Among the statements

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

$\left(S2\right):13{\left(13\right)}^n-11n-13 is divisible by 144 for infinitely many n\in \mathrm{\mathbb{N}}$

(A) only (S2) is correct

(B) both (S1) and (S2) are incorrect

(C) both (S1) and (S2) are correct

(D) only (S1) is correct

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

(A) both 14 and 34

(B) 34 but not by 14       

(C) 14 but not by 34         

(D) neither 14 nor 34  

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

(A) 20 

(B) 13

(C) 5

(D) 10

If $\frac{1}{n+1}{}^n{C}_n+\frac{1}{n}{}^n{C}_{n-1}+....+\frac{1}{2}{}^n{C}_1+{}^n{C}_0=\frac{1023}{10}, then n is equal to$

(A) 8                      (B) 9                   (C) 6                       (D) 7

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

(A) ${}^{101}{C}_{50}$                      (B) $-{}^{99}{C}_{49}$                   (C) $-{}^{101}{C}_{50}$                (D) ${}^{99}{C}_{49}$

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

(A) $\frac{1}{15}$                    (B) $\frac{4}{15}$                      (C) $\frac{14}{15}$                        (D) $\frac{8}{15}$

The value of $\frac{1}{1! 50!}+\frac{1}{3! 48!}+\frac{1}{5! 46!}+...+\frac{1}{49! 2!}+\frac{1}{51! 1!}$ is

(A) $\frac{2^{51}}{50!}$                    (B) $\frac{2^{50}}{50!}$                     (C) $\frac{2^{51}}{51!}$                     (D) $\frac{2^{50}}{51!}$

The value of $\sum _{r=0}^{22}{}^{22}{C}_r {}^{23}{C}_r$ is

(A) ${}^{45}{C}_{24}$                   (B) ${}^{45}{C}_{23}$                 (C) ${}^{44}{C}_{23}$                  (D) ${}^{44}{C}_{22}$

If ${\left({}^{30}{C}_1\right)}^2+2{\left({}^{30}{C}_2\right)}^2+3{\left({}^{30}{C}_3\right)}^2+...+30{\left({}^{30}{C}_{30}\right)}^2=\frac{\alpha 60!}{{(30!)}^2},$ then $\alpha$ is equal to

(A) 60                          (B) 10                      (C) 15                      (D) 30

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

(A) [x] is even but [y] is odd                        (B) [x] + [y] is even

(C) [x] and [y] are both odd                         (D) [x] is odd but [y] is even

The largest natural number n such that $3^n$ divides 66! is _______ .

The remainder, when $7^{103}$ is divided by 17, is ________ .

The remainder, when ${19}^{200}+{23}^{200}$ is divided by 49, is ________ .

Suppose $\sum _{r=0}^{2023} r^2 {}^{2023}{C}_r=2023\times \alpha \times 2^{2022}$. Then the value of $\alpha$ is ___________ . 

Let the sum of the coefficients of the first three terms in the expansion of ${(x-\frac{3}{x^2})}^n, x\ne 0,n\in N,$ be 376. 

Then the coefficient of $x^4$ is ___________ .

The remainder when ${\left(2023\right)}^{2023}$ is divided by 35 is __________ .

50th root of a number x is 12 and 50th root of another number y is 18. Then the remainder obtained on dividing (x + y) by 25 is _________ .

The remainder on dividing $5^{99}$ by 11 is _________ .

If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of

${(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}})}^n$ 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) $30\sqrt{3}$

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

(A) 64ab = 243                  (B) 729ab = 32                (C) 32ab = 729             (D) 243ab = 64

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) 1827                    (B) 5481                 (C) 2436                     (D) 3654

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

(A) ${10}^3-10$                 (B) ${12}^3-12$                  (C) ${13}^3-13$               (D) ${11}^3-11$

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

(A) 11                          (B) 44                         (C) 22                       (D) 33

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

(A) 66                       (B) 60                    (C) 63                       (D) 69

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

(A) 41                       (B) 92                      (C) 25                      (D) 63

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

(A) $\frac{26}{3}$                   (B) 9                         (C) 8                     (D) $\frac{80}{9}$

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

(A) 3025                   (B) 1210                   (C) 5445                    (D) 4895

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 ${(2+\frac{1}{\sqrt{2}})}^{200}$. 

