Question 1: Palette 1/228

JEE Mains 2026 MCQ

Let the arithmetic mean of $\frac{1}{a}$ and $\frac{1}{b}$ be $\frac{5}{16}$, $a > 2$. If $\alpha$ is such that $a$, $4$, $\alpha$, $b$ are in A.P., then the equation $\alpha x^2 - a x + 2(\alpha - 2b) = 0$ has :

A

one root in $(1, 4)$ and another in $(-2, 0)$

B

one root in $(0, 2)$ and another in $(-4, -2)$

C

both roots in the interval $(-2, 0)$

D

complex roots of magnitude less than $2$

Question 2: Palette 2/228

JEE Mains 2026 MCQ

$ \frac{6}{3^{26}} + \frac{10 \cdot 1}{3^{25}} + \frac{10 \cdot 2}{3^{24}} + \frac{10 \cdot 2^2}{3^{23}} + \ldots + \frac{10 \cdot 2^{24}}{3} $ is equal to :

A

$2^{26}$

B

$3^{25}$

C

$3^{26}$

D

$2^{25}$

Question 3: Palette 3/228

JEE Mains 2026 MCQ

The value of $\sum\limits_{k=1}^{\infty}(-1)^{k+1}\left(\frac{k(k+1)}{k!}\right)$ is

A

e/2

B

$\sqrt{e}$

C

2/e

D

1/e

Question 4: Palette 4/228

JEE Mains 2026 MCQ

The common difference of the A.P.: $a_1, a_2, \ldots, a_{\mathrm{m}}$ is 13 more than the common difference of the A.P.: $b_1, b_2, \ldots, b_n$. If $b_{31}=-277, b_{43}=-385$ and $a_{78}=327$, then $a_1$ is equal to

A

21

B

19

C

24

D

16

Question 5: Palette 5/228

JEE Mains 2026 MCQ

Let $a_1, a_2, a_3, a_4$ be an A.P. of four terms such that each term of the A.P. and its common difference $l$ are integers. If $a_1+a_2+a_3+a_4=48$ and $a_1 a_2 a_3 a_4+l^4=361$, then the largest term of the A.P. is equal to

A

27

B

24

C

23

D

21

Question 6: Palette 6/228

JEE Mains 2026 MCQ

$\left(\frac{1}{3}+\frac{4}{7}\right)+\left(\frac{1}{3^2}+\frac{1}{3} \times \frac{4}{7}+\frac{4^2}{7^2}\right)+\left(\frac{1}{3^3}+\frac{1}{3^2} \times \frac{4}{7}+\frac{1}{3} \times \frac{4^2}{7^2}+\frac{4^3}{7^3}\right)+\ldots$ upto infinite terms, is equal to

A

$\frac{7}{4}$

B

$\frac{4}{3}$

C

$\frac{6}{5}$

D

$\frac{5}{2}$

Question 7: Palette 7/228

JEE Mains 2026 MCQ

Let $729,81,9,1, \ldots$ be a sequence and $\mathrm{P}_n$ denote the product of the first $n$ terms of this sequence.

If $2 \sum\limits_{n=1}^{40}\left(\mathrm{P}_n\right)^{\frac{1}{n}}=\frac{3^\alpha-1}{3^\beta}$ and $\operatorname{gcd}(\alpha, \beta)=1$, then

$\alpha+\beta$ is equal to

A

73

B

74

C

75

D

76

Question 8: Palette 8/228

JEE Mains 2026 MCQ

Consider an A.P.: $a_1, a_2, \ldots, a_{\mathrm{n}} ; a_1>0$. If $a_2-a_1=\frac{-3}{4}, a_{\mathrm{n}}=\frac{1}{4} a_1$, and $\sum\limits_{\mathrm{i}=1}^{\mathrm{n}} a_{\mathrm{i}}=\frac{525}{2}$, then $\sum\limits_{\mathrm{i}=1}^{17} a_{\mathrm{i}}$ is equal to

A

238

B

136

C

476

D

952

Question 9: Palette 9/228

JEE Mains 2026 MCQ

Let $\sum\limits_{k=1}^n a_k=\alpha n^2+\beta n$. If $a_{10}=59$ and $a_6=7 a_1$, then $\alpha+\beta$ is equal to :

A

3

B

5

C

7

D

12

Question 10: Palette 10/228

JEE Mains 2026 MCQ

If the sum of the first four terms of an A.P. is 6 and the sum of its first six terms is 4 , then the sum of its first twelve terms is

A

-26

B

-20

C

-24

D

-22

Question 11: Palette 11/228

JEE Mains 2026 MCQ

The positive integer n, for which the solutions of the equation

$x(x+2) + (x+2)(x+4) + \cdots + (x+2n-2)(x+2n) = \frac{8n}{3}$ are two consecutive even integers, is :

A

3

B

6

C

9

D

12

Question 12: Palette 12/228

JEE Mains 2026 MCQ

Let $a_1, \frac{a_2}{2}, \frac{a_3}{2^2}, \ldots, \frac{a_{10}}{2^9}$ be a G.P. of common ratio $\frac{1}{\sqrt{2}}$. If $a_1 + a_2 + \ldots + a_{10} = 62$, then $a_1$ is equal to:

A

$\sqrt{2} - 1$

B

$2(\sqrt{2} - 1)$

C

$2 - \sqrt{2}$

D

$2(2 - \sqrt{2})$

Question 13: Palette 13/228

JEE Mains 2026 MCQ

Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing positive terms such that $a_2 \cdot a_3 \cdot a_4=64$ and $a_1+a_3+a_5=\frac{813}{7}$. Then $a_3+a_5+a_7$ is equal to :

A

3256

B

3252

C

3248

D

3244

Question 14: Palette 14/228

JEE Mains 2026 MCQ

Let $\alpha=3+4+8+9+13+14+\ldots$ upto 40 terms. If $(\tan \beta)^{\frac{\alpha}{1020}}$ is a root of the equation $x^2+x-2=0, \beta \in\left(0, \frac{\pi}{2}\right)$, then $\sin ^2 \beta+3 \cos ^2 \beta$ is equal to :

A

${ }2$

B

${\frac{7}{4}}$

C

$\frac{5}{2}$

D

$\frac{3}{2}$

Question 15: Palette 15/228

JEE Mains 2026 MCQ

Consider the quadratic equation $\left(n^2-2 n+2\right) x^2-3 x+\left(n^2-2 n+2\right)^2=0, n \in \mathbf{R}$. Let $\alpha$ be the minimum value of the product of its roots and $\beta$ be the maximum value of the sum of its roots. Then the sum of the first six terms of the G.P., whose first term is $\alpha$ and the common ratio is $\frac{\alpha}{\beta}$, is :

A

$\frac{61}{37}$

B

$\frac{121}{81}$

C

$\frac{364}{243}$

D

$\frac{1093}{729}$

Question 16: Palette 16/228

JEE Mains 2026 MCQ

The sum $1+\frac{1}{2}\left(1^2+2^2\right)+\frac{1}{3}\left(1^2+2^2+3^2\right)+\ldots$ upto 10 terms is equal to :

A

130

B

155

C

$\frac{315}{2}$

D

$\frac{325}{2}$

Question 17: Palette 17/228

JEE Mains 2026 MCQ

The value of $1^3-2^3+3^3-\ldots+15^3$ is:

A

1706

B

1856

C

1982

D

2403

Question 18: Palette 18/228

JEE Mains 2026 MCQ

The sum of the first ten terms of an A.P. is 160 and the sum of the first two terms of a G.P. is 8 . If the first term of the A.P. is equal to the common ratio of the G.P. and the first term of the G.P. is equal to common difference of the A.P., then the sum of all possible values of the first term of the G.P. is:

A

$\frac{34}{9}$

B

$\frac{34}{13}$

C

$\frac{32}{9}$

D

$\frac{32}{13}$

Question 19: Palette 19/228

JEE Mains 2026 MCQ

Let $\alpha, \beta$ be the roots of the equation $x^2-x+\mathrm{p}=0$ and $\gamma, \delta$ be the roots the equation $x^2-4 x+\mathrm{q}=0$; $p, q \in \mathbf{Z}$. If $\alpha, \beta, \gamma, \delta$ are in G.P., then $|p+q|$ equals :

A

16

B

32

C

34

D

38

Question 20: Palette 20/228

JEE Mains 2026 MCQ

If the sum of the first 10 terms of the series $\frac{1}{1+1^4 \times 4}+\frac{2}{1+2^4 \times 4}+\frac{3}{1+3^4 \times 4}+\frac{4}{1+4^4 \times 4}+\ldots \ldots$. is $\frac{m}{n}, \operatorname{gcd}(m, n)=1$, then $m+n$ is equal to :

A

256

B

264

C

276

D

284

Question 21: Palette 21/228

JEE Mains 2026 MCQ

Let $\mathrm{A}_1, \mathrm{~A}_2, \mathrm{~A}_3, \ldots \ldots . ., \mathrm{A}_{39}$ be 39 arithmetic means between the numbers 59 and 159. Then the mean of $\mathrm{A}_{25}, \mathrm{~A}_{28}, \mathrm{~A}_{31}$ and $\mathrm{A}_{36}$ is equal to :

A

129

B

136

C

131.50

D

134

Question 22: Palette 22/228

JEE Mains 2026 MCQ

Let the sum of the first $n$ terms of an A.P. be $3 n^2+5 n$. Then the sum of squares of the first 10 terms of the A.P. is:

A

10220

B

12860

C

15220

D

19780

Question 23: Palette 23/228

JEE Mains 2026 MCQ

$\sum_{n=1}^{10}\left(\frac{528}{n(n+1)(n+2)}\right)$ is equal to:

A

65

B

130

C

220

D

440

Question 24: Palette 24/228

JEE Mains 2026 MCQ

The first term of an A.P. of 30 non-negative terms is $\frac{10}{3}$. If the sum of this A.P. is the cube of its last term, then its common difference is:

A

$\frac{5}{87}$

B

$\frac{25}{83}$

C

$\frac{15}{29}$

D

$\frac{5}{29}$

Question 25: Palette 25/228

JEE Mains 2026 MCQ

Let $a_1, a_2, a_3, \ldots$ be an A.P. and $g_1 = a_1, g_2, g_3, \ldots$ be an increasing G.P. If $a_1 = a_2 + g_2 = 1$ and $a_3 + g_3 = 4$, then $a_{10} + g_5$ is equal to:

A

81

B

76

C

62

D

55

Question 26: Palette 26/228

JEE Mains 2026 MCQ

The sum $\frac{1^3}{1} + \frac{1^3 + 2^3}{1 + 3} + \frac{1^3 + 2^3 + 3^3}{1 + 3 + 5} + \ldots$ up to 8 terms, is :

A

70

B

71

C

72

D

73

Question 27: Palette 27/228

JEE Mains 2026 MCQ

Let A be the set of first 101 terms of an A.P., whose first term is 1 and the common difference is 5 and let B be the set of first 71 terms of an A.P., whose first term is 9 and the common difference is 7. Then the number of elements in $A \cap B$, which are divisible by 3, is :

A

4

B

5

C

6

D

7

Question 28: Palette 28/228

JEE Mains 2025 MCQ

If $ \frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \ldots \infty= \frac{\pi^4}{90} $,

$\frac{1}{1^4} + \frac{1}{3^4} + \frac{1}{5^4} + \ldots \infty= \alpha $,

$ \frac{1}{2^4} + \frac{1}{4^4} + \frac{1}{6^4} + \ldots \infty= \beta $,

then $ \frac{\alpha}{\beta} $ is equal to :

