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## Numerical

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Let a1, a2, a3, .... be a sequence of positive integers in arithmetic progression with common difference 2. Also, let b1...
JEE Advanced 2020 Paper 1 Offline
Let m be the minimum possible value of $${\log _3}({3^{{y_1}}} + {3^{{y_2}}} + {3^{{y_3}}})$$, where $${y_1},{y_2},{y_3}... JEE Advanced 2020 Paper 1 Offline Let AP(a; d) denote the set of all the terms of an infinite arithmetic progression with first term a and common differen... JEE Advanced 2019 Paper 1 Offline Let X be the set consisting of the first 2018 terms of the arithmetic progression 1, 6, 11, ...., and Y be the set consi... JEE Advanced 2018 Paper 1 Offline The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length... JEE Advanced 2017 Paper 1 Offline The coefficient of$${x^9}$$in the expansion of (1 + x) (1 +$${x^2)}$$(1 +$${x^3}$$) ....$$(1 + {x^{100}})$$is JEE Advanced 2015 Paper 2 Offline Suppose that all the terms of an arithmetic progression (A.P) are natural numbers. If the ratio of the sum of the first ... JEE Advanced 2015 Paper 2 Offline Let a, b, c be positive integers such that$${b \over a}$$is an integer. If a, b, c are in geometric progression and th... JEE Advanced 2014 Paper 1 Offline A pack contains$$n$$cards numbered from$$1$$to$$n.$$Two consecutive numbered cards are removed from the pack and t... JEE Advanced 2013 Paper 1 Offline A pack contains n cards numbered from 1 to n. Two consecutive numbered cards are removed from the pack and the sum of th... JEE Advanced 2013 Paper 1 Offline Let$${{a_1}}$$,$${{a_2}}$$,$${{a_3}}$$........$${{a_{100}}}$$be an arithmetic progression with$${{a_1}}$$= 3 and ... IIT-JEE 2011 Paper 1 Offline Let$${S_k}$$= 1, 2,....., 100, denote the sum of the infinite geometric series whose first term is$$\,{{k - 1} \over {...
IIT-JEE 2010 Paper 1 Offline
Let $${a_1},\,{a_{2\,}},\,{a_3}$$......,$${a_{11}}$$ be real numbers satisfying $${a_1} = 15,27 - 2{a_2} &gt; 0\,\,and\,... IIT-JEE 2010 Paper 2 Offline If$${\log _3}\,2\,,\,\,{\log _3}\,({2^x} - 5)\,,\,and\,\,{\log _3}\,\left( {{2^x} - {7 \over 2}} \right)$$are in arith... IIT-JEE 1990 ## MCQ (Single Correct Answer) More Let bi > 1 for I = 1, 2, ......, 101. Suppose logeb1, logeb2, ......., logeb101 are in Arithmetic Progression (A.P.) wit... JEE Advanced 2016 Paper 2 Offline Let$${a_1},{a_2},{a_3},.....$$be in harmonic progression with$${a_1} = 5$$and$${a_{20}} = 25.$$The least positive ... IIT-JEE 2012 Paper 2 Offline In the sum of first$$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 $${A_1}$$, $${G_1}$$, $${H_1}$$ denote the arithmetic, geometric and harmonic means, respectively, of two distinct p...
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 $$\,{V_r}$$ 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
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 &lt; b &lt; c$$and$$a + b + c = {3 \over ...
IIT-JEE 2002 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
Let the positive numbers $$a,b,c,d$$ be in A.P. Then $$abc,$$ $$abd,$$ $$acd,$$ $$bcd,$$ are
IIT-JEE 2001 Screening
Let $$\alpha$$, $$\beta$$ be the roots of $${x^2} - x + p = 0$$ and $$\gamma ,\delta$$ be the roots of $${x^2} - 4x +... IIT-JEE 2001 Screening Consider an infinite geometric series with first term a and common ratio$$r$$. If its sum is 4 and the second term is 3... IIT-JEE 2000 Screening The harmonic mean of the roots of the equation$$\left( {5 + \sqrt 2 } \right){x^2} - \left( {4 + \sqrt 5 } \right)x + 8...
