JEE MAIN Function Previous Year Question Paper and Solution with PDF

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JEE MAIN Previous Year Questions (PYQs)

JEE MAIN Function PYQ

Qus : 1 JEE MAIN PYQ

The position of a moving car at time t is given by f(t) = at² + bt + c, t > 0, where a, b and c are realnumbers greater than 1. Then the average speed of the car over the time interval [t₁, t₂] isattained at the point :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 6 September 2020 (Morning) PYQ

Solution

Qus : 2 JEE MAIN PYQ

Let the domains of the functions $f(x)=\log_{4}\big(\log_{3}\big(\log_{7}\big(8-\log_{2}(x^{2}+4x+5)\big)\big)\big)$ and $g(x)=\sin^{-1}\left(\dfrac{7x+10}{x-2}\right)$ be $(\alpha,\beta)$ and $[\gamma,\delta]$, respectively. Then $\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (4 April Evening Shift) PYQ

Solution

Qus : 3 JEE MAIN PYQ

If the maximum value of $a$, for which the function $f_a(x)=\tan^{-1}(2x)-3ax+7$ is non-decreasing in $\left(-\tfrac{\pi}{6},\,\tfrac{\pi}{6}\right)$, is $\bar a$, then $f_{\bar a}\!\left(\tfrac{\pi}{8}\right)$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (26 July Evening Shift) PYQ

Solution

Qus : 4 JEE MAIN PYQ

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 thevalue of${{f\left( 4 \right)} \over {f\left( 2 \right)}}$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 6 September 2020 (Morning) PYQ

Solution

Qus : 5 JEE MAIN PYQ

Let $f(x)=3\sqrt{x-2}+\sqrt{4-x}$ be a real-valued function. If $\alpha$ and $\beta$ are respectively the minimum and maximum values of $f$, then $\alpha^2+2\beta^2$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (4 April Evening Shift) PYQ

Solution

Qus : 6 JEE MAIN PYQ

Let f(x) = 5 – |x – 2| and g(x) = |x + 1|, x $ \in $ R. If f(x) attains maximum value at $\alpha $ and g(x) attains minimum value at $\beta $, then $\mathop {\lim }\limits_{x \to -\alpha \beta } {{\left( {x - 1} \right)\left( {{x^2} - 5x + 6} \right)} \over {{x^2} - 6x + 8}}$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 April Evening Shift) PYQ

Solution

Qus : 7 JEE MAIN PYQ

For a suitably chosen real constant a, let afunction, $f:R - \left\{ { - a} \right\} \to R$ be defined by$f(x) = {{a - x} \over {a + x}}$. Further suppose that for any realnumber $x \ne - a$ and $f(x) \ne - a$, (fof)(x) = x. Then $f\left( { - {1 \over 2}} \right)$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 6 September 2020 (Evening) PYQ

Solution

Qus : 8 JEE MAIN PYQ

Let the range of the function $f(x)=6+16\cos x\cdot \cos\!\left(\frac{\pi}{3}-x\right)\cdot \cos\!\left(\frac{\pi}{3}+x\right)\cdot \sin 3x\cdot \cos 6x,\ x\in\mathbb{R}$ be $[\alpha,\beta]$. Then the distance of the point $(\alpha,\beta)$ from the line $3x+4y+12=0$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (23 January Evening Shift) PYQ

Solution

Qus : 9 JEE MAIN PYQ

If $a\in\mathbb{R}$ and the equation $-3(x-[x])^{2}+2(x-[x])+a^{2}=0$ (where $[x]$ denotes the greatest integer $\le x$) has no integral solution, then all possible values of $a$ lie in the interval :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2014 (Offline) PYQ

Solution

Qus : 10 JEE MAIN PYQ

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:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (1 February Evening Shift) PYQ

Solution

Qus : 11 JEE MAIN PYQ

Let a relation $R$ on $\mathbb N\times\mathbb N$ be defined by $(x_1,y_1),R,(x_2,y_2)$ iff $x_1\le x_2$ or $y_1\le y_2$. Consider: (I) $R$ is reflexive but not symmetric. (II) $R$ is transitive. Which of the following is true?

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (4 April Evening Shift) PYQ

Solution

Qus : 12 JEE MAIN PYQ

Let $\mathbb{N}$ be the set of natural numbers and two functions $f$ and $g$ be defined as $f,g:\mathbb{N}\to\mathbb{N}$ such that $$ f(n)= \begin{cases} \dfrac{n+1}{2}, & \text{if $n$ is odd},\\[4pt] \dfrac{n}{2}, & \text{if $n$ is even}, \end{cases} \qquad g(n)=n-(-1)^n. $$ Then $f\circ g$ is –

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (10 January Evening Shift) PYQ

Solution

Qus : 13 JEE MAIN PYQ

$ \text{Let } f:\mathbb{R}\to\mathbb{R} \text{ be a function defined as } f(x)=a\sin\!\left(\frac{\pi\lfloor x\rfloor}{2}\right)+\lfloor 2-x\rfloor,\ a\in\mathbb{R}, \text{ where } \lfloor t\rfloor \text{ is the greatest integer } \le t. \text{ If } \lim_{x\to -1} f(x) \text{ exists, then the value of } \int_{0}^{4} f(x)\,dx \text{ is equal to:}$

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (27 July Morning Shift) PYQ

Solution

Qus : 14 JEE MAIN PYQ

Let a function $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as :

$f(x)= \begin{cases}\int\limits_{0}^{x}(5-|t-3|) d t, & x>4 \\ x^{2}+b x & , x \leq 4\end{cases}$

where $\mathrm{b} \in \mathbb{R}$. If $f$ is continuous at $x=4$, then which of the following statements is NOT true?

