1988
JEE Advanced
MSQ
IIT-JEE 1988
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 $a,b$ and $c$ respectively, then
A.
$a = b = c$
B.
$a \ge b \ge c$
C.
$a + c = b$
D.
$ac - {b^2} = 0$
2003
JEE Advanced
Numerical
IIT-JEE 2003
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, ${ - {c \over 2}}$ form a G.P.
Correct Answer: solve it
2002
JEE Advanced
Numerical
IIT-JEE 2002
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 in geometric progression and a, ${{H_1},{H_2}}$, b are in harmonic progression, show that $\,{{{G_1},{G_2}} \over {{H_1},{H_2}}} = {{{A_1} + {A_2}} \over {{H_1} + {H_2}}} = {{(2a + b)\,(a + 2b)} \over {9ab}}$.
Correct Answer: solve it.
2001
JEE Advanced
Numerical
IIT-JEE 2001
Let ${a_1}$, ${a_2}$,.....,${a_n}$ be positive real numbers in geometric progression. For each n, let ${A_n}$, ${G_n}$, ${H_n}$ be respectively, the arithmetic mean , geometric mean, and harmonic mean of ${a_1}$,${a_2}$......,${a_n}$. Find an expression for the geometric mean of ${G_1}$,${G_2}$,.....,${G_n}$ in terms of ${A_1}$,${A_2}$,.....,${A_n}$,${H_n}$,${H_1}$,${H_2}$,........,${H_n}$.
Correct Answer: $$G = {\left( {{A_1},{A_2}....{A_n}\,{H_1},\,{H_2}.....{H_n}} \right)^{{\raise0.5ex\hbox{$\scriptstyle 1$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle {2n}$}}}}$$
2000
JEE Advanced
Numerical
IIT-JEE 2000
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.
Correct Answer: solve it
1999
JEE Advanced
Numerical
IIT-JEE 1999
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
6u + 9v = 4
u + 2v + 3w = 6
4u + 5v + 6w = 12
6u + 9v = 4
then show that the roots of the equation $\left( {{1 \over u} + {1 \over v} + {1 \over w}} \right){x^2}$
$ + [{(b - c)^2} + {(c - a)^2} + {(d - b)^2}]x + u + v + w = 0$ and $20{x^2} + 10{(a - d)^2}x - 9 = 0$ are reciprocals of each other.
Correct Answer: solve it
1996
JEE Advanced
Numerical
IIT-JEE 1996
The real numbers ${x_1}$, ${x_2}$, ${x_3}$ satisfying the equation ${x^3} - {x^2} + \beta x + \gamma = 0$ are in AP. Find the intervals in which $\beta \,\,and\,\gamma $ lie.
Correct Answer: $$\beta \,\, \in \left( { - \infty ,\,{1 \over 3}} \right],\,\gamma \, \in \,\left[ { - {1 \over {27}},\infty } \right)$$
1991
JEE Advanced
Numerical
IIT-JEE 1991
If ${S_1}$, ${S_2}$, ${S_3}$,.............,${S_n}$ are the sums of infinite geometric series whose first terms are 1, 2, 3, ...................,n and whose common ratios are ${1 \over 2}$, ${1 \over 3}$, ${1 \over 4}$,....................$\,{1 \over {n + 1}}$ respectively, then find the values of ${S_1}^2 + {S_2}^2 + {S_3}^2 + ....... + {S^2}_{2n - 1}$
Correct Answer: $${{{}^n(2n + 1)\,(4n + 1) - 3} \over 3}$$
1991
JEE Advanced
Numerical
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 numbers. Show that q does not lie between p and $\,{\left( {{{n + 1} \over {n - 1}}} \right)^2}\,p$.
