Write the domain and range (principle value branch) of the following functions: $$f(x)=\tan ^{-1} x$$

(a) If $$f(x)=\left\{\begin{array}{ll}x^2, & \text { if } x \geq 1 \\ x, & \text { if } x<1\end{array}\right.$$, then show that $$f$$ is not differentiable at $$x=1$$.

OR

(b) Find the value(s) of '$$\lambda$$', if the function

$$f(x)=\left\{\begin{array}{cc} \frac{\sin ^2 \lambda x}{x^2}, & \text { if } x \neq 0 \\ 1, & \text { if } x=0 \end{array} \text { is continuous at } x=0 .\right.$$

Sketch the region bounded by the lines $$2 x+y=8, y=2, y=4$$ and the $$y$$-axis. Hence, obtain its area using integration.

(a) If the vectors $$\vec{a}$$ and $$\vec{b}$$ are such that $$|\vec{a}|=3,|\vec{b}|=\frac{2}{3}$$ and $$\vec{a} \times \vec{b}$$ is a unit vector, then find the angle between $$\vec{a}$$ and $$\vec{b}$$.

OR

(b) Find the area of a parallelogram whose adjacent side are determined by the vectors $$\vec{a}=\hat{i}-\hat{j}+3 \hat{k}$$ and $$\vec{b}=2 \hat{i}-7 \hat{j}+\hat{k}$$