If $\frac{{}^{200}{C}_{99}K}{a}=\frac{2^{l}m}{n}$, where $m$ and $n$ are odd numbers, then the ordered pair $(l,n)$ is equal to

(A) (50, 101)                 (B) (51, 101)                 (C) (51, 99)                 (D) (50, 51)

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

(A) ${}^{500}{C}_{301}$                   (B) ${}^{501}{C}_{200}$                 (C) ${}^{500}{C}_{300}$                (D) ${}^{501}{C}_{302}$

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

(A) a = 3b                    (B) a = b                     (C) ab = 1                   (D) ab = 3

The coefficient of $x^{18}$ in the expansion of ${(x^4-\frac{1}{x^3})}^{15}$ is __________ .

If the constant term in the expansion of ${(3x^2-\frac{1}{2x^5})}^7$ is $\alpha$, then $\left[\alpha \right]$ is equal to _______ .

The number of integral terms in the expansion of ${(3^{1/2}+5^{1/4})}^{680}$ is equal to ___________ .

Let $\alpha$ be the constant term in the binomial expansion of ${(\sqrt{x}-\frac{6}{x^{{\displaystyle \frac{3}{2}}}})}^n,n\le 15$. If the sum of the coefficients

of the remaining terms in the expansion is 649 and the coefficient of $x^{-n}$ is $\lambda \alpha$, then $\lambda$ is equal to _________ .

Let the sixth term in the binomial expansion of ${(\sqrt{2^{{\log }_2(10-3^x)}}+\sqrt[5]{2^{(x-2){\log }_{2}3}})}^m$, in the increasing powers of $2^{(x-2){\log }_{2}3}$, be 21.

If the binomial coefficients of the second, third and fourth terms in the expansion are respectively the first, third and fifth terms of an A.P.,

then the sum of the squares of all possible values of $x$ is _______ .

If the term without $x$ in the expansion of ${(x^{\frac{2}{3}}+\frac{\alpha }{x^3})}^{22}$ is 7315, then $\left|\alpha \right|$ is equal to ________ .

The constant term in the expansion of ${(2x+\frac{1}{x^7}+3x^2)}^5$ is _______.

Let the coefficients of three consecutive terms in the binomial expansion of ${(1+2x)}^n$ be in the ratio 2:5:8.Then the coefficient of the term,

which is in the middle of these three terms, is _________ .

If the co-efficient of $x^9$ in ${(\alpha x^3+\frac{1}{\beta x})}^{11}$ and the co-efficient of $x^{-9}$ in ${(\alpha x-\frac{1}{\beta x^3})}^{11}$ are equal, then ${\left(\alpha \beta \right)}^2$ is equal to ________ .

Let $\alpha >0$, be the smallest number such that the expansion of ${(x^{\frac{2}{3}}+\frac{2}{x^3})}^{30}$ has a term $\beta x^{-\alpha },\beta \in N.$ Then $\alpha$ is equal to ________ .

The coefficient of $x^{-6}$, in the expansion of ${(\frac{4x}{5}+\frac{5}{2x^2})}^9$, is __________ .

If the constant term in the binomial expansion of ${(\frac{x^{{\displaystyle \frac{5}{2}}}}{2}-\frac{4}{x^l})}^9$ is $-84$ and the coefficient of $x^{-3l}$ is $2^{\alpha }\beta$, where $\beta <0$ is an odd number, then $|\alpha l-\beta |$ is equal to __________ .

Let ${(a+bx+c{x}^2)}^{10}=\sum _{i=0}^{20}{p}_i{x}^i,a,b,c\in \mathrm{\mathbb{N}}$. If $p_1=20$ and $p_2=210,$ then $2(a+b+c)$ is equal to

(A) 8                     (B) 6                   (C) 12                    (D) 15

The coefficient of $x^7$ in ${(1-x+2x^3)}^{10}$ is _________ .