A

23

B

14

C

18

D

15

Question 29: Palette 29/228

JEE Mains 2025 MCQ

Let $a_n$ be the $n^{th}$ term of an A.P. If $S_n = a_1 + a_2 + a_3 + \ldots + a_n = 700$, $a_6 = 7$ and $S_7 = 7$, then $a_n$ is equal to :

A

65

B

56

C

70

D

64

Question 30: Palette 30/228

JEE Mains 2025 MCQ

If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the sum of its eighth, tenth and twelfth terms is 15309, then the sum of its first nine terms is :

A

757

B

755

C

750

D

760

Question 31: Palette 31/228

JEE Mains 2025 MCQ

Let $x_1, x_2, x_3, x_4$ be in a geometric progression. If $2,7,9,5$ are subtracted respectively from $x_1, x_2, x_3, x_4$, then the resulting numbers are in an arithmetic progression. Then the value of $\frac{1}{24}\left(x_1 x_2 x_3 x_4\right)$ is:

A

18

B

216

C

36

D

72

Question 32: Palette 32/228

JEE Mains 2025 MCQ

If the sum of the first 20 terms of the series $\frac{4 \cdot 1}{4+3 \cdot 1^2+1^4}+\frac{4 \cdot 2}{4+3 \cdot 2^2+2^4}+\frac{4 \cdot 3}{4+3 \cdot 3^2+3^4}+\frac{4 \cdot 4}{4+3 \cdot 4^2+4^4}+\ldots \cdot$ is $\frac{\mathrm{m}}{\mathrm{n}}$, where m and n are coprime, then $\mathrm{m}+\mathrm{n}$ is equal to :

A

423

B

421

C

422

D

420

Question 33: Palette 33/228

JEE Mains 2025 MCQ

Consider two sets A and B, each containing three numbers in A.P. Let the sum and the product of the elements of A be 36 and p respectively and the sum and the product of the elements of B be 36 and $q$ respectively. Let d and D be the common differences of $\mathrm{AP}^{\prime} \mathrm{s}$ in $A$ and $B$ respectively such that $D=d+3, d>0$. If $\frac{p+q}{p-q}=\frac{19}{5}$, then $\mathrm{p}-\mathrm{q}$ is equal to

A

540

B

450

C

600

D

630

Question 34: Palette 34/228

JEE Mains 2025 MCQ

Let $A=\{1,6,11,16, \ldots\}$ and $B=\{9,16,23,30, \ldots\}$ be the sets consisting of the first 2025 terms of two arithmetic progressions. Then $n(A \cup B)$ is

A

3814

B

4003

C

4027

D

3761

Question 35: Palette 35/228

JEE Mains 2025 MCQ

$1+3+5^2+7+9^2+\ldots$ upto 40 terms is equal to

A

40870

B

41880

C

43890

D

33980

Question 36: Palette 36/228

JEE Mains 2025 MCQ

The sum $1+\frac{1+3}{2!}+\frac{1+3+5}{3!}+\frac{1+3+5+7}{4!}+\ldots$ upto $\infty$ terms, is equal to

A

$3 e$

B

$2 e$

C

$4 e$

D

$6 e$

Question 37: Palette 37/228

JEE Mains 2025 MCQ

Let $a_1, a_2, a_3, \ldots$. be a G.P. of increasing positive numbers. If $a_3 a_5=729$ and $a_2+a_4=\frac{111}{4}$, then $24\left(a_1+a_2+a_3\right)$ is equal to

A

128

B

129

C

131

D

130

Question 38: Palette 38/228

JEE Mains 2025 MCQ

The sum $1+3+11+25+45+71+\ldots$ upto 20 terms, is equal to

A

7240

B

8124

C

7130

D

6982

Question 39: Palette 39/228

JEE Mains 2025 MCQ

The number of terms of an A.P. is even; the sum of all the odd terms is 24 , the sum of all the even terms is 30 and the last term exceeds the first by $\frac{21}{2}$. Then the number of terms which are integers in the A.P. is :

A

6

B

4

C

8

D

10

Question 40: Palette 40/228

JEE Mains 2025 MCQ

Let $a_1, a_2, a_3, \ldots$ be in an A.P. such that $\sum_\limits{k=1}^{12} a_{2 k-1}=-\frac{72}{5} a_1, a_1 \neq 0$. If $\sum_\limits{k=1}^n a_k=0$, then $n$ is :

A

18

B

17

C

11

D

10

Question 41: Palette 41/228

JEE Mains 2025 MCQ

Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its 11^th term is :

A

108

B

90

C

122

D

84

Question 42: Palette 42/228

JEE Mains 2025 MCQ

For positive integers $n$, if $4 a_n=\left(n^2+5 n+6\right)$ and $S_n=\sum\limits_{k=1}^n\left(\frac{1}{a_k}\right)$, then the value of $507 S_{2025}$ is :

A

540

B

675

C

1350

D

135

Question 43: Palette 43/228

JEE Mains 2025 MCQ

Let $\left\langle a_{\mathrm{n}}\right\rangle$ be a sequence such that $a_0=0, a_1=\frac{1}{2}$ and $2 a_{\mathrm{n}+2}=5 a_{\mathrm{n}+1}-3 a_{\mathrm{n}}, \mathrm{n}=0,1,2,3, \ldots$. Then $\sum\limits_{k=1}^{100} a_k$ is equal to

A

$3 a_{100}+100$

B

$3 a_{100}-100$

C

$3 a_{99}-100$

D

$3 a_{99}+100$

Question 44: Palette 44/228

JEE Mains 2025 MCQ

Let $\mathrm{T}_{\mathrm{r}}$ be the $\mathrm{r}^{\text {th }}$ term of an A.P. If for some $\mathrm{m}, \mathrm{T}_{\mathrm{m}}=\frac{1}{25}, \mathrm{~T}_{25}=\frac{1}{20}$, and $20 \sum\limits_{\mathrm{r}=1}^{25} \mathrm{~T}_{\mathrm{r}}=13$, then $5 \mathrm{~m} \sum\limits_{\mathrm{r}=\mathrm{m}}^{2 \mathrm{~m}} \mathrm{~T}_{\mathrm{r}}$ is equal to

A

98

B

126

C

112

D

142

Question 45: Palette 45/228

JEE Mains 2025 MCQ

In an arithmetic progression, if $\mathrm{S}_{40}=1030$ and $\mathrm{S}_{12}=57$, then $\mathrm{S}_{30}-\mathrm{S}_{10}$ is equal to :

A

525

B

505

C

510

D

515

Question 46: Palette 46/228

JEE Mains 2025 MCQ

If $7=5+\frac{1}{7}(5+\alpha)+\frac{1}{7^2}(5+2 \alpha)+\frac{1}{7^3}(5+3 \alpha)+\ldots \ldots \ldots \ldots \infty$, then the value of $\alpha$ is :

A

$\frac{1}{7}$

B

1

C

$\frac{6}{7}$

D

6

Question 47: Palette 47/228

JEE Mains 2025 MCQ

Let $S_n=\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\ldots$ upto $n$ terms. If the sum of the first six terms of an A.P. with first term -p and common difference p is $\sqrt{2026 \mathrm{~S}_{2025}}$, then the absolute difference betwen $20^{\text {th }}$ and $15^{\text {th }}$ terms of the A.P. is

A

20

B

45

C

90

D

25

Question 48: Palette 48/228

JEE Mains 2025 MCQ

If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to

A

$-120$

B

$-1200$

C

$-1080$

D

$-1020$

Question 49: Palette 49/228

JEE Mains 2025 MCQ

Suppose that the number of terms in an A.P. is $2 k, k \in N$. If the sum of all odd terms of the A.P. is 40 , the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27 , then k is equal to:

A

8

B

6

C

4

D

5

Question 50: Palette 50/228

JEE Mains 2025 MCQ

Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing positive terms. If $a_1 a_5=28$ and $a_2+a_4=29$, then $a_6$ is equal to:

A

812

B

784

C

628

D

526

Question 51: Palette 51/228

JEE Mains 2024 MCQ

Let $a, a r, a r^2$, ............ be an infinite G.P. If $\sum_\limits{n=0}^{\infty} a r^n=57$ and $\sum_\limits{n=0}^{\infty} a^3 r^{3 n}=9747$, then $a+18 r$ is equal to

A

27

B

38

C

31

D

46

Question 52: Palette 52/228

JEE Mains 2024 MCQ

If the sum of the series $\frac{1}{1 \cdot(1+\mathrm{d})}+\frac{1}{(1+\mathrm{d})(1+2 \mathrm{~d})}+\ldots+\frac{1}{(1+9 \mathrm{~d})(1+10 \mathrm{~d})}$ is equal to 5, then $50 \mathrm{~d}$ is equal to :

A

5

B

10

C

15

D

20

Question 53: Palette 53/228

JEE Mains 2024 MCQ

In an increasing geometric progression of positive terms, the sum of the second and sixth terms is $\frac{70}{3}$ and the product of the third and fifth terms is 49. Then the sum of the $4^{\text {th }}, 6^{\text {th }}$ and $8^{\text {th }}$ terms is equal to:

A

78

B

96

C

91

D

84

Question 54: Palette 54/228

JEE Mains 2024 MCQ

Let $A B C$ be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle $A B C$ and the same process is repeated infinitely many times. If $\mathrm{P}$ is the sum of perimeters and $Q$ is be the sum of areas of all the triangles formed in this process, then :

A

$\mathrm{P}^2=72 \sqrt{3} \mathrm{Q}$

B

$\mathrm{P}^2=36 \sqrt{3} \mathrm{Q}$

C

$\mathrm{P}=36 \sqrt{3} \mathrm{Q}^2$

D

$\mathrm{P}^2=6 \sqrt{3} \mathrm{Q}$

Question 55: Palette 55/228

JEE Mains 2024 MCQ

A software company sets up m number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took 8 more days to finish the assignment. The value of $\mathrm{m}$ is equal to:

A

125

B

160

C

150

D

180

Question 56: Palette 56/228

JEE Mains 2024 MCQ

For $x \geqslant 0$, the least value of $\mathrm{K}$, for which $4^{1+x}+4^{1-x}, \frac{\mathrm{K}}{2}, 16^x+16^{-x}$ are three consecutive terms of an A.P., is equal to :

A

10

B

4

C

8

D

16

Question 57: Palette 57/228

JEE Mains 2024 MCQ

If $\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\ldots+\frac{1}{\sqrt{99}+\sqrt{100}}=m$ and $\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\ldots+\frac{1}{99 \cdot 100}=\mathrm{n}$, then the point $(\mathrm{m}, \mathrm{n})$ lies on the line

A

$11(x-1)-100 y=0$

B

$11 x-100 y=0$

C

$11(x-1)-100(y-2)=0$

D

$11(x-2)-100(y-1)=0$

Question 58: Palette 58/228

JEE Mains 2024 MCQ

The value of $\frac{1 \times 2^2+2 \times 3^2+\ldots+100 \times(101)^2}{1^2 \times 2+2^2 \times 3+\ldots .+100^2 \times 101}$ is

A

$\frac{305}{301}$

B

$\frac{306}{305}$

C

$\frac{32}{31}$

D

$\frac{31}{30}$

Question 59: Palette 59/228

JEE Mains 2024 MCQ

Let three real numbers $a, b, c$ be in arithmetic progression and $a+1, b, c+3$ be in geometric progression. If $a>10$ and the arithmetic mean of $a, b$ and $c$ is 8, then the cube of the geometric mean of $a, b$ and $c$ is

