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JEE MAIN Previous Year Questions (PYQs)

JEE MAIN Continuity PYQ

JEE MAIN PYQ
Let $f:\mathbb{R}\to\mathbb{R}$ be a function given by $ f(x)= \begin{cases} \dfrac{1-\cos 2x}{x^2}, & x<0,\\[6pt] \alpha, & x=0,\\[6pt] \dfrac{\beta\sqrt{\,1-\cos x\,}}{x}, & x>0, \end{cases} $ where $\alpha,\beta\in\mathbb{R}$. If $f$ is continuous at $x=0$, then $\alpha^2+\beta^2$ is equal to:





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

Solution

JEE MAIN PYQ
If the function $ f(x)= \begin{cases} \dfrac{7^{x}-9^{x}-8^{x}+1}{\sqrt{2}-\sqrt{1+\cos^{2}x}}, & x\neq0,\\[6pt] a\log_{e}2\log_{e}3, & x=0 \end{cases} $ is continuous at $x=0$, then the value of $a^{2}$ is equal to:





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

Solution

JEE MAIN PYQ
Let $f:\left( { - {\pi \over 4},{\pi \over 4}} \right) \to R$ be defined as $f(x) = \left\{ {\matrix{ {{{(1 + |\sin x|)}^{{{3a} \over {|\sin x|}}}}} & , & { - {\pi \over 4} < x < 0} \cr b & , & {x = 0} \cr {{e^{\cot 4x/\cot 2x}}} & , & {0 < x < {\pi \over 4}} \cr } } \right.$ If f is continuous at x = 0, then the value of 6a + b2 is equal to :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2021 (27 July Morning Shift) PYQ

Solution

JEE MAIN PYQ
Consider the function $ f(x)= \begin{cases} \dfrac{a\,(7x-12-x^{2})}{\,b\,\lfloor x^{2}-7x+12\rfloor\,}, & x<3,\\[6pt] \dfrac{\sin(x-3)}{2^{\,x-1}}, & x>3,\\[6pt] b, & x=3, \end{cases} $ where $\lfloor x\rfloor$ denotes the greatest integer $\le x$. If $S$ denotes the set of all ordered pairs $(a,b)$ such that $f(x)$ is continuous at $x=3$, then the number of elements in $S$ is:





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

Solution

JEE MAIN PYQ
Let $f:[-1,2]\to\mathbb{R}$ be given by $f(x)=2x^{2}+x+\lfloor x^{2}\rfloor-\lfloor x\rfloor$, where $\lfloor t\rfloor$ denotes the greatest integer $\le t$. The number of points where $f$ is not continuous is:





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

Solution

JEE MAIN PYQ
Let f(x) = $\left\{ {\matrix{ {{{\left( {x - 1} \right)}^{{1 \over {2 - x}}}},} & {x > 1,x \ne 2} \cr {k\,\,\,\,\,\,\,\,\,\,\,\,\,\,} & {,x = 2} \cr } } \right.$

Thevaue of k for which f s continuous at x = 2 is :





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

Solution

JEE MAIN PYQ
The function $f : \mathbb{R} \to \mathbb{R}$ defined by $$ f(x) = \lim_{n \to \infty} \frac{\cos(2 \pi x) - x^{2n} \sin(x-1)}{1 + x^{2n+1} - x^{2n}} $$ is continuous for all $x$ in :





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

Solution

JEE MAIN PYQ
Let $[x]$ denote the greatest integer function, and let $m$ and $n$ respectively be the numbers of the points where the function $f(x) = [x] + |x - 2|$, $-2 < x < 3$, is not continuous and not differentiable. Then $m + n$ is equal to:





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

Solution

JEE MAIN PYQ
$\text { If } f(x)=\left\{\begin{array}{ll} x^3 \sin \left(\frac{1}{x}\right), & x \neq 0 \\ 0 & , x=0 \end{array}\right. \text {, then }$





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

Solution

JEE MAIN PYQ
If the function $f$ defined as $f(x) = \dfrac{1}{x} - \dfrac{kx - 1}{e^{2x} - 1}, ; x \ne 0$, is continuous at $x = 0$, then the ordered pair $(k, f(0))$ is equal to :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (16 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
Let the function $ f(x) = \begin{cases} \dfrac{\log_e(1+5x) - \log_e(1+\alpha x)}{x}, & x \neq 0 \\ 10, & x = 0 \end{cases} $ be continuous at $x=0$. Then $\alpha$ is equal to:





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

Solution

JEE MAIN PYQ
If the function $f$ defined as $f(x)=\dfrac{1}{x}-\dfrac{kx-1}{e^{2x}-1},\ x\ne0$, is continuous at $x=0$, then the ordered pair $(k,f(0))$ is equal to :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2018 (16 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
Let f : R $ \to $ R be defined as $f(x) = \left\{ \matrix{ 2\sin \left( { - {{\pi x} \over 2}} \right),if\,x < - 1 \hfill \cr |a{x^2} + x + b|,\,if - 1 \le x \le 1 \hfill \cr \sin (\pi x),\,if\,x > 1 \hfill \cr} \right.$ If f(x) is continuous on R, then a + b equals :





