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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 $t=2$.

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

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)]