Q 41 :

The coefficient of $x^{48}$ in $(1+x)+2{(1+x)}^2+3{(1+x)}^3+\cdots +100{(1+x)}^{100}$ is equal to             [2026]

• $100·{}^{100}{C}_{49}-{}^{100}{C}_{48}$

   

• ${}^{100}{C}_{50}+{}^{101}{C}_{49}$

   

• $100·{}^{101}{C}_{49}-{}^{101}{C}_{50}$

   

• $100·{}^{100}{C}_{49}-{}^{100}{C}_{50}$

   

Q 42 :

The sum of the coefficients of $x^{499} and x^{500}$ in ${(1+x\mathit{)}}^{1000}+x{(1+x\mathit{)}}^{999}+x^2{(1+x\mathit{)}}^{998}+\cdots +x^{1000}$ is:   [2026]

• ${}^{1002}C_{500}$

   

• ${}^{1000}C_{501}$

   

• ${}^{1001}C_{501}$

   

• ${}^{1002}C_{501}$

   

Q 43 :

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 :    [2026]

• Both Statement I and Statement II are true

   

• Statement I is false but Statement II is true

   

• Both Statement I and Statement II are false

   

• Statement I is true but Statement II is false

   

Q 44 :

If $\left((\frac{1}{{}^{15}{C}_0}+\frac{1}{{}^{15}{C}_1})\right)\left((\frac{1}{{}^{15}{C}_1}+\frac{1}{{}^{15}{C}_2})\right)\cdots \left((\frac{1}{{}^{15}{C}_{12}}+\frac{1}{{}^{15}{C}_{13}})\right)=\frac{{}^{13}{C}_{ a}}{{}^{14}{C}_0{ }^{14}{C}_1\cdots { }^{14}{C}_{12}},$ then $30 \alpha$ is equal to_____ [2026]

Q 45 :

Let $C_r$ denote the coefficient of $x^r$ in the binomial expansion of $(1+x{)}^n$, $n\in \mathrm{\mathbb{N}}$ $0\le r\le n$. If $P_n=C_0-C_1+\frac{2^2}{3} C_2-\frac{2^3}{4} C_3+\cdots +\frac{{(-2)}^n}{n+1} C_n,$ then the value of $\sum _{n=1}^{25}\frac{1}{P_{2n}}$ equals:    [2026]

• 675

   

• 525

   

• 650

   

• 580