A

120

B

316

C

312

D

128

Question 60: Palette 60/228

JEE Mains 2024 MCQ

Let the first three terms 2, p and q, with $q \neq 2$, of a G.P. be respectively the $7^{\text {th }}, 8^{\text {th }}$ and $13^{\text {th }}$ terms of an A.P. If the $5^{\text {th }}$ term of the G.P. is the $n^{\text {th }}$ term of the A.P., then $n$ is equal to:

A

151

B

177

C

163

D

169

Question 61: Palette 61/228

JEE Mains 2024 MCQ

Let $S_n$ denote the sum of the first $n$ terms of an arithmetic progression. If $S_{10}=390$ and the ratio of the tenth and the fifth terms is $15: 7$, then $\mathrm{S}_{15}-\mathrm{S}_5$ is equal to :

A

800

B

890

C

790

D

690

Question 62: Palette 62/228

JEE Mains 2024 MCQ

Let $3, a, b, c$ be in A.P. and $3, a-1, b+1, c+9$ be in G.P. Then, the arithmetic mean of $a, b$ and $c$ is :

A

-4

B

-1

C

13

D

11

Question 63: Palette 63/228

JEE Mains 2024 MCQ

Let $2^{\text {nd }}, 8^{\text {th }}$ and $44^{\text {th }}$ terms of a non-constant A. P. be respectively the $1^{\text {st }}, 2^{\text {nd }}$ and $3^{\text {rd }}$ terms of a G. P. If the first term of the A. P. is 1, then the sum of its first 20 terms is equal to -

A

990

B

980

C

960

D

970

Question 64: Palette 64/228

JEE Mains 2024 MCQ

For $0 < c < b < a$, let $(a+b-2 c) x^2+(b+c-2 a) x+(c+a-2 b)=0$ and $\alpha \neq 1$ be one of its root. Then, among the two statements

(I) If $\alpha \in(-1,0)$, then $b$ cannot be the geometric mean of $a$ and $c$

(II) If $\alpha \in(0,1)$, then $b$ may be the geometric mean of $a$ and $c$

A

only (II) is true

B

Both (I) and (II) are true

C

only (I) is true

D

Neither (I) nor (II) is true

Question 65: Palette 65/228

JEE Mains 2024 MCQ

The sum of the series $\frac{1}{1-3 \cdot 1^2+1^4}+\frac{2}{1-3 \cdot 2^2+2^4}+\frac{3}{1-3 \cdot 3^2+3^4}+\ldots$ up to 10 -terms is

A

$\frac{45}{109}$

B

$-\frac{55}{109}$

C

$\frac{55}{109}$

D

$-\frac{45}{109}$

Question 66: Palette 66/228

JEE Mains 2024 MCQ

Let $a$ and $b$ be be two distinct positive real numbers. Let $11^{\text {th }}$ term of a GP, whose first term is $a$ and third term is $b$, is equal to $p^{\text {th }}$ term of another GP, whose first term is $a$ and fifth term is $b$. Then $p$ is equal to

A

20

B

24

C

21

D

25

Question 67: Palette 67/228

JEE Mains 2024 MCQ

Let $S_n$ denote the sum of first $n$ terms of an arithmetic progression. If $S_{20}=790$ and $S_{10}=145$, then $\mathrm{S}_{15}-\mathrm{S}_5$ is :

A

405

B

390

C

410

D

395

Question 68: Palette 68/228

JEE Mains 2024 MCQ

If $\log _e \mathrm{a}, \log _e \mathrm{~b}, \log _e \mathrm{c}$ are in an A.P. and $\log _e \mathrm{a}-\log _e 2 \mathrm{~b}, \log _e 2 \mathrm{~b}-\log _e 3 \mathrm{c}, \log _e 3 \mathrm{c} -\log _e$ a are also in an A.P, then $a: b: c$ is equal to

A

$6: 3: 2$

B

$9: 6: 4$

C

$25: 10: 4$

D

$16: 4: 1$

Question 69: Palette 69/228

JEE Mains 2024 MCQ

If each term of a geometric progression $a_1, a_2, a_3, \ldots$ with $a_1=\frac{1}{8}$ and $a_2 \neq a_1$, is the arithmetic mean of the next two terms and $S_n=a_1+a_2+\ldots . .+a_n$, then $S_{20}-S_{18}$ is equal to

A

$-2^{15}$

B

$2^{15}$

C

$-2^{18}$

D

$2^{18}$

Question 70: Palette 70/228

JEE Mains 2024 MCQ

If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P, then the common ratio of the G.P. is equal to

A

7

B

6

C

5

D

4

Question 71: Palette 71/228

JEE Mains 2024 MCQ

In an A.P., the sixth term $a_6=2$. If the product $a_1 a_4 a_5$ is the greatest, then the common difference of the A.P. is equal to

A

$\frac{2}{3}$

B

$\frac{5}{8}$

C

$\frac{3}{2}$

D

$\frac{8}{5}$

Question 72: Palette 72/228

JEE Mains 2024 MCQ

$\text { The } 20^{\text {th }} \text { term from the end of the progression } 20,19 \frac{1}{4}, 18 \frac{1}{2}, 17 \frac{3}{4}, \ldots,-129 \frac{1}{4} \text { is : }$

A

$-115$

B

$-100$

C

$-110$

D

$-118$

Question 73: Palette 73/228

JEE Mains 2024 MCQ

The number of common terms in the progressions

$4,9,14,19, \ldots \ldots$, up to $25^{\text {th }}$ term and

$3,6,9,12, \ldots \ldots$, up to $37^{\text {th }}$ term is :

A

9

B

8

C

5

D

7

Question 74: Palette 74/228

JEE Mains 2023 MCQ

Let $A_{1}$ and $A_{2}$ be two arithmetic means and $G_{1}, G_{2}, G_{3}$ be three geometric

means of two distinct positive numbers. Then $G_{1}^{4}+G_{2}^{4}+G_{3}^{4}+G_{1}^{2} G_{3}^{2}$ is equal to :

A

$\left(A_{1}+A_{2}\right)^{2} G_{1} G_{3}$

B

$\left(A_{1}+A_{2}\right) G_{1}^{2} G_{3}^{2}$

C

$2\left(A_{1}+A_{2}\right) G_{1}^{2} G_{3}^{2}$

D

$2\left(A_{1}+A_{2}\right) G_{1} G_{3}$

Question 75: Palette 75/228

JEE Mains 2023 MCQ

Let a$_1$, a$_2$, a$_3$, .... be a G.P. of increasing positive numbers. Let the sum of its 6^th and 8^th terms be 2 and the product of its 3^rd and 5^th terms be $\frac{1}{9}$. Then $6(a_2+a_4)(a_4+a_6)$ is equal to

A

2$\sqrt2$

B

2

C

3$\sqrt3$

D

3

Question 76: Palette 76/228

JEE Mains 2023 MCQ

Let $s_{1}, s_{2}, s_{3}, \ldots, s_{10}$ respectively be the sum to 12 terms of 10 A.P. s whose first terms are $1,2,3, \ldots .10$ and the common differences are $1,3,5, \ldots \ldots, 19$ respectively. Then $\sum_\limits{i=1}^{10} s_{i}$ is equal to :

A

7360

B

7220

C

7260

D

7380

Question 77: Palette 77/228

JEE Mains 2023 MCQ

Let $< a_{\mathrm{n}} > $ be a sequence such that $a_{1}+a_{2}+\ldots+a_{n}=\frac{n^{2}+3 n}{(n+1)(n+2)}$. If $28 \sum_\limits{k=1}^{10} \frac{1}{a_{k}}=p_{1} p_{2} p_{3} \ldots p_{m}$, where $\mathrm{p}_{1}, \mathrm{p}_{2}, \ldots ., \mathrm{p}_{\mathrm{m}}$ are the first $\mathrm{m}$ prime numbers, then $\mathrm{m}$ is equal to

A

5

B

7

C

6

D

8

Question 78: Palette 78/228

JEE Mains 2023 MCQ

Let $a, b, c$ and $d$ be positive real numbers such that $a+b+c+d=11$. If the maximum value of $a^{5} b^{3} c^{2} d$ is $3750 \beta$, then the value of $\beta$ is

A

110

B

108

C

90

D

55

Question 79: Palette 79/228

JEE Mains 2023 MCQ

Let $x_{1}, x_{2}, \ldots, x_{100}$ be in an arithmetic progression, with $x_{1}=2$ and their mean equal to 200 . If $y_{i}=i\left(x_{i}-i\right), 1 \leq i \leq 100$, then the mean of $y_{1}, y_{2}, \ldots, y_{100}$ is :

A

10051.50

B

10049.50

C

10100

D

10101.50

Question 80: Palette 80/228

JEE Mains 2023 MCQ

If $\mathrm{S}_{n}=4+11+21+34+50+\ldots$ to $n$ terms, then $\frac{1}{60}\left(\mathrm{~S}_{29}-\mathrm{S}_{9}\right)$ is equal to :

A

227

B

226

C

220

D

223

Question 81: Palette 81/228

JEE Mains 2023 MCQ

Let the first term $\alpha$ and the common ratio r of a geometric progression be positive integers. If the sum of squares of its first three terms is 33033, then the sum of these three terms is equal to

A

241

B

231

C

220

D

210

Question 82: Palette 82/228

JEE Mains 2023 MCQ

Let $\mathrm{a}_{\mathrm{n}}$ be the $\mathrm{n}^{\text {th }}$ term of the series $5+8+14+23+35+50+\ldots$ and $\mathrm{S}_{\mathrm{n}}=\sum_\limits{k=1}^{n} a_{k}$. Then $\mathrm{S}_{30}-a_{40}$ is equal to :

A

11280

B

11290

C

11310

D

11260

Question 83: Palette 83/228

JEE Mains 2023 MCQ

Let $S_{K}=\frac{1+2+\ldots+K}{K}$ and $\sum_\limits{j=1}^{n} S_{j}^{2}=\frac{n}{A}\left(B n^{2}+C n+D\right)$, where $A, B, C, D \in \mathbb{N}$ and $A$ has least value. Then

A

$A+B+C+D$ is divisible by 5

B

$A+C+D$ is not divisible by $B$

C

$A+B=5(D-C)$

D

$A+B$ is divisible by $\mathrm{D}$

Question 84: Palette 84/228

JEE Mains 2023 MCQ

If $\operatorname{gcd}~(\mathrm{m}, \mathrm{n})=1$ and $1^{2}-2^{2}+3^{2}-4^{2}+\ldots . .+(2021)^{2}-(2022)^{2}+(2023)^{2}=1012 ~m^{2} n$ then $m^{2}-n^{2}$ is equal to :

A

220

B

200

C

240

D

180

Question 85: Palette 85/228

JEE Mains 2023 MCQ

The sum of the first $20$ terms of the series $5+11+19+29+41+\ldots$ is :

A

3420

B

3450

C

3250

D

3520

Question 86: Palette 86/228

JEE Mains 2023 MCQ

The sum $\sum\limits_{n = 1}^\infty {{{2{n^2} + 3n + 4} \over {(2n)!}}} $ is equal to :

