f(n)...

For some a, b, c $$\in\mathbb{N}$$, let $$f(x) = ax - 3$$ and $$\mathrm{g(x)=x^b+c,x\in\mathbb{R}}$$. If $${(fog)^{ - 1}}(x) = {\left( {{{x - 7} \over...

For $$\mathrm{p}, \mathrm{q} \in \mathbf{R}$$, consider the real valued function $$f(x)=(x-\mathrm{p})^{2}-\mathrm{q}, x \in \mathbf{R}$$ and $$\mathr...

The number of functions $$f$$, from the set $$\mathrm{A}=\left\{x \in \mathbf{N}: x^{2}-10 x+9 \leq 0\right\}$$ to the set $$\mathrm{B}=\left\{\mathrm...

Let $$f(x)=2 x^{2}-x-1$$ and $$\mathrm{S}=\{n \in \mathbb{Z}:|f(n)| \leq 800\}$$. Then, the value of $$\sum\limits_{n \in S} f(n)$$ is equal to _____...

Let $$f(x)$$ be a quadratic polynomial with leading coefficient 1 such that $$f(0)=p, p \neq 0$$, and $$f(1)=\frac{1}{3}$$. If the equations $$f(x)=0$...

The sum of the maximum and minimum values of the function $$f(x)=|5 x-7|+\left[x^{2}+2 x\right]$$ in the interval $$\left[\frac{5}{4}, 2\right]$$, whe...

Let f(x) and g(x) be two real polynomials of degree 2 and 1 respectively. If $$f(g(x)) = 8{x^2} - 2x$$ and $$g(f(x)) = 4{x^2} + 6x + 1$$, then the val...

Let c, k $$\in$$ R. If $$f(x) = (c + 1){x^2} + (1 - {c^2})x + 2k$$ and $$f(x + y) = f(x) + f(y) - xy$$, for all x, y $$\in$$ R, then the value of $$|2...

Let S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Define f : S $$\to$$ S as
$$f(n) = \left\{ {\matrix{
{2n} & , & {if\,n = 1,2,3,4,5} \cr
{2n - 11} & ,...

Let [t] denote the greatest integer $$\le$$ t and {t} denote the fractional part of t. The integral value of $$\alpha$$ for which the left hand limit ...

Let f : R $$\to$$ R be a function defined by $$f(x) = {{2{e^{2x}}} \over {{e^{2x}} + e}}$$. Then $$f\left( {{1 \over {100}}} \right) + f\left( {{2 \ov...

Let f : R $$\to$$ R satisfy $$f(x + y) = {2^x}f(y) + {4^y}f(x)$$, $$\forall$$x, y $$\in$$ R. If f(2) = 3, then $$14.\,{{f'(4)} \over {f'(2)}}$$ is equ...

Let $$f:R \to R$$ be a function defined by $$f(x) = {\left( {2\left( {1 - {{{x^{25}}} \over 2}} \right)(2 + {x^{25}})} \right)^{{1 \over {50}}}}$$. If...

The number of one-one functions f : {a, b, c, d} $$\to$$ {0, 1, 2, ......, 10} such
that 2f(a) $$-$$ f(b) + 3f(c) + f(d) = 0 is ___________....

The number of points where the function
$$f(x) = \left\{ {\matrix{
{|2{x^2} - 3x - 7|} & {if} & {x \le - 1} \cr
{[4{x^2} - 1]} & {if} & { - 1...

The number of 4-digit numbers which are neither multiple of 7 nor multiple of 3 is ____________.

If A = {x $$\in$$ R : |x $$-$$ 2| > 1}, B = {x $$\in$$ R : $$\sqrt {{x^2} - 3} $$ > 1}, C = {x $$\in$$ R : |x $$-$$ 4| $$\ge$$ 2} and Z is the s...

Let A = {n $$\in$$ N | n2 $$\le$$ n + 10,000}, B = {3k + 1 | k$$\in$$ N} an dC = {2k | k$$\in$$N}, then the sum of all the elements of the set A $$\ca...

Let S = {1, 2, 3, 4, 5, 6, 7}. Then the number of possible functions f : S $$\to$$ S such that f(m . n) = f(m) . f(n) for every m, n $$\in$$ S and m ....

Let A = {0, 1, 2, 3, 4, 5, 6, 7}. Then the number of bijective functions f : A $$\to$$ A such that f(1) + f(2) = 3 $$-$$ f(3) is equal to

If f(x) and g(x) are two polynomials such that the polynomial P(x) = f(x3) + x g(x3) is divisible by x2 + x + 1, then P(1) is equal to ___________....

If a + $$\alpha$$ = 1, b + $$\beta$$ = 2 and $$af(x) + \alpha f\left( {{1 \over x}} \right) = bx + {\beta \over x},x \ne 0$$, then the value of the e...

Suppose that a function f : R $$ \to $$ R satisfies
f(x + y) = f(x)f(y) for all x, y $$ \in $$ R and f(1) = 3. If $$\sum\limits_{i = 1}^n {f(i)} = 36...

Set A has m elements and set B has n elements. If the total number of subsets of A is 112 more
than the total number of subsets of B, then the value o...

Let A = {a, b, c} and B = {1, 2, 3, 4}. Then the
number of elements in the set C = {f : A $$ \to $$ B |
2 $$ \in $$ f(A) and f is not one-one} is ____...

The number of distinct solutions of the equation
$${\log _{{1 \over 2}}}\left| {\sin x} \right| = 2 - {\log _{{1 \over 2}}}\left| {\cos x} \right|$$ ...

Let X = {n $$ \in $$ N : 1 $$ \le $$ n $$ \le $$ 50}. If
A = {n $$ \in $$ X: n is a multiple of 2} and
B = {n $$ \in $$ X: n is a multiple of 7}, then...