Question 1: Palette 1/43

JEE Advanced 2022 MSQ

Let $a_{1}, a_{2}, a_{3}, \ldots$ be an arithmetic progression with $a_{1}=7$ and common difference 8. Let $T_{1}, T_{2}, T_{3}, \ldots$ be such that $T_{1}=3$ and $T_{n+1}-T_{n}=a_{n}$ for $n \geq 1$. Then, which of the following is/are TRUE ?

A

$T_{20}=1604$

B

$\sum\limits_{k=1}^{20} T_{k}=10510$

C

$T_{30}=3454$

D

$\sum\limits_{k=1}^{30} T_{k}=35610$

Question 2: Palette 2/43

JEE Advanced 2021 MSQ

For any positive integer n, let S_n : (0, $\infty$) $\to$ R be defined by ${S_n}(x) = \sum\nolimits_{k = 1}^n {{{\cot }^{ - 1}}\left( {{{1 + k(k + 1){x^2}} \over x}} \right)} $, where for any x $\in$ R, ${\cot ^{ - 1}}(x) \in (0,\pi )$ and ${\tan ^{ - 1}}(x) \in \left( { - {\pi \over 2},{\pi \over 2}} \right)$. Then which of the following statements is (are) TRUE?

A

${S_{10}}(x) = {\pi \over 2} - {\tan ^{ - 1}}\left( {{{1 + 11{x^2}} \over {10x}}} \right)$, for all x > 0

B

$\mathop {\lim }\limits_{n \to \infty } \cot ({S_n}(x)) = x$, for all x > 0

C

The equation ${S_3}(x) = {\pi \over 4}$ has a root in (0, $\infty$)

D

$tan({S_n}(x)) \le {1 \over 2}$, for all n $\ge$ 1 and x > 0

Question 3: Palette 3/43

JEE Advanced 2016 MCQ

Let b_i > 1 for I = 1, 2, ......, 101. Suppose log_eb₁, log_eb₂, ......., log_eb₁₀₁ are in Arithmetic Progression (A.P.) with the common difference log_e2. Suppose a₁, a₂, ......, a₁₀₁ are in A.P. such that a₁ = b₁ and a₅₁ = b₅₁. If t = b₁ + b₂ + .... + b₅₁ and s = a₁ + a₂ + ..... + a₅₁, then

A

s > t and a₁₀₁ > b₁₀₁

B

s > t and a₁₀₁ < b₁₀₁

C

s < t and a₁₀₁ > b₁₀₁

D

s < t and a₁₀₁ < b₁₀₁

Question 4: Palette 4/43

JEE Advanced 2013 MSQ

Let ${S_n} = {\sum\limits_{k = 1}^{4n} {\left( { - 1} \right)} ^{{{k\left( {k + 1} \right)} \over 2}}}{k^2}.$ Then ${S_n}$can take value(s)

A

1056

B

1088

C

1120

D

1332

Question 5: Palette 5/43

JEE Advanced 2012 MCQ

Let ${a_1},{a_2},{a_3},.....$ be in harmonic progression with ${a_1} = 5$ and ${a_{20}} = 25.$ The least positive integer $n$ for which ${a_n} < 0$ is

A

22

B

23

C

24

D

25

Question 6: Palette 6/43

JEE Advanced 2009 MCQ

If the sum of first $n$ terms of an A.P. is $c{n^2}$, then the sum of squares of these $n$ terms is

A

${{n\left( {4{n^2} - 1} \right){c^2}} \over 6}$

B

${{n\left( {4{n^2} + 1} \right){c^2}} \over 3}$

C

${{n\left( {4{n^2} - 1} \right){c^2}} \over 3}$

D

${{n\left( {4{n^2} + 1} \right){c^2}} \over 6}$

Question 7: Palette 7/43

JEE Advanced 2008 MCQ

Suppose four distinct positive numbers ${a_1},\,{a_{2\,}},\,{a_3},\,{a_4}\,$ are in G.P. Let ${b_1} = {a_1},{b_2} = {b_1} + {a_2},\,{b_3} = {b_2} + {a_{3\,\,}}\,\,\,and\,\,\,{b_4} = {b_3} + {a_4}$.