A

${{11e} \over 2} + {7 \over {2e}}$

B

${{13e} \over 4} + {5 \over {4e}} - 4$

C

${{11e} \over 2} + {7 \over {2e}} - 4$

D

${{13e} \over 4} + {5 \over {4e}}$

Question 87: Palette 87/228

JEE Mains 2023 MCQ

The sum of 10 terms of the series

${1 \over {1 + {1^2} + {1^4}}} + {2 \over {1 + {2^2} + {2^4}}} + {3 \over {1 + {3^2} + {3^4}}}\, + \,....$ is

A

${{58} \over {111}}$

B

${{56} \over {111}}$

C

${{55} \over {111}}$

D

${{59} \over {111}}$

Question 88: Palette 88/228

JEE Mains 2023 MCQ

Let $a_1, a_2, a_3, \ldots$ be an A.P. If $a_7=3$, the product $a_1 a_4$ is minimum and the sum of its first $n$ terms is zero, then $n !-4 a_{n(n+2)}$ is equal to :

A

24

B

$\frac{381}{4}$

C

9

D

$\frac{33}{4}$

Question 89: Palette 89/228

JEE Mains 2023 MCQ

If the sum and product of four positive consecutive terms of a G.P., are 126 and 1296 , respectively, then the sum of common ratios of all such GPs is

A

7

B

14

C

3

D

$\frac{9}{2}$

Question 90: Palette 90/228

JEE Mains 2023 MCQ

Let $a, b, c>1, a^3, b^3$ and $c^3$ be in A.P., and $\log _a b, \log _c a$ and $\log _b c$ be in G.P. If the sum of first 20 terms of an A.P., whose first term is $\frac{a+4 b+c}{3}$ and the common difference is $\frac{a-8 b+c}{10}$ is $-444$, then $a b c$ is equal to :

A

343

B

216

C

$\frac{343}{8}$

D

$\frac{125}{8}$

Question 91: Palette 91/228

JEE Mains 2023 MCQ

If ${a_n} = {{ - 2} \over {4{n^2} - 16n + 15}}$, then ${a_1} + {a_2}\, + \,....\, + \,{a_{25}}$ is equal to :

A

${{51} \over {144}}$

B

${{49} \over {138}}$

C

${{50} \over {141}}$

D

${{52} \over {147}}$

Question 92: Palette 92/228

JEE Mains 2023 MCQ

For three positive integers p, q, r, ${x^{p{q^2}}} = {y^{qr}} = {z^{{p^2}r}}$ and r = pq + 1 such that 3, 3 log$_yx$, 3 log$_zy$, 7 log$_xz$ are in A.P. with common difference $\frac{1}{2}$. Then r-p-q is equal to

A

12

B

$-$6

C

6

D

2

Question 93: Palette 93/228

JEE Mains 2022 MCQ

$ \begin{aligned} &\text { Let }\left\{a_{n}\right\}_{n=0}^{\infty} \text { be a sequence such that } a_{0}=a_{1}=0 \text { and } \\\\ &a_{n+2}=3 a_{n+1}-2 a_{n}+1, \forall n \geq 0 . \end{aligned} $

Then $a_{25} a_{23}-2 a_{25} a_{22}-2 a_{23} a_{24}+4 a_{22} a_{24}$ is equal to

A

483

B

528

C

575

D

624

Question 94: Palette 94/228

JEE Mains 2022 MCQ

Consider the sequence $a_{1}, a_{2}, a_{3}, \ldots$ such that $a_{1}=1, a_{2}=2$ and $a_{n+2}=\frac{2}{a_{n+1}}+a_{n}$ for $\mathrm{n}=1,2,3, \ldots .$ If $\left(\frac{\mathrm{a}_{1}+\frac{1}{\mathrm{a}_{2}}}{\mathrm{a}_{3}}\right) \cdot\left(\frac{\mathrm{a}_{2}+\frac{1}{\mathrm{a}_{3}}}{\mathrm{a}_{4}}\right) \cdot\left(\frac{\mathrm{a}_{3}+\frac{1}{\mathrm{a}_{4}}}{\mathrm{a}_{5}}\right) \ldots\left(\frac{\mathrm{a}_{30}+\frac{1}{\mathrm{a}_{31}}}{\mathrm{a}_{32}}\right)=2^{\alpha}\left({ }^{61} \mathrm{C}_{31}\right)$, then $\alpha$ is equal to :

A

$-$30

B

$-$31

C

$-$60

D

$-$61

Question 95: Palette 95/228

JEE Mains 2022 MCQ

Let the sum of an infinite G.P., whose first term is a and the common ratio is r, be 5 . Let the sum of its first five terms be $\frac{98}{25}$. Then the sum of the first 21 terms of an AP, whose first term is $10\mathrm{a r}, \mathrm{n}^{\text {th }}$ term is $\mathrm{a}_{\mathrm{n}}$ and the common difference is $10 \mathrm{ar}^{2}$, is equal to :

A

$21 \,\mathrm{a}_{11}$

B

$22 \,\mathrm{a}_{11}$

C

$15 \,\mathrm{a}_{16}$

D

$14 \,\mathrm{a}_{16}$

Question 96: Palette 96/228

JEE Mains 2022 MCQ

Suppose $a_{1}, a_{2}, \ldots, a_{n}$, .. be an arithmetic progression of natural numbers. If the ratio of the sum of first five terms to the sum of first nine terms of the progression is $5: 17$ and , $110 < {a_{15}} < 120$, then the sum of the first ten terms of the progression is equal to

A

290

B

380

C

460

D

510

Question 97: Palette 97/228

JEE Mains 2022 MCQ

Consider two G.Ps. 2, 2², 2³, ..... and 4, 4², 4³, .... of 60 and n terms respectively. If the geometric mean of all the 60 + n terms is ${(2)^{{{225} \over 8}}}$, then $\sum\limits_{k = 1}^n {k(n - k)} $ is equal to :

A

560

B

1540

C

1330

D

2600

Question 98: Palette 98/228

JEE Mains 2022 MCQ

The sum $\sum\limits_{n = 1}^{21} {{3 \over {(4n - 1)(4n + 3)}}} $ is equal to

A

$\frac{7}{87}$

B

$\frac{7}{29}$

C

$\frac{14}{87}$

D

$\frac{21}{29}$

Question 99: Palette 99/228

JEE Mains 2022 MCQ

The value of $1 + {1 \over {1 + 2}} + {1 \over {1 + 2 + 3}} + \,\,....\,\, + \,\,{1 \over {1 + 2 + 3 + \,\,.....\,\, + \,\,11}}$ is equal to:

A

${{20} \over {11}}$

B

${{11} \over {6}}$

C

${{241} \over {132}}$

D

${{21} \over {11}}$

Question 100: Palette 100/228

JEE Mains 2022 MCQ

The sum of the infinite series $1 + {5 \over 6} + {{12} \over {{6^2}}} + {{22} \over {{6^3}}} + {{35} \over {{6^4}}} + {{51} \over {{6^5}}} + {{70} \over {{6^6}}} + \,\,.....$ is equal to :

A

${{425} \over {216}}$

B

${{429} \over {216}}$

C

${{288} \over {125}}$

D

${{280} \over {125}}$

Question 101: Palette 101/228

JEE Mains 2022 MCQ

Let $\{ {a_n}\} _{n = 0}^\infty $ be a sequence such that ${a_0} = {a_1} = 0$ and ${a_{n + 2}} = 2{a_{n + 1}} - {a_n} + 1$ for all n $\ge$ 0. Then, $\sum\limits_{n = 2}^\infty {{{{a_n}} \over {{7^n}}}} $ is equal to:

A

${6 \over {343}}$

B

${7 \over {216}}$

C

${8 \over {343}}$

D

${{49} \over {216}}$

Question 102: Palette 102/228

JEE Mains 2022 MCQ

If n arithmetic means are inserted between a and 100 such that the ratio of the first mean to the last mean is 1 : 7 and a + n = 33, then the value of n is :

A

21

B

22

C

23

D

24

Question 103: Palette 103/228

JEE Mains 2022 MCQ

Let A₁, A₂, A₃, ....... be an increasing geometric progression of positive real numbers. If A₁A₃A₅A₇ = ${1 \over {1296}}$ and A₂ + A₄ = ${7 \over {36}}$, then the value of A₆ + A₈ + A₁₀ is equal to

A

33

B

37

C

43

D

47

Question 104: Palette 104/228

JEE Mains 2022 MCQ

Let $S = 2 + {6 \over 7} + {{12} \over {{7^2}}} + {{20} \over {{7^3}}} + {{30} \over {{7^4}}} + \,.....$. Then 4S is equal to

A

${\left( {{7 \over 3}} \right)^2}$

B

${{{7^3}} \over {{3^2}}}$

C

${\left( {{7 \over 3}} \right)^3}$

D

${{{7^2}} \over {{3^3}}}$

Question 105: Palette 105/228

JEE Mains 2022 MCQ

If a₁, a₂, a₃ ...... and b₁, b₂, b₃ ....... are A.P., and a₁ = 2, a₁₀ = 3, a₁b₁ = 1 = a₁₀b₁₀, then a₄ b₄ is equal to -

A

${{35} \over {27}}$

B

1

C

${{27} \over {28}}$

D

${{28} \over {27}}$

Question 106: Palette 106/228

JEE Mains 2022 MCQ

$x = \sum\limits_{n = 0}^\infty {{a^n},y = \sum\limits_{n = 0}^\infty {{b^n},z = \sum\limits_{n = 0}^\infty {{c^n}} } } $, where a, b, c are in A.P. and |a| < 1, |b| < 1, |c| < 1, abc $\ne$ 0, then :

A

x, y, z are in A.P.

B

x, y, z are in G.P.

C

${1 \over x}$, ${1 \over y}$, ${1 \over z}$ are in A.P.

D

${1 \over x}$ + ${1 \over y}$ + ${1 \over z}$ = 1 $-$ (a + b + c)

Question 107: Palette 107/228

JEE Mains 2022 MCQ

If $A = \sum\limits_{n = 1}^\infty {{1 \over {{{\left( {3 + {{( - 1)}^n}} \right)}^n}}}} $ and $B = \sum\limits_{n = 1}^\infty {{{{{( - 1)}^n}} \over {{{\left( {3 + {{( - 1)}^n}} \right)}^n}}}} $, then ${A \over B}$ is equal to :

A

${{11} \over 9}$

B

1

C

$-$${{11} \over 9}$

D

$-$${{11} \over 3}$

Question 108: Palette 108/228

JEE Mains 2022 MCQ

The sum 1 + 2 . 3 + 3 . 3² + ......... + 10 . 3⁹ is equal to :

A

${{2\,.\,{3^{12}} + 10} \over 4}$

B

${{19\,.\,{3^{10}} + 1} \over 4}$

C

$5\,.\,{3^{10}} - 2$

D

${{9\,.\,{3^{10}} + 1} \over 2}$

Question 109: Palette 109/228

JEE Mains 2022 MCQ

Let x, y > 0. If x³y² = 2¹⁵, then the least value of 3x + 2y is

A

30

B

32

C

36

D

40

Question 110: Palette 110/228

JEE Mains 2022 MCQ

If $\{ {a_i}\} _{i = 1}^n$, where n is an even integer, is an arithmetic progression with common difference 1, and $\sum\limits_{i = 1}^n {{a_i} = 192} ,\,\sum\limits_{i = 1}^{n/2} {{a_{2i}} = 120} $, then n is equal to :

A

48

B

96

C

92

D

104

Question 111: Palette 111/228

JEE Mains 2021 MCQ

Let S_n = 1 . (n $-$ 1) + 2 . (n $-$ 2) + 3 . (n $-$ 3) + ..... + (n $-$ 1) . 1, n $\ge$ 4.