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (27 July Morning Shift) PYQ

Solution

Qus : 15 JEE MAIN PYQ

Let $f:\mathbb{R}-{0}\to\mathbb{R}$ be a function such that $f(x) - 6f\left(\frac{1}{x}\right) = \frac{35}{3x} - \frac{5}{2}.$ If $\displaystyle \lim_{x\to 0}\left(\frac{1}{x} + f(x)\right) = \beta,\ \alpha, \beta \in \mathbb{R},$ then $\alpha + 2\beta$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (24 January Morning Shift) PYQ

Solution

Qus : 16 JEE MAIN PYQ

If for $p\ne q\ne 0$, the function $f(x)=\dfrac{\sqrt[7]{p(729+x)}-3}{\sqrt[3]{729+qx}-9}$ is continuous at $x=0$, then:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (27 July Evening Shift) PYQ

Solution

Qus : 17 JEE MAIN PYQ

Let $f(x) = \dfrac{9x^2 + 16}{2^{2x+1} + 2^{x+4} + 32}$. Then the value of $8 \big( f\left(\dfrac{1}{15}\right) + f\left(\dfrac{2}{15}\right) + \dots + f\left(\dfrac{50}{15}\right) \big)$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (24 January Morning Shift) PYQ

Solution

Qus : 18 JEE MAIN PYQ

Let $f(x)=2+|x|-|x-1|+|x+1|,;x\in\mathbb{R}$. Consider (S1): $f'!\left(-\tfrac{3}{2}\right)+f'!\left(-\tfrac{1}{2}\right)+f'!\left(\tfrac{1}{2}\right)+f'!\left(\tfrac{3}{2}\right)=2$ (S2): $\displaystyle \int_{-2}^{2} f(x),dx = 12$

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (27 July Evening Shift) PYQ

Solution

Qus : 19 JEE MAIN PYQ

Let $f\left( x \right) = \left\{ {\matrix{ { - 1} & { - 2 \le x < 0} \cr {{x^2} - 1,} & {0 \le x \le 2} \cr } } \right.$ and

$g(x) = \left| {f\left( x \right)} \right| + f\left( {\left| x \right|} \right).$

Then, in the interval (–2, 2), g is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (11 January Morning Shift) PYQ

Solution

Qus : 20 JEE MAIN PYQ

Let $f(x) = \cos \left( {2{{\tan }^{ - 1}}\sin \left( {{{\cot }^{ - 1}}\sqrt {{{1 - x} \over x}} } \right)} \right)$, 0 < x < 1. Then :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (26 August Morning Shift) PYQ

Solution

Qus : 21 JEE MAIN PYQ

Let f : R $\to$ R be defined as$f(x) = \left\{ {\matrix{ {{{\lambda \left| {{x^2} - 5x + 6} \right|} \over {\mu (5x - {x^2} - 6)}},} & {x < 2} \cr {{e^{{{\tan (x - 2)} \over {x - [x]}}}},} & {x > 2} \cr {\mu ,} & {x = 2} \cr } } \right.$ where [x] is the greatest integer is than or equal to x. If f is continuous at x = 2, then $\lambda$ + $\mu$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 February Morning Shift) PYQ

Solution

Qus : 22 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (11 January Morning Shift) PYQ

Solution

Qus : 23 JEE MAIN PYQ

Let $f, g: \mathbf{R} \rightarrow \mathbf{R}$ be defined as :

$f(x)=|x-1| \text { and } g(x)= \begin{cases}\mathrm{e}^x, & x \geq 0 \\ x+1, & x \leq 0 .\end{cases}$

Then the function $f(g(x))$ is

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (5 April Evening Shift) PYQ

Solution

Qus : 24 JEE MAIN PYQ

Let the sets A and B denote the domain and range respectively of the function $f(x)=\frac{1}{\sqrt{\lceil x\rceil-x}}$, where $\lceil x\rceil$ denotes the smallest integer greater than or equal to $x$. Then among the statements

(S1) : $A \cap B=(1, \infty)-\mathbb{N}$ and

(S2) : $A \cup B=(1, \infty)$

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (6 April Evening Shift) PYQ

Solution

Qus : 25 JEE MAIN PYQ

Let $f\colon A\to B$ be a function defined as $f(x)=\dfrac{x-1}{x-2}$, where $A=\mathbb{R}-{2}$ and $B=\mathbb{R}-{1}$. Then $f$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (15 April Evening Shift) PYQ

Solution

Qus : 26 JEE MAIN PYQ

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 $\in$ [$-$2, 2]. Then h is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (26 August Evening Shift) PYQ

Solution

Qus : 27 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (24 June Morning Shift) PYQ

Solution

Qus : 28 JEE MAIN PYQ

If the range of the function $f(x)=\dfrac{5-x}{x^2-3x+2}$, $x\ne1,2$, is $(-\infty,\alpha]\cup[\beta,\infty)$, then $\alpha^2+\beta^2$ is equal to:f

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (7 April Evening Shift) PYQ

Solution

Qus : 29 JEE MAIN PYQ

Let $f(x)=\dfrac{x}{\sqrt{a^{2}+x^{2}}}-\dfrac{d-x}{\sqrt{b^{2}+(d-x)^{2}}},\ x\in\mathbb{R}$, where $a,b,d$ are non-zero real constants. Then:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (11 January Evening Shift) PYQ

Solution

Qus : 30 JEE MAIN PYQ

Let $f:[0,\infty ) \to [0,\infty )$ be defined as $f(x) = \int_0^x {[y]dy} $ where [x] is the greatest integer less than or equal to x. Which of the following is true?

