jee-main

Papers (169)
2025
session1_22jan_shift1 25 session1_22jan_shift2 25 session1_23jan_shift1 25 session1_23jan_shift2 25 session1_24jan_shift1 25 session1_24jan_shift2 25 session1_28jan_shift1 25 session1_28jan_shift2 25 session1_29jan_shift1 29 session1_29jan_shift2 25
2024
session1_01feb_shift1 4 session1_01feb_shift2 22 session1_27jan_shift1 28 session1_27jan_shift2 30 session1_29jan_shift1 30 session1_29jan_shift2 23 session1_30jan_shift1 17 session1_30jan_shift2 30 session1_31jan_shift1 16 session1_31jan_shift2 15 session2_04apr_shift1 4 session2_04apr_shift2 30 session2_05apr_shift1 4 session2_05apr_shift2 30 session2_06apr_shift1 22 session2_06apr_shift2 30 session2_08apr_shift1 30 session2_08apr_shift2 30 session2_09apr_shift1 5 session2_09apr_shift2 30
2023
session1_01feb_shift1 24 session1_01feb_shift2 3 session1_24jan_shift1 13 session1_24jan_shift2 12 session1_25jan_shift1 28 session1_25jan_shift2 27 session1_29jan_shift1 29 session1_29jan_shift2 28 session1_30jan_shift1 2 session1_30jan_shift2 29 session1_31jan_shift1 28 session1_31jan_shift2 17 session2_06apr_shift1 5 session2_06apr_shift2 17 session2_08apr_shift1 29 session2_08apr_shift2 14 session2_10apr_shift1 29 session2_10apr_shift2 15 session2_11apr_shift1 5 session2_11apr_shift2 4 session2_12apr_shift1 26 session2_13apr_shift1 25 session2_13apr_shift2 20 session2_15apr_shift1 20
2022
session1_24jun_shift1 20 session1_24jun_shift2 25 session1_25jun_shift1 14 session1_25jun_shift2 17 session1_26jun_shift1 26 session1_26jun_shift2 23 session1_27jun_shift1 4 session1_27jun_shift2 29 session1_28jun_shift1 13 session1_29jun_shift1 20 session1_29jun_shift2 5 session2_25jul_shift1 29 session2_25jul_shift2 22 session2_26jul_shift1 29 session2_26jul_shift2 24 session2_27jul_shift1 26 session2_27jul_shift2 29 session2_28jul_shift1 12 session2_28jul_shift2 29 session2_29jul_shift1 18 session2_29jul_shift2 17
2021
session1_24feb_shift1 10 session1_24feb_shift2 7 session1_25feb_shift1 29 session1_25feb_shift2 29 session1_26feb_shift2 17 session2_16mar_shift1 29 session2_16mar_shift2 15 session2_17mar_shift1 20 session2_17mar_shift2 24 session2_18mar_shift1 12 session2_18mar_shift2 11 session3_20jul_shift1 30 session3_20jul_shift2 29 session3_22jul_shift1 7 session3_25jul_shift1 2 session3_25jul_shift2 15 session3_27jul_shift1 3 session3_27jul_shift2 4 session4_01sep_shift2 11 session4_26aug_shift1 5 session4_26aug_shift2 2 session4_27aug_shift1 3 session4_27aug_shift2 28 session4_31aug_shift1 28 session4_31aug_shift2 4
2020
session1_07jan_shift1 26 session1_07jan_shift2 17 session1_08jan_shift1 5 session1_08jan_shift2 12 session1_09jan_shift1 22 session1_09jan_shift2 18 session2_02sep_shift1 19 session2_02sep_shift2 17 session2_03sep_shift1 21 session2_03sep_shift2 9 session2_04sep_shift1 10 session2_04sep_shift2 24 session2_05sep_shift1 23 session2_05sep_shift2 27 session2_06sep_shift1 13 session2_06sep_shift2 10
2019
session1_09jan_shift1 6 session1_09jan_shift2 29 session1_10jan_shift1 30 session1_10jan_shift2 12 session1_11jan_shift1 6 session1_11jan_shift2 5 session1_12jan_shift1 10 session1_12jan_shift2 20 session2_08apr_shift1 29 session2_08apr_shift2 29 session2_09apr_shift1 29 session2_09apr_shift2 29 session2_10apr_shift1 2 session2_10apr_shift2 3 session2_12apr_shift1 3 session2_12apr_shift2 9