The sum $\sum\limits_{n = 4}^\infty {\left( {{{2{S_n}} \over {n!}} - {1 \over {(n - 2)!}}} \right)} $ is equal to :

A

${{e - 1} \over 3}$

B

${{e - 2} \over 6}$

C

${e \over 3}$

D

${e \over 6}$

Question 112: Palette 112/228

JEE Mains 2021 MCQ

Let a₁, a₂, ..........., a₂₁ be an AP such that $\sum\limits_{n = 1}^{20} {{1 \over {{a_n}{a_{n + 1}}}} = {4 \over 9}} $. If the sum of this AP is 189, then a₆a₁₆ is equal to :

A

57

B

72

C

48

D

36

Question 113: Palette 113/228

JEE Mains 2021 MCQ

Let a₁, a₂, a₃, ..... be an A.P. If ${{{a_1} + {a_2} + .... + {a_{10}}} \over {{a_1} + {a_2} + .... + {a_p}}} = {{100} \over {{p^2}}}$, p $\ne$ 10, then ${{{a_{11}}} \over {{a_{10}}}}$ is equal to :

A

${{19} \over {21}}$

B

${{100} \over {121}}$

C

${{21} \over {19}}$

D

${{121} \over {100}}$

Question 114: Palette 114/228

JEE Mains 2021 MCQ

The sum of 10 terms of the series

${3 \over {{1^2} \times {2^2}}} + {5 \over {{2^2} \times {3^2}}} + {7 \over {{3^2} \times {4^2}}} + ....$ is :

A

1

B

${{120} \over {121}}$

C

${{99} \over {100}}$

D

${{143} \over {144}}$

Question 115: Palette 115/228

JEE Mains 2021 MCQ

Three numbers are in an increasing geometric progression with common ratio r. If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference d. If the fourth term of GP is 3 r², then r² $-$ d is equal to :

A

7 $-$ 7$\sqrt 3 $

B

7 + $\sqrt 3 $

C

7 $-$ $\sqrt 3 $

D

7 + 3$\sqrt 3 $

Question 116: Palette 116/228

JEE Mains 2021 MCQ

If 0 < x < 1 and $y = {1 \over 2}{x^2} + {2 \over 3}{x^3} + {3 \over 4}{x^4} + ....$, then the value of e^{1 + y} at $x = {1 \over 2}$ is :

A

${1 \over 2}{e^2}$

B

2e

C

${1 \over 2}\sqrt e $

D

2e²

Question 117: Palette 117/228

JEE Mains 2021 MCQ

If 0 < x < 1, then ${3 \over 2}{x^2} + {5 \over 3}{x^3} + {7 \over 4}{x^4} + .....$, is equal to :

A

$x\left( {{{1 + x} \over {1 - x}}} \right) + {\log _e}(1 - x)$

B

$x\left( {{{1 - x} \over {1 + x}}} \right) + {\log _e}(1 - x)$

C

${{1 - x} \over {1 + x}} + {\log _e}(1 - x)$

D

${{1 + x} \over {1 - x}} + {\log _e}(1 - x)$

Question 118: Palette 118/228

JEE Mains 2021 MCQ

If for x, y $\in$ R, x > 0, y = log₁₀x + log₁₀x^1/3 + log₁₀x^1/9 + ...... upto $\infty$ terms

and ${{2 + 4 + 6 + .... + 2y} \over {3 + 6 + 9 + ..... + 3y}} = {4 \over {{{\log }_{10}}x}}$, then the ordered pair (x, y) is equal to :

A

(10⁶, 6)

B

(10⁴, 6)

C

(10², 3)

D

(10⁶, 9)

Question 119: Palette 119/228

JEE Mains 2021 MCQ

The sum of the series

${1 \over {x + 1}} + {2 \over {{x^2} + 1}} + {{{2^2}} \over {{x^4} + 1}} + ...... + {{{2^{100}}} \over {{x^{{2^{100}}}} + 1}}$ when x = 2 is :

A

$1 + {{{2^{101}}} \over {{4^{101}} - 1}}$

B

$1 + {{{2^{100}}} \over {{4^{101}} - 1}}$

C

$1 - {{{2^{100}}} \over {{4^{100}} - 1}}$

D

$1 - {{{2^{101}}} \over {{2^{400}} - 1}}$

Question 120: Palette 120/228

JEE Mains 2021 MCQ

If the sum of an infinite GP a, ar, ar², ar³, ....... is 15 and the sum of the squares of its each term is 150, then the sum of ar², ar⁴, ar⁶, ....... is :

A

${5 \over 2}$

B

${1 \over 2}$

C

${25 \over 2}$

D

${9 \over 2}$

Question 121: Palette 121/228

JEE Mains 2021 MCQ

Let S_n be the sum of the first n terms of an arithmetic progression. If S_3n = 3S_2n, then the value of ${{{S_{4n}}} \over {{S_{2n}}}}$ is :

A

6

B

4

C

2

D

8

Question 122: Palette 122/228

JEE Mains 2021 MCQ

Let S_n denote the sum of first n-terms of an arithmetic progression. If S₁₀ = 530, S₅ = 140, then S₂₀ $-$ S₆ is equal to:

A

1862

B

1842

C

1852

D

1872

Question 123: Palette 123/228

JEE Mains 2021 MCQ

If sum of the first 21 terms of the series ${\log _{{9^{1/2}}}}x + {\log _{{9^{1/3}}}}x + {\log _{{9^{1/4}}}}x + .......$, where x > 0 is 504, then x is equal to

A

243

B

9

C

7

D

81

Question 124: Palette 124/228

JEE Mains 2021 MCQ

Let S₁ be the sum of first 2n terms of an arithmetic progression. Let S₂ be the sum of first 4n terms of the same arithmetic progression. If (S₂ $-$ S₁) is 1000, then the sum of the first 6n terms of the arithmetic progression is equal to :

A

7000

B

1000

C

3000

D

5000

Question 125: Palette 125/228

JEE Mains 2021 MCQ

If $\alpha$, $\beta$ are natural numbers such that
100^$\alpha$ $-$ 199$\beta$ = (100)(100) + (99)(101) + (98)(102) + ...... + (1)(199), then the slope of the line passing through ($\alpha$, $\beta$) and origin is :

A

540

B

550

C

530

D

510

Question 126: Palette 126/228

JEE Mains 2021 MCQ

${1 \over {{3^2} - 1}} + {1 \over {{5^2} - 1}} + {1 \over {{7^2} - 1}} + .... + {1 \over {{{(201)}^2} - 1}}$ is equal to

A

${{101} \over {404}}$

B

${{25} \over {101}}$

C

${{101} \over {408}}$

D

${{99} \over {400}}$

Question 127: Palette 127/228

JEE Mains 2021 MCQ

The sum of the series

$\sum\limits_{n = 1}^\infty {{{{n^2} + 6n + 10} \over {(2n + 1)!}}} $ is equal to :

A

${{41} \over 8}e + {{19} \over 8}{e^{ - 1}} - 10$

B

${{41} \over 8}e - {{19} \over 8}{e^{ - 1}} - 10$

C

${{41} \over 8}e + {{19} \over 8}{e^{ - 1}} + 10$

D

$ - {{41} \over 8}e + {{19} \over 8}{e^{ - 1}} - 10$

Question 128: Palette 128/228

JEE Mains 2021 MCQ

The sum of the infinite series
$1 + {2 \over 3} + {7 \over {{3^2}}} + {{12} \over {{3^3}}} + {{17} \over {{3^4}}} + {{22} \over {{3^5}}} + ......$ is equal to :

A

${9 \over 4}$

B

${13 \over 4}$

C

${15 \over 4}$

D

${11 \over 4}$

Question 129: Palette 129/228

JEE Mains 2021 MCQ

In an increasing geometric series, the sum of the second and the sixth term is ${{25} \over 2}$ and the product of the third and fifth term is 25. Then, the sum of 4^th, 6^th and 8^th terms is equal to :

A

30

B

32

C

26

D

35

Question 130: Palette 130/228

JEE Mains 2021 MCQ

The minimum value of $f(x) = {a^{{a^x}}} + {a^{1 - {a^x}}}$, where a, $x \in R$ and a > 0, is equal to :

A

$a + {1 \over a}$

B

2a

C

a + 1

D

$2\sqrt a $

Question 131: Palette 131/228

JEE Mains 2021 MCQ

If $0 < \theta ,\phi < {\pi \over 2},x = \sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\theta } ,y = \sum\limits_{n = 0}^\infty {{{\sin }^{2n}}\phi } $ and $z = \sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\theta .{{\sin }^{2n}}\phi } $ then :

A

xy $-$ z = (x + y)z

B

xyz = 4

C

xy + z = (x + y)z

D

xy + yz + zx = z

Question 132: Palette 132/228

JEE Mains 2020 MCQ

The common difference of the A.P.
b₁, b₂, … , b_m is 2 more than the common
difference of A.P. a₁, a₂, …, a_n. If
a₄₀ = –159, a₁₀₀ = –399 and b₁₀₀ = a₇₀, then b₁ is equal to :

A

127

B

81

C

–127

D

-81

Question 133: Palette 133/228

JEE Mains 2020 MCQ

Let a , b, c , d and p be any non zero distinct real numbers such that
(a² + b² + c²)p² – 2(ab + bc + cd)p + (b² + c² + d²) = 0. Then :

A

a, c, p are in G.P.

B

a, b, c, d are in G.P.

C

a, b, c, d are in A.P.

D

a, c, p are in A.P.

Question 134: Palette 134/228

JEE Mains 2020 MCQ

If the sum of the first 20 terms of the series
${\log _{\left( {{7^{1/2}}} \right)}}x + {\log _{\left( {{7^{1/3}}} \right)}}x + {\log _{\left( {{7^{1/4}}} \right)}}x + ...$ is 460,
then x is equal to :

A

e²

B

7^1/2

C

7²

D

7^46/21

Question 135: Palette 135/228

JEE Mains 2020 MCQ

If the sum of the second, third and fourth terms of a positive term G.P. is 3 and the sum of its sixth, seventh and eighth terms is 243, then the sum of the first 50 terms of this G.P. is :

A

${2 \over {13}}\left( {{3^{50}} - 1} \right)$

B

${1 \over {13}}\left( {{3^{50}} - 1} \right)$

C

${1 \over {26}}\left( {{3^{49}} - 1} \right)$

D

${1 \over {26}}\left( {{3^{50}} - 1} \right)$

Question 136: Palette 136/228

JEE Mains 2020 MCQ

If ${3^{2\sin 2\alpha - 1}}$, 14 and ${3^{4 - 2\sin 2\alpha }}$ are the first three terms of an A.P. for some $\alpha $, then the sixth terms of this A.P. is:

A

66

B

81

C

65

D

78

Question 137: Palette 137/228

JEE Mains 2020 MCQ

If 2¹⁰ + 2⁹.3¹ + 2⁸ .3² +.....+ 2.3⁹ + 3¹⁰ = S - 2¹¹, then S is equal to :

A

${{{3^{11}}} \over 2} + {2^{10}}$

B

3¹¹ — 2¹²

C

2.3¹¹

D

3¹¹

Question 138: Palette 138/228

JEE Mains 2020 MCQ

Let a₁, a₂, ..., an be a given A.P. whose
common difference is an integer and
S_n = a₁ + a₂ + .... + a_n. If a₁ = 1, a_n = 300 and 15 $ \le $ n $ \le $ 50, then
the ordered pair (S_n-4, a_n–4) is equal to:

A

(2480, 249)

B

(2480, 248)

C

(2490, 248)

D

(2490, 249)

Question 139: Palette 139/228

JEE Mains 2020 MCQ

The minimum value of 2^sinx + 2^cosx is :

A

${2^{-1 + \sqrt 2 }}$

B

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

C

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

D

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

Question 140: Palette 140/228

JEE Mains 2020 MCQ

If 1+(1–2².1)+(1–4².3)+(1-6².5)+......+(1-20².19)= $\alpha $ - 220$\beta $,
then an ordered pair $\left( {\alpha ,\beta } \right)$ is equal to:

A

(11, 103)

B

(10, 103)

C

(10, 97)

D

(11, 97)

Question 141: Palette 141/228

JEE Mains 2020 MCQ

If the sum of the series

20 + 19${3 \over 5}$ + 19${1 \over 5}$ + 18${4 \over 5}$ + ...

upto nth term is 488 and the n^th term is negative, then :

A

n = 41

B

n = 60

C

n^th term is –4

D

n^th term is -4${2 \over 5}$

Question 142: Palette 142/228

JEE Mains 2020 MCQ

If the first term of an A.P. is 3 and the sum of its first 25 terms is equal to the sum of its next 15 terms, then the common difference of this A.P. is :

A

${1 \over 4}$

B

${1 \over 5}$

C

${1 \over 7}$

D

${1 \over 6}$

Question 143: Palette 143/228

JEE Mains 2020 MCQ

Let S be the sum of the first 9 terms of the series :
{x + k$a$} + {x² + (k + 2)$a$} + {x³ + (k + 4)$a$}
+ {x⁴ + (k + 6)$a$} + .... where a $ \ne $ 0 and x $ \ne $ 1.