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 February Morning Shift) PYQ

Solution

Qus : 31 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (11 January Evening Shift) PYQ

Solution

Qus : 32 JEE MAIN PYQ

In a bolt factory, machines $A, B$ and $C$ manufacture respectively $20 \%, 30 \%$ and $50 \%$ of the total bolts. Of their output 3, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found the defective, then the probability that it is manufactured by the machine $C$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (8 April Morning Shift) PYQ

Solution

Qus : 33 JEE MAIN PYQ

Let $$f(x) = \left\{ {\matrix{ {{{\sin (x - [x])} \over {x - [x]}}} & {,\,x \in ( - 2, - 1)} \cr {\max \{ 2x,3[|x|]\} } & {,\,|x| < 1} \cr 1 & {,\,otherwise} \cr } } \right.$$

where [t] denotes greatest integer $$\le$$ t. If m is the number of points where $$f$$ is not continuous and n is the number of points where $$f$$ is not differentiable, then the ordered pair (m, n) is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (24 June Evening Shift) PYQ

Solution

Qus : 34 JEE MAIN PYQ

The function $f:\,\mathbb{N}\setminus\{1\}\to\mathbb{N}$ defined by $f(n)=$ the highest prime factor of $n$, is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (27 January Morning Shift) PYQ

Solution

Qus : 35 JEE MAIN PYQ

A function f(x) is given by $f(x) = {{{5^x}} \over {{5^x} + 5}}$, then the sum of the series $f\left( {{1 \over {20}}} \right) + f\left( {{2 \over {20}}} \right) + f\left( {{3 \over {20}}} \right) + ....... + f\left( {{{39} \over {20}}} \right)$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 February Evening Shift) PYQ

Solution

Qus : 36 JEE MAIN PYQ

$ \text{Let } \alpha, \beta \text{ and } \gamma \text{ be three positive real numbers. Let } f(x) = \alpha x^{5} + \beta x^{3} + \gamma x,; x \in \mathbb{R} \text{ and } g : \mathbb{R} \to \mathbb{R} \text{ be such that } g(f(x)) = x \text{ for all } x \in \mathbb{R}. \text{ If } a_{1}, a_{2}, a_{3}, \ldots, a_{n} \text{ be in arithmetic progression with mean zero, then the value of } f!\left(g!\left(\frac{1}{n}\sum_{i=1}^{n} f(a_{i})\right)\right) \text{ is equal to:}$

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (28 July Morning Shift) PYQ

Solution

Qus : 37 JEE MAIN PYQ

Let $\alpha$ and $\beta$ be the roots of x² $-$ 6x $-$ 2 = 0. If a_n = $\alpha$$ⁿ $-$ $\beta$ⁿ for n $ \ge $ 1, then the value of ${{{a_{10}} - 2{a_8}} \over {3{a_9}}}$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 February Evening Shift) PYQ

Solution

Qus : 38 JEE MAIN PYQ

The number of functions $f$ from $\{1,2,3,\ldots,20\}$ onto $\{1,2,3,\ldots,20\}$ such that $f(k)$ is a multiple of $3$, whenever $k$ is a multiple of $4$, is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (11 January Evening Shift) PYQ

Solution

Qus : 39 JEE MAIN PYQ

Let $f:\mathbb{R}\setminus\{-\tfrac{1}{2}\}\to\mathbb{R}$ and $g:\mathbb{R}\setminus\{-\tfrac{5}{2}\}\to\mathbb{R}$ be defined as $f(x)=\dfrac{2x+3}{2x+1}$ and $g(x)=\dfrac{|x|+1}{2x+5}$. Then, the domain of the function $f\circ g$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (27 January Evening Shift) PYQ

Solution

Qus : 40 JEE MAIN PYQ

The minimum value of $f(x) = {a^{{a^x}}} + {a^{1 - {a^x}}}$, where a, $x \in R$ and a > 0, is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 February Evening Shift) PYQ

Solution

Qus : 41 JEE MAIN PYQ

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

$\sum\limits_{k = 1}^{10} {f(\alpha + k) = {{512} \over 3}({2^{20}} - 1)} $

holds, is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (25 June Morning Shift) PYQ

Solution

Qus : 42 JEE MAIN PYQ

Let $g(x)=3f\!\left(\dfrac{x}{3}\right)+f(3-x)$ and $f''(x)>0$ for all $x\in(0,3)$. If $g$ is decreasing in $(0,\alpha)$ and increasing in $(\alpha,3)$, then $8\alpha$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (27 January Evening Shift) PYQ

Solution

Qus : 43 JEE MAIN PYQ

The number of real solution(s) of the equation $x^2 + 3x + 2 = \min{|x - 3|,; |x + 2|}$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (24 January Evening Shift) PYQ

Solution

Qus : 44 JEE MAIN PYQ

The function $f:(-\infty,\infty)\to(-\infty,1)$, defined by $f(x)=\dfrac{2^x-2^{-x}}{2^x+2^{-x}}$, is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (24 January Evening Shift) PYQ

Solution

Qus : 45 JEE MAIN PYQ

If $f(x)=\dfrac{2^x}{,2^x+\sqrt{2},},; x\in\mathbb{R}$, then $\displaystyle \sum_{k=1}^{81} f!\left(\dfrac{k}{82}\right)$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (28 January Morning Shift) PYQ

Solution

Qus : 46 JEE MAIN PYQ

The sum of the squares of all the roots of the equation $x^2 + |2x - 3| - 4 = 0$ is

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (28 January Morning Shift) PYQ

Solution

Qus : 47 JEE MAIN PYQ

If $f(x)=\left\{\begin{array}{cc}2+2 x, & -1 \leq x < 0 \\ 1-\frac{x}{3}, & 0 \leq x \leq 3\end{array} ; g(x)=\left\{\begin{array}{cc}-x, & -3 \leq x \leq 0 \\ x, & 0 < x \leq 1\end{array}\right.\right.$, then range of $(f o g)(x)$ is

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (29 January Morning Shift) PYQ

Solution

Qus : 48 JEE MAIN PYQ

Consider the function $f:\left[\dfrac{1}{2},1\right]\to\mathbb{R}$ defined by $f(x)=4\sqrt{2}\,x^{3}-3\sqrt{2}\,x-1$. Consider the statements (I) The curve $y=f(x)$ intersects the $x$-axis exactly at one point. (II) The curve $y=f(x)$ intersects the $x$-axis at $x=\cos\!\left(\dfrac{\pi}{12}\right)$. Then