STATEMENT-1: The numbers ${b_1},\,{b_{2\,}},\,{b_3},\,{b_4}\,$ are neither in A.P. nor in G.P. and

STATEMENT-2 The numbers ${b_1},\,{b_{2\,}},\,{b_3},\,{b_4}\,$ are in H.P.

A

STATEMENT-1 is True, STATEMENT-2 is True;
STATEMENT-2 is a correct explanation for
STATEMENT-1

B

STATEMENT-1 is True, STATEMENT-2 is True;
STATEMENT-2 is NOT a correct explanation for
STATEMENT-1

C

STATEMENT-1 is True, STATEMENT-2 is False

D

STATEMENT-1 is False, STATEMENT-2 is True

Question 8: Palette 8/43

JEE Advanced 2008 MSQ

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 \over {{n^2} + kn + {k^2}}}} $ for $n$ $=1, 2, 3, ............$ Then,

A

${S_n} < {\pi \over {3\sqrt 3 }}$

B

${S_n} > {\pi \over {3\sqrt 3 }}$

C

${T_n} < {\pi \over {3\sqrt 3 }}$

D

${T_n} > {\pi \over {3\sqrt 3 }}$

Question 9: Palette 9/43

JEE Advanced 2007 MCQ

Let $\,{V_r}$ denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r-1). Let ${T_r} = \,{V_{r + 1}} - \,{V_r} - 2\,\,\,and\,\,\,{Q_r} = \,{T_{r + 1}} - \,{T_r}\,for\,r = 1,2,...$

${T_r}$ is always

A

an odd number

B

an even number

C

a prime number

D

a composite number

Question 10: Palette 10/43

JEE Advanced 2007 MCQ

Let $\,{V_r}$ denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r-1). Let ${T_r} = \,{V_{r + 1}} - \,{V_r} - 2\,\,\,and\,\,\,{Q_r} = \,{T_{r + 1}} - \,{T_r}\,for\,r = 1,2,...$

The sum ${V_1}$+${V_2}$ +...+${V_n}$ is

A

${1 \over {12}}n(n + 1)\,(3{n^2} - n + 1)$

B

${1 \over {12}}n(n + 1)\,(3{n^2} + n + 2)$

C

${1 \over 2}n(2{n^2} - n + 1)$

D

${1 \over 3}(2{n^3} - 2n + 3)$

Question 11: Palette 11/43

JEE Advanced 2007 MCQ

Let ${A_1}$, ${G_1}$, ${H_1}$ denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For $n \ge 2,\,Let\,{A_{n - 1}}\,\,and\,\,{H_{n - 1}}$ have arithmetic, geometric and harminic means as ${A_n},{G_n}\,,{H_n}$ repectively.

Which one of the following statements is correct ?

A

${H_1} > {H_2}\, > {H_3} > ...$

B

${H_1} < {H_2}\, < {H_3} < ...$

C

${H_1} > {H_2}\, > {H_3} > ...$ and ${H_1} < {H_2}\, < {H_3} < ...$

D

${H_1} < {H_2}\, < {H_3} < ...$ and ${H_1} > {H_2}\, > {H_3} > ...$

Question 12: Palette 12/43

JEE Advanced 2007 MCQ

Let ${A_1}$, ${G_1}$, ${H_1}$ denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For $n \ge 2,\,Let\,{A_{n - 1}}\,\,and\,\,{H_{n - 1}}$ have arithmetic, geometric and harminic means as ${A_n},{G_n}\,,{H_n}$ repectively.

Which one of the following statements is correct ?

A

${G_1} > {G_2}\, > {G_3} > ...$

B

${G_1} < {G_2}\, < {G_3} < ...$

C

${G_1} = {G_2}\, = {G_3} = ...$

D

${G_1} < {G_2}\, < {G_3} < ...$ and ${G_1} > {G_2}\, > {G_3} > ...$

Question 13: Palette 13/43

JEE Advanced 2007 MCQ

Let ${A_1}$, ${G_1}$, ${H_1}$ denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For $n \ge 2,\,Let\,{A_{n - 1}}\,\,and\,\,{H_{n - 1}}$ have arithmetic, geometric and harminic means as ${A_n},{G_n}\,,{H_n}$ repectively.

