Question 1: Palette 1/165

JEE Mains 2026 MCQ

Given below are two statements :

Statement I :

$25^{13} + 20^{13} + 8^{13} + 3^{13}$ is divisible by 7.

Statement II :

The integral part of $(7 + 4\sqrt{3})^{25}$ is an odd number.

In the light of the above statements, choose the correct answer from the options given below :

A

Statement I is false but Statement II is true

B

Both Statement I and Statement II are false

C

Both Statement I and Statement II are true

D

Statement I is true but Statement II is false

Question 2: Palette 2/165

JEE Mains 2026 MCQ

The sum of the coefficients of $x^{499}$ and $x^{500}$ in $(1 + x)^{1000} + x(1 + x)^{999} + x^2(1 + x)^{998} + \ldots + x^{1000}$ is :

A

${ }^{1002} C_{501}$

B

${ }^{1001} C_{501}$

C

${ }^{1000} C_{501}$

D

${ }^{1002} C_{500}$

Question 3: Palette 3/165

JEE Mains 2026 MCQ

Let $\mathrm{S}=\frac{1}{25!}+\frac{1}{3!23!}+\frac{1}{5!21!}+\ldots$ up to 13 terms. If $13 \mathrm{~S}=\frac{2^k}{n!}, k \in \mathrm{~N}$, then $n+k$ is equal to

A

50

B

52

C

49

D

51

Question 4: Palette 4/165

JEE Mains 2026 MCQ

The sum of all possible values of $\mathbf{n} \in \mathbf{N}$, so that the coefficients of $x, x^2$ and $x^3$ in the expansion of $\left(1+x^2\right)^2(1+x)^{\mathrm{n}}$, are in arithmetic progression is :

A

12

B

9

C

3

D

7

Question 5: Palette 5/165

JEE Mains 2026 MCQ

The value of $\frac{{ }^{100} \mathrm{C}_{50}}{51}+\frac{{ }^{100} \mathrm{C}_{51}}{52}+\ldots .+\frac{{ }^{100} \mathrm{C}_{100}}{101}$ is:

A

$\frac{2^{101}}{101}$

B

$\frac{2^{100}}{101}$

C

$\frac{2^{100}}{100}$

D

$\frac{2^{101}}{100}$

Question 6: Palette 6/165

JEE Mains 2026 MCQ

Let $\mathrm{C}_{\mathrm{r}}$ denote the coefficient of $x^{\mathrm{r}}$ in the binomial expansion of $(1+x)^{\mathrm{n}}, \mathrm{n} \in \mathrm{N}, 0 \leq \mathrm{r} \leq \mathrm{n}$. If

$P_n=C_0-C_1+\frac{2^2}{3} C_2-\frac{2^3}{4} C_3+\ldots . .+\frac{(-2)^n}{n+1} C_n$, then the value of $\sum\limits_{n=1}^{25} \frac{1}{P_{2 n}}$ equals.

A

675

B

580

C

525

D

650

Question 7: Palette 7/165

JEE Mains 2026 MCQ

The coefficient of $x^{48}$ in $(1+x)+2(1+x)^2+3(1+x)^3+\ldots+100(1+x)^{100}$ is equal to

A

$100 \cdot{ }^{100} \mathrm{C}_{49}-{ }^{100} \mathrm{C}_{48}$

B

$100 \cdot{ }^{101} \mathrm{C}_{49}-{ }^{101} \mathrm{C}_{50}$

C

${ }^{100} \mathrm{C}_{50}+{ }^{101} \mathrm{C}_{49}$

D

$100 \cdot{ }^{100} \mathrm{C}_{49}-{ }^{100} \mathrm{C}_{50}$

Question 8: Palette 8/165

JEE Mains 2026 MCQ

If the coefficient of $x$ in the expansion of $\left(a x^2+b x+c\right)(1-2 x)^{26}$ is -56 and the coefficients of $x^2$ and $x^3$ are both zero, then $\mathrm{a}+\mathrm{b}+\mathrm{c}$ is equal to :

A

1483

B

1300

C

1500

D

1403

Question 9: Palette 9/165

JEE Mains 2026 MCQ

If $26\left(\frac{2^3}{3}\left({ }^{12} \mathrm{C}_2\right)+\frac{2^5}{5}\left({ }^{12} \mathrm{C}_4\right)+\frac{2^7}{7}\left({ }^{12} \mathrm{C}_6\right)+\cdots+\frac{2^{13}}{13}\left({ }^{12} \mathrm{C}_{12}\right)\right)=3^{13}-\alpha$, then $\alpha$ is equal to :

A

45

B

48

C

51

D

54

Question 10: Palette 10/165

JEE Mains 2026 MCQ

If the coefficients of the middle terms in the binomial expansions of $(1+\alpha x)^{26}$ and $(1-\alpha x)^{28}, \alpha \neq 0$, are equal, then the value of $\alpha$ is:

A

1

B

$\frac{14}{13}$

C

$\frac{27}{7}$

D

$ \frac{7}{27} $

Question 11: Palette 11/165

JEE Mains 2026 MCQ

The coefficient of $x^2$ in the expansion of $\left(2 x^2+\frac{1}{x}\right)^{10}, x \neq 0$, is :

A

3240

B

3360

C

3480

D

3600

Question 12: Palette 12/165

JEE Mains 2026 MCQ

In the expansion of $\left(9 x-\frac{1}{3 \sqrt{x}}\right)^{18}, x>0$, if the term independent of $x$ is (221)k, then k is equal to:

A

84

B

78

C

168

D

198

Question 13: Palette 13/165

JEE Mains 2026 MCQ

Let the smallest value of $k \in \mathbb{N}$, for which the coefficient of $x^3$ in $(1+x)^3+(1+x)^4+(1+x)^5+\ldots+(1+x)^{99}+(1+k x)^{100}, x \neq 0$, is $\left(43 n+\frac{101}{4}\right)\left({ }^{100} \mathrm{C}_3\right)$ for some $n \in \mathrm{~N}$, be $p$. Then the value of $p+n$ is :

A

10

B

11

C

12

D

13

Question 14: Palette 14/165

JEE Mains 2026 MCQ

If for $3 \leq r \leq 30$, $\left({^{30}C_{30-r}}\right) + 3\left({^{30}C_{31-r}}\right) + 3\left({^{30}C_{32-r}}\right) + \left({^{30}C_{33-r}}\right) = {^mC_r}$, then m equals :

A

31

B

32

C

33

D

34

Question 15: Palette 15/165

JEE Mains 2025 MCQ

The number of integral terms in the expansion of $ \left( {5^\frac{1}{2}} + 7^\frac{1}{8} \right)^{1016} $ is:

A

127

B

128

C

130

D

129

Question 16: Palette 16/165

JEE Mains 2025 MCQ

The remainder when $\left((64)^{(64)}\right)^{(64)}$ is divided by 7 is equal to

A

4

B

6

C

3

D

1

Question 17: Palette 17/165

JEE Mains 2025 MCQ

If $1^2 \cdot\left({ }^{15} C_1\right)+2^2 \cdot\left({ }^{15} C_2\right)+3^2 \cdot\left({ }^{15} C_3\right)+\ldots+15^2 \cdot\left({ }^{15} C_{15}\right)=2^m \cdot 3^n \cdot 5^k$, where $m, n, k \in \mathbf{N}$, then $\mathrm{m}+\mathrm{n}+\mathrm{k}$ is equal to :

A

20

B

19

C

18

D

21

Question 18: Palette 18/165

JEE Mains 2025 MCQ

For an integer $n \geq 2$, if the arithmetic mean of all coefficients in the binomial expansion of $(x+y)^{2 n-3}$ is 16 , then the distance of the point $\mathrm{P}\left(2 n-1, n^2-4 n\right)$ from the line $x+y=8$ is

A

$\sqrt{2}$

B

$2 \sqrt{2}$

C

$5 \sqrt{2}$

D

$3 \sqrt{2}$

Question 19: Palette 19/165

JEE Mains 2025 MCQ

In the expansion of $\left(\sqrt[3]{2}+\frac{1}{\sqrt[3]{3}}\right)^n, n \in \mathrm{~N}$, if the ratio of $15^{\text {th }}$ term from the beginning to the $15^{\text {th }}$ term from the end is $\frac{1}{6}$, then the value of ${ }^n \mathrm{C}_3$ is

