Question

Let $f,g:N\to N$ such that $f\left(n+1\right)=f\left(n\right)+f\left(1\right) \forall n\in N$ and $g$ be any arbitrary function. Which of the following statements is NOT true?

JEE Main 2021 (25 Feb Shift 1)

Options

• A: If $f$ is onto, then $f\left(n\right)=n\forall n\in N$
• B: If $g$ is onto, then $fog$ is one-one
• C: $f$ is one-one
• D: If $fog$ is one-one, then $g$ is one-one

Question

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

JEE Main 2021 (25 Jul Shift 2)

Options

• A: $0$
• B: $4$
• C: $-2$
• D: $2$

Question

Let $A=\left\{1,2,3,\dots ,10\right\}$ and $f:A\to A$ be defined as

$f\left(k\right)=\left\{\begin{array}{cl}k+1& \mathrm{if} k \mathrm{is}odd\\ k& \mathrm{if} k \mathrm{is}even\end{array}\right.$

Then the number of possible functions $g:A\to A$ such that $gof=f$ is:

JEE Main 2021 (26 Feb Shift 2)

Options

• A: ${}^{10}\mathrm{C}_5$
• B: $5^5$
• C: $5!$
• D: ${10}^5$

Question

Let $f:R\to R$ be defined as $f\left(x+y\right)+f\left(x-y\right)=2f\left(x\right)f\left(y\right),f\left(\frac{1}{2}\right)=-1.$ Then the value of $\sum _{k=1}^{20}\frac{1}{\sin \left(k\right)\sin \left(k+f\left(k\right)\right)}$ is equal to :

JEE Main 2021 (27 Jul Shift 2)

Options

• A: ${cosec}^2\left(21\right)\cos \left(20\right)\cos \left(2\right)$
• B: ${\sec }^2\left(1\right)\sec \left(21\right)\cos \left(20\right)$
• C: ${cosec}^2\left(1\right)cosec\left(21\right)\sin \left(20\right)$
• D: ${\sec }^2\left(21\right)\sin \left(20\right)\sin \left(2\right)$

Question

Let $f\left(x\right)$ be a polynomial of degree $3$ such that $f\left(k\right)=-\frac{2}{k}$ for $k=2, 3, 4, 5.$ Then the value of $52-10 f\left(10\right)$ is equal to _____ .

JEE Main 2021 (01 Sep Shift 2)

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Question

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

JEE Main 2021 (27 Jul Shift 1)

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Question

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

JEE Main 2022 (25 Jul Shift 1)

Options

• A: $60$
• B: $90$
• C: $108$
• D: $126$

Question

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

JEE Main 2022 (26 Jul Shift 1)

Options

• A: $4$
• B: $10$
• C: $11$
• D: $16$

Question

Let $f\left(x\right)=\frac{x-1}{x+1}, x\in R-\{0,-1,1)$ . If $f^{n+1}\left(x\right)= f\left(f^n\left(x\right)\right)$ for all $n\in N$, then $f^6\left(6\right)+f^7\left(7\right)$ is equal to

JEE Main 2022 (26 Jun Shift 1)

Options

• A: $\frac{7}{6}$
• B: $-\frac{3}{2}$
• C: $\frac{7}{12}$
• D: $-\frac{11}{12}$

Question

Let $f:\mathrm{ℝ}\to \mathrm{ℝ}$ be defined as $f\left(x\right)=x-1$  and $g:R\to \left\{1,-1\right\}\to \mathrm{ℝ}$ be defined as $g\left(x\right)=\frac{x^2}{x^2-1}$. Then the function $fog$ is:

JEE Main 2022 (26 Jun Shift 2)

Options

• A: One-one but not onto
• B: onto but not one-one
• C: Both one-one and onto
• D: Neither one-one nor onto

Question

The domain of the function $f\left(x\right)={\sin }^{-1}\left[2x^2-3\right]+{\log }_2\left({\log }_{\frac{1}{2}}\left(x^2-5x+5\right)\right)$, where $\left[t\right]$ is the greatest integer function, is

JEE Main 2022 (27 Jul Shift 2)

Options

• A: $\left(-\sqrt{\frac{5}{2}},\frac{5-\sqrt{5}}{2}\right)$
• B: $\left(\frac{5-\sqrt{5}}{2},\frac{5+\sqrt{5}}{2}\right)$
• C: $\left(1,\frac{5-\sqrt{5}}{2}\right)$
• D: $\left[1,\frac{5+\sqrt{5}}{2}\right)$

Question

Let $\alpha ,\beta$ and $\gamma$ be three positive real numbers. Let $f\left(x\right)=\alpha {\mathrm{x}}^5+\beta {\mathrm{x}}^3+\gamma \mathrm{x},\mathrm{x}\in \mathrm{R}$ and $g:R\to R$ be such that $g\left(f\left(x\right)\right)=x$ for all $x\in R$. If $a_1,a_2,a_3,\dots ,a_n$ be in arithmetic progression with mean zero, then the value of $f\left(g\left(\frac{1}{n}\sum _{i=1}^{n}f\left(a_i\right)\right)\right)$ is equal to

JEE Main 2022 (28 Jul Shift 1)

Options

• A: $0$
• B: $3$
• C: $9$
• D: $27$

Question

Let a function $f:\mathrm{\mathbb{N}}\to \mathrm{\mathbb{N}}$ be defined by $f\left(n\right)=\left[\begin{array}{ll}2n,& n=2,4,6,8,\dots ..\\ n-1,& n=3,7,11,15,\dots ..\\ \frac{n+1}{2},& n=1,5,9,13,\dots ..\end{array}\right.$
then, $f$ is

JEE Main 2022 (28 Jun Shift 1)

