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25. (a) Find the vector equation of the line passing through the point $(2, 1, 3)$ and perpendicular to both the lines $\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3} ; \frac{x}{-3}=\frac{y}{2}=\frac{z}{5}$

Find the distance between the lines:$$\vec{r} = (\hat{i} + 2\hat{j} - 4\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 6\hat{k})$$$$\vec{r} = (3\hat{i} + 3\hat{j} - 5\hat{k}) + \mu(4\hat{i} + 6\hat{j} + 12\hat{k})$$

Find the coordinates of the foot of the perpendicular drawn from the point $P(0, 2, 3)$ to the line:$$\frac{x+3}{5} = \frac{y-1}{2} = \frac{z+4}{3}$$
OR
(b) Three vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ satisfy the condition $\vec{a} + \vec{b} + \vec{c} = \vec{0}$. Evaluate the quantity $\mu = \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}$, if $|\vec{a}| = 3$, $|\vec{b}| = 4$, and $|\vec{c}| = 2$.

Find the vector and the cartesian equations of a line that passes through the point A(1,2,-1) and parallel to the line 5x-25=14-7y=35z.

Find all the vectors of magnitude $3\sqrt{3}$ which are collinear to vector $\hat{i}+\hat{j}+\hat{k}.$

(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 sides are determined by the vectors $\vec{a} = \hat{i} - \hat{j} + 3\hat{k}$ and $\vec{b} = 2\hat{i} - 7\hat{j} + \hat{k}$.

If $x=a\sin 2t, y=a(\cos 2t+\log\tan t)$ then find $\frac{dy}{dx}$

If $y=x^{\frac{1}{x}}$ then find $\frac{dy}{dx}$ at $x=1$.

22. If $(x^{2}+y^{2})^{2}=xy$, then find $\frac{dy}{dx}$

(a) If $f(x) = \begin{cases} x^2, & \text{if } x \geq 1 \\ x, & \text{if } x < 1 \end{cases}$, then show that $f$ is not differentiable at $x=1$.
OR
(b) Find the value(s) of '$\lambda$', if the function $f(x) = \begin{cases} \frac{\sin^2 \lambda x}{x^2} & \text{if } x \neq 0 \\ 1 & \text{if } x=0 \end{cases}$ is continuous at $x=0$.

Draw the graph of $f(x)=\sin^{-1}x, x\in[-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}]$. Also, write range of $f(x)$.

Evaluate : $3\sin^{-1}(\frac{1}{\sqrt{2}})+2\cos^{-1}(\frac{\sqrt{3}}{2})+\cos^{-1}(0)$

21. (b) OR: Evaluate : $\cos^{-1}[\cos(-\frac{7\pi}{3})]$

21. (a) Find the domain of $y=\sin^{-1}(x^{2}-4)$.

Write the domain and range (principle value branch) of the following functions: f(x)=tan⁻¹x

$\vec{a}$ and $\vec{b}$ are two non-zero vectors such that the projection of $\vec{a}$ on $\vec{b}$ is 0. The angle between $\vec{a}$ and $\vec{b}$:

If a vector makes an angle of $\frac{\pi}{4}$ with the positive directions of both x-axis and y-axis, then the angle which it makes with positive z-axis is :

13. If $\theta$ is the angle between two vectors $\vec{a}$ and $\vec{b}$ then $\vec{a} \cdot \vec{b} \ge 0$ only when:

A unit vector along the vector $4\hat{i}-3\hat{k}$ is:

The corner points of the feasible region in the graphical representation of a linear programming problem are (2, 72), (15, 20) and (40, 15). If z=18x+9y be the objective function, then :