A

4960

B

2300

C

1040

D

4060

Question 20: Palette 20/165

JEE Mains 2025 MCQ

The sum of all rational terms in the expansion of $(2+\sqrt{3})^8$ is :

A

16923

B

18817

C

3763

D

33845

Question 21: Palette 21/165

JEE Mains 2025 MCQ

If $\sum\limits_{r=1}^9\left(\frac{r+3}{2^r}\right) \cdot{ }^9 C_r=\alpha\left(\frac{3}{2}\right)^9-\beta, \alpha, \beta \in \mathbb{N}$, then $(\alpha+\beta)^2$ is equal to

A

27

B

81

C

18

D

9

Question 22: Palette 22/165

JEE Mains 2025 MCQ

$If\,\sum\limits_{r = 0}^{10} {({{{{10}^{r + 1}} - 1} \over {{{10}^r}}}).{}^{11}{C_{r + 1}} = {{{}_\alpha 11 - {{11}^{11}}} \over {{{10}^{10}}}},\,then\,\,\alpha \,\,is\,\,equal\,\,to:} $

A

11

B

20

C

24

D

15

Question 23: Palette 23/165

JEE Mains 2025 MCQ

The largest $\mathrm{n} \in \mathbf{N}$ such that $3^{\mathrm{n}}$ divides 50 ! is :

A

22

B

20

C

21

D

23

Question 24: Palette 24/165

JEE Mains 2025 MCQ

The term independent of $x$ in the expansion of $\left(\frac{(x+1)}{\left(x^{2 / 3}+1-x^{1 / 3}\right)}-\frac{(x-1)}{\left(x-x^{1 / 2}\right)}\right)^{10}, x>1$, is :

A

240

B

120

C

150

D

210

Question 25: Palette 25/165

JEE Mains 2025 MCQ

The remainder, when $7^{103}$ is divided by 23, is equal to:

A

9

B

6

C

14

D

17

Question 26: Palette 26/165

JEE Mains 2025 MCQ

The least value of n for which the number of integral terms in the Binomial expansion of $(\sqrt[3]{7}+\sqrt[12]{11})^n$ is 183, is :

A

2184

B

2172

C

2196

D

2148

Question 27: Palette 27/165

JEE Mains 2025 MCQ

Let the coefficients of three consecutive terms $T_r$, $T_{r+1}$ and $T_{r+2}$ in the binomial expansion of $(a + b)^{12}$ be in a G.P. and let $p$ be the number of all possible values of $r$. Let $q$ be the sum of all rational terms in the binomial expansion of $(\sqrt[4]{3}+\sqrt[3]{4})^{12}$. Then $p + q$ is equal to:

A

295

B

283

C

299

D

287

Question 28: Palette 28/165

JEE Mains 2025 MCQ

Suppose $A$ and $B$ are the coefficients of $30^{\text {th }}$ and $12^{\text {th }}$ terms respectively in the binomial expansion of $(1+x)^{2 \mathrm{n}-1}$. If $2 \mathrm{~A}=5 \mathrm{~B}$, then n is equal to:

A

20

B

19

C

22

D

21

Question 29: Palette 29/165

JEE Mains 2025 MCQ

For some $\mathrm{n} \neq 10$, let the coefficients of the 5 th, 6 th and 7 th terms in the binomial expansion of $(1+\mathrm{x})^{\mathrm{n}+4}$ be in A.P. Then the largest coefficient in the expansion of $(1+\mathrm{x})^{\mathrm{n}+4}$ is:

A

10

B

35

C

70

D

20

Question 30: Palette 30/165

JEE Mains 2025 MCQ

If in the expansion of $(1+x)^{\mathrm{p}}(1-x)^{\mathrm{q}}$, the coefficients of $x$ and $x^2$ are 1 and -2 , respectively, then $\mathrm{p}^2+\mathrm{q}^2$ is equal to :

A

8

B

20

C

13

D

18

Question 31: Palette 31/165

JEE Mains 2025 MCQ

Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of

$\begin{aligned} & \left(x+\sqrt{x^3-1}\right)^5+\left(x-\sqrt{x^3-1}\right)^5, x>1 \text {. If } u \text { and } v \text { satisfy the equations } \\\\ & \alpha u+\beta v=18, \\\\ & \gamma u+\delta v=20, \end{aligned}$

then $\mathrm{u+v}$ equals :

A

4

B

3

C

5

D

8

Question 32: Palette 32/165

JEE Mains 2024 MCQ

The sum of the coefficient of $x^{2 / 3}$ and $x^{-2 / 5}$ in the binomial expansion of $\left(x^{2 / 3}+\frac{1}{2} x^{-2 / 5}\right)^9$ is

A

19/4

B

69/16

C

63/16

D

21/4

Question 33: Palette 33/165

JEE Mains 2024 MCQ

The coefficient of $x^{70}$ in $x^2(1+x)^{98}+x^3(1+x)^{97}+x^4(1+x)^{96}+\ldots+x^{54}(1+x)^{46}$ is ${ }^{99} \mathrm{C}_{\mathrm{p}}-{ }^{46} \mathrm{C}_{\mathrm{q}}$. Then a possible value of $\mathrm{p}+\mathrm{q}$ is :

A

61

B

83

C

55

D

68

Question 34: Palette 34/165

JEE Mains 2024 MCQ

If the term independent of $x$ in the expansion of $\left(\sqrt{\mathrm{a}} x^2+\frac{1}{2 x^3}\right)^{10}$ is 105 , then $\mathrm{a}^2$ is equal to :

A

6

B

4

C

2

D

9

Question 35: Palette 35/165

JEE Mains 2024 MCQ

If the constant term in the expansion of $\left(\frac{\sqrt[5]{3}}{x}+\frac{2 x}{\sqrt[3]{5}}\right)^{12}, x \neq 0$, is $\alpha \times 2^8 \times \sqrt[5]{3}$, then $25 \alpha$ is equal to :

A

724

B

742

C

693

D

639

Question 36: Palette 36/165

JEE Mains 2024 MCQ

If the coefficients of $x^4, x^5$ and $x^6$ in the expansion of $(1+x)^n$ are in the arithmetic progression, then the maximum value of $n$ is:

A

28

B

21

C

7

D

14

Question 37: Palette 37/165

JEE Mains 2024 MCQ

The sum of all rational terms in the expansion of $\left(2^{\frac{1}{5}}+5^{\frac{1}{3}}\right)^{15}$ is equal to :

A

633

B

6131

C

3133

D

931

Question 38: Palette 38/165

JEE Mains 2024 MCQ

Let $m$ and $n$ be the coefficients of seventh and thirteenth terms respectively

in the expansion of $\left(\frac{1}{3} x^{\frac{1}{3}}+\frac{1}{2 x^{\frac{2}{3}}}\right)^{18}$. Then $\left(\frac{\mathrm{n}}{\mathrm{m}}\right)^{\frac{1}{3}}$ is :

A

$\frac{1}{9}$

B

$\frac{1}{4}$

C

$\frac{4}{9}$

D

$\frac{9}{4}$

Question 39: Palette 39/165

JEE Mains 2024 MCQ

Let $a$ be the sum of all coefficients in the expansion of $\left(1-2 x+2 x^2\right)^{2023}\left(3-4 x^2+2 x^3\right)^{2024}$ and $b=\lim _\limits{x \rightarrow 0}\left(\frac{\int_0^x \frac{\log (1+t)}{t^{2024}+1} d t}{x^2}\right)$. If the equation $c x^2+d x+e=0$ and $2 b x^2+a x+4=0$ have a common root, where $c, d, e \in \mathbb{R}$, then $\mathrm{d}: \mathrm{c}:$ e equals

A

$2: 1: 4$

B

$1: 1: 4$

C

$1: 2: 4$

D

$4: 1: 4$

Question 40: Palette 40/165

JEE Mains 2024 MCQ

Suppose $2-p, p, 2-\alpha, \alpha$ are the coefficients of four consecutive terms in the expansion of $(1+x)^n$. Then the value of $p^2-\alpha^2+6 \alpha+2 p$ equals