If S = ${{{x^{10}} - x + 45a\left( {x - 1} \right)} \over {x - 1}}$, then k is equal to :

A

-3

B

1

C

-5

D

3

Question 144: Palette 144/228

JEE Mains 2020 MCQ

If the sum of first 11 terms of an A.P.,
a₁, a₂, a₃, .... is 0 (a $ \ne $ 0), then the sum of the A.P.,
a₁ , a₃ , a₅ ,....., a₂₃ is ka₁ , where k is equal to :

A

${{121} \over {10}}$

B

-${{121} \over {10}}$

C

${{72} \over 5}$

D

-${{72} \over 5}$

Question 145: Palette 145/228

JEE Mains 2020 MCQ

The sum of the first three terms of a G.P. is S and their product is 27. Then all such S lie in :

A

[-3, $\infty $)

B

(-$ \propto $, 9]

C

(-$ \propto $, -9] $ \cup $ [-3, $\infty $)

D

(-$ \propto $, -3] $ \cup $ [9, $\infty $)

Question 146: Palette 146/228

JEE Mains 2020 MCQ

If |x| < 1, |y| < 1 and x $ \ne $ y, then the sum to infinity of the following series

(x + y) + (x²+xy+y²) + (x³+x²y + xy²+y³) + ....

A

${{x + y - xy} \over {\left( {1 + x} \right)\left( {1 + y} \right)}}$

B

${{x + y - xy} \over {\left( {1 - x} \right)\left( {1 - y} \right)}}$

C

${{x + y + xy} \over {\left( {1 + x} \right)\left( {1 + y} \right)}}$

D

${{x + y + xy} \over {\left( {1 - x} \right)\left( {1 - y} \right)}}$

Question 147: Palette 147/228

JEE Mains 2020 MCQ

Let a_n be the n^th term of a G.P. of positive terms.

$\sum\limits_{n = 1}^{100} {{a_{2n + 1}} = 200} $ and $\sum\limits_{n = 1}^{100} {{a_{2n}} = 100} $,

then $\sum\limits_{n = 1}^{200} {{a_n}} $ is equal to :

A

150

B

175

C

225

D

300

Question 148: Palette 148/228

JEE Mains 2020 MCQ

The product ${2^{{1 \over 4}}}{.4^{{1 \over {16}}}}{.8^{{1 \over {48}}}}{.16^{{1 \over {128}}}}$ ... to $\infty $ is equal to :

A

${2^{{1 \over 4}}}$

B

${2^{{1 \over 2}}}$

C

1

D

2

Question 149: Palette 149/228

JEE Mains 2020 MCQ

If the 10^th term of an A.P. is ${1 \over {20}}$ and its 20^th term is ${1 \over {10}}$, then the sum of its first 200 terms is

A

100

B

$100{1 \over 2}$

C

$50{1 \over 4}$

D

50

Question 150: Palette 150/228

JEE Mains 2020 MCQ

Let ƒ : R $ \to $ R be such that for all x $ \in $ R
(2^1+x + 2^1–x), ƒ(x) and (3^x + 3^–x) are in A.P.,
then the minimum value of ƒ(x) is

A

2

B

0

C

3

D

4

Question 151: Palette 151/228

JEE Mains 2020 MCQ

If the sum of the first 40 terms of the series,
3 + 4 + 8 + 9 + 13 + 14 + 18 + 19 + ..... is (102)m, then m is equal to :

A

20

B

5

C

10

D

25

Question 152: Palette 152/228

JEE Mains 2020 MCQ

Let ${a_1}$ , ${a_2}$ , ${a_3}$ ,....... be a G.P. such that
${a_1}$ < 0, ${a_1}$ + ${a_2}$ = 4 and ${a_3}$ + ${a_4}$ = 16.
If $\sum\limits_{i = 1}^9 {{a_i}} = 4\lambda $, then $\lambda $ is equal to:

A

171

B

-171

C

-513

D

${{511} \over 3}$

Question 153: Palette 153/228

JEE Mains 2020 MCQ

Five numbers are in A.P. whose sum is 25 and product is 2520. If one of these five numbers is -${1 \over 2}$ , then the greatest number amongst them is:

A

${{21} \over 2}$

B

27

C

7

D

16

Question 154: Palette 154/228

JEE Mains 2019 MCQ

If a₁, a₂, a₃, ..... are in A.P. such that a₁ + a₇ + a₁₆ = 40, then the sum of the first 15 terms of this A.P. is :

A

120

B

200

C

150

D

280

Question 155: Palette 155/228

JEE Mains 2019 MCQ

For x $\varepsilon $ R, let [x] denote the greatest integer $ \le $ x, then the sum of the series $\left[ { - {1 \over 3}} \right] + \left[ { - {1 \over 3} - {1 \over {100}}} \right] + \left[ { - {1 \over 3} - {2 \over {100}}} \right] + .... + \left[ { - {1 \over 3} - {{99} \over {100}}} \right]$ is :

A

- 153

B

- 135

C

- 133

D

- 131

Question 156: Palette 156/228

JEE Mains 2019 MCQ

Let S_n denote the sum of the first n terms of an A.P. If S₄ = 16 and S₆= – 48, then S₁₀ is equal to :

A

- 320

B

- 380

C

- 460

D

- 210

Question 157: Palette 157/228

JEE Mains 2019 MCQ

Let a₁, a₂, a₃,......be an A.P. with a₆ = 2. Then the common difference of this A.P., which maximises the product a₁a₄a₅, is :

A

${3 \over 2}$

B

${6 \over 5}$

C

${8 \over 5}$

D

${2 \over 3}$

Question 158: Palette 158/228

JEE Mains 2019 MCQ

The sum
$1 + {{{1^3} + {2^3}} \over {1 + 2}} + {{{1^3} + {2^3} + {3^3}} \over {1 + 2 + 3}} + ...... + {{{1^3} + {2^3} + {3^3} + ... + {{15}^3}} \over {1 + 2 + 3 + ... + 15}}$$ - {1 \over 2}\left( {1 + 2 + 3 + ... + 15} \right)$ is equal to :

A

620

B

1240

C

1860

D

660

Question 159: Palette 159/228

JEE Mains 2019 MCQ

Let $a$, b and c be in G.P. with common ratio r, where $a$ $ \ne $ 0 and 0 < r $ \le $ ${1 \over 2}$ . If 3$a$, 7b and 15c are the first three terms of an A.P., then the 4^th term of this A.P. is :

A

$a$

B

${7 \over 3}a$

C

5$a$

D

${2 \over 3}a$

Question 160: Palette 160/228

JEE Mains 2019 MCQ

If a₁, a₂, a₃, ............... a_n are in A.P. and a₁ + a₄ + a₇ + ........... + a₁₆ = 114, then a₁ + a₆ + a₁₁ + a₁₆ is equal to :

A

38

B

98

C

76

D

64

Question 161: Palette 161/228

JEE Mains 2019 MCQ

The sum
${{3 \times {1^3}} \over {{1^3}}} + {{5 \times ({1^3} + {2^3})} \over {{1^2} + {2^2}}} + {{7 \times \left( {{1^3} + {2^3} + {3^3}} \right)} \over {{1^2} + {2^2} + {3^2}}} + .....$ upto 10 terms is:

A

600

B

660

C

680

D

620

Question 162: Palette 162/228

JEE Mains 2019 MCQ

Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If 99 more identical balls are addded to the total number of balls used in forming the equilaterial triangle, then all these balls can be arranged in a square whose each side contains exactly 2 balls less than the number of balls each side of the triangle contains. Then the number of balls used to form the equilateral triangle is :-

A

262

B

190

C

157

D

225

Question 163: Palette 163/228

JEE Mains 2019 MCQ

If the sum and product of the first three term in an A.P. are 33 and 1155, respectively, then a value of its 11^th term is :-

A

–25

B

–36

C

25

D

–35

Question 164: Palette 164/228

JEE Mains 2019 MCQ

The sum of the series 1 + 2 × 3 + 3 × 5 + 4 × 7 +.... upto 11th term is :-

A

945

B

916

C

915

D

946

Question 165: Palette 165/228

JEE Mains 2019 MCQ

Let the sum of the first n terms of a non-constant A.P., a₁, a₂, a₃, ..... be $50n + {{n(n - 7)} \over 2}A$, where A is a constant. If d is the common difference of this A.P., then the ordered pair (d, a₅₀) is equal to

A

(A, 50+45A)

B

(50, 50+45A)

C

(A, 50+46A)

D

(50, 50+46A)

Question 166: Palette 166/228

JEE Mains 2019 MCQ

If three distinct numbers a, b, c are in G.P. and the equations ax² + 2bx + c = 0 and dx² + 2ex + ƒ = 0 have a common root, then which one of the following statements is correct?

A

$d \over a$, $e \over b$, $f \over c$ are in G.P.

B

d, e, ƒ are in A.P

C

d, e, ƒ are in G.P

D

$d \over a$, $e \over b$, $f \over c$ are in A.P.