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (29 January Morning Shift) PYQ

Solution

Qus : 49 JEE MAIN PYQ

Let $f:\mathbb{R}\to\mathbb{R}$ be a function defined by $f(x)=(2+3a)x^{2}+\dfrac{a+2}{a-1}x+b$, $a\ne1$. If $f(x+y)=f(x)+f(y)+1-\dfrac{2}{7}xy$, then the value of $28\displaystyle\sum_{i=1}^{5}\lvert f(i)\rvert$ is

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (28 January Morning Shift) PYQ

Solution

Qus : 50 JEE MAIN PYQ

The function $f(x)=\dfrac{x^{2}+2x-15}{x^{2}-4x+9},\ x\in\mathbb{R}$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (6 April Morning Shift) PYQ

Solution

Qus : 51 JEE MAIN PYQ

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 ${f^6}(6) + {f^7}(7)$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (26 June Morning Shift) PYQ

Solution

Qus : 52 JEE MAIN PYQ

The remainder when (2021)²⁰²³ is divided by 7 is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (26 June Morning Shift) PYQ

Solution

Qus : 53 JEE MAIN PYQ

Let $f(x)=\dfrac{1}{7-\sin5x}$ be a function defined on $\mathbb{R}$. Then the range of the function $f(x)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (6 April Evening Shift) PYQ

Solution

Qus : 54 JEE MAIN PYQ

The function $f(x)=\dfrac{x}{x^{2}-6x-16}$, $x\in\mathbb{R}\setminus\{-2,8\}$:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (29 January Evening Shift) PYQ

Solution

Qus : 55 JEE MAIN PYQ

Let $f:\mathbb{R}-{0}\to(-\infty,1)$ be a polynomial of degree $2$, satisfying $f(x)f\left(\dfrac{1}{x}\right)=f(x)+f\left(\dfrac{1}{x}\right)$. If $f(K)=-2K$, then the sum of squares of all possible values of $K$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (28 January Evening Shift) PYQ

Solution

Qus : 56 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (29 July Evening Shift) PYQ

Solution

Qus : 57 JEE MAIN PYQ

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 the function fog is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (26 June Evening Shift) PYQ

Solution

Qus : 58 JEE MAIN PYQ

Let $f$ be a real valued continuous function defined on the positive real axis such that $g(x)=\int\limits_0^x t f(t) d t$. If $g\left(x^3\right)=x^6+x^7$, then value of $\sum\limits_{r=1}^{15} f\left(r^3\right)$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (28 January Evening Shift) PYQ

Solution

Qus : 59 JEE MAIN PYQ

et $f:[0,3]\to A$ be defined by $,f(x)=2x^3-15x^2+36x+7,$ and $g:[0,\infty)\to B$ be defined by $,g(x)=\dfrac{x^{2025}}{x^{2025}+1}.$ If both the functions are onto and $S={x\in\mathbb{Z},:,x\in A\ \text{or}\ x\in B},$ then $n(S)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (28 January Evening Shift) PYQ

Solution

Qus : 60 JEE MAIN PYQ

The range of a$\in$R for which the function f(x) = (4a $-$ 3)(x + log_e 5) + 2(a $-$ 7) cot$\left( {{x \over 2}} \right)$ sin²$\left( {{x \over 2}} \right)$, x $\ne$ 2n$\pi$, n$\in$N has critical points, is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (16 March Morning Shift) PYQ

Solution

Qus : 61 JEE MAIN PYQ

The sum of the solutions of the equation $\left|\sqrt{x}-2\right|+\sqrt{x},(\sqrt{x}-4)+2=0\ \ (x>0)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (8 April Morning Shift) PYQ

Solution

Qus : 62 JEE MAIN PYQ

The function $f:\mathbb{R}\to\left[-\dfrac12,\dfrac12\right]$ defined as $f(x)=\dfrac{x}{1+x^{2}}$, is

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (Offline) PYQ

Solution

Qus : 63 JEE MAIN PYQ

Let $f(x) = \lfloor x^2 - x \rfloor + | -x + \lfloor x \rfloor |$, where $x \in \mathbb{R}$ and $\lfloor t \rfloor$ denotes the greatest integer less than or equal to $t$. Then, $f$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (11 April Morning Shift) PYQ

Solution

Qus : 64 JEE MAIN PYQ

If $f(x)=\log_e\left(\dfrac{1-x}{1+x}\right),\ |x|<1$ then $f\left(\dfrac{2x}{1+x^2}\right)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (8 April Morning Shift) PYQ

Solution

Qus : 65 JEE MAIN PYQ

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}^{\left( {n - 1} \right)} {f\left( k \right)} $, n $ \in $ N then the value of n, for which g(n) = 20, is

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2 September 2020 (Evening) PYQ

Solution

Qus : 66 JEE MAIN PYQ

Let $f(x)$ be a function such that $f(x+y)=f(x)\cdot f(y)$ for all $x,y\in \mathbb{N}$. If $f(1)=3$ and $\displaystyle \sum_{k=1}^{n} f(k)=3279$, then the value of $n$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (24 January Morning Shift) PYQ

Solution

Qus : 67 JEE MAIN PYQ

Let f : (–1,$\infty $)$ \to $ R be defined by f(0) = 1 and
f(x) = ${1 \over x}{\log _e}\left( {1 + x} \right)$, x $ \ne $ 0. Then the function f :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2 September 2020 (Evening) PYQ

Solution

Qus : 68 JEE MAIN PYQ

Let $f(x)$ be a function such that $f(x+y)=f(x)\cdot f(y)$ for all $x,y\in \mathbb{N}$. If $f(1)=3$ and $\displaystyle \sum_{k=1}^{n} f(k)=3279$, then the value of $n$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (24 January Evening Shift) PYQ

