JEE Main
Mathematics
Functions
Previous Years Questions

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...
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