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If $\begin{bmatrix} 3 & -1 & \sin 3x \\ -7 & 4 & \cos 2x \\ -11 & 7 & 2 \end{bmatrix}$ is a singular matrix, then find all values of $x$
where $x \in \left[0, \frac{\pi}{2}\right]$.

If $A = \begin{bmatrix} 0 & 2 & 1 \\ -2 & -1 & -2 \\ 1 & -1 & 0 \end{bmatrix}$, find $A^{-1}$ and use it to solve the following system of equations :
$-2y + z = 7, 2x - y - z = 8, x - 2y = 10$

Check whether the lines given by $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $\frac{x-4}{5} = \frac{y-1}{2} = z$ are parallel or not. If parallel, find the distance between them, otherwise find their point of intersection, if the lines are intersecting.

Solve the following Linear Programming Problem graphically: Maximize $Z=\frac{2x}{5}+\frac{3y}{10}$ subject to constraints $2x+y\le 1000$, $x+y\le 800$, $x, y \ge 0$.

Solve the following Linear Programming Problem graphically: Maximise $Z=200x+120y$ subject to the constraints $x+y\le 300$, $3x+y\le 600$, $x-y\ge -100$, $x,y\ge 0$

Solve the following linear programming problem graphically: Maximize $Z=10500x+9000y$ Subject to constraints $x+y \le 50$, $2x+y \le 80$, $x, y \ge 0$

Find the shortest distance between the lines $\vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k}$ and $\vec{r}=(1+2\mu)\hat{i}+(2-5\mu)\hat{k}+(4\mu-1)\hat{j}$.

Find a point on the line $\frac{x-2}{3}=\frac{1-y}{2}=\frac{z-3}{2}$ at a distance of $\sqrt{2}$ units from the point (1, 2, 3).

If $\vec{a}$, $\vec{b}$ and $\vec{c}$ are unit vectors, then prove that $|\vec{a}-\vec{b}|^{2}+|\vec{b}-\vec{c}|^{2}+|\vec{c}-\vec{a}|^{2}\le 9$.

Let three toys A, B and C be placed in the same straight line. If the position vectors of A, B and C are $5\hat{i}-2\hat{j}$, $5\hat{i}+8\hat{j}$ and $a\hat{i}-52\hat{j}$ respectively, find the value of 'a'.

Solve the differential equation $(x+2y^{3})dy=y~dx$.

Find a particular solution of the differential equation $(x+1)\frac{dy}{dx}=2 e^{-y}-1$ given that $y=0$ when $x=0$.

Find the general solution of the differential equation $2x^{2}\frac{dy}{dx}=y^{2}+2xy.$

Find the general solution of the differential equation: $y\log y\frac{dx}{dy}+x=\frac{2}{y}$

Solve the following differential equation: $x\frac{dy}{dx}=y-x\sin^{2}(\frac{y}{x})$ given that $y(1)=\frac{\pi}{6}$

Evaluate: $\int_{0}^{2}\frac{1}{\sqrt{x^{2}+2x+3}}dx$

Evaluate: $\int_{0}^{\pi}\frac{\sin^{2026}x}{\sin^{2026}x+\cos^{2026}x}dx$

Evaluate: $\int_{-\frac{\pi}{6}}^{\frac{\pi}{2}}(\sin|x|+\cos|x|)dx$

Evaluate: $\int_{\frac{\pi}{12}}\frac{dx}{1+\sqrt{\cot x}}$

Find: $\int\frac{x-\sin x}{1-\cos x}dx$