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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.

Find the angle between the following pair of lines: $\frac{x-2}{3}=\frac{y+5}{2}=\frac{1-z}{-6}$ and $\frac{x-7}{1}=\frac{y}{2}=\frac{6-z}{-2}$.

Find the vector equation of a line passing through the origin and perpendicular to both the lines $\vec{r}=2\hat{i}-\hat{j}+2\hat{k}+\lambda(3\hat{i}+4\hat{j}+2\hat{k})$ and $\vec{r}=\mu(\hat{i}-\hat{j}+\hat{k})$.

If the lines $\frac{x-3}{1}=\frac{1-y}{1}=\frac{z+2}{p}$ and $\frac{2-x}{3}=\frac{y+1}{5}=\frac{z+56}{2p}$ are perpendicular to each other, then find the value(s) of p.

Find the co-ordinates of the point on the line $\vec{r}=-\hat{j}+3\hat{k}+\lambda(2\hat{i}-2\hat{j}+\hat{k})$ such that the sum of co-ordinates is 3.

If A, B and C be three non-collinear points such that $\vec{AB}=\hat{i}+2\hat{j}-\hat{k}$ and $\vec{AC}=2\hat{i}-3\hat{j}$, then find the area of $\Delta ABC$.

Three honey bees were found flying along the vectors $\vec{a}=2\hat{i}-3\hat{j}+\hat{k}$, $\vec{b}=4\hat{j}-2\hat{k}$ and $\vec{c}=3\hat{i}+2\hat{k}$ respectively. Find the value of $\lambda$ such that the path for $\vec{a}+\lambda\vec{b}$ is perpendicular to $\vec{c}$.

A unit vector $\vec{a}$ is such that it makes an angle $\frac{\pi}{4}$ with x-axis, $\frac{\pi}{3}$ with y-axis and an acute angle $\theta$ with z-axis. Find $\theta$ and the components of $\vec{a}$.

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.

If the position vectors of three points A, B and C are $3\hat{i}+\hat{j}$, $5\hat{i}+6\hat{j}-3\hat{k}$ and $4\hat{j}$ respectively, then show that they form an isosceles triangle.

If for two unit vectors $\vec{a}$ and $\vec{b}$, $|\vec{a}+2\vec{b}|=|2\vec{a}-\vec{b}|$, then find the angle between $\vec{a}$ and $\vec{b}$.

If $\vec{AB}=\hat{j}+\hat{k}$ and $\vec{AC}=3\hat{i}-\hat{j}+4\hat{k}$ represent the two vectors along the sides AB and AC of $\Delta ABC$, prove that the median $\vec{AD}=\frac{\vec{AB}+\vec{AC}}{2}$ where D is midpoint of BC. Hence, find the length of median AD.

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 $f(x)=\cos x+\sin^{2}x$, $x \in [0,\pi]$.

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 $t=2$.