Options

• A: One-one and onto
• B: One-one but not onto
• C: Onto but not one-one
• D: Neither one-one nor onto

Question

Let $S=\left\{1,2,3,4\right\}$. Then the number of elements in the set {$f:S\times S\to S:f$ is onto and $f\left(a,b\right)=f\left(b,a\right)$ $\ge a\forall \left(a,b\right)\in S\times S$} is

JEE Main 2022 (28 Jun Shift 2)

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Question

The equation $x^2–4x+\left[x\right]+3=x\left[x\right]$, where $\left[x\right]$ denotes the greatest integer function, has:

JEE Main 2023 (24 Jan Shift 1)

Options

• A: exactly two solutions in $(-\infty , \infty )$
• B: no solution
• C: a unique solution in $(-\infty , 1)$
• D: a unique solution in $(-\infty , \infty )$

Question

If $f\left(x\right)=x^3-x^2{f}^{\text{'}}\left(1\right)+x{f}^{"}\left(2\right)-f^{\text{'}\text{'}\text{'}}\left(3\right)$, $x\in \mathrm{R}$, then

JEE Main 2023 (24 Jan Shift 2)

Options

• A: $3f\left(1\right)+f\left(2\right)=f\left(3\right)$
• B: $f\left(3\right)-f\left(2\right)=f\left(1\right)$
• C: $2f\left(0\right)-f\left(1\right)+f\left(3\right)=f\left(2\right)$
• D: $f\left(1\right)+f\left(2\right)+f\left(3\right)=f\left(0\right)$

Question

Consider a function $\mathrm{f}:\mathrm{\mathbb{N}}\to \mathrm{ℝ}$, satisfying $f\left(1\right)+2f\left(2\right)+3f\left(3\right)+\dots +xf\left(x\right)=x\left(x+1\right)f\left(x\right) ;x\ge 2$ with $f\left(1\right)=1$. Then $\frac{1}{f\left(2022\right)}+\frac{{\displaystyle 1}}{{\displaystyle f\left(2028\right)}}$ is equal to

JEE Main 2023 (29 Jan Shift 2)

Options

• A: $8200$
• B: $8000$
• C: $8400$
• D: $8100$

Question

Let $A=\left\{1,2,3,4,5\right\}$ and $B=\left\{1,2,3,4,5,6\right\}$. Then the number of functions $f:A\to B$ satisfying $f\left(1\right)+f\left(2\right)=f\left(4\right)-1$ is equal to........

JEE Main 2023 (11 Apr Shift 2)

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Question

If the domain of the function $\sin ^{-1}\left(\frac{3 x-22}{2 x-19}\right)+\log _{\mathrm{e}}\left(\frac{3 x^2-8 x+5}{x^2-3 x-10}\right)$ is $(\alpha, \beta]$, then $3 \alpha+10 \beta$ is equal to:

JEE Main 2024 (04 Apr Shift 1)

Options

• A: 100
• B: 95
• C: 97
• D: 98

Question

Let the sum of the maximum and the minimum values of the function $f(x)=\frac{2 x^2-3 x+8}{2 x^2+3 x+8}$ be $\frac{\mathrm{m}}{\mathrm{n}}$, where $\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$. Then $\mathrm{m}+\mathrm{n}$ is equal to :

JEE Main 2024 (04 Apr Shift 1)

Options

• A: 195
• B: 201
• C: 217
• D: 182

Question

Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x)=\frac{2 x}{\sqrt{1+9 x^2}}$. If the composition of $f, \underbrace{(f \circ f \circ f \circ \cdots \circ f)}_{10 \text { times }}(x)=\frac{2^{10} x}{\sqrt{1+9 \alpha x^2}}$, then the value of $\sqrt{3 \alpha+1}$ is equal to ______

JEE Main 2024 (04 Apr Shift 2)

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Question

Let $A=\{1,3,7,9,11\}$ and $B=\{2,4,5,7,8,10,12\}$. Then the total number of one-one maps $f: \mathrm{A} \rightarrow \mathrm{B}$, such that $f(1)+f(3)=14$, is :

JEE Main 2024 (05 Apr Shift 1)

Options

• A: 480
• B: 240
• C: 120
• D: 180

Question

Let $f(x)=\frac{1}{7-\sin 5 x}$ be a function defined on $\mathbf{R}$. Then the range of the function $f(x)$ is equal to ;

JEE Main 2024 (06 Apr Shift 2)

Options

• A: $\left[\frac{1}{7}, \frac{1}{6}\right]$
• B: $\left[\frac{1}{8}, \frac{1}{5}\right]$
• C: $\left[\frac{1}{7}, \frac{1}{5}\right]$
• D: $\left[\frac{1}{8}, \frac{1}{6}\right]$

Question

Let $[t]$ be the greatest integer less than or equal to $t$. Let $A$ be the set of all prime factors of 2310 and $f: A \rightarrow \mathbb{Z}$ be the function $f(x)=\left[\log _2\left(x^2+\left[\frac{x^3}{5}\right]\right)\right]$. The number of one-to-one functions from $A$ to the range of $f$ is

JEE Main 2024 (08 Apr Shift 1)

Options

• A: 25
• B: 24
• C: 20
• D: 120

Question

If a function $f$ satisfies $f(\mathrm{~m}+\mathrm{n})=f(\mathrm{~m})+f(\mathrm{n})$ for all $\mathrm{m}, \mathrm{n} \in \mathbf{N}$ and $f(1)=1$, then the largest natural number $\lambda$ such that $\sum_{k=1}^{2022} f(\lambda+k) \leq(2022)^2$ is equal to _________

JEE Main 2024 (09 Apr Shift 1)

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