A

8

B

4

C

6

D

10

Question 41: Palette 41/165

JEE Mains 2024 MCQ

${ }^{n-1} C_r=\left(k^2-8\right){ }^n C_{r+1}$ if and only if :

A

$2 \sqrt{2}<\mathrm{k}<2 \sqrt{3}$

B

$2 \sqrt{2}<\mathrm{k} \leq 3$

C

$2 \sqrt{3}<\mathrm{k}<3 \sqrt{3}$

D

$2 \sqrt{3}<\mathrm{k} \leq 3 \sqrt{2}$

Question 42: Palette 42/165

JEE Mains 2024 MCQ

If A denotes the sum of all the coefficients in the expansion of $\left(1-3 x+10 x^2\right)^{\mathrm{n}}$ and B denotes the sum of all the coefficients in the expansion of $\left(1+x^2\right)^n$, then :

A

$\mathrm{B}=\mathrm{A}^3$

B

$3 \mathrm{A}=\mathrm{B}$

C

$A=3 B$

D

$\mathrm{A}=\mathrm{B}^3$

Question 43: Palette 43/165

JEE Mains 2023 MCQ

Let $\left(a+b x+c x^{2}\right)^{10}=\sum\limits_{i=0}^{20} p_{i} x^{i}, a, b, c \in \mathbb{N}$.

If $p_{1}=20$ and $p_{2}=210$, then $2(a+b+c)$ is equal to :

A

15

B

8

C

6

D

12

Question 44: Palette 44/165

JEE Mains 2023 MCQ

The coefficient of $x^{5}$ in the expansion of $\left(2 x^{3}-\frac{1}{3 x^{2}}\right)^{5}$ is :

A

$\frac{26}{3}$

B

$\frac{80}{9}$

C

9

D

8

Question 45: Palette 45/165

JEE Mains 2023 MCQ

Fractional part of the number $\frac{4^{2022}}{15}$ is equal to

A

$\frac{8}{15}$

B

$\frac{4}{15}$

C

$\frac{1}{15}$

D

$\frac{14}{15}$

Question 46: Palette 46/165

JEE Mains 2023 MCQ

If $\frac{1}{n+1}{ }^{n} \mathrm{C}_{n}+\frac{1}{n}{ }^{n} \mathrm{C}_{n-1}+\ldots+\frac{1}{2}{ }^{n} \mathrm{C}_{1}+{ }^{n} \mathrm{C}_{0}=\frac{1023}{10}$ then $n$ is equal to :

A

9

B

6

C

7

D

8

Question 47: Palette 47/165

JEE Mains 2023 MCQ

The sum, of the coefficients of the first 50 terms in the binomial expansion of $(1-x)^{100}$, is equal to

A

${ }^{99} \mathrm{C}_{49}$

B

${ }^{101} \mathrm{C}_{50}$

C

$-{ }^{99} \mathrm{C}_{49}$

D

$-{ }^{101} \mathrm{C}_{50}$

Question 48: Palette 48/165

JEE Mains 2023 MCQ

The sum of the coefficients of three consecutive terms in the binomial expansion of $(1+\mathrm{x})^{\mathrm{n}+2}$, which are in the ratio $1: 3: 5$, is equal to :

A

63

B

92

C

25

D

41

Question 49: Palette 49/165

JEE Mains 2023 MCQ

If the $1011^{\text {th }}$ term from the end in the binominal expansion of $\left(\frac{4 x}{5}-\frac{5}{2 x}\right)^{2022}$ is 1024 times $1011^{\text {th }}$R term from the beginning, then $|x|$ is equal to

A

$ \frac{5}{16} $

B

8

C

12

D

15

Question 50: Palette 50/165

JEE Mains 2023 MCQ

Let the number $(22)^{2022}+(2022)^{22}$ leave the remainder $\alpha$ when divided by 3 and $\beta$ when divided by 7. Then $\left(\alpha^{2}+\beta^{2}\right)$ is equal to :

A

13

B

10

C

20

D

5

Question 51: Palette 51/165

JEE Mains 2023 MCQ

If the coefficients of $x$ and $x^{2}$ in $(1+x)^{\mathrm{p}}(1-x)^{\mathrm{q}}$ are 4 and $-$5 respectively, then $2 p+3 q$ is equal to :

A

66

B

60

C

69

D

63

Question 52: Palette 52/165

JEE Mains 2023 MCQ

If the coefficient of ${x^7}$ in ${\left( {ax - {1 \over {b{x^2}}}} \right)^{13}}$ and the coefficient of ${x^{ - 5}}$ in ${\left( {ax + {1 \over {b{x^2}}}} \right)^{13}}$ are equal, then ${a^4}{b^4}$ is equal to :

A

22

B

33

C

44

D

11

Question 53: Palette 53/165

JEE Mains 2023 MCQ

$25^{190}-19^{190}-8^{190}+2^{190}$ is divisible by :

A

14 but not by 34

B

neither 14 nor 34

C

both 14 and 34

D

34 but not by 14

Question 54: Palette 54/165

JEE Mains 2023 MCQ

The absolute difference of the coefficients of $x^{10}$ and $x^{7}$ in the expansion of $\left(2 x^{2}+\frac{1}{2 x}\right)^{11}$ is equal to :

A

$11^{3}-11$

B

$13^{3}-13$

C

$12^{3}-12$

D

$10^{3}-10$

Question 55: Palette 55/165

JEE Mains 2023 MCQ

If the coefficients of three consecutive terms in the expansion of $(1+x)^{n}$ are in the ratio $1: 5: 20$, then the coefficient of the fourth term is

A

3654

B

1827

C

5481

D

2436

Question 56: Palette 56/165

JEE Mains 2023 MCQ

If the coefficient of ${x^7}$ in ${\left( {a{x^2} + {1 \over {2bx}}} \right)^{11}}$ and ${x^{ - 7}}$ in ${\left( {ax - {1 \over {3b{x^2}}}} \right)^{11}}$ are equal, then :

A

$243ab = 64$

B

$32ab = 729$

C

$64ab = 243$

D

$729ab = 32$

Question 57: Palette 57/165

JEE Mains 2023 MCQ

Among the statements :

(S1) : $2023^{2022}-1999^{2022}$ is divisible by 8

(S2) : $13(13)^{n}-12 n-13$ is divisible by 144 for infinitely many $n \in \mathbb{N}$

A

both (S1) and (S2) are incorrect

B

only (S1) is correct

C

only (S2) is correct

D

both (S1) and (S2) are correct

Question 58: Palette 58/165

JEE Mains 2023 MCQ

If ${ }^{2 n} C_{3}:{ }^{n} C_{3}=10: 1$, then the ratio $\left(n^{2}+3 n\right):\left(n^{2}-3 n+4\right)$ is :

A

$27: 11$

B

$2: 1$

C

$35: 16$

D

$65: 37$

Question 59: Palette 59/165

JEE Mains 2023 MCQ

If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of $\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)^{\mathrm{n}}$ is $\sqrt{6}: 1$, then the third term from the beginning is :

A

$30 \sqrt{2}$

B

$60 \sqrt{3}$

C

$60 \sqrt{2}$

D

$30 \sqrt{3}$

Question 60: Palette 60/165

JEE Mains 2023 MCQ

Let $x=(8 \sqrt{3}+13)^{13}$ and $y=(7 \sqrt{2}+9)^9$. If $[t]$ denotes the greatest integer $\leq t$, then :

A

$[x]$ is odd but $[y]$ is even

B

$[x]$ and $[y]$ are both odd

C

$[x]+[y]$ is even

D

$[x]$ is even but $[y]$ is odd

Question 61: Palette 61/165

JEE Mains 2023 MCQ

If the coefficient of $x^{15}$ in the expansion of $\left(\mathrm{a} x^{3}+\frac{1}{\mathrm{~b} x^{1 / 3}}\right)^{15}$ is equal to the coefficient of $x^{-15}$ in the expansion of $\left(a x^{1 / 3}-\frac{1}{b x^{3}}\right)^{15}$, where $a$ and $b$ are positive real numbers, then for each such ordered pair $(\mathrm{a}, \mathrm{b})$ :