Solution

Qus : 69 JEE MAIN PYQ

If $f(x)=\dfrac{2^{2x}}{2^{2x}+2},\ x\in\mathbb{R}$, then $f\!\left(\dfrac{1}{2023}\right)+f\!\left(\dfrac{2}{2023}\right)+\cdots+f\!\left(\dfrac{2022}{2023}\right)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (24 January Evening Shift) PYQ

Solution

Qus : 70 JEE MAIN PYQ

If the domain of the function $f(x)=\cos^{-1}!\left(\dfrac{2-|x|}{4}\right)+{\log_e(3-x)}^{-1}$ is $[-\alpha,\beta)-{\gamma}$, then $\alpha+\beta+\gamma$ equals:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (30 January Morning Shift) PYQ

Solution

Qus : 71 JEE MAIN PYQ

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (29 January Morning Shift) PYQ

Solution

Qus : 72 JEE MAIN PYQ

Let $f$ and $g$ be two functions defined by \[ f(x)= \begin{cases} x+1, & x<0,\\[2pt] |x-1|, & x\ge 0 \end{cases} \qquad\text{and}\qquad g(x)= \begin{cases} x+1, & x<0,\\[2pt] 1, & x\ge 0. \end{cases} \] Then $(g\circ f)(x)$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (11 April Evening Shift) PYQ

Solution

Qus : 73 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (8 April Morning Shift) PYQ

Solution

Qus : 74 JEE MAIN PYQ

The domain of the function \[ f(x)=\frac{1}{\sqrt{[x]^2-3[x]-10}} \] (where $[x]$ denotes the greatest integer less than or equal to $x$) is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (11 April Evening Shift) PYQ

Solution

Qus : 75 JEE MAIN PYQ

The value of $4 + {1 \over {5 + {1 \over {4 + {1 \over {5 + {1 \over {4 + ......\infty }}}}}}}}$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (17 March Morning Shift) PYQ

Solution

Qus : 76 JEE MAIN PYQ

Let $f:\mathbb{R}\setminus{0}\to\mathbb{R}$ satisfy $f!\left(\dfrac{x}{y}\right)=\dfrac{f(x)}{f(y)}$ for all $x,y$ with $f(y)\neq 0$. If $f'(1)=2024$, then which of the following is true?

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (30 January Evening Shift) PYQ

Solution

Qus : 77 JEE MAIN PYQ

Let $f(x)=\displaystyle\int_{0}^{x} g(t),dt$ where $g$ is a non-zero even function. If $f(x+5)=g(x)$, then $\displaystyle\int_{0}^{x} f(t),dt$ equals:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (8 April Evening Shift) PYQ

Solution

Qus : 78 JEE MAIN PYQ

Let $f(x)=a^{x}\ (a>0)$ be written as $f(x)=f_{1}(x)+f_{2}(x)$, where $f_{1}(x)$ is an even function and $f_{2}(x)$ is an odd function. Then $f_{1}(x+y)+f_{1}(x-y)$ equals:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (8 April Evening Shift) PYQ

Solution

Qus : 79 JEE MAIN PYQ

Let $f(x)=\left\{\begin{array}{ccc}-\mathrm{a} & \text { if } & -\mathrm{a} \leq x \leq 0 \\ x+\mathrm{a} & \text { if } & 0< x \leq \mathrm{a}\end{array}\right.$ where $\mathrm{a}> 0$ and $\mathrm{g}(x)=(f(|x|)-|f(x)|) / 2$. Then the function $g:[-a, a] \rightarrow[-a, a]$ is

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (8 April Evening Shift) PYQ

Solution

Qus : 80 JEE MAIN PYQ

The function $f(x)=\frac{x}{x^2-6 x-16}, x \in \mathbb{R}-\{-2,8\}$

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (29 January Evening Shift) PYQ

Solution

Qus : 81 JEE MAIN PYQ

If f : R $ \to $ R is a function defined by f(x)= [x - 1] $\cos \left( {{{2x - 1} \over 2}} \right)\pi $, where [.] denotes the greatestinteger function, then f is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (24 February Morning Shift) PYQ

Solution

Qus : 82 JEE MAIN PYQ

Let a function f : N $\to$ N be defined by

$f(n) = \left[ {\matrix{ {2n,} & {n = 2,4,6,8,......} \cr {n - 1,} & {n = 3,7,11,15,......} \cr {{{n + 1} \over 2},} & {n = 1,5,9,13,......} \cr } } \right.$ then, f is

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (28 June Morning Shift) PYQ

Solution

Qus : 83 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (25 January Evening Shift) PYQ

Solution

Qus : 84 JEE MAIN PYQ

If the domain of the function $f(x)=\log_e\!\left(\frac{2x+3}{4x^{2}+x-3}\right)+\cos^{-1}\!\left(\frac{2x-1}{x+2}\right)$ is $(\alpha,\beta)$, then the value of $5\beta-4\alpha$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (30 January Evening Shift) PYQ

Solution

Qus : 85 JEE MAIN PYQ

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 function $f(g(x))$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (24 February Morning Shift) PYQ

Solution

Qus : 86 JEE MAIN PYQ

The function $f(x) = {{4{x^3} - 3{x^2}} \over 6} - 2\sin x + \left( {2x - 1} \right)\cos x$ :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (24 February Morning Shift) PYQ

Solution

Qus : 87 JEE MAIN PYQ

The function $f : \mathbb{N} \to \mathbb{N}$ defined by $f(x) = x - 5\left\lfloor \dfrac{\pi x}{5} \right\rfloor$, where $\mathbb{N}$ is the set of natural numbers and $\lfloor x \rfloor$ denotes the greatest integer $\le x$, is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2017 (9 April Morning Shift) PYQ

Solution

Qus : 88 JEE MAIN PYQ

Let f : R $\to$ R be defined as

$f(x) = \left[ {\matrix{ {[{e^x}],} & {x < 0} \cr {a{e^x} + [x - 1],} & {0 \le x < 1} \cr {b + [\sin (\pi x)],} & {1 \le x < 2} \cr {[{e^{ - x}}] - c,} & {x \ge 2} \cr } } \right.$

where a, b, c $\in$ R and [t] denotes greatest integer less than or equal to t. Then, which of the following statements is true?

