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Find the vector and the Cartesian equations of a line passing through the point (1,2,-4) and parallel to the line joining the points A(3,3,-5) and B(1,0,-11). Hence, find the distance between the two lines. OR Find the equations of the line passing through the points A(1,2,3) and B(3,5,9). Hence, find the coordinates of the points on this line which are at a distance of 14 units from point B.

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 $\vec{b} = 2\hat{i} - 7\hat{j} + \hat{k}$.

Find the area of the region bounded by the curves x^{2}=y, y=x+2 and x-axis, using integration.

Find the area of the following region using integration: {(x,y): y² ≤ 2x and y ≥ x-4}

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

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 $\frac{dy}{dx}$

If $y=(x+\sqrt{x^{2}-1})^{2}$;, then show that $(x^{2}-1)(\frac{dy}{dx})^{2}=4y^{2}.$

(a) Differentiate $\text{sec}^{-1}\left(\frac{1}{\sqrt{1-x^2}}\right)$ w.r.t. $\sin^{-1}\left(2x\sqrt{1-x^2}\right)$.
OR
(b) If $y = \tan x + \sec x$, then prove that $\frac{d^2y}{dx^2} = \frac{\cos x}{(1-\sin x)^2}$.

(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