jee-advanced

Papers (53)
2025
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2024
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2023
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2022
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2020
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2019
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2018
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2017
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2016
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2015
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2014
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2013
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2012
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2011
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2010
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2009
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2008
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2007
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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
2001 mains

12 maths questions

Q3 Solving quadratics and applications Sufficient/necessary condition or logical relationship involving a quadratic View
3. Let- $1 < \mathrm { p } < 1$. Show that the equation $4 \mathrm { x } 2 - 3 \mathrm { x } - \mathrm { p } = 0$ has a unique root in the interval [1/2, 1] and identify it.
Q4 Circles Tangent Lines and Tangent Lengths View
4. Let $2 \times 2 + \underset { 2 } { 2 } - 3 \times y = 0$ be the equation of a pair of tangents drawn from the origin O to a circle of radius 3 with centre in the first quadrant. If A is one of the points of contact, find the length of OA .
Q5 Integration by Substitution Substitution to Evaluate a Definite Integral (Numerical Answer) View
5. Evaluate $\int \sin - 1 ( ( 2 \mathrm { x } + 2 ) / \sqrt { } ( 4 \mathrm { x } 2 + 8 \mathrm { x } + 13 ) ) d \mathrm { x }$.
Q6 Differential equations Qualitative Analysis of DE Solutions View
6. Let $\mathrm { f } ( \mathrm { x } ) , \mathrm { x } > 0$, be a nornegative continuous function, and let $\mathrm { F } ( \mathrm { x } ) = \int 0 \mathrm { x } \mathrm { f } ( \mathrm { t } ) \mathrm { dt } , \mathrm { x } > 0$. If for some $\mathrm { c } > 0 , \mathrm { f } ( \mathrm { x } ) < \mathrm { cF } ( \mathrm { x } )$ for all $\mathrm { x } > 0$, then show that $\mathrm { f } ( \mathrm { x } ) = 0$ for all $\mathrm { x } > 0$.
Q7 Areas by integration Integral Inequalities and Limit of Integral Sequences View
7. Let $\mathrm { b } ^ { 1 } 0$ and for $\mathrm { j } = 0,1,2$, $\_\_\_\_$ n , let Sj be the area of the region bounded by the y -axis and the curve xmy $= \sin$ by, $\quad \pi / \mathrm { b } \leq \mathrm { y } \leq ( ( \mathrm { j } + 1 ) \pi ) / \mathrm { b }$.
Show that S0, S1, S2, $\_\_\_\_$ Sn are in geometric progression. Also, find their sum for $\mathrm { a } = - 1$ and $\mathrm { b } = \pi$.
Q8 Differentiation from First Principles View
8. Let $\alpha$ Î R. Prove that a function $\mathrm { f } : \mathrm { R } - - > \mathrm { R }$ is differentiable at $\alpha$ if and only if there is a function $\mathrm { g } : \mathrm { R }$ --> $R$ which is continuous at $\alpha$ and satisfies $f ( x ) - f ( \alpha ) = f ( x ) ( x - \alpha )$ for all $x \hat { I } R$.
Q9 Circles Circle-Related Locus Problems View
9. Let Cl and C 2 be two circles with C 2 lying inside Cl . A circle C lying inside C1touches C1 internally and C2 externally. Identify the locus of the centre of C.
Q10 Vectors Introduction & 2D Section Ratios and Intersection via Vectors View
10. Show, by vector methods, that the angular bisectors of a triangle are concurrent and find an expression for the position vector of the point of concurrency in terms of the position vectors of the vertices.
Q11 Solving quadratics and applications Finding roots or coefficients of a quadratic using Vieta's relations View
11. (a) Let $\mathrm { a } , \mathrm { b } , \mathrm { c }$ be real numbers with $\mathrm { a } ^ { 1 } 0$ and let $\mathrm { a } , \mathrm { b }$ be the roots of the equation $\mathrm { ax } 2 + \mathrm { bx } + \mathrm { c } = 0$. Express the roots of $\mathrm { a } 2 \times 2 + \mathrm { abcx } + \mathrm { c } 3 = 0$ in terms of $\mathrm { a } , \mathrm { b }$.
(b) Let $\mathrm { a } , \mathrm { b } , \mathrm { c }$ be real numbers with $\mathrm { a } 2 + \mathrm { b } 2 + \mathrm { c } 2 = 1$. Show that the equation
$$\left| \begin{array} { c c c } a x - b y - c & b x + a y & c x + a \\ b x + a y & - a x + b y - c & c y + b \\ c x + a & c y + b & - a x - b y + c \end{array} \right| = 0 \text { represents a straight line. }$$
  1. (a) Let P be a point on the ellipse $\mathrm { x } 2 / \mathrm { a } 2 + \mathrm { y } 2 / \mathrm { b } 2 = 1,0 < \mathrm { b } < \mathrm { a }$. Let the line parallel to-qxis passing through P meet the circle $\mathrm { x } 2 + \mathrm { y } 2 = \mathrm { a } 2$ at the point Q such that P and Q are on the same side of - xxis . For two positive real numbers r and s , find the locus of the point R on PQ such that $\mathrm { PR } : \mathrm { RQ } = \mathrm { r } : \mathrm { s }$ as P varies over the ellipse.
    (b) If D is the area of a triangle with side lengths $\mathrm { a } , \mathrm { b } , \mathrm { c }$ then show that $\mathrm { D } < 1 / 4 \sqrt { } ( ( \mathrm { a } + \mathrm { b } + \mathrm { c } ) \mathrm { abc } )$.