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (28 June Morning Shift) PYQ

Solution

Qus : 89 JEE MAIN PYQ

Let $f:\mathbb{R}\to\mathbb{R}$ be a function defined by $f(x)=\log_{\sqrt{m}}\!\left(\sqrt{2}(\sin x-\cos x)+m-2\right)$, for some $m$, such that the range of $f$ is $[0,2]$. Then the value of $m$ is ______

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (25 January Evening Shift) PYQ

Solution

Qus : 90 JEE MAIN PYQ

$ \text{For } x\in\mathbb{R}, \text{ two real valued functions } f(x) \text{ and } g(x) \text{ are such that } g(x)=\sqrt{x}+1 \text{ and } (f\circ g)(x)=x+3-\sqrt{x}. \text{ Then } f(0) \text{ is equal to: } $

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (13 April Morning Shift) PYQ

Solution

Qus : 91 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (25 January Evening Shift) PYQ

Solution

Qus : 92 JEE MAIN PYQ

If $f(x)=\dfrac{4x+3}{6x-4}$, $x\ne\dfrac{2}{3}$, and $(f\circ f)(x)=g(x)$, where $g:\mathbb{R}-\left\{\dfrac{2}{3}\right\}\to\mathbb{R}-\left\{\dfrac{2}{3}\right\}$, then $(g\circ g\circ g)(4)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (31 January Morning Shift) PYQ

Solution

Qus : 93 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 April Morning Shift) PYQ

Solution

Qus : 94 JEE MAIN PYQ

Let $\displaystyle \sum_{k=1}^{10} f(a+k) = 16(2^{10} - 1)$ where the function $f$ satisfies $f(x+y) = f(x)f(y)$ for all natural numbers $x, y$ and $f(1) = 2$. Then the natural number $a$ is

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 April Morning Shift) PYQ

Solution

Qus : 95 JEE MAIN PYQ

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 defined for all x belonging to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (18 March Morning Shift) PYQ

Solution

Qus : 96 JEE MAIN PYQ

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 $-$ g, f/g, g/f, g $-$ f where $(f \pm g)(x) = f(x) \pm g(x),(f/g)x = {{f(x)} \over {g(x)}}$

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (18 March Morning Shift) PYQ

Solution

Qus : 97 JEE MAIN PYQ

For $x \in \mathbb{R}$, $f(x) = |\log 2 - \sin x|$ and $g(x) = f(f(x))$, then:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2016 (Offline) PYQ

Solution

Qus : 98 JEE MAIN PYQ

The domain of $$f(x)=\frac{\log_{(x+1)}(x-2)}{e^{2\log_e x}-(2x+3)},\quad x\in\mathbb{R}$$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (29 January Morning Shift) PYQ

Solution

Qus : 99 JEE MAIN PYQ

Let the range of the function $f(x)=\dfrac{1}{2+\sin3x+\cos3x},\ x\in\mathbb{R}$ be $[a,b]$. If $\alpha$ and $\beta$ are respectively the A.M. and the G.M. of $a$ and $b$, then $\dfrac{\alpha}{\beta}$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (9 April Evening Shift) PYQ

Solution

Qus : 100 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (29 January Morning Shift) PYQ

Solution

Qus : 101 JEE MAIN PYQ

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 sum of all the values of x for which f^$-$1(x) + g^$-$1(x) = ${{13} \over 2}$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (18 March Evening Shift) PYQ

Solution

Qus : 102 JEE MAIN PYQ

The domain of the function $f(x)=\dfrac{1}{4-x^{2}}+\log_{10}(x^{3}-x)$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 April Evening Shift) PYQ

Solution

Qus : 103 JEE MAIN PYQ

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 :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (31 August Evening Shift) PYQ

Solution

Qus : 104 JEE MAIN PYQ

Consider a function $f:\mathbb{N}\to\mathbb{R}$ satisfying \[ f(1)+2f(2)+3f(3)+\cdots+xf(x)=x(x+1)f(x),\quad x\ge 2, \] with $f(1)=1$. Then \[ \frac{1}{f(2022)}+\frac{1}{f(2028)} \] is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (29 January Evening Shift) PYQ

Solution

Qus : 105 JEE MAIN PYQ

$\displaystyle \int e^{\sec x},\big(\sec x\tan x,f(x)+\sec x\tan x+\sec^{2}x\big),dx ;=; e^{\sec x}f(x)+C$ Then a possible choice of $f(x)$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 April Evening Shift) PYQ

Solution

Qus : 106 JEE MAIN PYQ

If $f(x) + 2f\left(\dfrac{1}{x}\right) = 3x,; x \ne 0,$ and $S = {x \in \mathbb{R} : f(x) = f(-x)}$, then $S$ :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2016 (Offline) PYQ

Solution

Qus : 107 JEE MAIN PYQ

If the domain of the function $f(x) = \dfrac{1}{\sqrt{10 + 3x - x^2}} + \dfrac{1}{\sqrt{x + |x|}}$ is $(a, b)$, then $(1 + a)^2 + b^2$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (2 April Evening Shift) PYQ

Solution

Qus : 108 JEE MAIN PYQ

If the domain of the function $f(x) = \dfrac{1}{\sqrt{10 + 3x - x^2}} + \dfrac{1}{\sqrt{x + |x|}}$ is $(a, b)$, then $(1 + a)^2 + b^2$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (2 April Evening Shift) PYQ

Solution

Qus : 109 JEE MAIN PYQ

If the function $f:(-\infty,-1]\to(a,b]$ defined by $f(x)=e^{x^{3}-3x+1}$ is one–one and onto, then the distance of the point $P(2b+4,\ a+2)$ from the line $x+e^{-3}y=4$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (31 January Evening Shift) PYQ