Which one of the following statements is correct ?

A

${A_1} > {A_2}\, > {A_3} > ...$

B

${A_1} < {A_2}\, < {A_3} < ...$

C

${A_1} > {A_2}\, > {A_3} > ...$ and ${A_1} < {A_2}\, < {A_3} < ...$

D

${A_1} < {A_2}\, < {A_3} < ...$ and ${A_1} > {A_2}\, > {A_3} > ...$

Question 14: Palette 14/43

JEE Advanced 2007 MCQ

Let $\,{V_r}$ denote the sum of first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r-1). Let ${T_r} = \,{V_{r + 1}} - \,{V_r} - 2\,\,\,and\,\,\,{Q_r} = \,{T_{r + 1}} - \,{T_r}\,for\,r = 1,2,...$

Which one of the following is a correct statement?

A

${Q_1},\,\,{Q_2},\,\,{Q_3},...$ are A.P. with common difference 5

B

${Q_1},\,\,{Q_2},\,\,{Q_3},...$ are A.P. with common difference 6

C

${Q_1},\,\,{Q_2},\,\,{Q_3},...$ are A.P. with common difference 11

D

${Q_1} = \,\,{Q_2} = \,\,{Q_3} = ...$

Question 15: Palette 15/43

JEE Advanced 2007 MCQ

Which one of the following statements is correct?

A

$\mathrm{G}_{1} > \mathrm{G}_{2} > \mathrm{G}_{3} >\ldots$

B

$\mathrm{G_{1} < G_{2} < G_{3} < \ldots}$

C

$\mathrm{G}_{1}=\mathrm{G}_{2}=\mathrm{G}_{3}=\ldots$

D

$\mathrm{G}_{1} < \mathrm{G}_{3} < \mathrm{G}_{5}<\ldots$ and $\mathrm{G}_{2} > \mathrm{G}_{4} > \mathrm{G}_{6} > \ldots$

Question 16: Palette 16/43

JEE Advanced 2007 MCQ

Which one of the following statements is correct?

A

$A_{1} > A_{2} > A_{3} > \ldots$

B

$\mathrm{A}_{1} < \mathrm{A}_{2} < \mathrm{A}_{3} < \ldots$

C

$A_{1} > A_{3} > A_{5}>\ldots$ and $A_{2} < A_{4} < A_{6} < \ldots$

D

$A_{1} < A_{3} < A_{5} < \ldots$ and $A_{2}>A_{4} > A_{6} > \ldots$

Question 17: Palette 17/43

JEE Advanced 2007 MCQ

Which one of the following statements is correct?

A

$\mathrm{H}_{1} > \mathrm{H}_{2} > \mathrm{H}_{3} > \ldots$

B

$\mathrm{H}_{1} < \mathrm{H}_{2} < \mathrm{H}_{3} < \ldots$

C

$\mathrm{H}_{1}>\mathrm{H}_{3} > \mathrm{H}_{5} > \ldots$ and $\mathrm{H}_{2} < \mathrm{H}_{4} < \mathrm{H}_{6} < \ldots$

D

$\mathrm{H}_{1} < \mathrm{H}_{3} < \mathrm{H}_{5}< \ldots$ and $\mathrm{H}_{2} > \mathrm{H}_{4} > \mathrm{H}_{6} > \ldots$

Question 18: Palette 18/43

JEE Advanced 2007 MCQ

The sum V$_1$ + V$_2$ + ... + V$_n$ is

A

${1 \over {12}}n(n + 1)(3{n^2} - n + 1)$

B

${1 \over {12}}n(n + 1)(3{n^2} + n + 2)$

C

${1 \over 2}n(2{n^2} - n + 1)$

D

${1 \over 3}(2{n^3} - 2n + 3)$

Question 19: Palette 19/43

JEE Advanced 2007 MCQ

T$_r$ is always

A

an odd number

B

an even number

C

a prime number

D

a composite number

Question 20: Palette 20/43

JEE Advanced 2007 MCQ

Which one of the following is a correct statement?