A

a = 3b

B

ab = 1

C

ab = 3

D

a = b

Question 62: Palette 62/165

JEE Mains 2023 MCQ

The coefficient of ${x^{301}}$ in ${(1 + x)^{500}} + x{(1 + x)^{499}} + {x^2}{(1 + x)^{498}}\, + \,...\, + \,{x^{500}}$ is :

A

${}^{500}{C_{300}}$

B

${}^{501}{C_{200}}$

C

${}^{500}{C_{301}}$

D

${}^{501}{C_{302}}$

Question 63: Palette 63/165

JEE Mains 2023 MCQ

Let K be the sum of the coefficients of the odd powers of $x$ in the expansion of $(1+x)^{99}$. Let $a$ be the middle term in the expansion of ${\left( {2 + {1 \over {\sqrt 2 }}} \right)^{200}}$. If ${{{}^{200}{C_{99}}K} \over a} = {{{2^l}m} \over n}$, where m and n are odd numbers, then the ordered pair $(l,\mathrm{n})$ is equal to

A

(50, 101)

B

(50, 51)

C

(51, 101)

D

(51, 99)

Question 64: Palette 64/165

JEE Mains 2023 MCQ

If $a_r$ is the coefficient of $x^{10-r}$ in the Binomial expansion of $(1 + x)^{10}$, then $\sum\limits_{r = 1}^{10} {{r^3}{{\left( {{{{a_r}} \over {{a_{r - 1}}}}} \right)}^2}} $ is equal to

A

3025

B

4895

C

5445

D

1210

Question 65: Palette 65/165

JEE Mains 2023 MCQ

If ${({}^{30}{C_1})^2} + 2{({}^{30}{C_2})^2} + 3{({}^{30}{C_3})^2}\, + \,...\, + \,30{({}^{30}{C_{30}})^2} = {{\alpha 60!} \over {{{(30!)}^2}}}$ then $\alpha$ is equal to :

A

30

B

10

C

15

D

60

Question 66: Palette 66/165

JEE Mains 2023 MCQ

The value of $\sum\limits_{r = 0}^{22} {{}^{22}{C_r}{}^{23}{C_r}} $ is

A

${}^{44}{C_{23}}$

B

${}^{45}{C_{23}}$

C

${}^{44}{C_{22}}$

D

${}^{45}{C_{24}}$

Question 67: Palette 67/165

JEE Mains 2022 MCQ

$\sum\limits_{r=1}^{20}\left(r^{2}+1\right)(r !)$ is equal to

A

$22 !-21 !$

B

$22 !-2(21 !)$

C

$21 !-2(20 !)$

D

$21 !-20$ !

Question 68: Palette 68/165

JEE Mains 2022 MCQ

The remainder when $7^{2022}+3^{2022}$ is divided by 5 is :

A

0

B

2

C

3

D

4

Question 69: Palette 69/165

JEE Mains 2022 MCQ

The remainder when $(2021)^{2022}+(2022)^{2021}$ is divided by 7 is

A

0

B

1

C

2

D

6

Question 70: Palette 70/165

JEE Mains 2022 MCQ

$\sum\limits_{\matrix{ {i,j = 0} \cr {i \ne j} \cr } }^n {{}^n{C_i}\,{}^n{C_j}} $ is equal to

A

$2^{2 n}-{ }^{2 n} C_{n}$

B

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

C

$2^{2 n}-\frac{1}{2}{ }^{2 n} C_{n}$

D

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

Question 71: Palette 71/165

JEE Mains 2022 MCQ

The remainder when $(11)^{1011}+(1011)^{11}$ is divided by 9 is

A

1

B

4

C

6

D

8

Question 72: Palette 72/165

JEE Mains 2022 MCQ

For two positive real numbers a and b such that ${1 \over {{a^2}}} + {1 \over {{b^3}}} = 4$, then minimum value of the constant term in the expansion of ${\left( {a{x^{{1 \over 8}}} + b{x^{ - {1 \over {12}}}}} \right)^{10}}$ is :

A

${{105} \over 2}$

B

${{105} \over 4}$

C

${{105} \over 8}$

D

${{105} \over 16}$

Question 73: Palette 73/165

JEE Mains 2022 MCQ

Let n $\ge$ 5 be an integer. If 9ⁿ $-$ 8n $-$ 1 = 64$\alpha$ and 6ⁿ $-$ 5n $-$ 1 = 25$\beta$, then $\alpha$ $-$ $\beta$ is equal to

A

1 + ⁿC₂ (8 $-$ 5) + ⁿC₃ (8² $-$ 5²) + ...... + ⁿC_n (8^{n $-$ 1} $-$ 5^{n $-$ 1})

B

1 + ⁿC₃ (8 $-$ 5) + ⁿC₄ (8² $-$ 5²) + ...... + ⁿC_n (8^{n $-$ 2} $-$ 5^{n $-$ 2})

C

ⁿC₃ (8 $-$ 5) + ⁿC₄ (8² $-$ 5²) + ...... + ⁿC_n (8^{n $-$ 2} $-$ 5^{n $-$ 2})

D

ⁿC₄ (8 $-$ 5) + ⁿC₅ (8² $-$ 5²) + ...... + ⁿC_n (8^{n $-$ 3} $-$ 5^{n $-$ 3})

Question 74: Palette 74/165

JEE Mains 2022 MCQ

If the constant term in the expansion of

${\left( {3{x^3} - 2{x^2} + {5 \over {{x^5}}}} \right)^{10}}$ is 2^k.l, where l is an odd integer, then the value of k is equal to:

A

6

B

7

C

8

D

9

Question 75: Palette 75/165

JEE Mains 2022 MCQ

The term independent of x in the expansion of

$(1 - {x^2} + 3{x^3}){\left( {{5 \over 2}{x^3} - {1 \over {5{x^2}}}} \right)^{11}},\,x \ne 0$ is :

A

${7 \over {40}}$

B

${33 \over {200}}$

C

${39 \over {200}}$

D

${11 \over {50}}$

Question 76: Palette 76/165

JEE Mains 2022 MCQ

If

$\sum\limits_{k = 1}^{31} {\left( {{}^{31}{C_k}} \right)\left( {{}^{31}{C_{k - 1}}} \right) - \sum\limits_{k = 1}^{30} {\left( {{}^{30}{C_k}} \right)\left( {{}^{30}{C_{k - 1}}} \right) = {{\alpha (60!)} \over {(30!)(31!)}}} } $,

where $\alpha$ $\in$ R, then the value of 16$\alpha$ is equal to

A

1411

B

1320

C

1615

D

1855

Question 77: Palette 77/165

JEE Mains 2022 MCQ

The remainder when (2021)²⁰²³ is divided by 7 is :

A

1

B

2

C

5

D

6

Question 78: Palette 78/165

JEE Mains 2022 MCQ

The coefficient of x¹⁰¹ in the expression ${(5 + x)^{500}} + x{(5 + x)^{499}} + {x^2}{(5 + x)^{498}} + \,\,.....\,\, + \,\,{x^{500}}$, x > 0, is

A

⁵⁰¹C₁₀₁ (5)³⁹⁹

B

⁵⁰¹C₁₀₁ (5)⁴⁰⁰

C

⁵⁰¹C₁₀₀ (5)⁴⁰⁰

D

⁵⁰⁰C₁₀₁ (5)³⁹⁹

Question 79: Palette 79/165

JEE Mains 2022 MCQ

If ${1 \over {2\,.\,{3^{10}}}} + {1 \over {{2^2}\,.\,{3^9}}} + \,\,.....\,\, + \,\,{1 \over {{2^{10}}\,.\,3}} = {K \over {{2^{10}}\,.\,{3^{10}}}}$, then the remainder when K is divided by 6 is :

A

1

B

2

C

3

D

5

Question 80: Palette 80/165

JEE Mains 2022 MCQ

The remainder when 3²⁰²² is divided by 5 is :

A

1

B

2

C

3

D

4

Question 81: Palette 81/165

JEE Mains 2021 MCQ

$\sum\limits_{k = 0}^{20} {{{\left( {{}^{20}{C_k}} \right)}^2}} $ is equal to :

A

${}^{40}{C_{21}}$

B

${}^{40}{C_{19}}$

C

${}^{40}{C_{20}}$

D

${}^{41}{C_{20}}$

Question 82: Palette 82/165

JEE Mains 2021 MCQ

If ${{}^{20}{C_r}}$ is the co-efficient of x^r in the expansion of (1 + x)²⁰, then the value of $\sum\limits_{r = 0}^{20} {{r^2}.{}^{20}{C_r}} $ is equal to :