Also show that the equality occurs in the above inequality if and only if $\mathrm { a } = \mathrm { b } = \mathrm { c }$.
Q13 Differential equations Applied Modeling with Differential Equations View
13. A hemispherical tank of radius 2 metres is initially full of water and has an outlet of 12 cm 2 cross sectional are at the bottom. The outlet is opened at some instant.
The flow through the outlet is according to the law $\mathrm { v } ( \mathrm { t } ) = 0.6 \sqrt { } ( 2 \mathrm { gh } ( \mathrm { t } ) )$, where $\mathrm { v } ( \mathrm { t } )$ and $\mathrm { h } ( \mathrm { t } )$ are respectively the velocity of the flow through the outlet and the height of water level above the outlet at time $t$, and $g$ is the acceleration due to gravity. Find the time it takes to empty the tank. (Hint : Form a differential equation by relating the decrease of water level to the outflow).
Q14 Probability Definitions Conditional Probability and Bayes' Theorem View
14. (a) An urn contains $m$ white and $n$ black balls. A ball is drawn at random and is put back into the urn along with k additional balls of the same colour as that of the ball drawn. A ball is again drawn at random. What is the probability that the ball drawn now is white?
(b) An unbiased die, with faces numbered $1,2,3,4,5,6$ is thrown n times and the list of n numbers showing up is noted. What is the probability that, among the numbers $1,2,3,4,5,6$ only three numbers appear in this list?
Q15 Vectors Introduction & 2D Dot Product Computation View
15. (a) Find 3 -dimensional vectors $\overrightarrow { v _ { 1 } } , \overrightarrow { v _ { 2 } } , \overrightarrow { v _ { 3 } }$ satisfying
$$\overrightarrow { v _ { 1 } } \cdot \overrightarrow { v _ { 1 } } = 4 , \overrightarrow { v _ { 1 } } \cdot \overrightarrow { v _ { 2 } } = - 2 , \overrightarrow { v _ { 1 } } \cdot \overrightarrow { v _ { 3 } } = 6 , \overrightarrow { v _ { 2 } } \cdot \overrightarrow { v _ { 2 } } = 2 , \overrightarrow { v _ { 2 } } \cdot \overrightarrow { v _ { 3 } } = - 5 , \overrightarrow { v _ { 3 } } \cdot \overrightarrow { v _ { 3 } } = 29 .$$
(b) Let $\vec { A } ( t ) = f _ { 1 } ( t ) \hat { i } + f _ { 2 } ( t ) \hat { j }$ and $B ( t ) = g _ { 1 } ( t ) \hat { i } + g _ { 2 } ( t ) \hat { j } , t \in [ 0,1 ]$, where $f _ { 1 \text {, } } \mathrm { f } _ { 2 } , \mathrm {~g} _ { 1 } , \mathrm {~g} _ { 2 }$ are continuous functions. If $\vec { A } ( \mathrm { t } )$ and $\vec { B } ( \mathrm { t } )$ are non-zero vectors for all t and $\vec { A } ( 0 ) = 2 \hat { \imath } + 3 \hat { \jmath } , \vec { A } ( 1 ) = 6 \hat { \imath } + 2 \hat { \jmath } , \vec { B } ( 0 ) = 3 \hat { \imath } + 2 \hat { \jmath }$ and $\vec { B } ( 1 ) = 2 \hat { \imath } + 6 \hat { j }$. The show that $\vec { A } ( t )$ and $\vec { B } ( t )$ are parallel for some $t$.