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Evaluate: $\int_ 0 ^{π/2} [\sin 2x \tan⁻¹(\sin x)] dx$

(a) Evaluate: $\int_0^{2\pi} \frac{1}{1 + e^{\sin x}} dx $
OR
(b) Find: $\int \frac{x⁴} { ((x-1)(x²+1))}dx.$

If $A=\begin{bmatrix}3 & 2\\ 5 & -7\end{bmatrix}$, then find $A^{-1}$ and use it to solve the following system of equations : $3x+5y=11, 2x-7y=-3$.

If $A=\begin{bmatrix}1 & 0 & 2\\ 0 & 2 & 1\\ 2 & 0 & 3\end{bmatrix}$, then show that $A^{3}-6A^{2}+7A+2I=O$

32. Find the inverse of the matrix $A=\begin{bmatrix}1 & -1 & 2\\ 0 & 2 & -3\\ 3 & -2 & 4\end{bmatrix}$. Using the inverse, solve the system of linear equations $x-y+2z=1; 2y-3z=1; 3x-2y+4z=3$.

If $A=\begin{bmatrix}1 & 2 & 3\\ 3 & -2 & 1\\ 4 & 2 & 1\end{bmatrix}$, then show that A³ - 23A - 40I = O.

Find the equations of all the sides of the parallelogram ABCD whose vertices are $A(4,7,8), B(2,3,4), C(-1,-2,1)$ and $D(1,2,5)$. Also, find the coordinates of the foot of the perpendicular from A to CD.

Find the value of $b$ so that the lines $\frac{x-1}{2}=\frac{y-b}{3}=\frac{z-3}{4}$ and $\frac{x-4}{5}=\frac{y-1}{2}=z$ are intersecting lines. Also, find the point of intersection of these given lines.

If the angle between the lines $\frac{x-5}{\alpha}=\frac{y+2}{-5}=\frac{z+\frac{24}{5}}{\beta}$ and $\frac{x}{1}=\frac{y}{0}=\frac{z}{1}$ is $\frac{\pi}{4}$, find the relation between $\alpha$ and $\beta$.

35. (b) OR: Find the angle between the lines $2x=3y=-z$ and $6x=-y=-4z$.

35. (a) Show that the following lines do not intersect each other : $\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-1}{5};\frac{x+2}{4}=\frac{y-1}{3}=\frac{z+1}{-2}$

25. (b) OR: The equations of a line are $5x-3=15y+7=3-10z$. Write the direction cosines of the line and find the coordinates of a point through which it passes.

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 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.