A

$420 \times {2^{19}}$

B

$380 \times {2^{19}}$

C

$380 \times {2^{18}}$

D

$420 \times {2^{18}}$

Question 83: Palette 83/165

JEE Mains 2021 MCQ

A possible value of 'x', for which the ninth term in the expansion of ${\left\{ {{3^{{{\log }_3}\sqrt {{{25}^{x - 1}} + 7} }} + {3^{\left( { - {1 \over 8}} \right){{\log }_3}({5^{x - 1}} + 1)}}} \right\}^{10}}$ in the increasing powers of ${3^{\left( { - {1 \over 8}} \right){{\log }_3}({5^{x - 1}} + 1)}}$ is equal to 180, is :

A

0

B

$-$1

C

2

D

1

Question 84: Palette 84/165

JEE Mains 2021 MCQ

If the coefficients of x⁷ in ${\left( {{x^2} + {1 \over {bx}}} \right)^{11}}$ and x^$-$7 in ${\left( {{x} - {1 \over {bx^2}}} \right)^{11}}$, b $\ne$ 0, are equal, then the value of b is equal to :

A

2

B

$-$1

C

1

D

$-$2

Question 85: Palette 85/165

JEE Mains 2021 MCQ

The sum of all those terms which are rational numbers in the

expansion of (2^1/3 + 3^1/4)¹² is :

A

89

B

27

C

35

D

43

Question 86: Palette 86/165

JEE Mains 2021 MCQ

If the greatest value of the term independent of 'x' in the

expansion of ${\left( {x\sin \alpha + a{{\cos \alpha } \over x}} \right)^{10}}$ is ${{10!} \over {{{(5!)}^2}}}$, then the value of 'a' is equal to :

A

$-$1

B

1

C

$-$2

D

2

Question 87: Palette 87/165

JEE Mains 2021 MCQ

The lowest integer which is greater

than ${\left( {1 + {1 \over {{{10}^{100}}}}} \right)^{{{10}^{100}}}}$ is ______________.

A

3

B

4

C

2

D

1

Question 88: Palette 88/165

JEE Mains 2021 MCQ

If b is very small as compared to the value of a, so that the cube and other higher powers of ${b \over a}$ can be neglected in the identity ${1 \over {a - b}} + {1 \over {a - 2b}} + {1 \over {a - 3b}} + ..... + {1 \over {a - nb}} = \alpha n + \beta {n^2} + \gamma {n^3}$, then the value of $\gamma$ is :

A

${{{a^2} + b} \over {3{a^3}}}$

B

${{a + b} \over {3{a^2}}}$

C

${{{b^2}} \over {3{a^3}}}$

D

${{a + {b^2}} \over {3{a^3}}}$

Question 89: Palette 89/165

JEE Mains 2021 MCQ

For the natural numbers m, n, if ${(1 - y)^m}{(1 + y)^n} = 1 + {a_1}y + {a_2}{y^2} + .... + {a_{m + n}}{y^{m + n}}$ and ${a_1} = {a_2} = 10$, then the value of (m + n) is equal to :

A

88

B

64

C

100

D

80

Question 90: Palette 90/165

JEE Mains 2021 MCQ

The coefficient of x²⁵⁶ in the expansion of

(1 $-$ x)¹⁰¹ (x² + x + 1)¹⁰⁰ is :

A

${}^{100}{C_{16}}$

B

${}^{100}{C_{15}}$

C

$-$ ${}^{100}{C_{16}}$

D

$-$ ${}^{100}{C_{15}}$

Question 91: Palette 91/165

JEE Mains 2021 MCQ

Let (1 + x + 2x²)²⁰ = a₀ + a₁x + a₂x² + .... + a₄₀x⁴⁰. Then a₁ + a₃ + a₅ + ..... + a₃₇ is equal to

A

2²⁰(2²⁰ $-$ 21)

B

2¹⁹(2²⁰ $-$ 21)

C

2¹⁹(2²⁰ $+$ 21)

D

2²⁰(2²⁰ $+$ 21)

Question 92: Palette 92/165

JEE Mains 2021 MCQ

The value of $\sum\limits_{r = 0}^6 {\left( {{}^6{C_r}\,.\,{}^6{C_{6 - r}}} \right)} $ is equal to :

A

924

B

1024

C

1124

D

1324

Question 93: Palette 93/165

JEE Mains 2021 MCQ

If the fourth term in the expansion of ${(x + {x^{{{\log }_2}x}})^7}$ is 4480, then the value of x where x$\in$N is equal to :

A

3

B

1

C

4

D

2

Question 94: Palette 94/165

JEE Mains 2021 MCQ

If n is the number of irrational terms in the
expansion of ${\left( {{3^{1/4}} + {5^{1/8}}} \right)^{60}}$, then (n $-$ 1) is divisible by :

A

30

B

8

C

7

D

26

Question 95: Palette 95/165

JEE Mains 2021 MCQ

Let [ x ] denote greatest integer less than or equal to x. If for n$\in$N,

${(1 - x + {x^3})^n} = \sum\limits_{j = 0}^{3n} {{a_j}{x^j}} $,

then $\sum\limits_{j = 0}^{\left[ {{{3n} \over 2}} \right]} {{a_{2j}} + 4} \sum\limits_{j = 0}^{\left[ {{{3n - 1} \over 2}} \right]} {{a_{2j}} + 1} $ is equal to :

A

2^{n $-$ 1}

B

n

C

2

D

1

Question 96: Palette 96/165

JEE Mains 2021 MCQ

The maximum value of the term independent of 't' in the expansion
of ${\left( {t{x^{{1 \over 5}}} + {{{{(1 - x)}^{{1 \over {10}}}}} \over t}} \right)^{10}}$ where x$\in$(0, 1) is :

A

${{10!} \over {\sqrt 3 {{(5!)}^2}}}$

B

${{2.10!} \over {3\sqrt 3 {{(5!)}^2}}}$

C

${{10!} \over {3{{(5!)}^2}}}$

D

${{2.10!} \over {3{{(5!)}^2}}}$

Question 97: Palette 97/165

JEE Mains 2021 MCQ

If $n \ge 2$ is a positive integer, then the sum of the series ${}^{n + 1}{C_2} + 2\left( {{}^2{C_2} + {}^3{C_2} + {}^4{C_2} + ... + {}^n{C_2}} \right)$ is :

A

${{n(2n + 1)(3n + 1)} \over 6}$

B

${{n(n + 1)(2n + 1)} \over 6}$

C

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

D

${{n(n - 1)(2n + 1)} \over 6}$

Question 98: Palette 98/165

JEE Mains 2021 MCQ

The value of
-¹⁵C₁ + 2.¹⁵C₂ – 3.¹⁵C₃ + ... - 15.¹⁵C₁₅ + ¹⁴C₁ + ¹⁴C₃ + ¹⁴C₅ + ...+ ¹⁴C₁₁ is :

A

2¹³ - 13

B

2¹⁶ - 1

C

2¹⁴

D

2¹³ - 14

Question 99: Palette 99/165

JEE Mains 2020 MCQ

If the constant term in the binomial expansion of
${\left( {\sqrt x - {k \over {{x^2}}}} \right)^{10}}$ is 405, then |k| equals :

A

3

B

9

C

1

D

2

Question 100: Palette 100/165

JEE Mains 2020 MCQ

If {p} denotes the fractional part of the number p, then
$\left\{ {{{{3^{200}}} \over 8}} \right\}$, is equal to :

A

${5 \over 8}$

B

${7 \over 8}$

C

${1 \over 8}$

D

${3 \over 8}$

Question 101: Palette 101/165

JEE Mains 2020 MCQ

If for some positive integer n, the coefficients
of three consecutive terms in the binomial
expansion of (1 + x)^{n + 5} are in the ratio
5 : 10 : 14, then the largest coefficient in this expansion is :

A

330

B

792

C

252

D

462

Question 102: Palette 102/165

JEE Mains 2020 MCQ

The value of $\sum\limits_{r = 0}^{20} {{}^{50 - r}{C_6}} $ is equal to:

A

${}^{50}{C_6} - {}^{30}{C_6}$

B

${}^{51}{C_7} - {}^{30}{C_7}$

C

${}^{50}{C_7} - {}^{30}{C_7}$

D

${}^{51}{C_7} + {}^{30}{C_7}$

Question 103: Palette 103/165

JEE Mains 2020 MCQ

If the term independent of x in the expansion of
${\left( {{3 \over 2}{x^2} - {1 \over {3x}}} \right)^9}$ is k, then 18 k is equal to :

A

5

B

9

C

7

D

11

Question 104: Palette 104/165

JEE Mains 2020 MCQ

If the number of integral terms in the expansion
of (3^1/2 + 5^1/8)ⁿ is exactly 33, then the least value of n is :

A

264

B

256

C

128

D

248

Question 105: Palette 105/165

JEE Mains 2020 MCQ

Let $\alpha $ > 0, $\beta $ > 0 be such that
$\alpha $³ + $\beta $² = 4. If the maximum value of the term independent of x in
the binomial expansion of ${\left( {\alpha {x^{{1 \over 9}}} + \beta {x^{ - {1 \over 6}}}} \right)^{10}}$ is 10k,
then k is equal to :

A

176

B

336

C

352

D

84

Question 106: Palette 106/165

JEE Mains 2020 MCQ

In the expansion of ${\left( {{x \over {\cos \theta }} + {1 \over {x\sin \theta }}} \right)^{16}}$, if ${\ell _1}$ is the least value of the term independent of x when ${\pi \over 8} \le \theta \le {\pi \over 4}$ and ${\ell _2}$ is the least value of the term independent of x when ${\pi \over {16}} \le \theta \le {\pi \over 8}$, then the ratio ${\ell _2}$ : ${\ell _1}$ is equal to :

A

8 : 1

B

16 : 1

C

1 : 8

D

1 : 16

Question 107: Palette 107/165

JEE Mains 2020 MCQ

If $\alpha $ and $\beta $ be the coefficients of x⁴ and x² respectively in the expansion of
${\left( {x + \sqrt {{x^2} - 1} } \right)^6} + {\left( {x - \sqrt {{x^2} - 1} } \right)^6}$, then

A

$\alpha + \beta = 60$

B

$\alpha - \beta = 60$

C

$\alpha + \beta = -30$

D

$\alpha - \beta = -132$

Question 108: Palette 108/165

JEE Mains 2020 MCQ

The coefficient of x⁷ in the expression
(1 + x)¹⁰ + x(1 + x)⁹ + x²(1 + x)⁸ + ......+ x¹⁰ is:

A

120

B

330

C

420

D

210

Question 109: Palette 109/165

JEE Mains 2020 MCQ

The greatest positive integer k, for which 49^k + 1 is a factor of the sum
49¹²⁵ + 49¹²⁴ + ..... + 49² + 49 + 1, is:

A

32

B

60

C

63

D

65

Question 110: Palette 110/165

JEE Mains 2019 MCQ

If ²⁰C₁ + (2²) ²⁰C₂ + (3²) ²⁰C₃ + ..... + (20² ) ²⁰C₂₀ = A(2^{$\beta $}), then the ordered pair (A, $\beta $) is equal to :

A

(420, 19)

B

(420, 18)

C

(380, 18)

D

(380, 19)

Question 111: Palette 111/165

JEE Mains 2019 MCQ

The term independent of x in the expansion of
$\left( {{1 \over {60}} - {{{x^8}} \over {81}}} \right).{\left( {2{x^2} - {3 \over {{x^2}}}} \right)^6}$ is equal to :

A

36

B

- 108

C

- 36

D

- 72

Question 112: Palette 112/165

JEE Mains 2019 MCQ

The coefficient of x¹⁸ in the product
(1 + x) (1 – x)¹⁰ (1 + x + x²)⁹ is :

A

126

B

- 84

C

- 126

D

84

Question 113: Palette 113/165

JEE Mains 2019 MCQ

The smallest natural number n, such that the coefficient of x in the expansion of ${\left( {{x^2} + {1 \over {{x^3}}}} \right)^n}$ is ⁿC₂₃, is :

A

23

B

58

C

38

D

35

Question 114: Palette 114/165

JEE Mains 2019 MCQ

If the coefficients of x² and x³ are both zero, in the expansion of the expression (1 + ax + bx² ) (1 – 3x)¹⁵ in powers of x, then the ordered pair (a,b) is equal to :

A

(28, 861)

B

(28, 315)

C

(–21, 714)

D

(–54, 315)

Question 115: Palette 115/165

JEE Mains 2019 MCQ

If some three consecutive in the binomial expansion of (x + 1)ⁿ is powers of x are in the ratio 2 : 15 : 70, then the average of these three coefficient is :-

A

625

B

227

C

964

D

232

Question 116: Palette 116/165

JEE Mains 2019 MCQ

If the fourth term in the binomial expansion of ${\left( {{2 \over x} + {x^{{{\log }_8}x}}} \right)^6}$ (x > 0) is 20 × 8⁷, then a value of x is :

A

8^–2

B

8²

C

8³

D

8

Question 117: Palette 117/165

JEE Mains 2019 MCQ

If the fourth term in the binomial expansion of
${\left( {\sqrt {{x^{\left( {{1 \over {1 + {{\log }_{10}}x}}} \right)}}} + {x^{{1 \over {12}}}}} \right)^6}$ is equal to 200, and x > 1, then the value of x is :

A

100

B

10³

C

10

D

10⁴

Question 118: Palette 118/165

JEE Mains 2019 MCQ

The sum of the co-efficients of all even degree terms in x in the expansion of
${\left( {x + \sqrt {{x^3} - 1} } \right)^6}$ + ${\left( {x - \sqrt {{x^3} - 1} } \right)^6}$, (x > 1) is equal to:

A

32

B

26

C

29

D

24

Question 119: Palette 119/165

JEE Mains 2019 MCQ

The sum of the series

2.²⁰C₀ + 5.²⁰C₁ + 8.²⁰C₂ + 11.²⁰C₃ + ... +62.²⁰C₂₀ is equal to :

A

2²⁵

B

2²⁴

C

2²⁶

D

2²³

Question 120: Palette 120/165

JEE Mains 2019 MCQ

The total number of irrational terms in the binomial expansion of (7^1/5 – 3^1/10)⁶⁰ is :

A

54

B

55

C

49

D

48

Question 121: Palette 121/165

JEE Mains 2019 MCQ

A ratio of the 5th term from the beginning to the 5th term from the end in the binomial expansion of ${\left( {{2^{1/3}} + {1 \over {2{{\left( 3 \right)}^{1/3}}}}} \right)^{10}}$ is :

A

1 : 2(6)^1/3

B

1 : 4(6)^1/3

C

2(36)^1/3 : 1

D

4(36)^1/3 : 1

Question 122: Palette 122/165

JEE Mains 2019 MCQ

Let (x + 10)⁵⁰ + (x $-$ 10)⁵⁰ = a₀ + a₁x + a₂x² + . . . . + a₅₀x⁵⁰, for all x $ \in $ R; then ${{{a_2}} \over {{a_0}}}$ is equal to

A

12.25

B

12.75

C

12.00

D

12.50

Question 123: Palette 123/165

JEE Mains 2019 MCQ

Let S_n = 1 + q + q² + . . . . . + qⁿ and T_n = 1 + $\left( {{{q + 1} \over 2}} \right) + {\left( {{{q + 1} \over 2}} \right)^2}$ + . . . . . .+ ${\left( {{{q + 1} \over 2}} \right)^n}$ where q is a real number and q $ \ne $ 1. If   ¹⁰¹C₁ + ¹⁰¹C₂ . S₁ + .... + ¹⁰¹C₁₀₁ . S₁₀₀ = $\alpha $T₁₀₀ then $\alpha $ is equal to

A

202

B

200

C

2¹⁰⁰

D

2⁹⁹

Question 124: Palette 124/165

JEE Mains 2019 MCQ

The sum of the real values of x for which the middle term in the binomial expansion of ${\left( {{{{x^3}} \over 3} + {3 \over x}} \right)^8}$ equals 5670 is :

