## Fill in the Blanks

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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
JEE Advanced 2015 (Offline)
Suppose that all the terms of an arithmetic progression (A.P) are natural numbers. If the ratio of the sum of the first ...
JEE Advanced 2015 (Offline)
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...
JEE Advanced 2014 (Offline)
A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from the pack and the sum of th...
JEE Advanced 2013 (Offline)
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

## MCQ (Single Correct Answer)

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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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