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For any odd integer $$n$$ $$\ge 1,\,\,{n^3} - {\left( {n - 1} \right)^3} + .... + {\left( { - 1} \right)^{n - 1}}\,{1^3... IIT-JEE 1996 Let$$p$$and$$q$$be roots of the equation$${x^2} - 2x + A = 0$$and let$$r$$and$$s$$be the roots of the equation... IIT-JEE 1997 Let the harmonic mean and geometric mean of two positive numbers be the ratio 4 : 5. Then the two number are in the rati... IIT-JEE 1992 The sum of the first n terms of the series$${1^2} + {2.2^2} + {3^2} + {2.4^2} + {5^2} + {2.6^2} + .........$$is$$n\,...
IIT-JEE 1988
The solution of the equation $$lo{g_7}$$ $$lo{g_5}$$ $$\left( {\sqrt {x + 5} + \sqrt x } \right) = 0$$ is ................
IIT-JEE 1986
The sum of integers from 1 to 100 that are divisible by 2 or 5 is ............
IIT-JEE 1984

## Numerical

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The coefficient of $${x^9}$$ in the expansion of (1 + x) (1 + $${x^2)}$$ (1 + $${x^3}$$) ....$$(1 + {x^{100}})$$ is
Suppose that all the terms of an arithmetic progression (A.P) are natural numbers. If the ratio of the sum of the first ...
Let a, b, c be positive integers such that $${b \over a}$$ is an integer. If a, b, c are in geometric progression and th...
A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from the pack and the sum of th...
Let $${{a_1}}$$, $${{a_2}}$$, $${{a_3}}$$........ $${{a_{100}}}$$ be an arithmetic progression with $${{a_1}}$$ = 3 and ...
IIT-JEE 2011
Let $${S_k}$$= 1, 2,....., 100, denote the sum of the infinite geometric series whose first term is $$\,{{k - 1} \over {... IIT-JEE 2010 Let$${a_1},\,{a_{2\,}},\,{a_3}$$......,$${a_{11}}$$be real numbers satisfying$${a_1} = 15,27 - 2{a_2} > 0\,\,and\,\,{...
IIT-JEE 2010
If $${\log _3}\,2\,,\,\,{\log _3}\,({2^x} - 5)\,,\,and\,\,{\log _3}\,\left( {{2^x} - {7 \over 2}} \right)$$ are in arith...
IIT-JEE 1990

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The rational number, which equals the number $$2\overline {357}$$ with recurring decimal is
IIT-JEE 1983
In the quadratic equation $$\,\,a{x^2} + bx + c = 0,$$ $$\Delta$$ $$= {b^2} - 4ac$$ and $$\alpha + \beta ,\,{\alpha ^... IIT-JEE 2005 Screening An infinite G.P. has first term '$$x$$' and sum '$$5$$', then$$x$$belongs to IIT-JEE 2004 Screening Suppose$$a, b, c$$are in A.P. and$${a^2},{b^2},{c^2}$$are in G.P. If$$a ...
IIT-JEE 2002 Screening
Let $$\alpha$$, $$\beta$$ be the roots of $${x^2} - x + p = 0$$ and $$\gamma ,\delta$$ be the roots of $${x^2} - 4x +... IIT-JEE 2001 Screening Let the positive numbers$$a,b,c,d$$be in A.P. Then$$abc,abd,acd,bcd,$$are IIT-JEE 2001 Screening If the sum of the first$$2n$$terms of the A.P.$$2,5,8,......,$$is equal to the sum of the first$$n$$terms of the A.... IIT-JEE 2001 Screening Consider an infinite geometric series with first term a and common ratio$$r$$. If its sum is 4 and the second term is 3... IIT-JEE 2000 Screening Let$${a_1},{a_2},......{a_{10}}$$be in$$A,\,P,$$and$${h_1},{h_2},......{h_{10}}$$be in H.P. If$${a_1} = {h_1} = 2...
IIT-JEE 1999
The harmonic mean of the roots of the equation $$\left( {5 + \sqrt 2 } \right){x^2} - \left( {4 + \sqrt 5 } \right)x + 8... IIT-JEE 1999 Let$$n$$be an odd integer. If$$\sin n\theta = \sum\limits_{r = 0}^n {{b_r}{{\sin }^r}\theta ,} $$for every value of... IIT-JEE 1998 Let$${T_r}$$be the$${r^{th}}$$term of an A.P., for$$r=1, 2, 3, ....$$If for some positive integers$$m$$,$$n$$we... IIT-JEE 1998 If$$x > 1,y > 1,z > 1$$are in G.P., then$${1 \over {1 + In\,x}},{1 \over {1 + In\,y}},{1 \over {1 + In\,z}}$$are in ... IIT-JEE 1998 If$$In\left( {a + c} \right),In\left( {a - c} \right),In\left( {a - 2b + c} \right)$$are in A.P., then IIT-JEE 1994 The number$${\log _2}\,7$$is IIT-JEE 1990 Sum of the first n terms of the series$${1 \over 2} + {3 \over 4} + {7 \over 8} + {{15} \over {16}} + ............$$is... IIT-JEE 1988 If$$a,\,b,\,c$$are in GP., then the equations$$\,\,\alpha {x^2} + 2bx + c = 0$$and$$d{x^2} + 2ex + f = 0$$have a c... IIT-JEE 1985 If$$x,\,y$$and$$z$$are$$pth$$,$$qth$$and$$rth$$terms respectively of an A.P. and also of a G.P., then$${x^{y -...
IIT-JEE 1982
The third term of a geometric progression is 4. The product of the first five terms is
IIT-JEE 1982
Let $${a_1},{a_2},{a_3},.....$$ be in harmonic progression with $${a_1} = 5$$ and $${a_{20}} = 25.$$ The least positive ...
IIT-JEE 2012
Let $${b_i} > 1$$ for $$i = 1,\,2,......,101.$$ Suppose $${\log _e}{b_1},\,{\log _e}{b_2},.....,\,{\log _e}{b_{101}}$$ a...
IIT-JEE 2012
In the sum of first $$n$$ terms of an A.P. is $$c{n^\angle }$$, then the sum of squares of these $$n$$ terms is
IIT-JEE 2009
Suppose four distinct positive numbers $${a_1},\,{a_{2\,}},\,{a_3},\,{a_4}\,$$ are in G.P. Let $${b_1} = {a_1},{b_2} = {... IIT-JEE 2008 Let$$\,{V_r}$$denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common... IIT-JEE 2007 Let$$\,{V_r}$$denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common... IIT-JEE 2007 Let$$\,{V_r}$$denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common... IIT-JEE 2007 Let$${A_1}$$,$${G_1}$$,$${H_1}$$denote the arithmetic, geometric and harmonic means, respectively, of two distinct p... IIT-JEE 2007 Let$${A_1}$$,$${G_1}$$,$${H_1}$$denote the arithmetic, geometric and harmonic means, respectively, of two distinct p... IIT-JEE 2007 Let$${A_1}$$,$${G_1}$$,$${H_1}$$denote the arithmetic, geometric and harmonic means, respectively, of two distinct p... IIT-JEE 2007 ## MCQ (More than One Correct Answer) More For$$0 $$x =$$$$\sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\phi ,y = \sum\limits_{n = 0}^\infty {{{\sin }^{2n}}\phi ,... IIT-JEE 1993 If the first and the$$(2n-1)$$st terms of an A.P., a G.P. and an H.P. are equal and their$$n$$-th terms are$$a,b$$... IIT-JEE 1988 For a positive integer$$n$$, let$$a\left( n \right) = 1 + {1 \over 2} + {1 \over 3} + {1 \over 4} + .....\,{1 \over {...
IIT-JEE 1999
Let $${S_n} = {\sum\limits_{k = 1}^{4n} {\left( { - 1} \right)} ^{{{k\left( {k + 1} \right)} \over 2}}}{k^2}.$$ Then $${... JEE Advanced 2013 (Offline) Let$${S_n} = \sum\limits_{k = 1}^n {{n \over {{n^2} + kn + {k^2}}}} $$and$${T_n} = \sum\limits_{k = 0}^{n - 1} {{n \o...
IIT-JEE 2008
A straight line through the vertex p of a triangle PQR intersects the side QR at the point S and the circumcircle of the...
IIT-JEE 2008

