If $$\int \frac{\log \left(t+\sqrt{1+t^2}\right)}{\sqrt{1+t^2}} d t=\frac{1}{2}[g(t)]^2+c$$, (where $$c$$ is a constant of integration), then $$g(2)$$ is

If $$A=\left[\begin{array}{cc}1 & \tan x \\ -\tan x & 1\end{array}\right]$$, then $$A^T \cdot A^{-1}=$$

$$\mathbf{a}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{c}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}$$ are three vectors. For a vector $$\mathbf{r}$$ with $$\mathbf{r} \times \mathbf{a}=\mathbf{b}$$ and $$\mathbf{r} \cdot \mathbf{c}=3,|\mathbf{r}|$$ is

Let $$P Q R$$ be a right angled isosceles triangle, right angled at $$Q(2,1)$$. If the equation of the line $$P R$$ is $$2 x+y=3$$, then the combined equation representing the pair of lines $$P Q$$ and $$Q R$$ is