Let $$\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{2 \times 2}$$, where $$\mathrm{a}_{\mathrm{ij}} \neq 0$$ for all $$\mathrm{i}, \mathrm{j}$$ and $$\mathrm{A}^{2}=\mathrm{I}$$. Let a be the sum of all diagonal elements of $$\mathrm{A}$$ and $$\mathrm{b}=|\mathrm{A}|$$. Then $$3 a^{2}+4 b^{2}$$ is equal to :

Let $$5 f(x)+4 f\left(\frac{1}{x}\right)=\frac{1}{x}+3, x > 0$$. Then $$18 \int_\limits{1}^{2} f(x) d x$$ is equal to :

Let $$I(x)=\int \frac{x^{2}\left(x \sec ^{2} x+\tan x\right)}{(x \tan x+1)^{2}} d x$$. If $$I(0)=0$$, then $$I\left(\frac{\pi}{4}\right)$$ is equal to :

Let the position vectors of the points A, B, C and D be $$5 \hat{i}+5 \hat{j}+2 \lambda \hat{k}, \hat{i}+2 \hat{j}+3 \hat{k},-2 \hat{i}+\lambda \hat{j}+4 \hat{k}$$ and $$-\hat{i}+5 \hat{j}+6 \hat{k}$$. Let the set $$S=\{\lambda \in \mathbb{R}$$ : the points A, B, C and D are coplanar $$\}$$.

Then $$\sum_\limits{\lambda \in S}(\lambda+2)^{2}$$ is equal to :