Question : If $\cot B=9 $, then what will be the value of $\sec ^2 B$?

Option 1: $\frac{98}{81}$

Option 2: $\frac{92}{81}$

Option 3: $\frac{42}{81}$

Option 4: $\frac{82}{81}$

Question : The average of $n$ numbers is $a$. The first number is increased by 2, the second one is increased by 4, the third one is increased by 8, and so on. The average of the new numbers is:

Option 1: $a+\frac{2^{n–1}–1}{n}$

Option 2: $a+\frac{2(2n–1)}{n}$

Option 3: $a+\frac{2^{n–1}}{n}$

Option 4: $a+\frac{2^{n}–1}{n}$

Question : If $x=2-2^{\frac{1}{3}}+2^{\frac{2}{3}}$, then find the value of $x^3-6 x^2+18 x$.

Option 1: 40

Option 2: 33

Option 3: 45

Option 4: 22

Question : If $(x+y):(x-y)=11:1$, find the value of $(\frac{5x+3y}{x-2y})$.

Option 1: $\frac{45}{4}$

Option 2: $\frac{4}{45}$

Option 3: $-\frac{45}{4}$

Option 4: $-\frac{4}{45}$

Question : If the radius of a sphere is increased by 2.5 decimetres (dm), then its surface area increases by 110 dm$^2$. What is the volume (in dm$^3$ ) of the sphere?
(Take $\pi=\frac{22}{7}$)

Option 1: $\frac{13}{21}$

Option 2: $\frac{3}{7}$

Option 3: $\frac{4}{7}$

Option 4: $\frac{11}{21}$