IIT-JEE 1999
Let $${a_1},{a_2},......{a_{10}}$$ be in $$A,\,P,$$ and $${h_1},{h_2},......{h_{10}}$$ be in H.P. If $${a_1} = {h_1} = 2... IIT-JEE 1999 If$$x &gt; 1,y &gt; 1,z &gt; 1$$are in G.P., then$${1 \over {1 + In\,x}},{1 \over {1 + In\,y}},{1 \over {1 + In\,z}}$... 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 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 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 The rational number, which equals the number $$2\overline {357}$$ with recurring decimal is IIT-JEE 1983 If $$x,\,y$$ and $$z$$ are $$pth$$, $$qth$$ and $$rth$$ terms respectively of an A.P. and also of a G.P., then $${x^{y -... IIT-JEE 1982 The third term of a geometric progression is 4. The product of the first five terms is IIT-JEE 1982 ## MCQ (More than One Correct Answer) More Let$${S_n} = {\sum\limits_{k = 1}^{4n} {\left( { - 1} \right)} ^{{{k\left( {k + 1} \right)} \over 2}}}{k^2}.$$Then$${... JEE Advanced 2013 Paper 1 Offline Let $${S_n} = \sum\limits_{k = 1}^n {{n \over {{n^2} + kn + {k^2}}}}$$ and $${T_n} = \sum\limits_{k = 0}^{n - 1} {{n \o... IIT-JEE 2008 A straight line through the vertex p of a triangle PQR intersects the side QR at the point S and the circumcircle of the... IIT-JEE 2008 For a positive integer$$n$$, let$$a\left( n \right) = 1 + {1 \over 2} + {1 \over 3} + {1 \over 4} + .....\,{1 \over {... IIT-JEE 1999 For $$0 &lt; \phi &lt; \pi /2,$$ if $$x =$$$$\sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\phi ,y = \sum\limits_{n = 0}... IIT-JEE 1993 If the first and the$$(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$$... IIT-JEE 1988 ## Subjective More If$${a_n} = {3 \over 4} - {\left( {{3 \over 4}} \right)^2} + {\left( {{3 \over 4}} \right)^3} + ....{( - 1)^{n - 1}}{\l... IIT-JEE 2006 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, $${ - {c ... IIT-JEE 2003 Let a, b be positive real numbers. If a,$${{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}$$, ... IIT-JEE 2001 The fourth power of the common difference of an arithmatic progression with integer entries is added to the product of a... IIT-JEE 2000 Let a, b, c, d be real numbers in G.P. If u, v, w, satisfy the system of equations u + 2v + 3w = 6 4u + 5v + 6w = 12 ... IIT-JEE 1999 The real numbers$${x_1}$$,$${x_2}$$,$${x_3}$$satisfying the equation$${x^3} - {x^2} + \beta x + \gamma = 0$$are i... IIT-JEE 1996 If$${S_1}$$,$${S_2}$$,$${S_3}$$,.............,$${S_n}$$are the sums of infinite geometric series whose first terms a... IIT-JEE 1991 Let p be the first of the n arithmetic means between two numbers and q the first of n harmonic means between the same nu... IIT-JEE 1991 Solve for x the following equation:$${\log _{(2x + 3)}}(6{x^2} + 23x + 21) = 4 - {\log _{(3x + 7)}}(4{x^2} + 12x + 9)\... IIT-JEE 1987 Find the sum of the series : $$\sum\limits_{r = 0}^n {{{\left( { - 1} \right)}^r}\,{}^n{C_r}\left[ {{1 \over {{2^r}}} +... IIT-JEE 1985 If$$n$$is a natural number such that$$n = {p_1}{}^{{\alpha _1}}{p_2}{}^{{\alpha _2}}.{p_3}{}^{{\alpha _3}}........{p... IIT-JEE 1984 If $$a &gt; 0,\,b &gt; 0$$ and $$\,c &gt; 0,$$ prove that $$\,c &gt; 0,$$ prove that $$\left( {a + b + c} \right)\left( ... IIT-JEE 1984 Find three numbers$$a,b,c$$between$$2$$and$$18$$such that (i) their sum is$$25$$(ii) the numbers$$2,a, b$...
IIT-JEE 1983
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
The harmonic mean of two numbers is 4.Their arithmetic mean $$A$$ and the geometric mean $$G$$ satisfy the relation. $$2... IIT-JEE 1979 ## Fill in the Blanks More Let$$p$$and$$q$$be roots of the equation$${x^2} - 2x + A = 0$$and let$$r$$and$$s$$be the roots of the equation... IIT-JEE 1997 For any odd integer$$n \ge 1,\,\,{n^3} - {\left( {n - 1} \right)^3} + .... + {\left( { - 1} \right)^{n - 1}}\,{1^3...
IIT-JEE 1996
Let 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

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