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

Solution

JEE MAIN PYQ
If a function $f(x)$ defined by  $f(x) = \begin{cases} ae^x + be^{-x}, & -1 \leq x < 1 \\[6pt] cx^2, & 1 \leq x \leq 3 \\[6pt] ax^2 + 2cx, & 3 < x \leq 4 \end{cases} \\[10pt] $ be continuous for some $ a, b, c \in \mathbb{R} $ and $f'(0) + f'(2) = e,$  then the value of $a$ is





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

Solution

JEE MAIN PYQ
Let $\alpha$ $\in$ R be such that the function $f(x) = \left\{ {\matrix{ {{{{{\cos }^{ - 1}}(1 - {{\{ x\} }^2}){{\sin }^{ - 1}}(1 - \{ x\} )} \over {\{ x\} - {{\{ x\} }^3}}},} & {x \ne 0} \cr {\alpha ,} & {x = 0} \cr } } \right.$ is continuous at x = 0, where {x} = x $-$ [ x ] is the greatest integer less than or equal to x. Then :





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

Solution

JEE MAIN PYQ
$\mathrm{a}, \mathrm{b}>0$, let $f(x)= \begin{cases}\frac{\tan ((\mathrm{a}+1) x)+\mathrm{b} \tan x}{x}, & x< 0 \\ 3, & x=0 \\ \frac{\sqrt{\mathrm{a} x+\mathrm{b}^2 x^2}-\sqrt{\mathrm{a} x}}{\mathrm{~b} \sqrt{\mathrm{a}} x \sqrt{x}}, & x> 0\end{cases}$ be a continuous function at $x=0$. Then $\frac{\mathrm{b}}{\mathrm{a}}$ is equal to :





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

Solution

JEE MAIN PYQ
If the function $f(x) = \left\{ {\matrix{ {(1 + |\cos x|)^{\lambda \over {|\cos x|}}} & , & {0 < x < {\pi \over 2}} \cr \mu & , & {x = {\pi \over 2}} \cr e^{{{\cot 6x} \over {{}\cot 4x}}} & , & {{\pi \over 2} < x < \pi } \cr } } \right.$

is continuous at $x = {\pi \over 2}$, then $9\lambda + 6{\log _e}\mu + {\mu ^6} - {e^{6\lambda }}$ is equal to






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

Solution

JEE MAIN PYQ
If the function $f$ defined on $\left(\dfrac{\pi}{6}, \dfrac{\pi}{3}\right)$ by $f(x) = \begin{cases} \dfrac{\sqrt{2}\cos x - 1}{\cot x - 1}, & x \ne \dfrac{\pi}{4} \ k, & x = \dfrac{\pi}{4} \end{cases}$ is continuous, then $k$ is equal to





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

Solution

JEE MAIN PYQ
Let f, g : R $\to$ R be functions defined by

$f(x) = \left\{ {\matrix{ {[x]} & , & {x < 0} \cr {|1 - x|} & , & {x \ge 0} \cr } } \right.$ and $g(x) = \left\{ {\matrix{ {{e^x} - x} & , & {x < 0} \cr {{{(x - 1)}^2} - 1} & , & {x \ge 0} \cr } } \right.$ where [x] denote the greatest integer less than or equal to x. Then, the function fog is discontinuous at exactly :






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

Solution

JEE MAIN PYQ
A ray of light coming from the point $P(1,2)$ gets reflected from the point $Q$ on the $x$-axis and then passes through the point $R(4,3)$. If the point $S(h,k)$ is such that $PQRS$ is a parallelogram, then $hk^{2}$ is equal to:





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

Solution

JEE MAIN PYQ
The value of k for which the function

$f\left( x \right) = \left\{ {\matrix{ {{{\left( {{4 \over 5}} \right)}^{{{\tan \,4x} \over {\tan \,5x}}}}\,\,,} & {0 < x < {\pi \over 2}} \cr {k + {2 \over 5}\,\,\,,} & {x = {\pi \over 2}} \cr } } \right.$

is continuous at x = ${\pi \over 2},$ is :





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

Solution

JEE MAIN PYQ
Let \; f:\mathbb{R}\to\mathbb{R} \text{ be defined as} \[ f(x) = \begin{cases} \dfrac{\sin\!\big((a+1)x\big)+\sin 2x}{2x}, & x<0 \\[8pt] b, & x=0 \\[8pt] \dfrac{\sqrt{x+bx^{3}}-\sqrt{x}}{b\,x^{5/2}}, & x>0 \end{cases} \] If f is continuous at x = 0, then the value of a + b is equal to :





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

Solution

JEE MAIN PYQ
If the function $f(x) = \left\{ {\matrix{ {{1 \over x}{{\log }_e}\left( {{{1 + {x \over a}} \over {1 - {x \over b}}}} \right)} & , & {x < 0} \cr k & , & {x = 0} \cr {{{{{\cos }^2}x - {{\sin }^2}x - 1} \over {\sqrt {{x^2} + 1} - 1}}} & , & {x > 0} \cr } } \right.$ is continuous at x = 0, then ${1 \over a} + {1 \over b} + {4 \over k}$ is equal to :





