Paper Generator

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 $t=\frac{\pi}{4}$.

Find whether the function $f(x)=\begin{cases}x-1, & x<2 \\ 2x-3, & x\ge 2\end{cases}$ at $x=2$ is differentiable or not.

Show that the function $f(x)=\begin{cases}\frac{\cos x}{-x+\frac{\pi}{2}}, & x\ne\frac{\pi}{2} \\ 1, & x=\frac{\pi}{2}\end{cases}$ is continuous at $x=\frac{\pi}{2}$.

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

Evaluate : $\tan^{-1}(-\frac{1}{\sqrt{3}})+\cot^{-1}(\frac{1}{\sqrt{3}})+\tan^{-1}(\sin(-\frac{\pi}{2}))+\tan^{-1}(\tan\frac{2\pi}{3})$

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.

Check whether $f:R-\{3\} \rightarrow R$ defined as $f(x)=\frac{x-2}{x-3}$ is onto or not.

A box contains 4 red, 5 blue and 1 green marble. A child randomly takes out a marble from the box, notes down the colour and puts it back in the box. If the activity is repeated 3 times, what is the probability that at least one marble is red?

If $3P(A)=P(B)=\frac{3}{5}$ and $P(A|B)=\frac{2}{3}$ then $P(A\cup B)$ is:

For two events A and B such that $P(A) \ne 0$ and $P(B) \ne 1$, $P(A^{\prime}/B^{\prime})=$

If E and F are two independent events such that $P(E)=\frac{3}{10}$, $P(E\cup F)=\frac{1}{2}$ then $P(E|F)-P(F|E)$ is equal to:

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)]