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
2014 06apr

28 maths questions

Q61 Solving quadratics and applications Optimization or extremal value of an expression via completing the square View
If $a \in R$ and the equation $- 3 ( x - [ x ] ) ^ { 2 } + 2 ( x - [ x ] ) + a ^ { 2 } = 0$ (where $[ x ]$ denotes the greatest integer $\leq x )$ has no integral solution, then all possible values of $a$ lie in the interval
(1) $( - 2 , - 1 )$
(2) $( - \infty , - 2 ) \cup ( 2 , \infty )$
(3) $( - 1,0 ) \cup ( 0,1 )$
(4) $( 1,2 )$
Q62 Arithmetic Sequences and Series Find Specific Term from Given Conditions View
Let $\alpha$ and $\beta$ be the roots of equation $p x ^ { 2 } + q x + r = 0 , p \neq 0$. If $p , q , r$ are in A.P. and $\frac { 1 } { \alpha } + \frac { 1 } { \beta } = 4$, then the value of $| \alpha - \beta |$ is
(1) $\frac { \sqrt { 34 } } { 9 }$
(2) $\frac { 2 \sqrt { 13 } } { 9 }$
(3) $\frac { \sqrt { 61 } } { 9 }$
(4) $\frac { 2 \sqrt { 17 } } { 9 }$
Q63 Complex Numbers Arithmetic Modulus Inequalities and Bounds (Proof-Based) View
If $z$ is a complex number such that $| z | \geq 2$, then the minimum value of $\left| z + \frac { 1 } { 2 } \right|$:
(1) Is strictly greater than $\frac { 5 } { 2 }$
(2) Is strictly greater than $\frac { 3 } { 2 }$ but less than $\frac { 5 } { 2 }$
(3) Is equal to $\frac { 5 } { 2 }$
(4) Lies in the interval $( 1,2 )$
Q64 Arithmetic Sequences and Series Summation of Derived Sequence from AP View
If $( 10 ) ^ { 9 } + 2 ( 11 ) ^ { 1 } ( 10 ) ^ { 8 } + 3 ( 11 ) ^ { 2 } ( 10 ) ^ { 7 } + \ldots\ldots + 10 ( 11 ) ^ { 9 } = k ( 10 ) ^ { 9 }$, then $k$ is equal to:
(1) 100
(2) 110
(3) $\frac { 121 } { 10 }$
(4) $\frac { 441 } { 100 }$
Q65 Geometric Sequences and Series Arithmetic-Geometric Sequence Interplay View
Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. Then the common ratio of the G.P. is:
(1) $2 - \sqrt { 3 }$
(2) $2 + \sqrt { 3 }$
(3) $\sqrt { 2 } + \sqrt { 3 }$
(4) $3 + \sqrt { 2 }$
Q66 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
If the coefficients of $x ^ { 3 }$ and $x ^ { 4 }$ in the expansion of $\left( 1 + a x + b x ^ { 2 } \right) ( 1 - 2 x ) ^ { 18 }$ in powers of $x$ are both zero, then $( a , b )$ is equal to
(1) $\left( 14 , \frac { 272 } { 3 } \right)$
(2) $\left( 16 , \frac { 272 } { 3 } \right)$
(3) $\left( 16 , \frac { 251 } { 3 } \right)$
(4) $\left( 14 , \frac { 251 } { 3 } \right)$
Q67 Reciprocal Trig & Identities View
Let $f _ { k } ( x ) = \frac { 1 } { k } \left( \sin ^ { k } x + \cos ^ { k } x \right)$ where $x \in R$ and $k \geq 1$. Then $f _ { 4 } ( x ) - f _ { 6 } ( x )$ equals
(1) $\frac { 1 } { 4 }$
(2) $\frac { 1 } { 12 }$
(3) $\frac { 1 } { 6 }$
(4) $\frac { 1 } { 3 }$
Q68 Straight Lines & Coordinate Geometry Line Equation and Parametric Representation View
Let $P S$ be the median of the triangle with vertices $P ( 2,2 ) , Q ( 6 , - 1 )$ and $R ( 7,3 )$. The equation of the line passing through $( 1 , - 1 )$ and parallel to $P S$ is
(1) $4 x + 7 y + 3 = 0$
(2) $2 x - 9 y - 11 = 0$
(3) $4 x - 7 y - 11 = 0$
(4) $2 x + 9 y + 7 = 0$
Q69 Straight Lines & Coordinate Geometry Collinearity and Concurrency View
Let $a , b , c$ and $d$ be non-zero numbers. If the point of intersection of the lines $4 a x + 2 a y + c = 0$ \& $5 b x + 2 b y + d = 0$ lies in the fourth quadrant and is equidistant from the two axes then
(1) $3 b c - 2 a d = 0$
(2) $3 b c + 2 a d = 0$
(3) $2 b c - 3 a d = 0$
(4) $2 b c + 3 a d = 0$
Q70 Circles Circles Tangent to Each Other or to Axes View
