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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)]

The corner points of the feasible region determined by the system of linear constraints are (0,0), (0, 40), (20, 40), (60, 20) and (60, 0). If the objective function of an LPP is $Z=4x+3y$, then the maximum value is :

Direction ratios of lines $l_{1}$ and $l_{2}$ are <12,-3, 9> and <4, q,-p> respectively. The values of p and q for which $l_{1}$ and $l_{2}$ are parallel are respectively:

Direction cosines of the line given by equations $\frac{2x-1}{4}=\frac{1-y}{3}=\frac{-z}{6}$ are

If $l_{1}$, $m_{1}$, $n_{1}$ and $l_{2}$, $m_{2}$, $n_{2}$ are direction cosines of lines $L_{1}$ and $L_{2}$ respectively and $\theta$ is the acute angle between them, then :