A

0

B

8

C

6

D

4

Question 125: Palette 125/165

JEE Mains 2019 MCQ

The value of r for which ²⁰C_r ²⁰C₀ + ²⁰C_r$-$1 ²⁰C₁ + ²⁰C_r$-$2 ²⁰C₂ + . . . . .+ ²⁰C₀ ²⁰C_r  is maximum, is

A

20

B

15

C

10

D

11

Question 126: Palette 126/165

JEE Mains 2019 MCQ

The positive value of $\lambda $ for which the co-efficient of x² in the expression x² ${\left( {\sqrt x + {\lambda \over {{x^2}}}} \right)^{10}}$ is 720, is -

A

4

B

$2\sqrt 2 $

C

3

D

$\sqrt 5 $

Question 127: Palette 127/165

JEE Mains 2019 MCQ

If  ${\sum\limits_{i = 1}^{20} {\left( {{{{}^{20}{C_{i - 1}}} \over {{}^{20}{C_i} + {}^{20}{C_{i - 1}}}}} \right)} ^3} = {k \over {21}}$  then k is equal to

A

100

B

200

C

50

D

400

Question 128: Palette 128/165

JEE Mains 2019 MCQ

If the third term in the binomial expansion
of ${\left( {1 + {x^{{{\log }_2}x}}} \right)^5}$ equals 2560, then a possible value of x is -

A

$2\sqrt 2 $

B

$4\sqrt 2 $

C

${1 \over 8}$

D

${1 \over 4}$

Question 129: Palette 129/165

JEE Mains 2019 MCQ

The coefficient of t⁴ in the expansion of ${\left( {{{1 - {t^6}} \over {1 - t}}} \right)^3}$ is :

A

14

B

15

C

10

D

12

Question 130: Palette 130/165

JEE Mains 2019 MCQ

If the fractional part of the number $\left\{ {{{{2^{403}}} \over {15}}} \right\} is \, {k \over {15}}$, then k is equal to :

A

8

B

14

C

6

D

1

Question 131: Palette 131/165

JEE Mains 2018 MCQ

The coefficient of x² in the expansion of the product
(2$-$x²) .((1 + 2x + 3x²)⁶ + (1 $-$ 4x²)⁶) is :

A

107

B

106

C

108

D

155

Question 132: Palette 132/165

JEE Mains 2018 MCQ

The sum of the co-efficients of all odd degree terms in the expansion of

${\left( {x + \sqrt {{x^3} - 1} } \right)^5} + {\left( {x - \sqrt {{x^3} - 1} } \right)^5}$, $\left( {x > 1} \right)$ is

A

2

B

-1

C

0

D

1

Question 133: Palette 133/165

JEE Mains 2018 MCQ

The coefficien of x¹⁰ in the expansion of (1 + x)²(1 + x²)³(1 + x³)⁴ is equal to :

A

52

B

56

C

50

D

44

Question 134: Palette 134/165

JEE Mains 2018 MCQ

If n is the degree of the polynomial,

${\left[ {{2 \over {\sqrt {5{x^3} + 1} - \sqrt {5{x^3} - 1} }}} \right]^8} + $ ${\left[ {{2 \over {\sqrt {5{x^3} + 1} + \sqrt {5{x^3} - 1} }}} \right]^8}$

and m is the coefficient of xⁿ in it, then the ordered pair (n, m) is equal to :

A

(24, (10)⁸)

B

(8, 5(10)⁴)

C

(12, (20)⁴)

D

(12, 8(10)⁴)

Question 135: Palette 135/165

JEE Mains 2017 MCQ

The coefficient of x⁻⁵ in the binomial expansion of

${\left( {{{x + 1} \over {{x^{{2 \over 3}}} - {x^{{1 \over 3}}} + 1}} - {{x - 1} \over {x - {x^{{1 \over 2}}}}}} \right)^{10}},$ where x $ \ne $ 0, 1, is :

A

1

B

4

C

$-$ 4

D

$-$ 1

Question 136: Palette 136/165

JEE Mains 2017 MCQ

If (27)⁹⁹⁹ is divided by 7, then the remainder is :

A

1

B

2

C

3

D

6

Question 137: Palette 137/165

JEE Mains 2017 MCQ

The value of $\left( {{}^{21}{C_1} - {}^{10}{C_1}} \right) + \left( {{}^{21}{C_2} - {}^{10}{C_2}} \right) + \left( {{}^{21}{C_3} - {}^{10}{C_3}} \right)$
$\left( {{}^{21}{C_4} - {}^{10}{C_4}} \right)$$ + .... + \left( {{}^{21}{C_{10}} - {}^{10}{C_{10}}} \right)$ is

A

${2^{21}} - {2^{10}}$

B

${2^{20}} - {2^{9}}$

C

${2^{20}} - {2^{10}}$

D

${2^{21}} - {2^{11}}$

Question 138: Palette 138/165

JEE Mains 2016 MCQ

If the coefficients of x⁻² and x⁻⁴ in the expansion of ${\left( {{x^{{1 \over 3}}} + {1 \over {2{x^{{1 \over 3}}}}}} \right)^{18}},\left( {x > 0} \right),$ are m and n respectively, then ${m \over n}$ is equal to :

A

182

B

${4 \over 5}$

C

${5 \over 4}$

D

27

Question 139: Palette 139/165

JEE Mains 2016 MCQ

For x $ \in $ R, x $ \ne $ -1,

if (1 + x)²⁰¹⁶ + x(1 + x)²⁰¹⁵ + x²(1 + x)²⁰¹⁴ + . . . . + x²⁰¹⁶ =

$\sum\limits_{i = 0}^{2016} {{a_i}} \,{x^i},\,\,$ then a₁₇ is equal to :

A

${{2017!} \over {17!\,\,\,2000!}}$

B

${{2016!} \over {17!\,\,\,1999!}}$

C

${{2017!} \over {2000!}}$

D

${{2016!} \over {16!}}$

Question 140: Palette 140/165

JEE Mains 2016 MCQ

If the number of terms in the expansion of ${\left( {1 - {2 \over x} + {4 \over {{x^2}}}} \right)^n},\,x \ne 0,$ is 28, then the sum of the coefficients of all the terms in this expansion, is :

A

243

B

729

C

64

D

2187

Question 141: Palette 141/165

JEE Mains 2015 MCQ

The sum of coefficients of integral power of $x$ in the binomial expansion ${\left( {1 - 2\sqrt x } \right)^{50}}$ is :

A

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

B

${1 \over 2}\left( {{2^{50}} + 1} \right)$

C

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

D

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

Question 142: Palette 142/165

JEE Mains 2014 MCQ

If the coefficints of ${x^3}$ and ${x^4}$ in the expansion of $\left( {1 + ax + b{x^2}} \right){\left( {1 - 2x} \right)^{18}}$ in powers of $x$ are both zero, then $\left( {a,\,b} \right)$ is equal to:

A

$\left( {14,{{272} \over 3}} \right)$

B

$\left( {16,{{272} \over 3}} \right)$

C

$\left( {16,{{251} \over 3}} \right)$

D

$\left( {14,{{251} \over 3}} \right)$

Question 143: Palette 143/165

JEE Mains 2013 MCQ

The term independent of $x$ in expansion of
${\left( {{{x + 1} \over {{x^{2/3}} - {x^{1/3}} + 1}} - {{x - 1} \over {x - {x^{1/2}}}}} \right)^{10}}$ is

A

4

B

120

C

210

D

310

Question 144: Palette 144/165

JEE Mains 2012 MCQ

If $n$ is a positive integer, then ${\left( {\sqrt 3 + 1} \right)^{2n}} - {\left( {\sqrt 3 - 1} \right)^{2n}}$ is :

A

an irrational number

B

an odd positive integer

C

an even positive integer

D

a rational number other than positive integers

Question 145: Palette 145/165

JEE Mains 2011 MCQ

The coefficient of ${x^7}$ in the expansion of ${\left( {1 - x - {x^2} + {x^3}} \right)^6}$ is

A

$-132$

B

$-144$

C

$132$

D

$144$

Question 146: Palette 146/165

JEE Mains 2010 MCQ

Let ${s_1} = \sum\limits_{j = 1}^{10} {j\left( {j - 1} \right){}^{10}} {C_j}$,

${{s_2} = \sum\limits_{j = 1}^{10} {} } j.{}^{10}{C_j}$ and

${{s_3} = \sum\limits_{j = 1}^{10} {{j^2}.{}^{10}{C_j}.} }$

Statement-1 : ${{S_3} = 55 \times {2^9}}$.
Statement-2 : ${{S_1} = 90 \times {2^8}}$ and ${{S_2} = 10 \times {2^8}}$.