Let $C$ be the circle with center at $( 1,1 )$ and radius $= 1$. If $T$ is the circle centered at $( 0 , y )$, passing through the origin and touching the circle $C$ externally, then the radius of $T$ is equal to
(1) $\frac { 1 } { 2 }$
(2) $\frac { 1 } { 4 }$
(3) $\frac { \sqrt { 3 } } { \sqrt { 2 } }$
(4) $\frac { \sqrt { 3 } } { 2 }$
Q71 Circles Circle-Related Locus Problems View
The locus of the foot of perpendicular drawn from the centre of the ellipse $x ^ { 2 } + 3 y ^ { 2 } = 6$ on any tangent to it is
(1) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 6 x ^ { 2 } + 2 y ^ { 2 }$
(2) $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 6 x ^ { 2 } - 2 y ^ { 2 }$
(3) $\left( x ^ { 2 } - y ^ { 2 } \right) ^ { 2 } = 6 x ^ { 2 } + 2 y ^ { 2 }$
(4) $\left( x ^ { 2 } - y ^ { 2 } \right) ^ { 2 } = 6 x ^ { 2 } - 2 y ^ { 2 }$
Q72 Chain Rule Limit Evaluation Involving Composition or Substitution View
$\lim _ { x \rightarrow 0 } \frac { \sin \left( \pi \cos ^ { 2 } x \right) } { x ^ { 2 } }$ is equal to
(1) $- \pi$
(2) $\pi$
(3) $\frac { \pi } { 2 }$
(4) 1
Q74 Measures of Location and Spread View
The variance of the first 50 even natural numbers is:
(1) 437
(2) $\frac { 437 } { 4 }$
(3) $\frac { 833 } { 4 }$
(4) 833
Q75 Sine and Cosine Rules Heights and distances / angle of elevation problem View
A bird is sitting on the top of a vertical pole 20 m high and its elevation from a point $O$ on the ground is $45 ^ { \circ }$. It flies off horizontally straight away from the point $O$. After one second, the elevation of the bird from $O$ is reduced to $30 ^ { \circ }$. Then the speed (in m/s) of the bird is
(1) $20 \sqrt { 2 }$
(2) $20 ( \sqrt { 3 } - 1 )$
(3) $40 ( \sqrt { 2 } - 1 )$
(4) $40 ( \sqrt { 3 } - \sqrt { 2 } )$
Q77 Matrices Matrix Algebra and Product Properties View
If $A$ is a $3 \times 3$ non-singular matrix such that $A A ^ { \prime } = A ^ { \prime } A$ and $B = A ^ { - 1 } A ^ { \prime }$, then $B B ^ { \prime }$ equals, where $X ^ { \prime }$ denotes the transpose of the matrix $X$.
(1) $B ^ { - 1 }$
(2) $\left( B ^ { - 1 } \right) ^ { \prime }$
(3) $I + B$
(4) $I$
Q78 Matrices Determinant and Rank Computation View
If $\alpha , \beta \neq 0 , f ( n ) = \alpha ^ { n } + \beta ^ { n }$ and $$\begin{vmatrix} 3 & 1 + f(1) & 1 + f(2) \\ 1 + f(1) & 1 + f(2) & 1 + f(3) \\ 1 + f(2) & 1 + f(3) & 1 + f(4) \end{vmatrix} = K ( 1 - \alpha ) ^ { 2 } ( 1 - \beta ) ^ { 2 } ( \alpha - \beta ) ^ { 2 }$$, then $K$ is equal to
(1) 1
(2) - 1
(3) $\alpha \beta$
(4) $\frac { 1 } { \alpha \beta }$
Q79 Composite & Inverse Functions Derivative of an Inverse Function View
If $g$ is the inverse of a function $f$ and $f ^ { \prime } ( x ) = \frac { 1 } { 1 + x ^ { 5 } }$, then $g ^ { \prime } ( x )$ is equal to
(1) $\frac { 1 } { 1 + \{ g ( x ) \} ^ { 5 } }$
(2) $1 + \{ g ( x ) \} ^ { 5 }$
(3) $1 + x ^ { 5 }$
(4) $5 x ^ { 4 }$
Q80 Stationary points and optimisation Determine parameters from given extremum conditions View
If $f$ \& $g$ are differentiable functions in $[ 0,1 ]$ satisfying $f ( 0 ) = 2 = g ( 1 ) , g ( 0 ) = 0$ \& $f ( 1 ) = 6$, then for some $c \in ] 0,1 [$
(1) $f ^ { \prime } ( c ) = g ^ { \prime } ( c )$
(2) $f ^ { \prime } ( c ) = 2 g ^ { \prime } ( c )$
(3) $2 f ^ { \prime } ( c ) = g ^ { \prime } ( c )$
(4) $2 f ^ { \prime } ( c ) = 3 g ^ { \prime } ( c )$
Q81 Stationary points and optimisation Determine parameters from given extremum conditions View
If $x = - 1$ and $x = 2$ are extreme points of $f ( x ) = \alpha \log | x | + \beta x ^ { 2 } + x$, then
(1) $\alpha = 2 , \beta = - \frac { 1 } { 2 }$
