jee-main

Papers (191)
2026
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2025
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2024
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2023
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2022
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2021
session1_24feb_shift1 9 session1_24feb_shift2 4 session1_25feb_shift1 29 session1_25feb_shift2 29 session1_26feb_shift2 15 session2_16mar_shift1 29 session2_16mar_shift2 18 session2_17mar_shift1 21 session2_17mar_shift2 27 session2_18mar_shift1 18 session2_18mar_shift2 9 session3_20jul_shift1 29 session3_20jul_shift2 29 session3_22jul_shift1 9 session3_25jul_shift1 8 session3_25jul_shift2 14 session3_27jul_shift1 4 session3_27jul_shift2 7 session4_01sep_shift2 14 session4_26aug_shift1 5 session4_26aug_shift2 2 session4_27aug_shift1 3 session4_27aug_shift2 29 session4_31aug_shift1 28 session4_31aug_shift2 4
2020
session1_07jan_shift1 28 session1_07jan_shift2 20 session1_08jan_shift1 5 session1_08jan_shift2 11 session1_09jan_shift1 26 session1_09jan_shift2 16 session2_02sep_shift1 18 session2_02sep_shift2 16 session2_03sep_shift1 23 session2_03sep_shift2 8 session2_04sep_shift1 14 session2_04sep_shift2 27 session2_05sep_shift1 22 session2_05sep_shift2 29 session2_06sep_shift1 11 session2_06sep_shift2 10
2019
session1_09jan_shift1 6 session1_09jan_shift2 29 session1_10jan_shift1 29 session1_10jan_shift2 14 session1_11jan_shift1 6 session1_11jan_shift2 5 session1_12jan_shift1 10 session1_12jan_shift2 29 session2_08apr_shift1 29 session2_08apr_shift2 29 session2_09apr_shift1 29 session2_09apr_shift2 29 session2_10apr_shift1 2 session2_10apr_shift2 5 session2_12apr_shift1 3 session2_12apr_shift2 9
2018
08apr 30 15apr 28 15apr_shift1 28 15apr_shift2 6 16apr 19
2017
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2016
03apr 28 09apr 29 10apr 30
2015
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2014
06apr 28 09apr 28 11apr 4 12apr 5 19apr 29
2013
07apr 29 09apr 12 22apr 5 23apr 14 25apr 13
2012
07may 17 12may 21 19may 14 26may 17 offline 30
2011
jee-main_2011.pdf 18
2010
jee-main_2010.pdf 6
2009
jee-main_2009.pdf 2
2008
jee-main_2008.pdf 4
2007
jee-main_2007.pdf 38
2006
jee-main_2006.pdf 15
2005
jee-main_2005.pdf 25
2004
jee-main_2004.pdf 22
2003
jee-main_2003.pdf 8
2002
jee-main_2002.pdf 12
2022 session2_25jul_shift1

29 maths questions

Q61 Roots of polynomials Vieta's formulas: compute symmetric functions of roots View
If $\alpha , \beta , \gamma , \delta$ are the roots of the equation $x ^ { 4 } + x ^ { 3 } + x ^ { 2 } + x + 1 = 0$, then $\alpha ^ { 2021 } + \beta ^ { 2021 } + \gamma ^ { 2021 } + \delta ^ { 2021 }$ is equal to
(1) 4
(2) 1
(3) - 4
(4) - 1
Q62 Complex Numbers Argand & Loci Intersection of Loci and Simultaneous Geometric Conditions View
For $n \in N$, let $S _ { n } = \left\{ z \in C : \left| z - 3 + 2i \right| = \frac { n } { 4 } \right\}$ and $T _ { n } = \left\{ z \in C : \left| z - 2 + 3i \right| = \frac { 1 } { n } \right\}$. Then the number of elements in the set $\left\{ n \in N : S _ { n } \cap T _ { n } = \phi \right\}$ is
(1) 0
(2) 2
(3) 3
(4) 4
Q63 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
The number of solutions of $\cos x = \sin x$, such that $- 4 \pi \leq x \leq 4 \pi$ is
(1) 4
(2) 6
(3) 8
(4) 12
Q64 Straight Lines & Coordinate Geometry Line Equation and Parametric Representation View