Question 167: Palette 167/228

JEE Mains 2019 MCQ

The sum $\sum\limits_{k = 1}^{20} {k{1 \over {{2^k}}}} $ is equal to

A

$2 - {11 \over {{2^{19}}}}$

B

$2 - {3 \over {{2^{17}}}}$

C

$1 - {11 \over {{2^{20}}}}$

D

$2 - {21 \over {{2^{20}}}}$

Question 168: Palette 168/228

JEE Mains 2019 MCQ

The sum of all natural numbers 'n' such that 100 < n < 200 and H.C.F. (91, n) > 1 is :

A

3221

B

3121

C

3203

D

3303

Question 169: Palette 169/228

JEE Mains 2019 MCQ

If sin⁴$\alpha $ + 4 cos⁴$\beta $ + 2 = 4$\sqrt 2 $ sin $\alpha $ cos $\beta $; $\alpha $, $\beta $ $ \in $ [0, $\pi $],
then cos($\alpha $ + $\beta $) $-$ cos($\alpha $ $-$ $\beta $) is equal to :

A

$ - \sqrt 2 $

B

0

C

$-$ 1

D

$\sqrt 2 $

Question 170: Palette 170/228

JEE Mains 2019 MCQ

If the sum of the first 15 terms of the series ${\left( {{3 \over 4}} \right)^3} + {\left( {1{1 \over 2}} \right)^3} + {\left( {2{1 \over 4}} \right)^3} + {3^3} + {\left( {3{3 \over 4}} \right)^3} + ....$ is equal to 225 k, then k is equal to :

A

9

B

108

C

27

D

54

Question 171: Palette 171/228

JEE Mains 2019 MCQ

If   ⁿC₄, ⁿC₅ and ⁿC₆ are in A.P., then n can be :

A

11

B

12

C

9

D

14

Question 172: Palette 172/228

JEE Mains 2019 MCQ

Let  S_k = ${{1 + 2 + 3 + .... + k} \over k}.$ If   $S_1^2 + S_2^2 + .....\, + S_{10}^2 = {5 \over {12}}$A,  then A is equal to :

A

283

B

156

C

301

D

303

Question 173: Palette 173/228

JEE Mains 2019 MCQ

The product of three consecutive terms of a G.P. is 512. If 4 is added to each of the first and the second of these terms, the three terms now form an A.P. Then the sum of the original three terms of the given G.P. is :

A

36

B

28

C

32

D

24

Question 174: Palette 174/228

JEE Mains 2019 MCQ

Let x, y be positive real numbers and m, n positive integers. The maximum value of the expression ${{{x^m}{y^n}} \over {\left( {1 + {x^{2m}}} \right)\left( {1 + {y^{2n}}} \right)}}$ is :

A

${1 \over 2}$

B

${1 \over 4}$

C

${{m + n} \over {6mn}}$

D

1

Question 175: Palette 175/228

JEE Mains 2019 MCQ

If 19^th term of a non-zero A.P. is zero, then its (49^th term) : (29^th term) is :

A

2 : 1

B

4 : 1

C

1 : 3

D

3 : 1

Question 176: Palette 176/228

JEE Mains 2019 MCQ

The sum of an infinite geometric series with positive terms is 3 and the sum of the cubes of its terms is ${{27} \over {19}}$.Then the common ratio of this series is :

A

${4 \over 9}$

B

${1 \over 3}$

C

${2 \over 3}$

D

${2 \over 9}$

Question 177: Palette 177/228

JEE Mains 2019 MCQ

Let a₁, a₂, . . . . . ., a₁₀ be a G.P.    If ${{{a_3}} \over {{a_1}}} = 25,$ then ${{{a_9}} \over {{a_5}}}$ equals

A

5³

B

2(5²)

C

4(5²)

D

5⁴

Question 178: Palette 178/228

JEE Mains 2019 MCQ

Let a₁, a₂, a₃, ..... a₁₀ be in G.P. with a_i > 0 for i = 1, 2, ….., 10 and S be the set of pairs (r, k), r, k $ \in $ N (the set of natural numbers) for which

$\left| {\matrix{ {{{\log }_e}\,{a_1}^r{a_2}^k} & {{{\log }_e}\,{a_2}^r{a_3}^k} & {{{\log }_e}\,{a_3}^r{a_4}^k} \cr {{{\log }_e}\,{a_4}^r{a_5}^k} & {{{\log }_e}\,{a_5}^r{a_6}^k} & {{{\log }_e}\,{a_6}^r{a_7}^k} \cr {{{\log }_e}\,{a_7}^r{a_8}^k} & {{{\log }_e}\,{a_8}^r{a_9}^k} & {{{\log }_e}\,{a_9}^r{a_{10}}^k} \cr } } \right|$ $=$ 0.

Then the number of elements in S, is -

A

10

B

4

C

2

D

infinitely many

Question 179: Palette 179/228

JEE Mains 2019 MCQ

The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is -

A

1356

B

1256

C

1365

D

1465

Question 180: Palette 180/228

JEE Mains 2019 MCQ

The sum of the following series

$1 + 6 + {{9\left( {{1^2} + {2^2} + {3^2}} \right)} \over 7} + {{12\left( {{1^2} + {2^2} + {3^2} + {4^2}} \right)} \over 9}$

       $ + {{15\left( {{1^2} + {2^2} + ... + {5^2}} \right)} \over {11}} + .....$ up to 15 terms, is :

A

7520

B

7510

C

7830

D

7820

Question 181: Palette 181/228

JEE Mains 2019 MCQ

Let a, b and c be the 7^th, 11^th and 13^th terms respectively of a non-constant A.P. If these are also three consecutive terms of a G.P., then ${a \over c}$ equal to :

A

2

B

${1 \over 2}$

C

${7 \over 13}$

D

4

Question 182: Palette 182/228

JEE Mains 2019 MCQ

If a, b, c be three distinct real numbers in G.P. and a + b + c = xb , then x cannot be

A

2

B

-3

C

4

D

-2

Question 183: Palette 183/228

JEE Mains 2019 MCQ

Let ${a_1},{a_2},.......,{a_{30}}$ be an A.P.,

$S = \sum\limits_{i = 1}^{30} {{a_i}} $ and $T = \sum\limits_{i = 1}^{15} {{a_{\left( {2i - 1} \right)}}} $.

If $a_5$ = 27 and S - 2T = 75, then $a_{10}$ is equal to :

A

47

B

42

C

52

D

57

Question 184: Palette 184/228

JEE Mains 2018 MCQ

Let ${1 \over {{x_1}}},{1 \over {{x_2}}},...,{1 \over {{x_n}}}\,\,$ (x_i $ \ne $ 0 for i = 1, 2, ..., n) be in A.P. such that x₁=4 and x₂₁ = 20. If n is the least positive integer for which ${x_n} > 50,$ then $\sum\limits_{i = 1}^n {\left( {{1 \over {{x_i}}}} \right)} $ is equal to :

A

${1 \over 8}$

B

3

C

${{13} \over 8}$

D

${{13} \over 4}$

Question 185: Palette 185/228

JEE Mains 2018 MCQ

The sum of the first 20 terms of the series

$1 + {3 \over 2} + {7 \over 4} + {{15} \over 8} + {{31} \over {16}} + ...,$ is :

A

$38 + {1 \over {{2^{19}}}}$

B

$38 + {1 \over {{2^{20}}}}$

C

$39 + {1 \over {{2^{20}}}}$

D

$39 + {1 \over {{2^{19}}}}$

Question 186: Palette 186/228

JEE Mains 2018 MCQ

Let A be the sum of the first 20 terms and B be the sum of the first 40 terms of the series
1² + 2.2² + 3² + 2.4² + 5² + 2.6² ...........
If B - 2A = 100$\lambda $, then $\lambda $ is equal to

A

496

B

232

C

248

D

464

Question 187: Palette 187/228

JEE Mains 2018 MCQ

Let ${a_1}$, ${a_2}$, ${a_3}$, ......... ,${a_{49}}$ be in A.P. such that

$\sum\limits_{k = 0}^{12} {{a_{4k + 1}}} = 416$ and ${a_9} + {a_{43}} = 66$.

$a_1^2 + a_2^2 + ....... + a_{17}^2 = 140m$, then m is equal to

A

33

B

66

C

68

D

34

Question 188: Palette 188/228

JEE Mains 2018 MCQ

Let    A_n = $\left( {{3 \over 4}} \right) - {\left( {{3 \over 4}} \right)^2} + {\left( {{3 \over 4}} \right)^3}$ $-$. . . . . + ($-$1)^n-1 ${\left( {{3 \over 4}} \right)^n}$    and    B_n = 1 $-$ A_n.
Then, the least dd natural numbr p, so that B_n > A_n , for all n$ \ge $ p, is :

A

9

B

7

C

11

D

5

Question 189: Palette 189/228

JEE Mains 2018 MCQ

If  a,   b,   c  are in A.P. and  a²,  b²,  c² are in G.P. such that
a < b < c and   a + b + c = ${3 \over 4},$ then the value of a is :

A

${1 \over 4} - {1 \over {4\sqrt 2 }}$

B

${1 \over 4} - {1 \over {3\sqrt 2 }}$

C

${1 \over 4} - {1 \over {2\sqrt 2 }}$

D

${1 \over 4} - {1 \over {\sqrt 2 }}$

Question 190: Palette 190/228

JEE Mains 2018 MCQ

If x₁, x₂, . . ., x_n and ${1 \over {{h_1}}}$, ${1 \over {{h_2}}}$, . . . , ${1 \over {{h_n}}}$ are two A.P..s such that x₃ = h₂ = 8 and x₈ = h₇ = 20, then x₅.h₁₀ equals :

A

2560

B

2650

C

3200

D

1600

Question 191: Palette 191/228

JEE Mains 2018 MCQ

If b is the first term of an infinite G.P. whose sum is five, then b lies in the interval :

A

($-$ $\infty $, $-$10]

B

($-$10, 0)

C

(0, 10)

D

[10, $\infty $)

Question 192: Palette 192/228

JEE Mains 2017 MCQ

Let

S_n = ${1 \over {{1^3}}}$$ + {{1 + 2} \over {{1^3} + {2^3}}} + {{1 + 2 + 3} \over {{1^3} + {2^3} + {3^3}}} + ......... + {{1 + 2 + ....... + n} \over {{1^3} + {2^3} + ...... + {n^3}}}.$

If 100 S_n = n, then n is equal to :

A

199

B

99

C

200

D

19

Question 193: Palette 193/228

JEE Mains 2017 MCQ

If three positive numbers a, b and c are in A.P. such that abc = 8, then the minimum possible value of b is :

A

2

B

4${^{{1 \over 3}}}$

C

4${^{{2 \over 3}}}$

D

4

Question 194: Palette 194/228

JEE Mains 2017 MCQ

If the sum of the first n terms of the series $\,\sqrt 3 + \sqrt {75} + \sqrt {243} + \sqrt {507} + ......$ is $435\sqrt 3 ,$ then n equals :

A

18

B

15

C

13

D

29

Question 195: Palette 195/228

JEE Mains 2017 MCQ

If the arithmetic mean of two numbers a and b, a > b > 0, is five times their geometric mean, then ${{a + b} \over {a - b}}$ is equal to :

A

${{\sqrt 6 } \over 2}$

B

${{3\sqrt 2 } \over 4}$

C

${{7\sqrt 3 } \over {12}}$

D

${{5\sqrt 6 } \over {12}}$

Question 196: Palette 196/228

JEE Mains 2017 MCQ

For any three positive real numbers a, b and c,

9(25${a^2}$ + b²) + 25(c² - 3$a$c) = 15b(3$a$ + c).
Then

A

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

B

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

C

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

D

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

Question 197: Palette 197/228

JEE Mains 2016 MCQ

If   A > 0, B > 0   and    A + B = ${\pi \over 6}$,

then the minimum value of tanA + tanB is :

A

$\sqrt 3 - \sqrt 2 $

B

$2 - \sqrt 3 $

C

$4 - 2\sqrt 3 $

D

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

Question 198: Palette 198/228

JEE Mains 2016 MCQ

Let z = 1 + ai be a complex number, a > 0, such that z³ is a real number.

