jee-advanced

Papers (53)
2025
paper1 16 paper2 16
2024
paper1 17 paper2 17
2023
paper1 17 paper2 17
2022
paper1 18 paper2 18
2021
paper1 23 paper2 19
2020
paper1 18 paper2 18
2019
paper1 18 paper2 18
2018
paper1 18 paper2 18
2017
paper1 18 paper2 18
2016
paper1 18 paper2 18
2015
paper1 20 paper2 20
2014
paper1 20 paper2 20
2013
paper1 20 paper2 20
2012
paper1 1 paper2 2
2011
paper1 6 paper2 9
2010
paper1 28 paper2 19
2009
paper1 19 paper2 17
2008
paper1 23 paper2 22
2007
paper1 27 paper2 5
2006
jee-advanced_2006.pdf 26
2005
mains 10 screening 16
2004
mains 18 screening 17
2003
mains 20 screening 17
2002
mains 4 screening 27
2001
mains 12 screening 28
2000
mains 5 screening 30
1999
jee-advanced_1999.pdf 33
1998
jee-advanced_1998.pdf 25
2003 mains

20 maths questions

Q1 Complex Numbers Arithmetic Modulus Inequalities and Bounds (Proof-Based) View
If $z _ { 1 }$ and $z _ { 2 }$ are two complex numbers such that $\left| \mathrm { z } _ { 1 } \right| < 1 < \left| \mathrm { z } _ { 1 } \right|$ then prove that $$\left| \frac { 1 - z _ { 1 } \bar { z } _ { 2 } } { z _ { 1 } - z _ { 2 } } \right| < 1 .$$
Q2 Straight Lines & Coordinate Geometry Full function study (variation table, limits, asymptotes) View
Find a point on the curve $x ^ { 2 } + 2 y ^ { 2 } = 6$ whose distance from the line $x + y = 7$, is minimum.
Q3 3x3 Matrices Determinant and Rank Computation View
If matrix $$A = \left[ \begin{array} { l l l } a & b & c \\ b & c & a \\ c & a & b \end{array} \right]$$ where $\mathrm { a } , \mathrm { b } , \mathrm { c }$ are real positive numbers, $\mathrm { abc } = 1$ and $\mathrm { A } ^ { \mathrm { T } } \mathrm { A } = \mathrm { I }$, then find the value of $\mathrm { a } ^ { 3 } + \mathrm { b } ^ { 3 } + \mathrm { c } ^ { 3 }$.
Q4 Binomial Theorem (positive integer n) Prove a Binomial Identity or Inequality View
Prove that $$\begin{aligned} & 2 ^ { k } \binom { n } { 0 } \binom { n } { k } - 2 ^ { n - 1 } \binom { n } { 1 } \binom { n - 1 } { k - 1 } + 2 ^ { k - 2 } \\ & \binom { n } { 2 } \binom { n - 2 } { k - 2 } - \ldots \ldots \ldots ( - 1 ) ^ { k } \binom { n } { k } \binom { n - k } { 0 } = \binom { n } { k } . \end{aligned}$$
Q5 Proof Integral Equation with Symmetry or Substitution View
If f is an even function then prove that $$\int _ { 0 } ^ { \pi / 2 } f ( \cos 2 x ) \cos x d x = \sqrt { 2 } \int _ { 0 } ^ { \pi / 4 } f ( \sin 2 x ) \cos x d x$$
Q6 Probability Definitions Probability Using Set/Event Algebra View
For a student to qualify, he must pass at least two out of three exams. The probability that he will pass the 1st exam is P . If he fails in one of the exams then the probability of his passing in the next exam is $\mathrm { P } / 2$ otherwise it remains the same. Find the probability that he will qualify.
Q7 Circles Area and Geometric Measurement Involving Circles View
For the circle $x ^ { 2 } + y ^ { 2 } = t ^ { 2 }$, find the value of $r$ for which the area enclosed by the tangents drawn from the point $\mathrm { P } ( 6,8 )$ to the circle and the chord of contact is maximum.
Q8 Complex numbers 2 Modulus Inequalities and Bounds (Proof-Based) View
Prove that there exists no complex number z such that $| \mathrm { z } | < 1 / 3$ and $$\sum _ { \mathrm { r } = 1 } { } ^ { \mathrm { n } } \text { at } 2 = 1 \quad \text { where } \mathrm { r } _ { \mathrm { r } } \text { as } 2 .$$
Q9 Conditional Probability Bayes' Theorem with Production/Source Identification View
A is targeting to $\mathrm { B } , \mathrm { B }$ and C are targeting to A . Probability of hitting the target by $\mathrm { A } , \mathrm { B }$ and C are $2 / 3,1 / 2$ and $1 / 3$ respectively. If A is hit then find the probability that B hits the target and C does not.
Q10 Composite & Inverse Functions Symmetry, Periodicity, and Parity from Composition Conditions View
If a function $\mathrm { f } : [ - 2 \mathrm { a } , 2 \mathrm { a } ] - - > \mathrm { R }$ is an odd function such that $\mathrm { f } ( \mathrm { x } ) = \mathrm { f } ( 2 \mathrm { a } - \mathrm { x } )$ for $\mathrm { x } \hat { \mathrm { I } } [ \mathrm { a } , 2 \mathrm { a } [$ and the left hand derivative at $\mathrm { x } = \mathrm { a }$ is 0 then find the left hand derivative at $\mathrm { x } = - \mathrm { a }$.
