jee-advanced

Papers (53)
2025
paper1 16 paper2 16
2024
paper1 17 paper2 17
2023
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2022
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2021
paper1 23 paper2 19
2020
paper1 18 paper2 18
2019
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2018
paper1 18 paper2 18
2017
paper1 18 paper2 18
2016
paper1 18 paper2 18
2015
paper1 20 paper2 20
2014
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2013
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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
2005 screening

16 maths questions

Q7 Standard Integrals and Reverse Chain Rule Integral Equation to Determine a Function Value View
7. If $\int \sin x ^ { 1 } t ^ { 2 } f ( t ) d t = 1 - \sin x \forall x \hat { I } [ 0 , \Pi / 2 ]$ then $f ( 1 / \sqrt { } 3 )$ is:
(a) 3
(b) $\sqrt { } 3$
(c) $1 / 3$
(d) none of these
Q8 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View
8.
$$\binom { 30 } { 0 } \binom { 30 } { 10 } - \binom { 30 } { 1 } \binom { 30 } { 11 } + \ldots \ldots \binom { 30 } { 20 } \binom { 30 } { 30 } =$$
(a) $\quad { } ^ { 30 } \mathrm { C } _ { 11 }$
(b) $\quad { } ^ { 60 } \mathrm { C } _ { 10 }$
(c) $\quad { } ^ { 30 } \mathrm { C } _ { 10 }$
(d) $\quad { } ^ { 65 } \mathrm { C } _ { 55 }$
Q9 Vectors 3D & Lines Triangle Properties and Special Points View
9. A variable plane $x / a + y / b + z / c = 1$ at a unit distance from origin cuts the coordinate axes at $A , B$ and $C$. Centroid $( x , y , z )$ satisfies the equation $1 / x ^ { 2 } + 1 / y ^ { 2 } + 1 / z ^ { 2 } = K$. The value of $K$ is :
(a) 9
(b) 3
(c) $1 / 9$
(d) $1 / 3$
Q10 Geometric Sequences and Series Nature of roots given coefficient constraints View
10. Let $f ( x ) = a x ^ { 2 } + b x + c$, $a ^ { 1 } 0$ and $D = b ^ { 2 } - 4 a c$. If $a + b , a ^ { 2 } + b ^ { 2 }$ and $a ^ { 3 } + b ^ { 3 }$ are in G.P., then :
(a) $\quad \mathrm { D } ^ { 1 } 0$
(b) $\quad \mathrm { bD } ^ { 1 } 0$
(c) $\quad \mathrm { cD } ^ { 1 } 0$
(d) $\quad b c ^ { 1 } 0$
Q11 Circles Optimization in a triangle View
11. Tangent at a point of the ellipse $x ^ { 2 } / a ^ { 2 } + y ^ { 2 } / b ^ { 2 } = 1$ is drawn which cuts the coordinate axes at $A$ and $B$. The minimum area of the triangle $O A B$ is ( $O$ being the origin) :
(a) $a b$
(b) $\left( a ^ { 3 } + a b + b ^ { 3 } \right) / 3$
(c) $a ^ { 2 } + b ^ { 2 }$
(d) $\left( \left( a ^ { 2 } + b ^ { 2 } \right) \right) / 4$
Q12 Probability Definitions Finite Equally-Likely Probability Computation View
12. A fair die is rolled. The probability that the first time 1 occurs at the even throw is:
(a) $1 / 6$ ... Powered By IITians
(b) $5 / 11$
(c) $6 / 11$
(d) $5 / 36$
Q13 Differential equations Solving Separable DEs with Initial Conditions View
13. If $x d y = y ( d x + y d y ) , y ( 1 ) = 1$ and $y ( x ) > 0$. Then $y ( - 3 ) = :$
(a) 3
(b) 2
(c) 1
(d) 0
Q14 Composite & Inverse Functions Injectivity, Surjectivity, or Bijectivity Classification View
14. $f ( x ) = \left\{ \begin{array} { l } x , \quad \text { if } x \text { is rational } \\ 0 , \quad \text { if } x \text { is irrational } \end{array} \right.$ and
$$g ( x ) = \left\{ \begin{array} { l } 0 , \quad \text { if } x \text { is rational } \\ x , \quad \text { if } x \text { is irrational. } \end{array} \text { Then } \mathrm { f } - \mathrm { g } \right. \text { is: }$$