Then the sum 1 + z + z² + . . . . .+ z¹¹ is equal to :

A

$ - 1250\,\sqrt 3 \,i$

B

$ 1250\,\sqrt 3 \,i$

C

$1365\,\sqrt 3 i$

D

$-$ $1365\,\sqrt 3 i$

Question 199: Palette 199/228

JEE Mains 2016 MCQ

Let a₁, a₂, a₃, . . . . . . . , a_n, . . . . . be in A.P.

If a₃ + a₇ + a₁₁ + a₁₅ = 72,

then the sum of its first 17 terms is equal to :

A

306

B

153

C

612

D

204

Question 200: Palette 200/228

JEE Mains 2016 MCQ

Let x, y, z be positive real numbers such that x + y + z = 12 and x³y⁴z⁵ = (0.1) (600)³. Then x³ + y³ + z³is equal to :

A

270

B

258

C

342

D

216

Question 201: Palette 201/228

JEE Mains 2016 MCQ

If the ${2^{nd}},{5^{th}}\,and\,{9^{th}}$ terms of a non-constant A.P. are in G.P., then the common ratio of this G.P. is :

A

1

B

${7 \over 4}$

C

${8 \over 5}$

D

${4 \over 3}$

Question 202: Palette 202/228

JEE Mains 2016 MCQ

If the sum of the first ten terms of the series ${\left( {1{3 \over 5}} \right)^2} + {\left( {2{2 \over 5}} \right)^2} + {\left( {3{1 \over 5}} \right)^2} + {4^2} + {\left( {4{4 \over 5}} \right)^2} + .......is\,{{16} \over 5}m,$ then m is equal to :

A

100

B

99

C

102

D

101

Question 203: Palette 203/228

JEE Mains 2015 MCQ

The sum of first 9 terms of the series.

${{{1^3}} \over 1} + {{{1^3} + {2^3}} \over {1 + 3}} + {{{1^3} + {2^3} + {3^3}} \over {1 + 3 + 5}} + ......$

A

142

B

192

C

71

D

96

Question 204: Palette 204/228

JEE Mains 2015 MCQ

If m is the A.M. of two distinct real numbers l and n $(l,n > 1)$ and ${G_1},{G_2}$ and ${G_3}$ are three geometric means between $l$ and n, then $G_1^4\, + 2G_2^4\, + G_3^4$ equals:

A

$4\,lm{n^2}$

B

$4\,{l^2}{m^2}{n^2}$

C

$4\,{l^2}m\,n$

D

$4\,l\,{m^2}n$

Question 205: Palette 205/228

JEE Mains 2014 MCQ

Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. then the common ratio of the G.P. is :

A

$2 - \sqrt 3 $

B

$2 + \sqrt 3 $

C

$\sqrt 2 + \sqrt 3 $

D

$3 + \sqrt 2 $

Question 206: Palette 206/228

JEE Mains 2014 MCQ

If ${(10)^9} + 2{(11)^1}\,({10^8}) + 3{(11)^2}\,{(10)^7} + ......... + 10{(11)^9} = k{(10)^9},$, then k is equal to :

A

100

B

110

C

${{121} \over {10}}$

D

${{441} \over {100}}$

Question 207: Palette 207/228

JEE Mains 2013 MCQ

The sum of first 20 terms of the sequence 0.7, 0.77, 0.777,........,is

A

${7 \over {81}}\left( {179 - {{10}^{ - 20}}} \right)$

B

$\,{7 \over 9}\left( {99 - {{10}^{ - 20}}} \right)$

C

${7 \over {81}}\left( {179 + {{10}^{ - 20}}} \right)$

D

${7 \over 9}\left( {99 + {{10}^{ - 20}}} \right)$

Question 208: Palette 208/228

JEE Mains 2012 MCQ

Statement-1: The sum of the series 1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) +.....+ (361 + 380 + 400) is 8000.

Statement-2: $\sum\limits_{k = 1}^n {\left( {{k^3} - {{(k - 1)}^3}} \right)} = {n^3}$, for any natural number n.

A

Statement-1 is false, Statement-2 is true.

B

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

C

Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

D

Statement-1 is true, Statement-2 is false.

Question 209: Palette 209/228

JEE Mains 2011 MCQ

A man saves ₹ 200 in each of the first three months of his service. In each of the subsequent months his saving increases by ₹ 40 more than the saving of immediately previous month. His total saving from the start of service will be ₹ 11040 after

A

19 months

B

20 months

C

21 months

D

18 months

Question 210: Palette 210/228

JEE Mains 2010 MCQ

A person is to count 4500 currency notes. Let ${a_n}$ denote the number of notes he counts in the ${n^{th}}$ minute. If ${a_1}$ = ${a_2}$ = ....= ${a_{10}}$= 150 and ${a_{10}}$, ${a_{11}}$,.... are in an AP with common difference - 2, then the time taken by him to count all notes is

A

34 minutes

B

125 minutes

C

135 minutes

D

24 minutes

Question 211: Palette 211/228

JEE Mains 2009 MCQ

The sum to infinite term of the series $1 + {2 \over 3} + {6 \over {{3^2}}} + {{10} \over {{3^3}}} + {{14} \over {{3^4}}} + .....$ is

A

3

B

4

C

6

D

2

Question 212: Palette 212/228

JEE Mains 2008 MCQ

The first two terms of a geometric progression add up to 12. the sum of the third and the fourth terms is 48. If the terms of the geometric progression are alternately positive and negative, then the first term is

A

- 4

B

- 12

C

12

D

4

Question 213: Palette 213/228

JEE Mains 2007 MCQ

In a geometric progression consisting of positive terms, each term equals the sum of the next two terns. Then the common ratio of its progression is equals

A

${\sqrt 5 }$

B

$\,{1 \over 2}\left( {\sqrt 5 - 1} \right)$

C

${1 \over 2}\left( {1 - \sqrt 5 } \right)$

D

${1 \over 2}\sqrt 5 $.

Question 214: Palette 214/228

JEE Mains 2007 MCQ

The sum of series ${1 \over {2!}} - {1 \over {3!}} + {1 \over {4!}} - .......$ upto infinity is

A

${e^{ - {1 \over 2}}}$

B

${e^{ + {1 \over 2}}}$

C

${e^{ - 2}}$

D

${e^{ - 1}}$

Question 215: Palette 215/228

JEE Mains 2006 MCQ

If ${{a_1},{a_2},....{a_n}}$ are in H.P., then the expression ${{a_1}\,{a_2} + \,{a_2}\,{a_3}\, + .... + {a_{n - 1}}\,{a_n}}$ is equal to

A

$n({a_1}\, - {a_n})$

B

$(n - 1)({a_1}\, - {a_n})$

C

$n{a_1}{a_n}$

D

$(n - 1)\,\,{a_1}{a_n}$

Question 216: Palette 216/228

JEE Mains 2006 MCQ

Let ${a_1}$, ${a_2}$, ${a_3}$.....be terms on A.P. If ${{{a_1} + {a_2} + .....{a_p}} \over {{a_1} + {a_2} + .....{a_q}}} = {{{p^2}} \over {{q^2}}},\,p \ne q,\,then\,{{{a_6}} \over {{a_{21}}}}\,$ equals

A

${{41} \over {11}}$

B

${7 \over 2}$

C

${2 \over 7}$

D

${{11} \over {41}}$

Question 217: Palette 217/228

JEE Mains 2005 MCQ

The sum of the series $1 + {1 \over {4.2!}} + {1 \over {16.4!}} + {1 \over {64.6!}} + .......$ ad inf. is

A

${{e - 1} \over {\sqrt e }}\,$

B

${{e + 1} \over {\sqrt e }}$

C

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

D

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

Question 218: Palette 218/228

JEE Mains 2005 MCQ

If $x = \sum\limits_{n = 0}^\infty {{a^n},\,\,y = \sum\limits_{n = 0}^\infty {{b^n},\,\,z = \sum\limits_{n = 0}^\infty {{c^n},} } } \,\,$ where a, b, c are in A.P and $\,\left| a \right| < 1,\,\left| b \right| < 1,\,\left| c \right| < 1$ then x, y, z are in

A

G.P.

B

A.P.

C

Arithmetic-Geometric Progression

D

H.P.

Question 219: Palette 219/228

JEE Mains 2004 MCQ

The sum of series ${1 \over {2\,!}} + {1 \over {4\,!}} + {1 \over {6\,!}} + ........$ is

A

${{\left( {{e^2} - 2} \right)} \over e}\,$

B

${{{{\left( {e - 1} \right)}^2}} \over {2e}}$

C

${{\left( {{e^2} - 1} \right)} \over {2e}}\,$

D

${{\left( {{e^2} - 1} \right)} \over 2}$

Question 220: Palette 220/228

JEE Mains 2004 MCQ

Let ${{T_r}}$ be the rth term of an A.P. whose first term is a and common difference is d. If for some positive integers m, n, $m \ne n,\,\,{T_m} = {1 \over n}\,\,and\,{T_n} = {1 \over m},\,$ then a - d equals

A

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

B

1

C

${1 \over {m\,n}}$

D

0

Question 221: Palette 221/228

JEE Mains 2004 MCQ

The sum of the first n terms of the series ${1^2} + {2.2^2} + {3^2} + {2.4^2} + {5^2} + {2.6^2} + ....\,is\,{{n{{(n + 1)}^2}} \over 2}$ when n is even. When n is odd the sum is

A

${\left[ {{{n(n + 1)} \over 2}} \right]^2}$

B

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

C

${{n{{(n + 1)}^2}} \over 4}$

D

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

Question 222: Palette 222/228

JEE Mains 2003 MCQ

The sum of the serier ${1 \over {1.2}} - {1 \over {2.3}} + {1 \over {3.4}}..............$ up to $\infty $ is equal to

A

$\log {\,_e}\left( {{4 \over e}} \right)\,\,$

B

$2\,\log {\,_e}2$

C

$\log {\,_e}2 - 1\,$

D

$\log {\,_e}2$

Question 223: Palette 223/228

JEE Mains 2002 MCQ

The value of $\,{2^{1/4}}.\,\,{4^{1/8}}.\,{8^{1/16}}...\infty $ is

A

1

B

2

C

3/2

D

4

Question 224: Palette 224/228

JEE Mains 2002 MCQ

${1^3} - \,\,{2^3} + {3^3} - {4^3} + ... + {9^3} = $

A

425

B

- 425

C

475

D

- 475

Question 225: Palette 225/228

JEE Mains 2002 MCQ

l, m, n are the ${p^{th}}$, ${q^{th}}$ and ${r^{th}}$ term of a G.P all positive, $then\,\left| {\matrix{ {\log \,l} & p & 1 \cr {\log \,m} & q & 1 \cr {\log \,n} & r & 1 \cr } } \right|\,equals$

A

- 1

B

2

C

1

D

0

Question 226: Palette 226/228

JEE Mains 2002 MCQ

Sum of infinite number of terms of GP is 20 and sum of their square is 100. The common ratio of GP is

A

5

B

3/5

C

8/5

D

1/5

Question 227: Palette 227/228

JEE Mains 2002 MCQ

If 1, ${\log _9}\,\,({3^{1 - x}} + 2),\,\,{\log _3}\,\,({4.3^x} - 1)$ are in A.P. then x equals

A

${\log _3}\,4\,\,\,$

B

$1 - \,{\log _3}\,4\,$

C

$1 - \,{\log _4}\,3$

D

${\log _4}\,3$

Question 228: Palette 228/228

JEE Mains 2002 MCQ

Fifth term of a GP is 2, then the product of its 9 terms is

A

256

B

512

C

1024

D

none of these