Correct Answer: solve it
1985
JEE Advanced
Numerical
IIT-JEE 1985
Find the sum of the series :
$$\sum\limits_{r = 0}^n {{{\left( { - 1} \right)}^r}\,{}^n{C_r}\left[ {{1 \over {{2^r}}} + {{{3^r}} \over {{2^{2r}}}} + {{{7^r}} \over {{2^{3r}}}} + {{{{15}^r}} \over {{2^{4r}}}}..........up\,\,to\,\,m\,\,terms} \right]} $$
Correct Answer: $${{{2^{mn}} - 1} \over {{2^{mn}}\left( {{2^n} - 1} \right)}}$$
1984
JEE Advanced
Numerical
IIT-JEE 1984
If $n$ is a natural number such that
$n = {p_1}{}^{{\alpha _1}}{p_2}{}^{{\alpha _2}}.{p_3}{}^{{\alpha _3}}........{p_k}{}^{{\alpha _k}}$ and ${p_1},{p_2},\,\,......,\,{p_k}$ are distinct primes, then show that $In$ $n \ge k$ $in$ 2
$n = {p_1}{}^{{\alpha _1}}{p_2}{}^{{\alpha _2}}.{p_3}{}^{{\alpha _3}}........{p_k}{}^{{\alpha _k}}$ and ${p_1},{p_2},\,\,......,\,{p_k}$ are distinct primes, then show that $In$ $n \ge k$ $in$ 2
Correct Answer: Solve it.
1984
JEE Advanced
Numerical
IIT-JEE 1984
If $a > 0,\,b > 0$ and $\,c > 0,$ prove that $\,c > 0,$ prove that $\left( {a + b + c} \right)\left( {{1 \over a} + {1 \over b} + {1 \over c}} \right) \ge 9$
Correct Answer: Solve it.
1983
JEE Advanced
Numerical
IIT-JEE 1983
Find three numbers $a,b,c$ between $2$ and $18$ such that
(i) their sum is $25$
(ii) the numbers $2,$ $a, b$ are consecutive terms of an A.P. and
(iii) the numbers $b,c,18$ are consecutive terms of a G.P.
(i) their sum is $25$
(ii) the numbers $2,$ $a, b$ are consecutive terms of an A.P. and
(iii) the numbers $b,c,18$ are consecutive terms of a G.P.
Correct Answer: $$5, 8, 12$$
1982
JEE Advanced
Numerical
IIT-JEE 1982
Does there exist a geometric progression containing $27, 8$ and $12$ as three of its terms? If it exits, how many such progressions are possible ?
Correct Answer: $$ \Rightarrow $$ yes infinite
1980
JEE Advanced
Numerical
IIT-JEE 1980
The interior angles of a polygon are in arithmetic progression. The smallest angle is ${120^ \circ }$, and the common difference is ${5^ \circ }$, Find the number of sides of the polygon.
Correct Answer: 9
1979
JEE Advanced
Numerical
IIT-JEE 1979
The harmonic mean of two numbers is 4.Their arithmetic mean $A$ and the geometric mean $G$ satisfy the relation. $2A + {G^2} = 27$
Correct Answer: $$3$$ and $$6$$ or $$6$$ and $$3$$
1997
JEE Advanced
Numerical
IIT-JEE 1997
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 ${x^2} - 18x + B = 0.$ If $p < q < r < s$ are in arithmetic progression, then $A = \,..........$ and $B = \,..........$
Correct Answer: $$-3,77$$
1996
JEE Advanced
Numerical
IIT-JEE 1996
For any odd integer $n$ $ \ge 1,\,\,{n^3} - {\left( {n - 1} \right)^3} + .... + {\left( { - 1} \right)^{n - 1}}\,{1^3} = ........$
Correct Answer: $${1 \over 4}{\left( {n + 1} \right)^2}\left( {2n - 1} \right)$$
1992
JEE Advanced
Numerical
IIT-JEE 1992
Let the harmonic mean and geometric mean of two positive numbers be the ratio 4 : 5. Then the two number are in the ratio .........
Correct Answer: 4 : 1 or 1 : 4
1988
JEE Advanced
Numerical
IIT-JEE 1988
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
$n\,\,{\left( {n + 1} \right)^2}/2,$ when $n$ is even. When $n$ is odd, the sum is .............
$n\,\,{\left( {n + 1} \right)^2}/2,$ when $n$ is even. When $n$ is odd, the sum is .............
Correct Answer: $${{{n^2}\left( {n + 1} \right)} \over 2}$$
1986
JEE Advanced
Numerical
IIT-JEE 1986
The solution of the equation $lo{g_7}$ $lo{g_5}$ $\left( {\sqrt {x + 5} + \sqrt x } \right) = 0$ is .............
Correct Answer: 4
1984
JEE Advanced
Numerical
IIT-JEE 1984
The sum of integers from 1 to 100 that are divisible by 2 or 5 is ............
Correct Answer: 3050