A

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

B

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

C

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

D

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

Question 147: Palette 147/165

JEE Mains 2009 MCQ

The remainder left out when ${8^{2n}} - {\left( {62} \right)^{2n + 1}}$ is divided by 9 is :

A

2

B

7

C

8

D

0

Question 148: Palette 148/165

JEE Mains 2008 MCQ

Statement - 1 : $\sum\limits_{r = 0}^n {\left( {r + 1} \right)\,{}^n{C_r} = \left( {n + 2} \right){2^{n - 1}}.} $
Statement - 2 : $\sum\limits_{r = 0}^n {\left( {r + 1} \right)\,{}^n{C_r}{x^r} = {{\left( {1 + x} \right)}^n} + nx{{\left( {1 + x} \right)}^{n - 1}}.} $

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 149: Palette 149/165

JEE Mains 2007 MCQ

The sum of the series ${}^{20}{C_0} - {}^{20}{C_1} + {}^{20}{C_2} - {}^{20}{C_3} + .....\, - \,.....\, + {}^{20}{C_{10}}$ is

A

$0$

B

${}^{20}{C_{10}}$

C

$ - {}^{20}{C_{10}}$

D

${1 \over 2}{}^{20}{C_{10}}$

Question 150: Palette 150/165

JEE Mains 2007 MCQ

In the binomial expansion of ${\left( {a - b} \right)^n},\,\,\,n \ge 5,$ the sum of ${5^{th}}$ and ${6^{th}}$ terms is zero, then $a/b$ equals

A

${{n - 5} \over 6}$

B

${{n - 4} \over 5}$

C

${5 \over {n - 4}}$

D

${6 \over {n - 5}}$

Question 151: Palette 151/165

JEE Mains 2006 MCQ

For natural numbers $m$ , $n$, if ${\left( {1 - y} \right)^m}{\left( {1 + y} \right)^n}\,\, = 1 + {a_1}y + {a_2}{y^2} + ..........$ and ${a_1} = {a_2} = 10,$ then $\left( {m,\,n} \right)$ is

A

$\left( {20,\,45} \right)$

B

$\left( {35,\,20} \right)$

C

$\left( {45,\,35} \right)$

D

$\left( {35,\,45} \right)$

Question 152: Palette 152/165

JEE Mains 2006 MCQ

If the expansion in powers of $x$ of the function ${1 \over {\left( {1 - ax} \right)\left( {1 - bx} \right)}}$ is ${a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^3}.....$ then ${a_n}$ is

A

${{{b^n} - {a^n}} \over {b - a}}$

B

${{{a^n} - {b^n}} \over {b - a}}$

C

${{{a^{n + 1}} - {b^{n + 1}}} \over {b - a}}$

D

${{{b^{n + 1}} - {a^{n + 1}}} \over {b - a}}$

Question 153: Palette 153/165

JEE Mains 2005 MCQ

The value of $\,{}^{50}{C_4} + \sum\limits_{r = 1}^6 {^{56 - r}} {C_3}$ is

A

${}^{55}{C_4}$

B

${}^{55}{C_3}$

C

${}^{56}{C_3}$

D

${}^{56}{C_4}$

Question 154: Palette 154/165

JEE Mains 2005 MCQ

If the coefficients of r^th, (r+1)^th, and (r + 2)^th terms in the binomial expansion of ${{\rm{(1 + y )}}^m}$ are in A.P., then m and r satisfy the equation

A

${m^2} - m(4r - 1) + 4\,{r^2} - 2 = 0$

B

${m^2} - m(4r + 1) + 4\,{r^2} + 2 = 0$

C

${m^2} - m(4r + 1) + 4\,{r^2} - 2 = 0$

D

${m^2} - m(4r - 1) + 4\,{r^2} + 2 = 0$

Question 155: Palette 155/165

JEE Mains 2005 MCQ

If $x$ is so small that ${x^3}$ and higher powers of $x$ may be neglected, then ${{{{\left( {1 + x} \right)}^{{3 \over 2}}} - {{\left( {1 + {1 \over 2}x} \right)}^3}} \over {{{\left( {1 - x} \right)}^{{1 \over 2}}}}}$ may be approximated as

A

$1 - {3 \over 8}{x^2}$

B

$3x + {3 \over 8}{x^2}$

C

$ - {3 \over 8}{x^2}$

D

${x \over 2} - {3 \over 8}{x^2}$

Question 156: Palette 156/165

JEE Mains 2005 MCQ

If the coefficient of ${x^7}$ in ${\left[ {a{x^2} + \left( {{1 \over {bx}}} \right)} \right]^{11}}$ equals the coefficient of ${x^{ - 7}}$ in ${\left[ {ax - \left( {{1 \over {b{x^2}}}} \right)} \right]^{11}}$, then $a$ and $b$ satisfy the relation

A

$a - b = 1$

B

$a + b = 1$

C

${a \over b} = 1$

D

$ab = 1$

Question 157: Palette 157/165

JEE Mains 2004 MCQ

If ${S_n} = \sum\limits_{r = 0}^n {{1 \over {{}^n{C_r}}}} \,\,and\,\,{t_n} = \sum\limits_{r = 0}^n {{r \over {{}^n{C_r}}},\,} $then ${{{t_{ n}}} \over {{S_n}}}$ is equal to

A

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

B

${1 \over 2}n - 1$

C

n - 1

D

${1 \over 2}n$

Question 158: Palette 158/165

JEE Mains 2004 MCQ

The coefficient of ${x^n}$ in expansion of $\left( {1 + x} \right){\left( {1 - x} \right)^n}$ is

A

${\left( { - 1} \right)^{n - 1}}n$

B

${\left( { - 1} \right)^n}\left( {1 - n} \right)$

C

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

D

$\left( {n - 1} \right)$

Question 159: Palette 159/165

JEE Mains 2004 MCQ

The coefficient of the middle term in the binomial expansion in powers of $x$ of ${\left( {1 + \alpha x} \right)^4}$ and ${\left( {1 - \alpha x} \right)^6}$ is the same if $\alpha $ equals

A

${3 \over 5}$

B

${10 \over 3}$

C

${{ - 3} \over {10}}$

D

${{ - 5} \over {3}}$

Question 160: Palette 160/165

JEE Mains 2003 MCQ

If $x$ is positive, the first negative term in the expansion of ${\left( {1 + x} \right)^{27/5}}$ is

A

6th term

B

7th term

C

5th term

D

8th term.

Question 161: Palette 161/165

JEE Mains 2003 MCQ

The number of integral terms in the expansion of ${\left( {\sqrt 3 + \root 8 \of 5 } \right)^{256}}$ is

A

35

B

32

C

33

D

34

Question 162: Palette 162/165

JEE Mains 2002 MCQ

The positive integer just greater than ${\left( {1 + 0.0001} \right)^{10000}}$ is

A

4

B

5

C

2

D

3

Question 163: Palette 163/165

JEE Mains 2002 MCQ

If the sum of the coefficients in the expansion of $\,{\left( {a + b} \right)^n}$ is 4096, then the greatest coefficient in the expansion is

A

1594

B

792

C

924

D

2924

Question 164: Palette 164/165

JEE Mains 2002 MCQ

$r$ and $n$ are positive integers $\,r > 1,\,n > 2$ and coefficient of $\,{\left( {r + 2} \right)^{th}}$ term and $3{r^{th}}$ term in the expansion of ${\left( {1 + x} \right)^{2n}}$ are equal, then $n$ equals

A

$3r$

B

$3r + 1$

C

$2r$

D

$2r + 1$

Question 165: Palette 165/165

JEE Mains 2002 MCQ

The coefficients of ${x^p}$ and ${x^q}$ in the expansion of ${\left( {1 + x} \right)^{p + q}}$ are

A

equal

B

equal with opposite signs

C

reciprocals of each other

D

none of these