Solution

Qus : 110 JEE MAIN PYQ

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| - 2} \over {\left| {[x]} \right| - 3}}} $ is ($-$ $\infty$, a) $]\cup$ [b, c) $\cup$ [4, $\infty$), a < b < c, then the value of a + b + c is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (20 July Morning Shift) PYQ

Solution

Qus : 111 JEE MAIN PYQ

If $A=\begin{bmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{bmatrix}$, then the matrix $A^{-50}$ when $\theta=\dfrac{\pi}{12}$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 January Morning Shift) PYQ

Solution

Qus : 112 JEE MAIN PYQ

For $x \in \mathbb{R}, x \ne 0$, let $f_{0}(x) = \dfrac{1}{1 - x}$ and $f_{n+1}(x) = f_{0}(f_{n}(x)),; n = 0,1,2,\ldots$ Then the value of $f_{100}(3) + f_{1}\left(\dfrac{2}{3}\right) + f_{2}\left(\dfrac{3}{2}\right)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2016 (9 April Morning Shift) PYQ

Solution

Qus : 113 JEE MAIN PYQ

Let f be a real valued continuous function on [0, 1] and $f(x) = x + \int\limits_0^1 {(x - t)f(t)dt} $.

Then, which of the following points (x, y) lies on the curve y = f(x) ?

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (29 June Evening Shift) PYQ

Solution

Qus : 114 JEE MAIN PYQ

If the domain of the function $f(x) = \log_e\left(\dfrac{2x - 3}{5 + 4x}\right) + \sin^{-1}\left(\dfrac{4 + 3x}{2 - x}\right)$ is $[\alpha, \beta]$, then $\alpha^2 + 4\beta$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (3 April Morning Shift) PYQ

Solution

Qus : 115 JEE MAIN PYQ

$ \text{Suppose } f:\mathbb{R}\to(0,\infty) \text{ be a differentiable function such that } 5f(x+y)=f(x)\cdot f(y),\ \forall x,y\in\mathbb{R}. $ $ \text{If } f(3)=320,\ \text{then } \displaystyle \sum_{n=0}^{5} f(n) \text{ is equal to:} $

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (30 January Morning Shift) PYQ

Solution

Qus : 116 JEE MAIN PYQ

Let $f(x)=e^{x}-x$ and $g(x)=x^{2}-x,\ \forall x\in\mathbb{R}$. Then the set of all $x\in\mathbb{R}$ where the function $h(x)=(f\circ g)(x)$ is increasing, is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (10 April Morning Shift) PYQ

Solution

Qus : 117 JEE MAIN PYQ

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 which (fof)(x) = x, for all $x \in R - \left\{ {{\alpha \over 6}} \right\}$, is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (20 July Evening Shift) PYQ

Solution

Qus : 118 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (1 September Evening Shift) PYQ

Solution

Qus : 119 JEE MAIN PYQ

Let $f:\mathbf{R}\rightarrow\mathbf{R}$ and $g:\mathbf{R}\rightarrow\mathbf{R}$ be defined as $ f(x)= \begin{cases} \log_e x, & x>0,\\[4pt] e^{-x}, & x\le 0 \end{cases} $ and $ g(x)= \begin{cases} x, & x\ge 0,\\[4pt] e^{x}, & x<0. \end{cases} $ Then, $g\circ f:\mathbf{R}\rightarrow\mathbf{R}$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (1 February Morning Shift) PYQ

Solution

Qus : 120 JEE MAIN PYQ

The range of the function $f(x)=\sqrt{\,3-x\,}+\sqrt{\,2+x\,}$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (30 January Evening Shift) PYQ

Solution

Qus : 121 JEE MAIN PYQ

Let f(x) = log_e(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)'($\alpha $) and b = (fog)($\alpha $), then :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (10 April Evening Shift) PYQ

Solution

Qus : 122 JEE MAIN PYQ

The function $f(x) = \left\{ {\matrix{ {{\pi \over 4} + {{\tan }^{ - 1}}x,} & {\left| x \right| \le 1} {{1 \over 2}\left( {\left| x \right| - 1} \right),} & {\left| x \right| > 1} } } \right.$is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 4 September 2020 (Evening) PYQ

Solution

Qus : 123 JEE MAIN PYQ

Let $x=(8\sqrt{3}+13)^{13}$ and $y=(7\sqrt{2}+9)^{9}$. If $[t]$ denotes the greatest integer $\le t$, then:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (30 January Evening Shift) PYQ

Solution

Qus : 124 JEE MAIN PYQ

If the domain of the function $f(x)=\sqrt{\dfrac{x^{2}-25}{4-x^{2}}}+\log_{10}(x^{2}+2x-15)$ is $(-\infty,\alpha)\cup[\beta,\infty)$, then $\alpha^{2}+\beta^{3}$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (1 February Evening Shift) PYQ

Solution

Qus : 125 JEE MAIN PYQ

The number of real roots of the equation $5+\lvert 2^{x}-1\rvert=2^{x},(2^{x}-2)$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (10 April Evening Shift) PYQ

Solution

Qus : 126 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (22 July Evening Shift) PYQ

Solution

Qus : 127 JEE MAIN PYQ

For any real number $x$, let $[x]$ denote the largest integer less than equal to $x$. Let $f$ be a real valued function defined on the interval $[-10,10]$ by $f(x)=\left\{\begin{array}{l}x-[x], \text { if }[x] \text { is odd } \\ 1+[x]-x, \text { if }[x] \text { is even } .\end{array}\right.$Then the value of $\frac{\pi^{2}}{10} \int_{-10}^{10} f(x) \cos \pi x \,d x$ is :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (25 July Morning Shift) PYQ