A

Q$_1$, Q$_2$, Q$_3$, ... are in A.P. with common difference 5

B

Q$_1$, Q$_2$, Q$_3$, ... are in A.P. with common difference 6

C

Q$_1$, Q$_2$, Q$_3$, ... are in A.P. with common difference 11

D

Q$_1$ = Q$_2$ = Q$_3$, ...

Question 21: Palette 21/43

JEE Advanced 2005 MCQ

In the quadratic equation $\,\,a{x^2} + bx + c = 0,$ $\Delta $ $ = {b^2} - 4ac$ and $\alpha + \beta ,\,{\alpha ^2} + {\beta ^2},\,{\alpha ^3} + {\beta ^3},$ are in G.P. where $\alpha ,\beta $ are the root of $\,\,a{x^2} + bx + c = 0,$ then

A

$\Delta \ne 0$

B

$b\Delta = 0$

C

$c\Delta = 0$

D

$\Delta = 0$

Question 22: Palette 22/43

JEE Advanced 2005 MCQ

If total number of runs scored in $n$ matches is $\left(\frac{n+1}{4}\right)\left(2^{n+1}-n-2\right)$ where $n > 1$, and the runs scored in the $k^{\text {th }}$ match are given by $k .2^{n+1-k}$, where $1 \leq k \leq n$. Find, $n$.

A

5

B

7

C

15

D

1

Question 23: Palette 23/43

JEE Advanced 2004 MCQ

An infinite G.P. has first term '$x$' and sum '$5$', then $x$ belongs to

A

$x < - 10$

B

$ - 10 < x < 0$

C

$0 < x < 10$

D

$x > 10$

Question 24: Palette 24/43

JEE Advanced 2002 MCQ

Suppose $a, b, c$ are in A.P. and ${a^2},{b^2},{c^2}$ are in G.P. If $a < b < c$ and $a + b + c = {3 \over 2},$ then the value of $a$ is

A

${1 \over {2\sqrt 2 }}$

B

${1 \over {2\sqrt 3 }}$

C

${1 \over 2} - {1 \over {\sqrt 3 }}$

D

${1 \over 2} - {1 \over {\sqrt 2 }}$

Question 25: Palette 25/43

JEE Advanced 2001 MCQ

Let the positive numbers $a,b,c,d$ be in A.P. Then $abc,$ $abd,$ $acd,$ $bcd,$ are

A

NOT in A.P./GP./H.P.

B

inA.P.

C

in GP.

D

in H.P.

Question 26: Palette 26/43

JEE Advanced 2001 MCQ

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.P.$57,59,61,.....,$ then $n$ equals

A

10

B

12

C

11

D

13

Question 27: Palette 27/43

JEE Advanced 2001 MCQ

Let $\alpha $, $\beta $ be the roots of ${x^2} - x + p = 0$ and $\gamma ,\delta $ be the roots of ${x^2} - 4x + q = 0.$ If $\alpha ,\beta ,\gamma ,\delta $ are in G.P., then the integral values of $p$ and $q$ respectively, are

A

$-2,-32$

B

$-2,3$

C

$-6,3$

D

$-6,-32$

Question 28: Palette 28/43

JEE Advanced 2000 MCQ

Consider an infinite geometric series with first term a and common ratio $r$. If its sum is 4 and the second term is 3/4, then

A

$a = {4 \over 7},r = {3 \over 7}\,\,\,\,$

B

$a = 2,\,r = {3 \over 8}$

C

$a = {3 \over 2},r = {1 \over 2}$

D

$a = 3,\,r = {1 \over 4}$

Question 29: Palette 29/43

JEE Advanced 1999 MCQ

The harmonic mean of the roots of the equation $\left( {5 + \sqrt 2 } \right){x^2} - \left( {4 + \sqrt 5 } \right)x + 8 + 2\sqrt 5 = 0$ is

A

2

B

4

C

6

D

8

Question 30: Palette 30/43

JEE Advanced 1999 MCQ

Let ${a_1},{a_2},......{a_{10}}$ be in $A,\,P,$ and ${h_1},{h_2},......{h_{10}}$ be in H.P. If ${a_1} = {h_1} = 2$ and ${a_{10}} = {h_{10}} = 3,$ then ${a_4}{h_7}$ is