A line, with the slope greater than one, passes through the point $A(4,3)$ and intersects the line $x - y - 2 = 0$ at the point $B$. If the length of the line segment $AB$ is $\frac { \sqrt { 29 } } { 3 }$, then $B$ also lies on the line
(1) $2x + y = 9$
(2) $3x - 2y = 7$
(3) $x + 2y = 6$
(4) $2x - 3y = 3$
Q65 Circles Circle-Related Locus Problems View
Let the locus of the centre $(\alpha , \beta),\ \beta > 0$, of the circle which touches the circle $x ^ { 2 } + (y - 1) ^ { 2 } = 1$ externally and also touches the $x$-axis be $L$. Then the area bounded by $L$ and the line $y = 4$ is
(1) $\frac { 32 \sqrt { 2 } } { 3 }$
(2) $\frac { 40 \sqrt { 2 } } { 3 }$
(3) $\frac { 64 } { 3 }$
(4) $\frac { 32 } { 3 }$
Q66 Sequences and series, recurrence and convergence Convergence proof and limit determination View
If $\lim _ { n \rightarrow \infty } \left( \sqrt { n ^ { 2 } - n - 1 } + n\alpha + \beta \right) = 0$ then $8\alpha + \beta$ is equal to
(1) 4
(2) - 8
(3) - 4
(4) 8
Q68 Sine and Cosine Rules Heights and distances / angle of elevation problem View
A tower $PQ$ stands on a horizontal ground with base $Q$ on the ground. The point $R$ divides the tower in two parts such that $QR = 15$ m. If from a point $A$ on the ground the angle of elevation of $R$ is $60 ^ { \circ }$ and the part $PR$ of the tower subtends an angle of $15 ^ { \circ }$ at $A$, then the height of the tower is
(1) $5(2\sqrt { 3 } + 3)$ m
(2) $5(\sqrt { 3 } + 3)$ m
(3) $10(\sqrt { 3 } + 1)$ m
(4) $10(2\sqrt { 3 } + 1)$ m
Q69 3x3 Matrices Linear System Existence and Uniqueness via Determinant View
The number of $\theta \in (0,4\pi)$ for which the system of linear equations $3(\sin 3\theta) x - y + z = 2$ $3(\cos 2\theta) x + 4y + 3z = 3$ $6x + 7y + 7z = 9$ has no solution is
(1) 6
(2) 7
(3) 8
(4) 9
Q70 Permutations & Arrangements Counting Functions with Constraints View
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
(1) 60
(2) 90
(3) 108
(4) 126
Q71 Stationary points and optimisation Find absolute extrema on a closed interval or domain View
If the absolute maximum value of the function $f(x) = (x ^ { 2 } - 2x + 7) e ^ { (4x ^ { 3 } - 12x ^ { 2 } - 180x + 31)}$ in the interval $[-3,0]$ is $f(\alpha)$, then
(1) $\alpha = 0$
(2) $\alpha = - 3$
(3) $\alpha \in (-1,0)$
(4) $\alpha \in (-3,-1)$
Q72 Stationary points and optimisation Find critical points and classify extrema of a given function View
The curve $y(x) = ax ^ { 3 } + bx ^ { 2 } + cx + 5$ touches the $x$-axis at the point $P(-2,0)$ and cuts the $y$-axis at the point $Q$, where $y'$ is equal to 3. Then the local maximum value of $y(x)$ is
(1) $\frac { 27 } { 4 }$
(2) $\frac { 29 } { 4 }$
(3) $\frac { 37 } { 4 }$
(4) $\frac { 9 } { 2 }$
Q73 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
For any real number $x$, let $\lfloor x \rfloor$ denote the largest integer less than or equal to $x$. Let $f$ be a real-valued function defined on the interval $[-10,10]$ by $f(x) = \begin{cases} x - \lfloor x \rfloor, & \text{if } \lfloor x \rfloor \text{ is odd} \\ 1 + \lfloor x \rfloor - x, & \text{if } \lfloor x \rfloor \text{ is even} \end{cases}$ Then, the value of $\frac { \pi ^ { 2 } } { 10 } \int _ { - 10 } ^ { 10 } f(x) \cos(\pi x)\, dx$ is
(1) 4
(2) 2
(3) 1
(4) 0
Q74 Areas by integration View
The area of the region given by $A = \{(x,y) : x ^ { 2 } \leq y \leq \min(x + 2, 4 - 3x)\}$ is
(1) $\frac { 31 } { 8 }$
(2) $\frac { 17 } { 6 }$
(3) $\frac { 19 } { 6 }$
