Q 31 :

If $f\left(x\right)$ satisfies the relation $f\left(x\right)=e^x+{\int }_0^1(y+x{e}^x)f\left(y\right)dy,$ then $e+f\left(0\right)$ is equal to _______ .           [2026]

Q 32 :

Let $I\left(x\right)=\int \frac{3 dx}{(4x+6)\left(\sqrt{4x^2+8x+3}\right)}$ and $I\left(0\right)=\frac{\sqrt{3}}{4}+20.$ If $I \left(\frac{{\displaystyle 1}}{{\displaystyle 2}}\right)=\frac{a\sqrt{2}}{b}+c, \text{where }a,b,c\in \mathrm{\mathbb{N}}, \gcd (a,b)=1,$then a+b+c is equal to.  [2026]

• 31

   

• 30

   

• 29

   

• 28

   

Q 33 :

If $\int \left(\left(\frac{1-5{\cos }^{2}x}{{\sin }^{5}x {\cos }^{2}x}\right)\right) dx=f\left(x\right)+C,$ where C is the constant of integration, then $f \left(\frac{\pi }{6}\right)-f \left(\frac{\pi }{4}\right)$ is equal to   [2026]

• $\frac{4}{\sqrt{3}}\left(8-\sqrt{6}\right)$

   

• $\frac{2}{\sqrt{3}}\left(4+\sqrt{6}\right)$

   

• $\frac{1}{\sqrt{3}}\left(26+\sqrt{3}\right)$

   

• $\frac{1}{\sqrt{3}}\left(26-\sqrt{3}\right)$