Paper Generator

Find a vector of magnitude 4 units perpendicular to each of the vectors $2\hat{i}-\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}-\hat{k}$ and hence verify your answer.

Solve the following differential equation $x^{2}dy+y(x+y)dx=0$

Find the particular solution of the differential equation $\frac{dy}{dx}-2xy=3x^{2}e^{x^{2}};y(0)=5.$

Find: $\int\frac{2x+1}{(x+1)^{2}(x-1)}dx$

Evaluate : $\int_{0}^{\pi}\frac{e^{cos~x}}{e^{cos~x}+e^{-cos~x}}d~x$

If $x=a~sin^{3}\theta$, $y=b~cos^{3}\theta$ then find $\frac{d^{2}y}{dx^{2}}$ at $\theta=\frac{\pi}{4}$

If $\sqrt{1-x^{2}}+\sqrt{1-y^{2}}=a(x-y)$, prove that $\frac{dy}{dx}=\sqrt{\frac{1-y^{2}}{1-x^{2}}}$

Find $\frac{dy}{dx}$ , if $(cos~x)^{y}=(cos~y)^{x}.$

Find the vector equation of the line passing through the point (2, 3, -5) and making equal angles with the co-ordinate axes.

Find: $\int\frac{1}{5+4x-x^{2}}dx$

Find: $\int cos^{3}x~e^{log~sin~x}dx$

The area of the circle is increasing at a uniform rate of $2~cm^{2}/sec$. How fast is the circumference of the circle increasing when the radius $r=5$ cm?

Check for differentiability of the function f defined by $f(x)=|x-5|$, at the point $x=5$.

Verify whether the function f defined by $f(x)=\begin{cases}x~sin(\frac{1}{x}),x\ne0\\ 0&,x=0\end{cases}$ is continuous at $x=0$ or not.

Find value of k if $sin^{-1}[k~tan(2~cos^{-1}\frac{\sqrt{3}}{2})]=\frac{\pi}{3}.$

If the lines $\frac{x-1}{-3}=\frac{y-2}{2k}=\frac{z-3}{2}$ and $\frac{x-1}{3k}=\frac{y-1}{1}=\frac{z-6}{-7}$ are perpendicular to each other, find the value of k and hence write the vector equation of a line perpendicular to these two lines and passing through the point (3, -4, 7).

Find the distance between the line $\frac{x}{2}=\frac{2y-6}{4}=\frac{1-z}{-1}$ and another line parallel to it passing through the point (4, 0, -5).

If $A=[\begin{bmatrix}2&1&-3\\ 3&2&1\\ 1&2&-1\end{bmatrix}],$ find $A^{-1}$ and hence solve the following system of equations: $2x+y-3z=13$, $3x+2y+z=4$, $x+2y-z=8$

Check whether the relation S in the set of real numbers R defined by $S=\{(a,b)$: where $a-b+\sqrt{2}$ is an irrational number is reflexive, symmetric or transitive.

Let $A=R-\{5\}$ and $B=R-\{1\}$. Consider the function $f:A\rightarrow B$, defined by $f(x)=\frac{x-3}{x-5}$ Show that f is one-one and onto.