(a) one-one and into
(b) neither one-one nor onto
(c) many one and onto
(d) one-one and onto
Q15 Combinations & Selection Lattice Path Counting View
15. A rectangle with sides ( $2 n - 1$ ) and ( $2 m - 1$ ) is divided into squares of unit length. The number of rectangle which can be formed with sides of odd length is :
(a) $m ^ { 2 } n ^ { 2 }$
(b) $m n ( m + 1 ) ( n + 1 )$
(c) $4 ^ { m + n - 1 }$
(d) none of these
Q16 Complex Numbers Arithmetic Roots of Unity and Cyclotomic Expressions View
16. The minimum value of $\left| a + b w + c w ^ { 2 } \right|$, where $a , b$ and $c$ are all not equal integers and $w \left( \begin{array} { l l } 1 & 1 \end{array} \right)$ is a cube root of unity, is:
(a) $\sqrt { } 3$
(b) $1 / 3$
(c) 1
(d) 0
III askllTians ||
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  1. If $P = \left[ \begin{array} { c c } \sqrt { 3 } / 2 & 1 / 2 \\ - 1 / 2 & \sqrt { 3 } / 2 \end{array} \right] , A = \left[ \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right]$ and $\mathrm { Q } = \mathrm { PAP } ^ { \top }$, then $\mathrm { P } ^ { \top } \mathrm { Q } ^ { 2005 } \mathrm { P }$ is:
    (a) $\left[ \begin{array} { c c } 1 & 2005 \\ 0 & 1 \end{array} \right]$
    (b) $\left[ \begin{array} { c c } 1 & 2005 \\ 2005 & 1 \end{array} \right]$
    (c) $\left[ \begin{array} { c c } 1 & 0 \\ 2005 & 1 \end{array} \right] ^ { 1 }$
    (d) $\left[ \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right]$
  2. The shaded region, where $P \equiv ( - 1,0 ) , Q \equiv ( - 1 + \sqrt { } 2 , \sqrt { } 2 )$ $R \equiv ( - 1 + \sqrt { } 2 , - \sqrt { } 2 ) , S \equiv ( 1,0 )$ is represented by:
    (a) $| z + 1 | > 2 , | \arg ( z + 1 ) | < n / 4$
    (b) $| z + 1 | < 2 , | \arg ( z + 1 ) | < n / 2$
    (c) $| z - 1 | > 2 , | \arg ( z + 1 ) | > \pi / 4$
    (d) $| z - 1 | < 2 , | \arg ( z + 1 ) | > n / 2$
  3. The number of ordered pairs ( $a , \beta$ ), where $a , \beta \hat { I } ( - \Pi , \Pi )$ satisfying $\cos ( a - \beta ) = 1$ and $\cos ( a + \beta ) = 1 / e$ is :
    (a) 0
    (b) 1
    (c) 2
    (d) 4
  4. Let $f ( x ) = | x | - 1$, then points where $f ( x )$ is not differentiable is/(are):
    (a) $0 , + 1$
    (b) + 1
    (c) 0
    (d) 1
  5. The second degree polynomial $f ( x )$, satisfying $f ( 0 ) = 0 , f ( 1 ) = 1 , f ^ { \prime } ( x ) > 0$ for all $x \hat { I }$ ( 0 , 1) :
    (a) $f ( x ) = f$
    (b) $f ( x ) = a x + ( 1 - a ) x ^ { 2 } ; \forall a \hat { I } ( 0 , ¥ )$
    (c) $f ( x ) = a x + ( 1 - a ) x ^ { 2 } ; \forall a \hat { I } ( 0,2 )$
    (d) no such polynomial
  6. If $f$ is a differentiable function satisfying $f ( 1 / n ) = 0$ for all $n > 1 , n \hat { I } I$, then :
    (a) $f ( x ) = 0 , x \hat { I } ( 0,1 ]$
    (b) $f ^ { \prime } ( 0 ) = 0 = f ( 0 )$

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(c) $f ( 0 ) = 0$ but $f ^ { \prime } ( 0 )$ not necessarily zero
(d) $| f ( x ) | < 1 , x \hat { I } ( 0,1 ]$
Q23 3x3 Matrices Matrix Power Computation and Application View
23. If $A = \left[ \begin{array} { c c c } 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & - 2 & 4 \end{array} \right]$, $6 A ^ { - 1 } = A ^ { 2 } + c A + d I$, then $( c , d )$ is:
(a) $( - 6,11 )$
(b) $( - 11,6 )$
(c) $( 11,6 )$
(d) $( 6,11 )$
Q24 Sine and Cosine Rules Determine an angle or side from a trigonometric identity/equation View
24. In a $\triangle \mathrm { ABC }$, among the following which one is true?
(a) $( b + c ) \cos A / 2 = a \sin ( ( B + C ) / 2 )$
(b) $( b + c ) \cos ( ( B + C ) / 2 ) = a \sin A / 2$
(c) $( b - c ) \cos ( ( B - C ) / 2 ) = a \cos ( A / 2 )$
(d) $( b - c ) \cos A / 2 = a \cos ( ( B - C ) / 2 )$
Q25 Vectors 3D & Lines True/False or Multiple-Statement Verification View
25. If $\vec { a } , \vec { b } , \vec { c }$ are three non zero, non coplanar vectors and $\vec { b } _ { 1 } = \vec { b } - \frac { \vec { b } \cdot \vec { a } } { | \vec { a } | ^ { 2 } } \vec { a }$,
$$\begin{aligned} & \vec { b } _ { 2 } = \vec { b } + \frac { \vec { b } \cdot \vec { a } } { | \vec { a } | ^ { 2 } } \vec { a } , \text { And } \vec { c } _ { 1 } = \vec { c } - \frac { \vec { c } \cdot \vec { a } } { | \vec { a } | ^ { 2 } } \vec { a } - \frac { \vec { c } \cdot \vec { b } } { | \vec { b } | ^ { 2 } } \vec { b } , \quad \vec { c } _ { 2 } = \vec { c } - \frac { \vec { c } \cdot \vec { a } } { | \vec { a } | ^ { 2 } } \vec { a } - \frac { \vec { c } \cdot \vec { b } } { \left| \vec { b } _ { 1 } \right| ^ { 2 } } \vec { b } _ { 1 } \\ & \vec { c } _ { 3 } = \vec { c } - \frac { \vec { c } \cdot \vec { a } } { | \vec { a } | ^ { 2 } } \vec { a } - \frac { \vec { c } \cdot \vec { b } } { | \vec { b } | ^ { 2 } } \vec { b } _ { 2 } , \vec { c } _ { 4 } = \vec { a } - \frac { \vec { c } \cdot \vec { a } } { | \vec { a } | ^ { 2 } } \vec { a } \end{aligned}$$
Then which of the following is a set of mutually orthogonal vectors:
(a) $\left( \vec { a } , \vec { b } _ { 1 } , \vec { c } _ { 1 } \right)$
(b) $\left( \vec { a } , \vec { b } _ { 1 } , \vec { c } _ { 2 } \right)$
(c) $\quad \left( \vec { a } , \vec { b } _ { 2 } , \vec { c } _ { 3 } \right)$
(d) $\left( \vec { a } , \vec { b } _ { 2 } , \vec { c } _ { 4 } \right)$
Q26 Tangents, normals and gradients Chain Rule with Composition of Explicit Functions View
26. If $y = f ( x )$ and $y \cos x + x \cos y = \Pi$, then the value of $f ^ { \prime } ( 0 )$ is :
(a) $\sqcap$
(b) $- \Pi$
(c) 0
(d) $2 \Pi$
Q27 Proof Existence or properties of extrema via abstract/theoretical argument View
27. Let $f$ be twice differentiable function satisfying $f ( 1 ) = 1 , f ( 2 ) = 4 , f ( 3 ) = 9$, then :
(a) $f ^ { \prime } ( x ) = 2 , \forall x \hat { I } ( R )$
(b) $f ^ { \prime } ( x ) = 5 = f ^ { \prime \prime } ( x )$, for some $x \hat { I } ( 1,3 )$
(c) There exists at least one $x \hat { I } ( 1,3 )$ such that $f ^ { \prime } ( x ) = 2$
(d) none of these
Q28 Composite & Inverse Functions Existence or Properties of Functions and Inverses (Proof-Based) View
28. If $X$ and $Y$ are two non-empty sets where $f : X - - > Y$ is function is defined such that $\mathrm { f } ( \mathrm { c } ) = \{ \mathrm { f } ( \mathrm { x } ) : \mathrm { x }$ ÎC $\}$ for C ÍX and $f ^ { - 1 } ( D ) = \{ x : f ( x )$ Î $D \}$ for $D$ Í $y$, for any A Í X and B Í Y then :
(a) $f ^ { - 1 } ( f ( A ) ) = A$
(b) $f ^ { - 1 } ( f ( A ) ) = A$ only if $f ( X ) = Y$
(c) $f \left( f ^ { - 1 } ( B ) \right) = B$ only if B Í $f ( x )$
(d) $f \left( f ^ { - 1 } ( B ) \right) = B$