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The maximum distance between the points $(a\cos \alpha ,a\sin \alpha )$ and $(a\cos \beta ,a\sin \beta )$ is

The sum of the roots of the equation $\sqrt{5x^2-6x+8}-\sqrt{5x^2-6x-7}=1$ is

If the Boolean expression $(p\oplus q)\wedge (\sim p\odot q)$ is equivalent to $p\wedge q$, where $\oplus ,\odot \in \{\wedge ,\vee \}$, then the ordered pair $(\oplus ,\odot )$ is:

If $x,y,z\in R$ and x+y+z =5, xy +yz+zx=3, probability for x to be positive only is

The value of ${\int }_0^{\pi }\frac{1}{5+4\cos x}\mathrm{dx}$ is

Find the equation to the straight line

(i) cutting off intercepts 3 and 2 from the axes.

(ii) cutting off intercepts - 5 and 6 from the axes.

Find f + g, f - g, $cf(cϵR,c\ne 0)$. fg. $\frac{1}{f}$ and $\frac{f}{g}$ in each of the following :
(i) $f\left(x\right)=x^3+1$ and $g\left(x\right)=x+1$
(ii) $f\left(x\right)=\sqrt{x-1}$ and $g\left(x\right)=\sqrt{x+1}$

If the area of $△$ on the argand plane, whose vertices are -z,iz,z-iz, is 600 sq. units, then |z| =

If a fair coin is tossed n times, then the total number of possible outcomes are

Which among the following represents the graph of $y=\sqrt{x+1}$

Prove the following by using the principle of mathematical induction for all n ∈ N: n (n + 1) (n + 5) is a multiple of 3.

Balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If $669$ more balls are added, then all the balls can be arranged in the shape of a square such that each of its sides contains $8$ balls less than each side of triangle had. The initial number of balls lies in the interval

The value of $\log \tan {17}^{\circ }+\log \tan {37}^{\circ }+\log \tan {53}^{\circ }+\log \tan {73}^{\circ }$ is

If $\left[x\right]$ is the greatest integer less than or equal to $x$, for all $x\in R$, then evaluate $\underset{x\to 1^{+}}{lim}\frac{1-|x|+\sin |1-x|}{|1-x|[1-x]}$

In $\Delta ABC,\overrightarrow{AB}=\hat{i}+3\hat{j}-2\hat{k},\overrightarrow{AC}=3\hat{i}-\hat{j}-2\hat{k}.$
If the bisector of $\angle BAC$ meets $BC$ at $D$ and $G$ is the centroid of $\Delta ABC$, then $|\overrightarrow{GD}|=$

Probability that A speaks truth is.
A coin is tossed. A reports that a head appears. The probability that
actually there was head is

A.

B.

C.

D.

If the area of a rectangular field is $3600$ square units and the length and breadth of the field are coprime to each other, then the possible value of length(s) is (are)

Solve the inequality

If $z=(1+i)(1-i\sqrt{3})(-2-2i)\left(i\right)\left(3\right),$ then

The coefficients of 5th, 6th and 7th terms in the expansion of $(1+x{)}^n$ are in A.P., find n.

A single letter is selected at random from the word "PROBABILITY”. The probability that the selected letter is a vowel is

[MNR 1986; UPSEAT 2000]

Show that the given differential equation is homogeneous and then solve it.

$(x^2+xy)dy=(x^2+y^2)dx.$

Write the value of $cos{1}^{\circ }+cos{2}^{\circ }+...+cos{180}^{\circ }.$

If $f\left(x\right)=cos\left(logx\right)$, then $f\left(x\right)f\left(y\right)-\frac{1}{2}\{f\left(\frac{x}{y}\right)+f(xy\left)\right\}$ has the value

If $(5+2\sqrt{6}{)}^n=m+f$, where n and m are positive integers and 0 $\le$ f < 1, then $\frac{1}{1-f}-f$ is equal to: