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Find the domain of \(f(x)=\sin^{-1}(-x^{2})\).

Solve the differential equation given by:$$x \, dy - y \, dx - \sqrt{x^{2} + y^{2}} \, dx = 0$$

Solve graphically the following linear programming problem : Maximise \(z = 6x + 3y\), subject to the constraints\begin{align}
4x + y &\ge 80 \\
3x + 2y &\le 150 \\
x + 5y &\ge 115 \\
x, y &\ge 0
\end{align}

Solve the following Linear Programming Problem graphically: Maximize: \(P = 70x + 40y\) subject to: \(3x + 2y ≤ 9, 3x + y ≤ 9, x ≥ 0, y ≥ 0\)

If \(f(x)=a(\tan x-\cot x)\), where \(a>0\), then find whether \(f(x)\) is increasing or decreasing function in its domain.

Find the maximum and minimum values of the function given by \(f(x)=5+\sin 2x\).

The area of the region bounded by the line \(y=mx (m>0)\), the curve \(x^{2}+y^{2}=4\) and the \(x\)-axis in the first quadrant is \(\frac{\pi}{2}\) units. Using integration, find the value of m.

33. Using integration, find the area of the region bounded by the parabola $y^{2}=4ax$ and its latus rectum.

Determine graphically the minimum value of the following objective function : $z=500x+400y$ subject to constraints $x+y\le200, x\ge20, y\ge4x, y\ge0$

30. Solve the following linear programming problem graphically: Minimise: $z=-3x+4y$ subject to the constraints $x+2y\le8, 3x+2y\le12, x,y\ge0$

If $\vec{r}=3\hat{i}-2\hat{j}+6\hat{k}$, find the value of $(\vec{r}\times\hat{j})\cdot(\vec{r}\times\hat{k})-12$

24. If the projection of the vector $\hat{i}+\hat{j}+\hat{k}$ on the vector $p\hat{i}+\hat{j}-2\hat{k}$ is $\frac{1}{3}$, then find the value(s) of $p$.

If $A=\begin{bmatrix}1&2&-2\\ -1&3&0\\ 0&-2&1\end{bmatrix}$ and $B^{-1}=\begin{bmatrix}3&-1&1\\ -15&6&-5\\ 5&-2&2\end{bmatrix},$ find $(AB)^{-1}$.

OR Solve the following system of equations by matrix method :$ x+2y+3z=6, 2x-y+z=2, 3x+2y-2z=3.$

Show that a function $f:\mathbb{R}\rightarrow\mathbb{R}$ defined as $f(x)=\frac{5x-3}{4}$ is both one-one and onto.

Let $f : \mathbb{R} - \left\{ \frac{4}{3} \right\} \to \mathbb{R}$ be a function defined as:$$f(x) = \frac{4x}{3x+4}$$Show that $f$ is a one-one function. Also, check whether $f$ is an onto function or not.

34. (a) If N denotes the set of all natural numbers and R is the relation on $N \times N$ defined by $(a, b) R (c, d)$, if $ad(b+c)=bc(a+d)$. Show that R is an equivalence relation.

A relation $R$ is defined on a set of real numbers $\mathbb{R}$ as:$$R = \{(x, y) : x \cdot y \text{ is an irrational number}\}$$Check whether $R$ is reflexive, symmetric, and transitive or not.

Solve the following differential equation : $xe^{\frac{y}{x}}-y+x\frac{dy}{dx}=0$

Find the general solution of the differential equation : $\frac{d}{dx}(xy^{2})=2y(1+x^{2})$

Find the general solution of the differential equation \(e^{x}\tan y~dx+(1-e^{x})\sec^{2}y~dy=0\).