## Subjective

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If $$a > 0,\,b > 0$$ and $$\,c > 0,$$ prove that $$\,c > 0,$$ prove that $$\left( {a + b + c} \right)\left( {{1 \over a}... IIT-JEE 1984 If$$n$$is a natural number such that$$n = {p_1}{}^{{\alpha _1}}{p_2}{}^{{\alpha _2}}.{p_3}{}^{{\alpha _3}}........{p...
IIT-JEE 1984
Find three numbers $$a,b,c$$ between $$2$$ and $$18$$ such that (i) their sum is $$25$$ (ii) the numbers $$2,$$ $$a, b... IIT-JEE 1983 The harmonic mean of two numbers is 4.Their arithmetic mean$$A$$and the geometric mean$$G$$satisfy the relation.$$2...
IIT-JEE 1979
The real numbers $${x_1}$$, $${x_2}$$, $${x_3}$$ satisfying the equation $${x^3} - {x^2} + \beta x + \gamma = 0$$ are i...
IIT-JEE 1996
Solve for x the following equation: $${\log _{(2x + 3)}}(6{x^2} + 23x + 21) = 4 - {\log _{(3x + 7)}}(4{x^2} + 12x + 9)\... IIT-JEE 1987 Does there exist a geometric progression containing$$27, 8$$and$$12$$as three of its terms? If it exits, how many su... IIT-JEE 1982 The interior angles of a polygon are in arithmetic progression. The smallest angle is$${120^ \circ }$$, and the common ... IIT-JEE 1980 If a, b, c are in A.P.,$${a^2}$$,$${b^2}$$,$${c^2}$$are in H.P., then prove that either a = b = c or a, b,$${ - {c ...
IIT-JEE 2003
The fourth power of the common difference of an arithmatic progression with integer entries is added to the product of a...
IIT-JEE 2000
Let p be the first of the n arithmetic means between two numbers and q the first of n harmonic means between the same nu...
IIT-JEE 1991
If $${S_1}$$, $${S_2}$$, $${S_3}$$,.............,$${S_n}$$ are the sums of infinite geometric series whose first terms a...
IIT-JEE 1991
Let a, b be positive real numbers. If a, $${{A_1},{A_2}}$$, b are in arithmetic progression, a, $${{G_1},{G_2}}$$, b are...
IIT-JEE 2002
Let $${a_1}$$, $${a_2}$$,.....,$${a_n}$$ be positive real numbers in geometric progression. For each n, let $${A_n}$$, \$...
IIT-JEE 2001
Find the sum of the series : $$\sum\limits_{r = 0}^n {{{\left( { - 1} \right)}^r}\,{}^n{C_r}\left[ {{1 \over {{2^r}}} +... IIT-JEE 1985 If$${a_n} = {3 \over 4} - {\left( {{3 \over 4}} \right)^2} + {\left( {{3 \over 4}} \right)^3} + ....{( - 1)^{n - 1}}{\l...
IIT-JEE 2006
Let a, b, c, d be real numbers in G.P. If u, v, w, satisfy the system of equations u + 2v + 3w = 6 4u + 5v + 6w = 12 ...
IIT-JEE 1999

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