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

Solution

JEE MAIN PYQ
Let a function f : R $\to$ R be defined as $f(x) = \left\{ {\matrix{ {\sin x - {e^x}} & {if} & {x \le 0} \cr {a + [ - x]} & {if} & {0 < x < 1} \cr {2x - b} & {if} & {x \ge 1} \cr } } \right.$ where [ x ] is the greatest integer less than or equal to x. If f is continuous on R, then (a + b) is equal to:





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

Solution

JEE MAIN PYQ
Let f : R $ \to $ R be a function defined as
$f(x) = \left\{ {\matrix{ 5 & ; & {x \le 1} \cr {a + bx} & ; & {1 < x < 3} \cr {b + 5x} & ; & {3 \le x < 5} \cr {30} & ; & {x \ge 5} \cr } } \right.$ Then, f is





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

Solution

JEE MAIN PYQ
Let $\quad f(x)= \begin{cases}(1+a x)^{1 / x} & , x<0 \\ 1+b, & x=0 \\ \frac{(x+4)^{1 / 2}-2}{(x+c)^{1 / 3}-2}, & x>0\end{cases}$ be continuous at $x=0$. Then $e^a b c$ is equal to:





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

Solution

JEE MAIN PYQ
Let $f:\mathbf{R}\rightarrow\mathbf{R}$ be defined as $ f(x)= \begin{cases} \dfrac{a - b\cos 2x}{x^2}, & x < 0, \\[6pt] x^2 + cx + 2, & 0 \le x \le 1, \\[6pt] 2x + 1, & x > 1. \end{cases} $ If $f$ is continuous everywhere in $\mathbf{R}$ and $m$ is the number of points where $f$ is **not differentiable**, then $m + a + b + c$ equals :





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

Solution

JEE MAIN PYQ
If$f(x) = \left\{ {\matrix{ {{{\sin (p + 1)x + \sin x} \over x}} & {,x < 0} \cr q & {,x = 0} \cr {{{\sqrt {x + {x^2}} - \sqrt x } \over {{x^{{\raise0.5ex\hbox{$\scriptstyle 3$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}}}}}} & {,x > 0} \cr } } \right.$
is continuous at x = 0, then the ordered pair (p, q) is equal to





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

Solution

JEE MAIN PYQ
Let f : R $\to$ R be defined as $f(x) = \left\{ {\matrix{ {{{{x^3}} \over {{{(1 - \cos 2x)}^2}}}{{\log }_e}\left( {{{1 + 2x{e^{ - 2x}}} \over {{{(1 - x{e^{ - x}})}^2}}}} \right),} & {x \ne 0} \cr {\alpha ,} & {x = 0} \cr } } \right.$ If f is continuous at x = 0, then $\alpha$ is equal to :





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

Solution

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 July Morning Shift) PYQ

Solution

JEE MAIN PYQ
Let $a,b\in\mathbb{R}$, $(a\neq 0)$. If the function $f$ defined as $f(x)= \begin{cases} \dfrac{2x^{2}}{a}, & 0\le x<1 \\ a, & 1\le x<\sqrt{2} \\ \dfrac{2b^{2}-4b}{x^{3}}, & \sqrt{2}\le x<\infty \end{cases}$ is continuous in the interval $[0,\infty)$, then an ordered pair $(a,b)$ is :





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JEE MAIN JEE Mains PYQ JEE MAIN JEE Main 2016 (10 April Morning Shift) PYQ

Solution

JEE MAIN PYQ
If the function $f(x) = \begin{cases} \dfrac{\log_e(1 - x + x^{2}) + \log_e(1 + x + x^{2})}{\sec x - \cos x}, & x \in \left( -\tfrac{\pi}{2}, \tfrac{\pi}{2} \right) \setminus \{0\} \\ k, & x = 0 \end{cases}$ is continuous at $x=0$, then $k$ is equal to:





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

Solution

JEE MAIN PYQ
If $f(x)= \begin{cases} x+a, & x\le 0\\ |x-4|, & x>0 \end{cases} \quad\text{and}\quad g(x)= \begin{cases} x+1, & x<0\\ (x-4)^{2}+b, & x\ge 0 \end{cases}$ are continuous on $\mathbb{R}$, then $(g\circ f)(2)+(f\circ g)(-2)$ is equal to:





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

Solution

JEE MAIN PYQ
$ f(x)=\left\{\begin{array}{l} \frac{2}{x}\left\{\sin \left(k_1+1\right) x+\sin \left(k_2-1\right) x\right\}, \quad x<0 \\ 4, \quad x=0 \\ \frac{2}{x} \log _e\left(\frac{2+k_1 x}{2+k_2 x}\right), \quad x>0 \end{array}\right. $

is continuous at $x=0$, then $k_1^2+k_2^2$ is equal to :





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

Solution


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