jee-advanced

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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
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2003
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2002
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2001
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2000
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1999
jee-advanced_1999.pdf 33
1998
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1999 jee-advanced_1999.pdf

33 maths questions

Q3 Proof by induction Prove a Binomial Identity or Inequality View
3. Let n be any positive integer. Prove that:
$$\sum _ { k = 0 } ^ { m } \frac { \binom { 2 n - k } { k } } { \binom { 2 n - k } { n } } \cdot \frac { ( 2 n - 4 k + 1 ) } { ( 2 n - 2 k + 1 ) } 2 ^ { n - 2 k } = \frac { \binom { n } { m } } { \binom { 2 n - 2 m } { n - m } } 2 ^ { n - 2 m }$$
for each nonnegative integer $\mathrm { m } \leq \mathrm { n }$. (Here $\left. \binom { p } { q } = \square ^ { p } C _ { q } \right)$
Q4 Trig Proofs Triangle Trigonometric Relation View
4. Let $A B C$ be a triangle having $O$ and $I$ as its circumcentre and incentre respectively. If $R$ and $r$ are the circumradius and the inradius, respectively, then prove that (IO) $2 = R 2 - 2 R r$. Further show that the triangle BIO is a right-angled triangle if and only if $b$ is the arithmetic mean of a and c.
Q5 Circles Tangent Lines and Tangent Lengths View
5. Let $\mathrm { T } 1 , \mathrm {~T} 2$ be two tangents drawn from $( - 2,0 )$ onto the circle $\mathrm { C } : \mathrm { x } 2 + \mathrm { y } 2 = 1$. Determine the circles touching C and having T1, T2 as their pair of tangents. Further, find the equations of all possible common tangents to these circles, when taken two at a time.
Q6 Circles Circle-Related Locus Problems View
6. Consider the family of circles $x 2 + y 2 = r 2,2 < r < 5$. If in the first quadrant, the common tangent to a circle of this family and the ellipse $4 \times 2 + 25 y 2 = 100$ meets the coordinate axes at A and B , then find the equation of the locus of the mid point of AB .
Q7 Integration by Parts Substitution Combined with Symmetry or Companion Integral View
7. Integrate
(A) $\int \frac { x ^ { 3 } + 3 x + 2 } { \left( x ^ { 2 } + 1 \right) ^ { 2 } ( x + 1 ) } \mathrm { dx }$.
(B) $\quad \int _ { 0 } ^ { \pi } \frac { e ^ { \cos x } } { e ^ { \cos x } + e ^ { - \cos x } } \mathrm { dx }$. 8. Let $\mathrm { f } ( \mathrm { x } )$ be a continuous function given by
$$f ( x ) = \left\{ \begin{array} { c c } 2 x , & | x | \leq 1 \\ x ^ { 2 } + a x + b , & | x | > 1 \end{array} \right.$$
Find the area of the region in the third quadrant bounded by the curves $x = - 2 y 2$ and $y = \mathrm { f } ( \mathrm { x } )$ lying on the left on the line $8 \mathrm { x } + 1 = 0$. 9. Find the co-ordinates of all the P on the ellipse $\mathrm { x } 2 / \mathrm { a } 2 + \mathrm { y } 2 / \mathrm { b } 2 = 1$, for which the area of the triangle PON is maximum, where O denotes the origin and N , the foot of the perpendicular from O to the tangent at P . 10. A curve passing through the point $( 1,1 )$ has the property that the perpendicular distance of the origin from the normal at any point P of the curve is equal to the distance of P from the x -axis. Determine the equation of the curve. 11. Eight players $\mathrm { P } 1 , \mathrm { P } 2 , \ldots \ldots . \mathrm { P } 8$ play a knock-out tournament. It is known that whenever the players Pi and Pj play, the play Pi will win if $\mathrm { i } < \mathrm { j }$. Assuming that the players are paired at random in each round, what is the probability that the player P4 reaches the final? 12. Let $\vec { u }$ and $\vec { v }$ be unit vectors. If $\vec { w }$ is a vector such that $\vec { w } + ( \vec { w } \times \vec { u } ) = \vec { v }$, then prove that $| ( \vec { u } \times \vec { v } ) \cdot \vec { w } | \leq \frac { 1 } { 2 }$ and that the equality holds if and only if $\vec { u }$ is perpendicular to $\vec { v }$.
Q8 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View
8. If for a real number $y , [ y ]$ is the greatest integer less than or equal to $y$, then the value of the integral $\int \sqcap / 23 \sqcap / 2 [ 2 \sin x . d x ]$ is:
(A) $- \pi$
(B) 0
(C) $- \pi / 2$
(D) $\pi / 2$
Q9 Arithmetic Sequences and Series Arithmetic-Geometric Hybrid Problem View
9. Let $\mathrm { a } 1 , \mathrm { a } 2 , \ldots \ldots , \mathrm { a } 10$ be in A.P. and h1, h2, ……, h10 be in H.P. If a1 $= \mathrm { h } 1 = 2$ and a10 = $\mathrm { h } 10 = 3$, then a4 h7 is :
(A) 2
(B) 3
(C) 5
(D) 6
Q10 Vectors Introduction & 2D Magnitude of Vector Expression View
10. Let $\vec { a } = 2 \hat { \imath } + \hat { \jmath } - 2 \hat { k }$ and $\hat { b } = \hat { \imath } + \hat { \jmath }$. If $\hat { c }$ is a vector such that $\vec { a } \cdot \vec { c } = | \vec { c } - \vec { a } | = 2 \sqrt { 2 }$ and the angle between $( \vec { a } \times \vec { b } )$ and $\vec { c }$ is $30 ^ { \circ }$, then $| ( \vec { a } \times \vec { b } ) \times \vec { c } | =$
(A) $\frac { 3 } { 2 }$
(B) $\frac { 3 } { 2 }$
(C) 2
(D) 3
Q11 Standard trigonometric equations Inverse trigonometric equation View
11. The number of real solutions of $\tan - 1 \sqrt { } ( x ( x + 1 ) ) + \sin - 1 \sqrt { } ( x 2 + x + 1 ) = \pi / 2$ is:
(A) zero
(B) one
(C) two
(D) infinite
Q12 Conic sections Tangent and Normal Line Problems View
12. Let $\mathrm { P } ( \mathrm { a } \sec \mathrm { q } , \mathrm { b } \tan \mathrm { q } )$ and $\mathrm { Q } ( \mathrm { a } \sec \mathrm { q } , \mathrm { b } \tan \mathrm { q } )$ where $\mathrm { q } + \mathrm { q } = \pi / 2$, be two points on the hyperbola $\times 2 / \mathrm { a } 2 - \mathrm { y } 2 / \mathrm { b } 2 =$ 1. If ( $\mathrm { h } , \mathrm { k }$ ) is the point of intersection of the normals at P and Q , then K is equal to :
(A) $( a 2 + b 2 ) / a$
(B) $- ( ( a 2 + b 2 ) / a )$
(C) $( a 2 + b 2 ) / b$
(D) $- ( ( a 2 + b 2 ) / b )$
Q13 Straight Lines & Coordinate Geometry Slope and Angle Between Lines View
13. Let $P Q R$ be a right angled isosceles triangle, right angled at $P ( 2,1 )$. If the equation of the line $Q R$ is $2 x + y = 3$, then the equation representing the pair of lines $P Q$ and $P R$ is :
(A) $3 x 2 - 3 y 2 + 8 x y + 20 x + 10 y + 25 = 0$
(B) $3 x 2 - 3 y 2 + 8 x y - 20 x - 10 y + 25 = 0$
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(C) $3 x 2 - 3 y 2 + 8 x y + 10 x + 15 y + 20 = 0$
(D) $3 x 2 - 3 y 2 - 8 x y - 10 x - 15 y - 20 = 0$
Q14 Factor & Remainder Theorem Polynomial Construction from Root/Value Conditions View
14. If $\mathrm { f } ( \mathrm { x } ) \left| \begin{array} { c c c } 1 & x & x + 1 \\ 2 x & x ( x - 1 ) & ( x + 1 ) x \\ 3 x ( x - 1 ) & x ( x - 1 ) ( x - 2 ) & ( x + 1 ) x ( x - 1 ) \end{array} \right|$ then $\mathrm { f } ( 100 )$ is equal to :
(A) 0
(B) 1
(C) 100
(D) $\quad - 100$
Q15 Composite & Inverse Functions Continuity and Discontinuity Analysis of Piecewise Functions View
15. The function $f ( x ) = [ x ] 2 - [ x 2 ]$ (where [y] is the greatest integer less than or equal to $y$ ), is discontinuous at :
(A) all integers
(B) all integers except 0 and 1
(C) all integers except 0
(D) all integers except 1
Q16 Circles Chord Length and Chord Properties View
16. If two distinct chords, drawn from the point ( $p , q$ ) on the circle $x 2 + y 2 = p x + q y$ (where $p q { } ^ { 1 } 0$ ) are bisected by the $x$-axis, then :
(A) $\mathrm { p } 2 = \mathrm { q } 2$
(B) $p 2 = 8 q 2$
(C) $p 2 < 8 q 2$
(D) $p 2 > 8 q 2$
Q17 Modulus function Composite or piecewise function extremum analysis View
17. The function $f ( x ) = ( x 2 - 1 ) | x 2 - 3 x + 2 |$ is NOT differentiable at:
(A) - 1
(B) 0
(C) 1
(D) 2
Q18 Discriminant and conditions for roots Optimization or extremal value of an expression via completing the square View
18. If the roots of the equation $x 2 - 2 a x + a 2 + a - 3 = 0$ are real and less than 3,then:
(A) $a < 2$
(B) $2 < a < 3$
(C) $3 < a < 4$
(D) a $> 4$
Q19 Differential equations Verification that a Function Satisfies a DE View
19. A solution of the differential equation $( d x / d y ) 2 - x ( d x / d y ) + y = 0$ is :
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(A) $y = 2$
(B) $y = 2 x$
(B) $y = 2 x - 4$
(D) $y = 2 \times 2 - 4$
Q20 Curve Sketching Limit Evaluation Involving Composition or Substitution View
20. $\lim x \rightarrow 0 ( x \tan 2 x - 2 x \tan x ) / ( 1 - \cos 2 x ) 2$ is:
(A) $y = 2$
(B) $y = 2 x$
(C) $y = 2 x - 4$
(D) $y = 2 \times 2 - 4$
Q21 Vectors 3D & Lines Perpendicularity or Parallel Condition View
21. Let $\vec { a } = \overrightarrow { 2 } \hat { \jmath } + \hat { k } , \vec { b } = \hat { \imath } + \overrightarrow { 2 } \hat { \jmath } - \hat { k }$ and a unit vector $\vec { c }$ be coplanar. If $\vec { c }$ is perpendicular to $\vec { a }$, then $\vec { c } =$
(A) $\frac { 1 } { \sqrt { 2 } } ( - \hat { \jmath } + \hat { k } )$
(B) $\frac { 1 } { \sqrt { 3 } } ( - \hat { \imath } - \hat { \jmath } - \hat { k } )$
(C) $\frac { 1 } { \sqrt { 5 } } ( \hat { \imath } - 2 \hat { \jmath } )$
(D) $\frac { 1 } { \sqrt { 3 } } ( \hat { \imath } - \hat { \jmath } - \hat { k } )$
Q22 Binomial Theorem (positive integer n) Determine Parameters from Conditions on Coefficients or Terms View
22. If in the expansion of $( 1 + x ) n$, the coefficients of $x$ and $x 2$ are 3 and - 6 respectively, then m is :
(A) 6
(B) 9
(C) 12
(D) 24
Q23 Standard Integrals and Reverse Chain Rule Definite Integral Evaluation (Computational) View
23. $\int \sqcap / 33 \pi / 4 \mathrm { dx } / ( 1 + \cos \mathrm { x } )$ is equal to :
(A) 2
(B) - 2
(C) $1 / 2$
(D) $- 1 / 2$
Q24 Conic sections Tangent and Normal Line Problems View
24. If $x = 9$ is the chord of contact of the hyperbola $x 2 - y 2 = 0$, then the equation of the corresponding pair of tangents is :
(A) $9 x 2 - 8 y 2 + 18 x - 9 = 0$
(B) $9 x 2 - 8 y 2 - 18 x + 9 = 0$
(C) $9 x 2 - 8 y 2 - 18 x - 9 = 0$
(D) $9 x 2 - 8 y 2 + 18 x + 9 = 0$
Q25 3 marks Probability Definitions Probability Involving Algebraic or Number-Theoretic Conditions View
25. If the integers $m$ and $n$ are chosen at random between 1 and 100 , then the probability that a number of the form $7 m + 7 n$ is divisible by 5 equals :
(A) $1 / 4$
(B) $1 / 7$
(C) $1 / 8$
(D) $1 / 49$
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DIRECTIONS : Question numbers $26 - 35$ carry 3 marks each and may have more than one correct answers. All correct answers must be marked to get any credit in these questions.
Q26 Circles Chord Length and Chord Properties View
26. Let L1 be a straight line passing through the origin and L2 be the straight line $x + y =$ 1. If the intercepts made by the circle $x 2 + y 2 - x + 3 y = 0$ on L1 and L2 are equal, then which of the following equations can represent L1?
(A) $x + y = 0$
(B) $x - y = 0$
(B) $x + 7 y = 0$
(D) $x - 7 y = 0$
Q27 Vectors Introduction & 2D Magnitude of Vector Expression View
27. Let $\vec { a }$ and $\vec { b }$ be two non-collinear unit vectors. If $\vec { u } = \vec { a } - ( \vec { a } , \vec { b } ) \vec { b }$ and $\vec { v } = \vec { a } \times \vec { b }$, then $| \vec { v } |$ is:
(A) $| \vec { u } |$
(B) $\quad | \vec { u } | + | \vec { u } \cdot \vec { a } |$
(C) $| \vec { u } | + | \vec { u } , \vec { b } |$
(D) $| \vec { u } | + \vec { u } \cdot ( \vec { a } + \vec { b } )$
Q28 Sequences and series, recurrence and convergence Compute Partial Sum of an Arithmetic Sequence View
28. For a positive integer n , let $\mathrm { a } ( \mathrm { n } ) = 1 + 1 / 2 + 1 / 3 + 1 / 4 + \ldots \ldots + 1 / ( ( 2 \mathrm { n } ) - 1 )$. Then :
(A) a (100) £ 100
(B) a $( 100 ) > ( 100 )$
(C) a (200) $\pounds 100$
(D) a $( 200 ) > 100$
Q29 Stationary points and optimisation Composite or piecewise function extremum analysis View
29. The function $f ( x ) = \int - 1 x t$ (et-1)(t-1)(t-2)3(t-3)5dt has a local minimum at $x =$
(A) 0
(B) 1
(C) 2
(D) 3
Q30 Circles Tangent Lines and Tangent Lengths View
30. On the ellipse $4 x 2 + 9 y 2 = 1$, the points at which the tangents are parallel to the line $8 \mathrm { x } = 9 \mathrm { y }$ are :
(A) $( 2 / 5,1 / 5 )$
B) $( - 2 / 5,1 / 5 )$
(C) $( - 2 / 5 , - 1 / 5 )$
(D) $( 2 / 5 , - 1 / 5 )$
Q31 Modelling and Hypothesis Testing Probability Using Set/Event Algebra View
31. The probabilities that a student passes in Mathematics, Physics and Chemistry are $\mathrm { m } , \mathrm { p }$ and c , respectively. Of these subjects, the student has a $75 \%$ chance of passing in atleast one, a $50 \%$ chance of passing in atleast two, and a $40 \%$ chance of passing in exactly two. Which of the following relations are true?
(A) $\mathrm { p } + \mathrm { m } + \mathrm { c } = 19 / 20$
(B) $p + m + c = 27 / 20$
(C) $\mathrm { pmc } = 1 / 10$
(D) $\mathrm { pmc } = 1 / 4$
Q32 Differential equations Higher-Order and Special DEs (Proof/Theory) View
32. The differential equation representing the family of curves $y 2 = 2 c ( x + \sqrt { } c )$, where $c$ is a positive parameter, is of :
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(A) order 1
(B) order 2
(C) degree 3
(D) 10
Q33 Geometric Sequences and Series Fractal/Iterative Geometric Construction (Area, Length, or Perimeter Series) View
33. Let $S 1 , S 2 \ldots$ be squares such that for each $n ^ { 3 } 1$, the length of a side of Snequals the length of a diagonal of $\mathrm { Sn } + 1$. If the length of a side of S 1 is 10 cm , then for which of the following values of $n$ is the area of $S n$ less than $1 \mathrm { sq } . \mathrm { cm }$ ?
(A) 7
(B) 8
(C) 9
(D) 10
Q34 Areas by integration View
34. For which of the following values of $m$ is the area of the region bounded by the curve $y = x - x 2$ and he line $y = m x$ equals 9/2?
(A) - 4
(B) - 2
(C) 2
(D) 4
Q35 Trig Proofs Trigonometric Identity Simplification View
35. For a positive integer n ., let $\mathrm { f } \_ \mathrm { n } ( \theta ) = ( \tan \theta / 2 ) ( 1 + \sec \theta ) ( 1 + \sec 2 \theta ) ( 1 + \sec 4 \theta )$... $( 1 + \sec 2 \mathrm { n } \theta )$. Then
(A) $\mathrm { f } 2 ( \Pi / 16 ) = 1$;
(B) $f 3 ( \pi / 32 ) = 1$
(C) $\mathrm { f } 4 ( \pi / 16 ) = 1$
(D) f5 $( \sqcap / 128 ) = 1$
SECTION II
Instructions
There 12 questions in the section. Attempt ALL questions. At the end of the anwers to a question, draw a horizontal line and start answer to the next question. The corresponding question number must be written in the left margin. Answer all parts of a question at one place only. The use of Arabic numerals ( $0,1,2 , \ldots \ldots . .9$ ) only is allowed in answering the questions irrespective of the language in which you answer.
  1. For complex numbers z and q , prove that $| \mathrm { z } | 2 \mathrm { w } - | \mathrm { w } | 2 \mathrm { z } = \mathrm { z } - \mathrm { w }$ if and only if $\mathrm { z } = \mathrm { w }$ or z $\mathrm { w } - = 1$.
  2. Let $a , b , c d$ be real numbers in G.P. If $u , v , w$ satisfy the system of equations

$$\begin{aligned} & u + 2 v + 3 w = 6 \\ & 4 u + 5 v + 6 w = 12 \\ & 6 u + 9 v = 4 \end{aligned}$$
Then slow that the roots of the equation :
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... Powered By IITians $( 1 / u + 1 / v + 1 / w ) \times 2 + [ ( b - c ) 2 + ( c - a ) 2 + ( d - b ) 2 ] x + u + v + w = 0$ and $20 \times 2 + 10 ( a - d ) 2 x - 9 = 0$ are reciprocals of each other.