Paper Generator

The probability of hitting the target by a trained sniper is three times the probability of not hitting the target on a stormy day due to high wind speed. The sniper fired two shots on the target on a stormy day when wind speed was very high. Find the probability that (i) target is hit (ii) atleast one shot misses the target.

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 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: $y\log y\frac{dx}{dy}+x=\frac{2}{y}$

Find: $\int\frac{x+2}{\sqrt{9x-x^{2}}}dx$

A spherical balloon loses its volume due to escape of air from it in such a way that decrease of volume at any instant is proportional to its surface area. Show that the radius is decreasing at a constant rate.

Let n be a fixed positive integer. A relation R is defined in set Z such that $R=\{(x,y):(x-y) \text{ is divisible by } n, x, y\in Z\}$. Determine if R is an equivalence relation.

Let $A=\mathbb{R}-\{3\}$ and $B=\mathbb{R}-\{1\}$. A function $f:A\rightarrow B$ is defined by $f(x)=(\frac{x-2}{x-3})$. Find whether f is one-one and onto.

Let two rods placed on the ground be represented by vectors $4\hat{i}-\hat{j}+3\hat{k}$ and $-2\hat{i}+\hat{j}-2\hat{k}$. Find a vector representing a flag-post of height 5 m that has to be erected perpendicular to both the rods.

Determine the values of x for which $f(x)=\frac{x-3}{x+1}$, $x\ne -1$ is an increasing function.

Find the values of x for which $f(x)=x^{x}$, $x>0$ is increasing.

Find the sub-interval(s) of $(0,\frac{\pi}{2})$ in which $f(x)=\tan x-4x$ is increasing.

If the volume of a solid hemisphere increases at a uniform rate, prove that its surface area varies inversely as its radius.

Find the absolute maximum value of $f(x)=\cos x+\sin^{2}x$, $x \in [0,\pi]$.

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

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