2018
08apr 29 15apr 28 15apr_shift1 28 15apr_shift2 2 16apr 15
2017
02apr 28 08apr 29 09apr 30
2016
03apr 30 09apr 30 10apr 28
2015
04apr 29 10apr 30
2014
06apr 28 09apr 28 11apr 4 12apr 5 19apr 29
2013
07apr 29 09apr 14 22apr 5 23apr 14 25apr 13
2012
07may 18 12may 22 19may 13 26may 17 offline 30
2011
jee-main_2011.pdf 13
2010
jee-main_2010.pdf 1
2009
jee-main_2009.pdf 1
2008
jee-main_2008.pdf 1
2007
jee-main_2007.pdf 38
2005
jee-main_2005.pdf 19
2004
jee-main_2004.pdf 11
2003
jee-main_2003.pdf 9
2002
jee-main_2002.pdf 8
2019 session1_09jan_shift2

29 maths questions

Q61 Discriminant and conditions for roots Nature of roots given coefficient constraints View
The number of all possible positive integral value of $\alpha$ for which the roots of the quadratic equation $6x^2 - 11x + \alpha = 0$ are rational numbers is:
(1) 5
(2) 3
(3) 4
(4) 2
Q62 Inequalities Simultaneous/Compound Quadratic Inequalities View
If both the roots of the quadratic equation $x^2 - mx + 4 = 0$ are real and distinct and they lie in the interval $(1,5)$, then $m$ lies in the interval: Note: In the actual JEE paper interval was $[1,5]$
(1) $(-5,-4)$
(2) $(3,4)$
(3) $(5,6)$
(4) $(4,5)$
Q63 Complex Numbers Arithmetic Roots of Unity and Cyclotomic Expressions View
Let $z_0$ be a root of quadratic equation, $x^2 + x + 1 = 0$. If $z = 3 + 6iz_0^{81} - 3iz_0^{93}$, then $\arg(z)$ is equal to:
(1) 0
(2) $\frac{\pi}{4}$
(3) $\frac{\pi}{6}$
(4) $\frac{\pi}{3}$
Q64 Permutations & Arrangements Forming Numbers with Digit Constraints View
The number of natural numbers less than 7000 which can be formed by using the digits $0,1,3,7,9$ (repetition of digits allowed) is equal to:
(1) 375
(2) 250
(3) 374
(4) 372
Q65 Arithmetic Sequences and Series Telescoping or Non-Standard Summation Involving an AP View
The sum of the following series $1 + 6 + \frac{9\left(1^2 + 2^2 + 3^2\right)}{7} + \frac{12\left(1^2 + 2^2 + 3^2 + 4^2\right)}{9} + \frac{15\left(1^2 + 2^2 + \ldots + 5^2\right)}{11} + \ldots$ up to 15 terms, is:
(1) 7520
(2) 7510
(3) 7830
(4) 7820
Q66 Arithmetic Sequences and Series Arithmetic-Geometric Hybrid Problem View
Let $a, b$ and $c$ be the $7^{\text{th}}, 11^{\text{th}}$ and $13^{\text{th}}$ terms respectively of a non-constant A.P. If these are also the three consecutive terms of a G.P., then $\frac{a}{c}$ is equal to:
(1) 2
(2) $\frac{7}{13}$
(3) $\frac{1}{2}$
(4) 4
Q67 Binomial Theorem (positive integer n) Find a Specific Coefficient in a Single Binomial Expansion View
The coefficient of $t^4$ in the expansion of $\left(\frac{1-t^6}{1-t}\right)^3$ is
(1) 10
(2) 14
(3) 15
(4) 12
Q68 Standard trigonometric equations Solve trigonometric equation for solutions in an interval View
If $0 \leq x < \frac{\pi}{2}$, then the number of values of $x$ for which $\sin x - \sin 2x + \sin 3x = 0$, is:
(1) 4
(2) 3
(3) 2
(4) 1
Q69 Geometric Probability View
Let $S$ be the set of all triangles in the $xy$-plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in $S$ has area 50 sq. units, then the number of elements in the set $S$ is:
