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If $y=P \cos ux+Q \sin ux$, show that $\frac{d^{2}y}{dx^{2}}+u^{2}y=0$.

Differentiate $x^{x}$ with respect to $x \log x$.

If $x=t+\frac{1}{t}$ and $y=t-\frac{1}{t}$, find $\frac{dy}{dx}$ at $t=2$.

If $x=a\sin^{3}t$, $y=b\cos^{3}t$, then find $\frac{dy}{dx}$ at $t=\frac{\pi}{4}$.

Evaluate $\sin[\tan^{-1}\tan(\frac{3\pi}{4})]$.

Find the value of $\sin[\cot^{-1}\sqrt{2}(\cos(\tan^{-1}1))]$.

A relation R on $A=\{1,2,3\}$ is defined as $R=\{(1,1),(3,3),(1,2)\}$. Is R a symmetric relation? Justify. Write the smallest relation set $R_{1}$ such that $R\cup R_{1}$ becomes an equivalence relation on the set {1, 2, 3}.

Check whether $f:Z\times Z \rightarrow Z\times Z$ (where Z is the set of integers) defined as $f(x,y)=(2y,3x)$ is injective or not.

The region represented by the system of inequations $3x+y\ge 3$, $2x-y\ge -5$, $x, y\ge 0$ is:

In a linear programming problem, the linear function which has to be maximized or minimized is called

For the feasible region shown below, the non-trivial constraints of the linear programming problem are

In the graph, the feasible region representing the Linear Programming Problem for maximising objective function $Z=px+qy$, $p, q>0$ is shaded. If all points on segment AB give max (Z), then which of the following is true? [Graph shows A at (0, 5) and B at (3, 4)]

If $l_{1}$, $m_{1}$, $n_{1}$ and $l_{2}$, $m_{2}$, $n_{2}$ are direction cosines of lines $L_{1}$ and $L_{2}$ respectively and $\theta$ is the acute angle between them, then :

For any two vectors $\vec{a}$ and $\vec{b}$, which of the following statements is always true ?

The order and degree of the differential equation $1+(\frac{d^{3}y}{dx^{3}})^{3}=\lambda\frac{d^{2}y}{dx^{2}}$ is:

The general solution for the differential equation $\frac{dy}{dx}=e^{3x-y}$ is:

The area of the region bounded by the curve $y=x$ and x-axis, between $x=0$ and $x=2$ is:

The value of $\int_{-5}^{-1}\frac{1}{x}dx$ is equal to:

$\int\frac{1}{\sqrt{1+\cos 2x}}dx$ is equal to:

$\int\frac{dx}{\sqrt{25-16x^{2}}}$ is equal to: