2025
JEE Advanced 2025 Paper 2 OnlineJEE Advanced 2025 Paper 1 Online
2024
JEE Advanced 2024 Paper 2 OnlineJEE Advanced 2024 Paper 1 Online
2023
JEE Advanced 2023 Paper 2 OnlineJEE Advanced 2023 Paper 1 Online
2022
JEE Advanced 2022 Paper 2 OnlineJEE Advanced 2022 Paper 1 Online
2021
JEE Advanced 2021 Paper 2 OnlineJEE Advanced 2021 Paper 1 Online
2020
JEE Advanced 2020 Paper 2 OfflineJEE Advanced 2020 Paper 1 Offline
2019
JEE Advanced 2019 Paper 2 OfflineJEE Advanced 2019 Paper 1 Offline
2018
JEE Advanced 2018 Paper 2 OfflineJEE Advanced 2018 Paper 1 Offline
2017
JEE Advanced 2017 Paper 2 OfflineJEE Advanced 2017 Paper 1 Offline
2016
JEE Advanced 2016 Paper 2 OfflineJEE Advanced 2016 Paper 1 Offline
2015
JEE Advanced 2015 Paper 2 OfflineJEE Advanced 2015 Paper 1 Offline
2014
JEE Advanced 2014 Paper 2 OfflineJEE Advanced 2014 Paper 1 Offline
2013
JEE Advanced 2013 Paper 2 OfflineJEE Advanced 2013 Paper 1 Offline
2012
IIT-JEE 2012 Paper 2 OfflineIIT-JEE 2012 Paper 1 Offline
2011
IIT-JEE 2011 Paper 1 OfflineIIT-JEE 2011 Paper 2 Offline
2010
IIT-JEE 2010 Paper 2 OfflineIIT-JEE 2010 Paper 1 Offline
2009
IIT-JEE 2009 Paper 2 OfflineIIT-JEE 2009 Paper 1 Offline
2008
IIT-JEE 2008 Paper 2 OfflineIIT-JEE 2008 Paper 1 Offline
2007
IIT-JEE 2007IIT-JEE 2007 Paper 2 Offline
2006
IIT-JEE 2006IIT-JEE 2006 Screening
2005
IIT-JEE 2005 ScreeningIIT-JEE 2005
2004
IIT-JEE 2004IIT-JEE 2004 Screening
2003
IIT-JEE 2003IIT-JEE 2003 Screening
2002
IIT-JEE 2002IIT-JEE 2002 Screening
2001
IIT-JEE 2001IIT-JEE 2001 Screening
2000
IIT-JEE 2000 ScreeningIIT-JEE 2000
1999
IIT-JEE 1999 ScreeningIIT-JEE 1999
1998
IIT-JEE 1998IIT-JEE 1998 Screening
1997
IIT-JEE 1997
1996
IIT-JEE 1996
1995
IIT-JEE 1995 ScreeningIIT-JEE 1995
1994
IIT-JEE 1994
1993
IIT-JEE 1993
1992
IIT-JEE 1992
1991
IIT-JEE 1991
1990
IIT-JEE 1990
1989
IIT-JEE 1989
1988
IIT-JEE 1988
1987
IIT-JEE 1987
1986
IIT-JEE 1986
1985
IIT-JEE 1985
1984
IIT-JEE 1984
1983
IIT-JEE 1983
1982
IIT-JEE 1982
1981
IIT-JEE 1981
1980
IIT-JEE 1980
1979
IIT-JEE 1979
1978
IIT-JEE 1978
IIT-JEE 2000
Paper was held on Tue, Apr 11, 2000 9:00 AM
Practice Questions
Mathematics
1
If $$\alpha ,\,\beta $$ are the roots of $$a{x^2} + bx + c = 0$$, $$\,\left( {a \ne 0} \right)$$ and $$\alpha + \delta ,\,\,\beta + \delta $$ are the roots of $$A{x^2} + Bx + c = 0,$$ $$\left( {A \ne 0\,} \right)\,$$ for some contant $$\delta $$, then prove that $${{{b^2} - 4ac} \over {{a^2}}} = {{{B^2} - 4Ac} \over {{A^2}}}$$.
2
A coin has probability $$p$$ of showing head when tossed. It is tossed $$n$$ times. Let $${p_n}$$ denote the probability that no two (or more) consecutive heads occur. Prove that $${p_1} = 1,{p_2} = 1 - {p^2}$$ and $${p_n} = \left( {1 - p} \right).\,\,{p_{n - 1}} + p\left( {1 - p} \right){p_{n - 2}}$$ for all $$n \ge 3.$$
3
For $$x>0,$$ let $$f\left( x \right) = \int\limits_e^x {{{\ln t} \over {1 + t}}dt.} $$ Find the function
$$f\left( x \right) + f\left( {{1 \over x}} \right)$$ and show that $$f\left( e \right) + f\left( {{1 \over e}} \right) = {1 \over 2}.$$
Here, $$\ln t = {\log _e}t$$.
4
Suppose $$p\left( x \right) = {a_0} + {a_1}x + {a_2}{x^2} + .......... + {a_n}{x^n}.$$ If
$$\left| {p\left( x \right)} \right| \le \left| {{e^{x - 1}} - 1} \right|$$ for all $$x \ge 0$$, prove that
$$\left| {{a_1} + 2{a_2} + ........ + n{a_n}} \right| \le 1$$.
5
Let $$ABC$$ be a triangle with incentre $$I$$ and inradius $$r$$. Let $$D,E,F$$ be the feet of the perpendiculars from $$I$$ to the sides $$BC$$, $$CA$$ and $$AB$$ respectively. If $${r_1},{r_2}$$ and $${r_3}$$ are the radii of circles inscribed in the quadrilaterals $$AFIE$$, $$BDIF$$ and $$CEID$$ respectively, prove that $$${{{r_1}} \over {r - {r_1}}} + {{{r_2}} \over {r - {r_2}}} + {{{r_3}} \over {r - {r_3}}} = {{{r_1}{r_2}{r_3}} \over {\left( {e - {r_1}} \right)\left( {r - {r_2}} \right)\left( {r - {r_3}} \right)}}$$$
6
If $${x^2} + {y^2} = 1$$ then
7
Let $${C_1}$$ and $${C_2}$$ be respectively, the parabolas $${x^2} = y - 1$$ and $${y^2} = x - 1$$. Let $$P$$ be any point on $${C_1}$$ and $$Q$$ be any point on $${C_2}$$. Let $${P_1}$$ and $${Q_1}$$ be the reflections of $$P$$ and $$Q$$, respectively, with respect to the line $$y=x$$. Prove that $${P_1}$$ lies on $${C_2}$$, $${Q_1}$$ lies on $${C_1}$$ and $$PQ \ge $$ min $$\left\{ {P{P_1},Q{Q_1}} \right\}$$. Hence or otherwise determine points $${P_0}$$ and $${Q_0}$$ on the parabolas $${C_1}$$ and $${C_2}$$ respectively such that $${P_0}{Q_0} \le PQ$$ for all pairs of points $$(P,Q)$$ with $$P$$ on $${C_1}$$ and $$Q$$ on $${C_2}$$.
8
Let $$ABC$$ be an equilateral triangle inscribed in the circle $${x^2} + {y^2} = {a^2}$$. Suppose perpendiculars from $$A, B, C$$ to the major axis of the ellipse $$x.{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$, $$(a>b)$$ meets the ellipse respectively, at $$P, Q, R$$. so that $$P, Q, R$$ lie on the same side of the major axis as $$A, B, C$$ respectively. Prove that the normals to the ellipse drawn at the points $$P, Q$$ and $$R$$ are concurrent.
9
Let $$ABC$$ and $$PQR$$ be any two triangles in the same plane. Assume that the prependiculars from the points $$A, B, C$$ to the sides $$QR, RP, PQ$$ respectively are concurrent. Using vector methods or otherwise, prove that the prependiculars from $$P, Q, R $$ to $$BC,$$ $$CA$$, $$AB$$ respectively are also concurrent.
10
For points $$P\,\,\, = \left( {{x_1},\,{y_1}} \right)$$ and $$Q\,\,\, = \left( {{x_2},\,{y_2}} \right)$$ of the co-ordinate plane, a new distance $$d\left( {P,\,Q} \right)$$ is defined by $$d\left( {P,\,Q} \right)$$$$ = \left( {{x_2},\,{y_2}} \right)\left| {{x_1} - {x_2}} \right| + \left| {{y_1} - {y_2}} \right|.$$ Let $$O = (0, 0)$$ and $$A = (3, 2)$$. Prove that the set of points in the first quadrant which are equidistant (with respect to the new distance) from $$O$$ and $$A$$ consists of the union of a line segment of finite length and an infinite ray. Sketch this set in a labelled diagram.
11
The fourth power of the common difference of an arithmatic progression with integer entries is added to the product of any four consecutive terms of it. Prove that the resulting sum is the square of an integer.
12
A coin probability $$p$$ of showing head when tossed. It is tossed $$n$$ times. Let $${p_n}$$ denote the probability that no two (or more) consecutive heads occur. Prove that $${p_1} = 1,\,\,{p_2} = 1 - {p^2}$$ and $${p_n} = \left( {1 - p} \right).\,\,{p_{n - 1}} + p\left( {1 - p} \right){p_{n - 2}}$$ for all $$n \ge 3.$$

Prove by induction on, that $${p_n} = A{\alpha ^n} + B{\beta ^n}$$ for all $$n \ge 1,$$ where $$\alpha $$ and $$\beta $$ are the roots of quadratic equation $${x^2} - \left( {1 - p} \right)x - p\left( {1 - p} \right) = 0$$ and $$A = {{{p^2} + \beta - 1} \over {\alpha \beta - {\alpha ^2}}},B = {{{p^2} + \alpha - 1} \over {\alpha \beta - {\beta ^2}}}.$$

13
Let $$a,\,b,\,c$$ be possitive real numbers such that $${b^2} - 4ac > 0$$ and let $${\alpha _1} = c.$$ Prove by induction that $${\alpha _{n + 1}} = {{a\alpha _n^2} \over {\left( {{b^2} - 2a\left( {{\alpha _1} + {\alpha _2} + ... + {\alpha _n}} \right)} \right)}}$$ is well-defined and
$${\alpha _{n + 1}} < {{{\alpha _n}} \over 2}$$ for all $$n = 1,2,....$$ (Here, 'well-defined' means that the denominator in the expression for $${\alpha _{n + 1}}$$ is not zero.)
14
For every possitive integer $$n$$, prove that
$$\sqrt {\left( {4n + 1} \right)} < \sqrt n + \sqrt {n + 1} < \sqrt {4n + 2}.$$
Hence or otherwise, prove that $$\left[ {\sqrt n + \sqrt {\left( {n + 1} \right)} } \right] = \left[ {\sqrt {4n + 1} \,\,} \right],$$
where $$\left[ x \right]$$ denotes the gratest integer not exceeding $$x$$.
15
For any positive integer $$m$$, $$n$$ (with $$n \ge m$$), let $$\left( {\matrix{ n \cr m \cr } } \right) = {}^n{C_m}$$
Prove that $$\left( {\matrix{ n \cr m \cr } } \right) + \left( {\matrix{ {n - 1} \cr m \cr } } \right) + \left( {\matrix{ {n - 2} \cr m \cr } } \right) + ........ + \left( {\matrix{ m \cr m \cr } } \right) = \left( {\matrix{ {n + 1} \cr {m + 2} \cr } } \right)$$

Hence or otherwise, prove that $$\left( {\matrix{ n \cr m \cr } } \right) + 2\left( {\matrix{ {n - 1} \cr m \cr } } \right) + 3\left( {\matrix{ {n - 2} \cr m \cr } } \right) + ........ + \left( {n - m + 1} \right)\left( {\matrix{ m \cr m \cr } } \right) = \left( {\matrix{ {n + 2} \cr {m + 2} \cr } } \right).$$.

16
In any triangle $$ABC,$$ prove that $$$\cot {A \over 2} + \cot {B \over 2} + \cot {C \over 2} = \cot {A \over 2}\cot {B \over 2}\cot {C \over 2}.$$$