A

2

B

3

C

5

D

6

Question 31: Palette 31/43

JEE Advanced 1999 MSQ

For a positive integer $n$, let
$a\left( n \right) = 1 + {1 \over 2} + {1 \over 3} + {1 \over 4} + .....\,{1 \over {\left( {{2^n}} \right) - 1}}$. Then

A

$a\left( {100} \right) \le 100$

B

$a\left( {100} \right) > 100$

C

$a\left( {200} \right) \le 100$

D

$a\left( {200} \right) > 100$

Question 32: Palette 32/43

JEE Advanced 1998 MCQ

Let $n$ be an odd integer. If $\sin n\theta = \sum\limits_{r = 0}^n {{b_r}{{\sin }^r}\theta ,} $ for every value of $\theta ,$ then

A

${b_0} = 1,\,b = 3$

B

${b_0} = 0,\,{b_1} = n$

C

${b_0} = - 1,\,{b_1} = n$

D

${b_0} = 0,\,{b_1} = {n^2} - 3n + 3$

Question 33: Palette 33/43

JEE Advanced 1998 MCQ

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

A

A.P.

B

H.P.

C

G.P.

D

None of these

Question 34: Palette 34/43

JEE Advanced 1998 MCQ

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 have
${T_m} = {1 \over n}$ and ${T_n} = {1 \over m},$ then ${T_n} = {1 \over m},$ equals

A

${1 \over {mn}}$

B

${1 \over {mn}} + {1 \over n}$

C

$1$

D

$0$

Question 35: Palette 35/43

JEE Advanced 1994 MCQ

If $In\left( {a + c} \right),In\left( {a - c} \right),In\left( {a - 2b + c} \right)$ are in A.P., then

A

$a,\,b,\,c$ are in A.P.

B

${a^2},\,{b^2},\,{c^2}$ are in A.P.

C

$a,\,b,\,c$ are in G.P.

D

$a,\,b,\,c$ are in H.P.

Question 36: Palette 36/43

JEE Advanced 1993 MSQ

For $0 < \phi < \pi /2,$ if
$x = $$\sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\phi ,y = \sum\limits_{n = 0}^\infty {{{\sin }^{2n}}\phi ,\,\,\,\,z = \sum\limits_{n = 0}^{} {{{\cos }^{2n}}\phi {{\sin }^{2n}}\phi } } } \infty $ then

A

$xyz = xz + y$

B

$xyz = xy + z$

C

$xyz = x + y + z$

D

$xyz = yz + x$

Question 37: Palette 37/43

JEE Advanced 1990 MCQ

The number ${\log _2}\,7$ is

A

an integer

B

a rational number

C

an irrational number

D

a prime number

Question 38: Palette 38/43

JEE Advanced 1988 MCQ

Sum of the first n terms of the series ${1 \over 2} + {3 \over 4} + {7 \over 8} + {{15} \over {16}} + ............$ is equal to

A

${2^n} - n - 1$

B

$1 - {2^{ - n}}$

C

$n + {2^{ - n}} - 1$

D

${2^n} + 1$

Question 39: Palette 39/43

JEE Advanced 1988 MSQ

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

$ac - {b^2} = 0$

Question 40: Palette 40/43

JEE Advanced 1985 MCQ

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 common root if ${d \over a},\,{e \over b},{f \over c}$ are in ________.

A

A.P.

B

GP.

C

H.P.

D

none of these

Question 41: Palette 41/43

JEE Advanced 1983 MCQ

The rational number, which equals the number $2\overline {357} $ with recurring decimal is

A

${{2355} \over {1001}}$

B

${{2379} \over {997}}$

C

${{2355} \over {999}}$

D

none of these

Question 42: Palette 42/43

JEE Advanced 1982 MCQ

The third term of a geometric progression is 4. The product of the first five terms is

A

4³

B

4⁵

C

4⁴

D

none of these

Question 43: Palette 43/43

JEE Advanced 1982 MCQ

If $x,\,y$ and $z$ are $pth$, $qth$ and $rth$ terms respectively of an A.P. and also of a G.P., then ${x^{y - z}}\,{y^{z - x}}\,{z^{x - y}}$ is equal to :

A

$xyz$

B

$0$

C

$1$

D

None of these