Q 111 :

If ${\int }_0^{1}4{\cot }^{-1}(1-2x+4x^2) dx=a {\tan }^{-1}\left(2\right)-b{\log }_e\left(5\right),$ where $a,b\in \mathrm{\mathbb{N}}, then (2a + b)$ is equal to _______.  [2026]

Q 112 :

Let $[\cdot ]$ denote the greatest integer function and $f\left(x\right)=\underset{n\to \infty }{\lim }\frac{1}{n^3}\sum _{k=1}^n\left[\left[\frac{k^2}{3^x}\right]\right].$ Then $12\sum _{j=1}^{\infty }f\left(j\right)$  is equal to ________.   [2026]

Q 113 :

Let $[·]$ be the greatest integer function. If $\alpha ={\int }_0^{64}\left(x^{1/3}-\left[x^{1/3}\right]\right) dx,$ then $\frac{1}{\pi }{\int }_0^{\alpha \pi }\left(\frac{{\sin }^2\theta }{{\sin }^6\theta +{\cos }^6\theta }\right) d\theta$ is equal to_______. [2026]

Q 114 :

Let f be a polynomial function such that $f(x^2+1)=x^4+5x^2+2, \text{for all }x\in \mathrm{ℝ}.$ Then ${\int }_0^{3}f\left(x\right) dx$ is equal to:   [2026]

• $\frac{27}{2}$

   

• $\frac{33}{2}$

   

• $\frac{41}{3}$

   

• $\frac{5}{3}$

   

Q 115 :

The value of $\underset{r=1}{\overset{20}{\sum (}}\left|\left|\sqrt[\pi ]{\left(({\int }_0^{r}x \left|\sin \left(\pi x\right)\right| dx)\right)}\right|\right|)$ is     [2026]