(2) $\alpha = 2 , \beta = \frac { 1 } { 2 }$
(3) $\alpha = - 6 , \beta = \frac { 1 } { 2 }$
(4) $\alpha = - 6 , \beta = - \frac { 1 } { 2 }$
Q82 Tangents, normals and gradients Common tangent line to two curves View
The slope of the line touching both the parabolas $y ^ { 2 } = 4 x$ and $x ^ { 2 } = - 32 y$ is
(1) $\frac { 1 } { 8 }$
(2) $\frac { 2 } { 3 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 3 } { 2 }$
Q83 Integration by Substitution Substitution to Transform Integral Form (Show Transformed Expression) View
The integral $\int \left( 1 + x - \frac { 1 } { x } \right) e ^ { x + \frac { 1 } { x } } d x$, is equal to
(1) $( x + 1 ) e ^ { x + \frac { 1 } { x } } + c$
(2) $- x e ^ { x + \frac { 1 } { x } } + c$
(3) $( x - 1 ) e ^ { x + \frac { 1 } { x } } + c$
(4) $x e ^ { x + \frac { 1 } { x } } + c$
Q84 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
The integral $\int _ { 0 } ^ { \pi } \sqrt { 1 + 4 \sin ^ { 2 } \frac { x } { 2 } - 4 \sin \frac { x } { 2 } } \, d x$ equals
(1) $4 \sqrt { 3 } - 4$
(2) $4 \sqrt { 3 } - 4 - \frac { \pi } { 3 }$
(3) $\pi - 4$
(4) $\frac { 2 \pi } { 3 } - 4 - 4 \sqrt { 3 }$
Q85 Areas by integration View
The area (in sq. unit) of the region described by $A = \left\{ ( x , y ) : x ^ { 2 } + y ^ { 2 } \leq 1 \text{ and } y ^ { 2 } \leq 1 - x \right\}$ is
(1) $\frac { \pi } { 2 } - \frac { 2 } { 3 }$
(2) $\frac { \pi } { 2 } + \frac { 2 } { 3 }$
(3) $\frac { \pi } { 2 } + \frac { 4 } { 3 }$
(4) $\frac { \pi } { 2 } - \frac { 4 } { 3 }$
Q86 Differential equations Applied Modeling with Differential Equations View
Let the population of rabbits surviving at a time $t$ be governed by the differential equation $\frac { d p ( t ) } { d t } = \frac { 1 } { 2 } \{ p ( t ) - 400 \}$. If $p ( 0 ) = 100$, then $p ( t )$ equals
(1) $600 - 500 e ^ { \frac { t } { 2 } }$
(2) $400 - 300 e ^ { \frac { - t } { 2 } }$
(3) $400 - 300 e ^ { t / 2 }$
(4) $300 - 200 e ^ { \frac { - t } { 2 } }$
Q87 Vectors Introduction & 2D Dot Product Computation View
If $[ \vec { a } \times \vec { b } \quad \vec { b } \times \vec { c } \quad \vec { c } \times \vec { a } ] = \lambda \left[ \vec { a } \quad \vec { b } \quad \vec { c } \right] ^ { 2 }$ then $\lambda$ is equal to
(1) 0
(2) 1
(3) 2
(4) 3
Q88 Vectors: Lines & Planes MCQ: Identify Correct Equation or Representation View
The image of the line $\frac { x - 1 } { 3 } = \frac { y - 3 } { 1 } = \frac { z - 4 } { - 5 }$ in the plane $2 x - y + z + 3 = 0$ is the line
(1) $\frac { x - 3 } { 3 } = \frac { y + 5 } { 1 } = \frac { z - 2 } { - 5 }$
(2) $\frac { x - 3 } { - 3 } = \frac { y + 5 } { - 1 } = \frac { z - 2 } { 5 }$
(3) $\frac { x + 3 } { 3 } = \frac { y - 5 } { 1 } = \frac { z - 2 } { - 5 }$
(4) $\frac { x + 3 } { - 3 } = \frac { y - 5 } { - 1 } = \frac { z + 2 } { 5 }$
Q89 Vectors: Lines & Planes Dihedral Angle or Angle Between Planes/Lines View
The angle between the lines whose direction cosines satisfy the equations $l + m + n = 0$ and $l ^ { 2 } = m ^ { 2 } + n ^ { 2 }$ is
(1) $\frac { \pi } { 6 }$
(2) $\frac { \pi } { 2 }$
(3) $\frac { \pi } { 3 }$
(4) $\frac { \pi } { 4 }$
Q90 Independent Events View
Let $A$ and $B$ be two events such that $P ( \overline { A \cup B } ) = \frac { 1 } { 6 } , P ( A \cap B ) = \frac { 1 } { 4 }$ and $P ( \bar { A } ) = \frac { 1 } { 4 }$, where $\bar { A }$ stands for the complement of the event $A$. Then the events $A$ and $B$ are
(1) Independent but not equally likely.
(2) Independent and equally likely.
(3) Mutually exclusive and independent.
(4) Equally likely but not independent.