(1) 36
(2) 32
(3) 9
(4) 18
Q70 Straight Lines & Coordinate Geometry Triangle Properties and Special Points View
Let the equations of two sides of a triangle be $3x - 2y + 6 = 0$ and $4x + 5y - 20 = 0$. If the orthocenter of this triangle is at $(1,1)$ then the equation of its third side is:
(1) $122y + 26x + 1675 = 0$
(2) $26x - 122y - 1675 = 0$
(3) $26x + 61y + 1675 = 0$
(4) $122y - 26x - 1675 = 0$
Q71 Circles Intersection of Circles or Circle with Conic View
If the circles $x^2 + y^2 - 16x - 20y + 164 = r^2$ and $(x-4)^2 + (y-7)^2 = 36$ intersect at two distinct points, then:
(1) $r > 11$
(2) $0 < r < 1$
(3) $1 < r < 11$
(4) $r = 11$
Q72 Areas by integration View
Let $A(4,-4)$ and $B(9,6)$ be points on the parabola, $y^2 = 4x$. Let $C$ be chosen on the arc $AOB$ of the parabola, where $O$ is the origin, such that the area of $\triangle ACB$ is maximum. Then, the area (in sq. units) of $\triangle ACB$, is:
(1) 32
(2) $31\frac{3}{4}$
(3) $30\frac{1}{2}$
(4) $31\frac{1}{4}$
Q73 Conic sections Eccentricity or Asymptote Computation View
A hyperbola has its centre at the origin, passes through the point $(4,2)$ and has transverse axis of length 4 along the $x$-axis. Then the eccentricity of the hyperbola is:
(1) $\sqrt{3}$
(2) $\frac{3}{2}$
(3) $\frac{2}{\sqrt{3}}$
(4) 2
Q74 Sign Change & Interval Methods View
For each $x \in R$, let $[x]$ be the greatest integer less than or equal to $x$. Then $\lim_{x \rightarrow 0^-} \frac{x([x] + |x|)\sin[x]}{|x|}$ is equal to
(1) 1
(2) 0
(3) $-\sin 1$
(4) $\sin 1$
Q76 Measures of Location and Spread View
A data consists of $n$ observations: $x_1, x_2, \ldots, x_n$. If $\sum_{i=1}^{n}(x_i + 1)^2 = 9n$ and $\sum_{i=1}^{n}(x_i - 1)^2 = 5n$, then the standard deviation of this data is
(1) 5
(2) $\sqrt{7}$
(3) $\sqrt{5}$
(4) 2
Q77 Matrices Determinant and Rank Computation View
If $A = \left[\begin{array}{ccc} e^t & e^{-t}\cos t & e^{-t}\sin t \\ e^t & -e^{-t}\cos t - e^{-t}\sin t & -e^{-t}\sin t + e^{-t}\cos t \\ e^t & 2e^{-t}\sin t & -2e^{-t}\cos t \end{array}\right]$, then $A$ is:
(1) Invertible only if $t = \pi$
(2) Not invertible for any $t \in R$
(3) Invertible only if $t = \frac{\pi}{2}$
(4) Invertible for all $t \in R$
Q78 Simultaneous equations View
If the system of linear equations $x - 4y + 7z = g$; $3y - 5z = h$; $-2x + 5y - 9z = k$ is consistent, then:
(1) $g + h + 2k = 0$
(2) $g + 2h + k = 0$
(3) $2g + h + k = 0$
(4) $g + h + k = 0$
Q79 Standard trigonometric equations Inverse trigonometric equation View
If $x = \sin^{-1}(\sin 10)$ and $y = \cos^{-1}(\cos 10)$, then $y - x$ is equal to:
(1) 10
(2) $\pi$
(3) 0
(4) $7\pi$
Q80 Differential equations Integral Equations Reducible to DEs View
Let $f:[0,1] \rightarrow R$ be such that $f(xy) = f(x) \cdot f(y)$, for all $x,y \in [0,1]$, and $f(0) \neq 0$. If $y = y(x)$ satisfies the differential equation, $\frac{dy}{dx} = f(x)$ with $y(0) = 1$ then $y\left(\frac{1}{4}\right) + y\left(\frac{3}{4}\right)$ is equal to:
(1) 5
(2) 2
(3) 3
(4) 4
Q81 Composite & Inverse Functions Injectivity, Surjectivity, or Bijectivity Classification View
Let $A = \{x \in R : x$ is not a positive integer$\}$. Define a function $f: A \rightarrow R$ as $f(x) = \frac{2x}{x-1}$, then $f$ is:
(1) Injective but not surjective
(2) Not injective
(3) Surjective but not injective
(4) Neither injective nor surjective
Q82 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
Let $f$ be a differentiable function from $R$ to $R$ such that $|f(x) - f(y)| \leq 2|x-y|^{3/2}$, for all $x,y \in R$. If $f(0) = 1$ then $\int_0^1 f^2(x)\,dx$ is equal to
(1) 0
(2) 1
(3) 2
(4) $\frac{1}{2}$
Q83 Parametric differentiation View
If $x = 3\tan t$ and $y = 3\sec t$, then the value of $\frac{d^2y}{dx^2}$ at $t = \frac{\pi}{4}$, is:
(1) $\frac{1}{6}$
(2) $\frac{1}{6\sqrt{2}}$
(3) $\frac{1}{3\sqrt{2}}$
(4) $\frac{3}{2\sqrt{2}}$
Q84 Integration by Substitution Substitution to Compute an Indefinite Integral with Initial Condition View
If $f(x) = \int \frac{\left(5x^8 + 7x^6\right)}{\left(x^2 + 1 + 2x^7\right)^2}\,dx,\,(x \geq 0)$, and $f(0) = 0$, then the value of $f(1)$ is
(1) $\frac{-1}{4}$
(2) $\frac{1}{2}$
(3) $\frac{1}{4}$
(4) $-\frac{1}{2}$
Q85 Integration by Substitution Substitution to Evaluate a Definite Integral (Numerical Answer) View
If $\int_0^{\pi/3} \frac{\tan\theta}{\sqrt{2k\sec\theta}}\,d\theta = 1 - \frac{1}{\sqrt{2}},\,(k > 0)$, then the value of $k$ is
(1) $\frac{1}{2}$
(2) 1
(3) 2
(4) 4
Q86 Areas by integration View
The area of the region $A = \{(x,y): 0 \leq y \leq x|x| + 1$ and $-1 \leq x \leq 1\}$ in sq. units, is
(1) $\frac{4}{3}$
(2) 2
(3) $\frac{1}{3}$
(4) $\frac{2}{3}$
Q87 Vectors 3D & Lines Vector Algebra and Triple Product Computation View
Let $\vec{a} = \hat{\mathrm{i}} + \hat{\mathrm{j}} + \sqrt{2}\hat{\mathrm{k}},\, \vec{b} = b_1\hat{\mathrm{i}} + b_2\hat{\mathrm{j}} + \sqrt{2}\hat{\mathrm{k}}$ and $\vec{c} = 5\hat{\mathrm{i}} + \hat{\mathrm{j}} + \sqrt{2}\hat{\mathrm{k}}$ be three vectors such that the projection vector of $\vec{b}$ on $\vec{a}$ is $|\vec{a}|$. If $\vec{a} + \vec{b}$ is perpendicular to $\vec{c}$, then $|\vec{b}|$ is equal to:
(1) $\sqrt{22}$
(2) $\sqrt{32}$
(3) 6
(4) 4
Q88 Vectors: Lines & Planes Prove Perpendicularity/Orthogonality of Line and Plane View
If the lines $x = ay + b,\, z = cy + d$ and $x = a'z + b',\, y = c'z + d'$ are perpendicular, then
(1) $cc' + a + a' = 0$
(2) $aa' + c + c' = 0$
(3) $bb' + cc' + 1 = 0$
(4) $ab' + bc' + 1 = 0$
Q89 Vectors: Lines & Planes Find Cartesian Equation of a Plane View
The equation of the plane containing the straight line $\frac{x}{2} = \frac{y}{3} = \frac{z}{4}$ and perpendicular to the plane containing the straight lines $\frac{x}{3} = \frac{y}{4} = \frac{z}{2}$ and $\frac{x}{4} = \frac{y}{2} = \frac{z}{3}$ is:
(1) $3x + 2y - 3z = 0$
(2) $x + 2y - 2z = 0$
(3) $x - 2y + z = 0$
(4) $5x + 2y - 4z = 0$
Q90 Probability Definitions Conditional Probability and Bayes' Theorem View
An urn contains 5 red and 2 green balls. A ball is drawn at random from the urn. If the drawn ball is green, then a red ball is added to the urn and if the drawn ball is red, then a green ball is added to the urn; the original ball is not returned to the urn. Now, a second ball is drawn at random from it. The probability that the second ball is red, is:
(1) $\frac{21}{49}$
(2) $\frac{26}{49}$
(3) $\frac{32}{49}$
(4) $\frac{27}{49}$