Q11 Proof Solve trigonometric inequality View
Using the relation $2 ( 1 - \cos x ) < x ^ { 2 } , x ^ { 1 } 0$ or otherwise, prove that $\sin ( \tan x ) > x \forall x \hat { \mathrm { I } } [ 0 , \pi / 4 ]$.
Q12 Arithmetic Sequences and Series Arithmetic-Geometric Hybrid Problem View
If $\mathrm { a } , \mathrm { b } , \mathrm { c }$ are in A.P., $\mathrm { a } ^ { 2 } , \mathrm { b} ^ { 2 } , \mathrm { c } ^ { 2 }$ are in H.P., then prove that either $\mathrm { a } = \mathrm { b } = \mathrm { c }$ or $\mathrm { a } , \mathrm { b } , - \mathrm { c } / 2$ form a G.P.
Q13 Discriminant and conditions for roots Parameter range for specific root conditions (location/count) View
If $x ^ { 2 } + ( a - b ) x + ( 1 - a - b ) = 0$ where $a , b$ Î $R$ then find the values of $a$ for which equation has unequal real roots for all values of $b$.
Q14 Circles Locus and Trajectory Derivation View
Normals are drawn from the point $P$ with slopes $m _ { 1 } , m _ { 2 } , m _ { 3 }$ to the parabola $y ^ { 2 } = 4 x$. If locus of $P$ with $\mathrm { m } _ { 1 } \mathrm { m} _ { 2 } = \mathrm { a }$ is a part of the parabola itself then finda.
Q15 Indefinite & Definite Integrals Definite Integral Evaluation (Computational) View
If the function $\mathrm { f } : [ 0,4 ] - - > \mathrm { R }$ is differentiable then show that (i) For $\mathrm { a } , \mathrm { b } \hat { \mathrm { I } } ( 0,4 ) , ( \mathrm { f } ( 4 ) ) ^ { 2 } = \mathrm { f } ( \mathrm { a } ) \mathrm { f } ( \mathrm { b } )$ (ii) $\left. \int _ { 0 } ^ { 4 } \mathrm { f } ( \mathrm { t } ) \mathrm { dt } = 2 \left[ \alpha \mathrm { f } \left( \alpha ^ { 2 } \right) + \beta \mathrm { f } ( \beta ) ^ { 2 } \right] \right] \forall 0 < \alpha , \beta < 2$.
Q16 Vectors 3D & Lines Perpendicular/Orthogonal Projection onto a Plane View
(i) Find the equation of the plane passing through the points $( 2,1,0 ) , ( 5,0,1 )$ and $( 4,1,1 )$ (ii) If P is the point $( 2,1,6 )$ then the point Q such that PQ is perpendicular to the plane in (i) and the mid point of PQ lies on it.
Q17 Proof Qualitative Analysis of DE Solutions View
If $\mathrm { P } ( 1 ) = 0$ and $( \mathrm { dP } ( \mathrm { x } ) ) / \mathrm { dx } > \mathrm { P } ( \mathrm { x } )$ for all $\mathrm { x } > 1$ then prove that $\mathrm { P } ( \mathrm { x } ) > 0$ for all $\mathrm { x } > 1$.
Q18 Proof Direct Proof of a Stated Identity or Equality View
If In is the area of $n$ sided regular polygon inscribed in a circle of unit radius and On be the area of the polygon circumscribing the given circle, prove that $$I _ { n } = \frac { O _ { n } } { 2 } \left( 1 + \sqrt { 1 - \left( \frac { 2 l _ { n } } { n } \right) ^ { 2 } } \right)$$
Q19 Vectors 3D & Lines Vector Algebra and Triple Product Computation View
If \overrightarrow { \mathrm { u } } , \mathrm { v } ^ { \rightarrow } , \mathrm { w } ^ { \rightarrow } are three noncoplanar unit vectors and $\alpha , \beta , \gamma$ are the angles between $\mathrm { u } \rightarrow$ are and $\mathrm { v } ^ { \rightarrow } , \mathrm { v } ^ { \rightarrow } are and $\overrightarrow { \mathrm { w } } , \mathrm { w } ^ { \rightarrow } and $\overrightarrow { \mathrm { u } }$ are respectively and $\overrightarrow { \mathrm { x } } , \mathrm { y } ^ { \rightarrow } , \mathrm { z }$ are unit vectors along the bisectors of the angles $\mathrm { a } , \mathrm { b } , \mathrm { g }$ respectively. Prove that $$\left[ \begin{array} { l l l } \vec { x } \times \vec { y } & \vec { y } \times \vec { z } & \vec { z } \times \vec { x } \end{array} \right] = \frac { 1 } { 16 } \left[ \begin{array} { l l l } \vec { u } & \vec { v } & \vec { w } \end{array} \right] ^ { 2 } \sec ^ { 2 } \frac { \alpha } { 2 } \sec ^ { 2 } \frac { \beta } { 2 } \sec ^ { 2 } \frac { \gamma } { 2 }$$
Q20 Differential equations Inverse/implicit relationship between position and time View
A right circular cone with radius R and height H contains a liquid which evaporates at a rate proportional to its surface area in contact with air (proportionality constant $= \mathrm { k } > 0$ ). Find the time after which the cone is empty.