(4) $\frac { 27 } { 8 }$
Q75 Integration by Substitution Substitution to Compute an Indefinite Integral with Initial Condition View
The slope of the tangent to a curve $C : y = y(x)$ at any point $(x,y)$ on it is $\frac { 2e ^ { 2x } - 6e ^ { -x } + 9 } { 2 + 9e ^ { -2x } }$. If $C$ passes through the points $\left(0, \frac{1}{2} + \frac{\pi}{2\sqrt{2}}\right)$ and $\left(\alpha, \frac{1}{2} e ^ { 2\alpha }\right)$ then $e ^ { \alpha }$ is equal to
(1) $\frac { 3 + \sqrt { 2 } } { 3 - \sqrt { 2 } }$
(2) $\frac { 3 } { \sqrt { 2 } } \cdot \frac { 3 + \sqrt { 2 } } { 3 - \sqrt { 2 } }$
(3) $\frac { 1 } { \sqrt { 2 } } \cdot \frac { \sqrt { 2 } + 1 } { \sqrt { 2 } - 1 }$
(4) $\frac { \sqrt { 2 } + 1 } { \sqrt { 2 } - 1 }$
Q76 Differential equations First-Order Linear DE: General Solution View
The general solution of the differential equation $(x - y^2)dx + y(5x + y^2)dy = 0$ is
(1) $y ^ { 2 } + x ^ { 4 } = C(y ^ { 2 } + 2x ^ { 3 })$
(2) $y ^ { 2 } + 2x ^ { 4 } = C(y ^ { 2 } + x ^ { 3 })$
(3) $y ^ { 2 } + x ^ { 3 } = C(2y ^ { 2 } + x ^ { 4 })$
(4) $y ^ { 2 } + 2x ^ { 3 } = C(2y ^ { 2 } + x ^ { 4 })$
Q77 Vectors Introduction & 2D True/False or Multiple-Statement Verification View
Let $ABC$ be a triangle such that $\overrightarrow { BC } = \vec { a }$, $\overrightarrow { CA } = \vec { b }$, $\overrightarrow { AB } = \vec { c }$, $|\vec{a}| = 6\sqrt{2}$, $|\vec{b}| = 2\sqrt{3}$ and $\vec{b} \cdot \vec{c} = 12$. Consider the statements: S1: $|\vec{a} \times \vec{b} + \vec{c} \times \vec{b}| - |\vec{c}| = 6(2\sqrt{2} - 1)$ S2: $\angle ABC = \cos^{-1}\sqrt{\frac{2}{3}}$. Then
(1) both $S1$ and $S2$ are true
(2) only $S1$ is true
(3) only $S2$ is true
(4) both $S1$ and $S2$ are false
Q78 Vectors: Lines & Planes Distance Computation (Point-to-Plane or Line-to-Line) View
Let $P$ be the plane containing the straight line $\frac { x - 3 } { 9 } = \frac { y + 4 } { - 1 } = \frac { z - 7 } { - 5 }$ and perpendicular to the plane containing the straight lines $\frac { x } { 2 } = \frac { y } { 3 } = \frac { z } { 5 }$ and $\frac { x } { 3 } = \frac { y } { 7 } = \frac { z } { 8 }$. If $d$ is the distance of $P$ from the point $(2,-5,11)$, then $d ^ { 2 }$ is equal to
(1) $\frac { 147 } { 2 }$
(2) 96
(3) $\frac { 32 } { 3 }$
(4) 54
Q79 Binomial Distribution Find Parameters from Moment Conditions View
If the sum and the product of mean and variance of a binomial distribution are 24 and 128 respectively, then the probability of one or two successes is:
(1) $\frac { 33 } { 2 ^ { 32 } }$
(2) $\frac { 33 } { 2 ^ { 29 } }$
(3) $\frac { 33 } { 2 ^ { 28 } }$
(4) $\frac { 33 } { 2 ^ { 27 } }$
Q80 Discriminant and conditions for roots Probability involving discriminant conditions View
If the numbers appeared on the two throws of a fair six faced die are $\alpha$ and $\beta$, then the probability that $x ^ { 2 } + \alpha x + \beta > 0$, for all $x \in R$, is
(1) $\frac { 17 } { 36 }$
(2) $\frac { 4 } { 9 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 19 } { 36 }$
Q81 Arithmetic Sequences and Series Arithmetic-Geometric Hybrid Problem View
Let $a, b$ be two non-zero real numbers. If $p$ and $r$ are the roots of the equation $x ^ { 2 } - 8ax + 2a = 0$ and $q$ and $s$ are the roots of the equation $x ^ { 2 } + 12bx + 6b = 0$, such that $\frac { 1 } { p }, \frac { 1 } { q }, \frac { 1 } { r }, \frac { 1 } { s }$ are in A.P., then $a ^ { - 1 } - b ^ { - 1 }$ is equal to $\_\_\_\_$.