Solution

Qus : 128 JEE MAIN PYQ

$ \text{If the domain of } f(x)=\dfrac{\lfloor x\rfloor}{1+x^{2}},\ \text{where } \lfloor x\rfloor \text{ is greatest integer } \le x,\ \text{is } [2,6),\ \text{then its range is:} $

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (31 January Morning Shift) PYQ

Solution

Qus : 129 JEE MAIN PYQ

Let $f$ be a function such that $f(x)+3f\left(\dfrac{24}{x}\right)=4x,; x\ne0$. Then $f(3)+f(8)$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (3 April Evening Shift) PYQ

Solution

Qus : 130 JEE MAIN PYQ

Let $A=\{x\in\mathbb{R}:\ x\ \text{is not a positive integer}\}$. Define a function $f:A\to\mathbb{R}$ as $f(x)=\dfrac{2x}{x-1}$. Then $f$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (9 January Evening Shift) PYQ

Solution

Qus : 131 JEE MAIN PYQ

The remainder when $(11)^{1011} + (1011)^{11}$ is divided by $9$ is

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (25 July Evening Shift) PYQ

Solution

Qus : 132 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (31 January Evening Shift) PYQ

Solution

Qus : 133 JEE MAIN PYQ

If the domain of the function $f(x)=\log_{7}!\big(1-\log_{4}(x^{2}-9x+18)\big)$ is $(\alpha,\beta)\cup(\gamma,\delta)$, then $\alpha+\beta+\gamma+\delta$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (3 April Evening Shift) PYQ

Solution

Qus : 134 JEE MAIN PYQ

Let g : N $\to$ N be defined as g(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 is true?

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 July Morning Shift) PYQ

Solution

Qus : 135 JEE MAIN PYQ

Let $A=\{1,2,3,4\}$ and $B=\{1,4,9,16\}$. Then the number of many-one functions $f:A\to B$ such that $1\in f(A)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (22 January Evening Shift) PYQ

Solution

Qus : 136 JEE MAIN PYQ

Let $f:[0,\infty ) \to [0,\infty )$ be defined as $f(x) = \int_0^x {[y]dy} $ where [x] is the greatest integer less than or equal to x. Which of the following is true?

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 July Morning Shift) PYQ

Solution

Qus : 137 JEE MAIN PYQ

Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function satisfying $f(0)=1$ and $f(2x)-f(x)=x$ for all $x\in\mathbb{R}$. If $\lim_{n\to\infty}{f(x)-f\left(\dfrac{x}{2^{n}}\right)}=G(x)$, then $\displaystyle \sum_{r=1}^{10} G(r^{2})$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (4 April Morning Shift) PYQ

Solution

Qus : 138 JEE MAIN PYQ

For $x \in (0, 3/2)$, let $f(x) = \sqrt{x}$, $g(x) = \tan x$ and $h(x) = \dfrac{1 - x^2}{1 + x^2}$. If $\phi(x) = (h \circ f \circ g)(x)$, then $\phi\left(\dfrac{\pi}{3}\right)$ is equal to :

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2019 (12 April Morning Shift) PYQ

Solution

Qus : 139 JEE MAIN PYQ

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

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (26 July Morning Shift) PYQ

Solution

Qus : 140 JEE MAIN PYQ

Let $f(x)=x^5+2e^{x/4}$ for all $x\in\mathbb R$. Consider a function $g(x)$ such that $(g\circ f)(x)=x$ for all $x\in\mathbb R$. Then the value of $8g'(2)$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (4 April Morning Shift) PYQ

Solution

Qus : 141 JEE MAIN PYQ

Let $f(x)=\log_e x$ and $g(x)=\dfrac{x^{4}-2x^{3}+3x^{2}-2x+2}{2x^{2}-2x+1}$. Then the domain of $f\circ g$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (23 January Morning Shift) PYQ

Solution

Qus : 142 JEE MAIN PYQ

If $f(x) = \begin{cases} \int_{0}^{x} \left( 5 + |1 - t| \right) dt, & x > 2 \\ 5x + 1, & x \leq 2 \end{cases}$, then

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (25 July Evening Shift) PYQ

Solution

Qus : 143 JEE MAIN PYQ

If the domain of the function $\sin^{-1}\!\left(\dfrac{3x-22}{2x-19}\right)+\log_e\!\left(\dfrac{3x^2-8x+5}{x^2-3x-10}\right)$ is $(\alpha,\beta)$, then $3\alpha+10\beta$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2024 (4 April Morning Shift) PYQ

Solution

Qus : 144 JEE MAIN PYQ

Let $R$ be a relation on $\mathbb{R}$, given by $R=\{(a,b):\,3a-3b+\sqrt{7}\text{ is an irrational number}\}$. Then $R$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (1 February Morning Shift) PYQ

Solution

Qus : 145 JEE MAIN PYQ

Let $f(x)= \begin{cases} x^{3}-x^{2}+10x-7, & x\le 1,\\ -2x+\log_{2}(b^{2}-4), & x>1. \end{cases}$ Then the set of all values of $b$ for which $f(x)$ has maximum value at $x=1$ is:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2022 (26 July Morning Shift) PYQ

Solution

Qus : 146 JEE MAIN PYQ

Let $S=\left\{x:\ x\in\mathbb{R}\ \text{and}\ (\sqrt{3}+\sqrt{2})^{\,x^{2}-4}+(\sqrt{3}-\sqrt{2})^{\,x^{2}-4}=10\right\}$. Then $n(S)$ is equal to:

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2023 (1 February Morning Shift) PYQ

Solution

Qus : 147 JEE MAIN PYQ

Let $f,g:(1,\infty)\to\mathbb{R}$ be defined as $f(x)=\dfrac{2x+3}{5x+2}$ and $g(x)=\dfrac{2-3x}{1-x}$. If the range of the function $f\circ g:[2,4]\to\mathbb{R}$ is $[\alpha,\beta]$, then $\dfrac{1}{\beta-\alpha}$ is equal to

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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2025 (4 April Morning Shift) PYQ

Solution

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