Q82 Permutations & Arrangements Dictionary Order / Rank of a Permutation View
The letters of the word 'MANKIND' are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word 'MANKIND' is $\_\_\_\_$.
Q83 Arithmetic Sequences and Series Summation of Derived Sequence from AP View
Let $a _ { 1 } = b _ { 1 } = 1$, $a _ { n } = a _ { n - 1 } + 2$ and $b _ { n } = a _ { n } + b _ { n - 1 }$ for every natural number $n \geq 2$. Then $\sum _ { n = 1 } ^ { 15 } a _ { n } \cdot b _ { n }$ is equal to $\_\_\_\_$.
Q84 Binomial Theorem (positive integer n) Find the Largest Term or Coefficient in a Binomial Expansion View
If the maximum value of the term independent of $t$ in the expansion of $\left( t ^ { 2 } x ^ { \frac { 1 } { 5 } } + \frac { 1 - x ^ { \frac { 1 } { 10 } } } { t } \right)^{10}$, $x \geq 0$, is $K$, then $8K$ is equal to $\_\_\_\_$.
Q85 Circles Circles Tangent to Each Other or to Axes View
The sum of diameters of the circles that touch (i) the parabola $75x ^ { 2 } = 64(5y - 3)$ at the point $\left(\frac { 8 } { 5 }, \frac { 6 } { 5 }\right)$ and (ii) the $y$-axis, is equal to $\_\_\_\_$.
Q86 Conic sections Circle Equation Derivation View
Let the equation of two diameters of a circle $x ^ { 2 } + y ^ { 2 } - 2x + 2fy + 1 = 0$ be $2px - y = 1$ and $2x + py = 4p$. Then the slope $m \in (0,\infty)$ of the tangent to the hyperbola $3x ^ { 2 } - y ^ { 2 } = 3$ passing through the centre of the circle is equal to $\_\_\_\_$.
Q87 3x3 Matrices Matrix Power Computation and Application View
Let $A = \begin{pmatrix} 2 & -1 & -1 \\ 1 & 0 & -1 \\ 1 & -1 & 0 \end{pmatrix}$ and $B = A - I$. If $\omega = \frac { \sqrt { 3 }\, i - 1 } { 2 }$, then the number of elements in the set $\left\{ n \in \{1,2,\ldots,100\} : A ^ { n } + \omega B ^ { n } = A + B \right\}$ is equal to $\_\_\_\_$.
Q88 Curve Sketching Continuity and Discontinuity Analysis of Piecewise Functions View
Let $f(x) = \begin{cases} \left\lfloor 4x ^ { 2 } - 8x + 5 \right\rfloor, & \text{if } 8x ^ { 2 } - 6x + 1 \geq 0 \\ \left\lfloor 4x ^ { 2 } - 8x + 5 \right\rfloor, & \text{if } 8x ^ { 2 } - 6x + 1 < 0 \end{cases}$, where $\lfloor \alpha \rfloor$ denotes the greatest integer less than or equal to $\alpha$. Then the number of points in $R$ where $f$ is not differentiable is $\_\_\_\_$.
Q89 Sequences and series, recurrence and convergence Convergence proof and limit determination View
If $\lim _ { n \rightarrow \infty } \frac { (n+1)^{k-1} } { n ^ { k + 1 } } \left[ (nk+1) + (nk+2) + \ldots + (nk+n) \right] = 33 \cdot \lim _ { n \rightarrow \infty } \frac { 1 } { n ^ { k + 1 } } \cdot \left( 1 ^ { k } + 2 ^ { k } + 3 ^ { k } + \ldots + n ^ { k } \right)$, then the integral value of $k$ is equal to $\_\_\_\_$.
Q90 Vectors: Lines & Planes Dihedral Angle or Angle Between Planes/Lines View
The line of shortest distance between the lines $\frac { x - 2 } { 0 } = \frac { y - 1 } { 1 } = \frac { z } { 1 }$ and $\frac { x - 3 } { 2 } = \frac { y - 5 } { 2 } = \frac { z - 1 } { 1 }$ makes an angle of $\sin ^ { - 1 } \sqrt { \frac { 2 } { 27 } }$ with the plane $P : ax - y - z = 0$, $a > 0$. If the image of the point $(1,1,-5)$ in the plane $P$ is $(\alpha, \beta, \gamma)$, then $\alpha + \beta - \gamma$ is equal to $\_\_\_\_$.