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If the locus of the circumcentre of of variable triangle having sides $y-$axis, $y=2$ and $lx+my=1$, where $(l,m)$ lies on the parabola $y^2=4ax$ is a curve $C$, then the curve $C$ is symmetric about the line

If $(a-d{)}^2$ , $a^2$ , $(a+d{)}^2$ are in GP, then $\frac{d^2}{a^2}$ equals to [a ≠ 0,d ≠ 0]

Match the following for sets:$A=\{12,14,15,16,19,20\}B=\{18,21,20,15,17,13\}C=\{21,23,12,16,14,18\}$

If $A,B,C\ne \frac{(2n+1)\pi }{2}$, then the numerical value of $\frac{\sin (B-C)}{\cos B\cos C}+\frac{\sin (C-A)}{\cos C\cos A}+\frac{\sin (A-B)}{\cos A\cos B}$ is

The perimeter of a triangle is 20 and the points (-2, -3) and (-2, 3) are two of the vertices of it. Then the locus of third vertex is :

A man alternately tosses a coin and throws a die beginning with coin. The probability that he gets a head in the coin before he gets 5 or 6 on the die is

The period of $\left|\sin \frac{x}{2}\right|+\left|\cos (\frac{x}{4}-\frac{\pi }{6})\right|$ is

If $\frac{3+i\sin \theta }{4-i\cos \theta },\theta \in [0,2\pi ]$, is a real number, then an argument of $\sin \theta +i\cos \theta$ is :

Find the value of
$ta{n}^2\left(se{c}^{-1}\left(3\right)\right)$+$co{t}^2\left(cose{c}^{-1}\left(4\right)\right)$

If ${\cos }^{-1}\left(\frac{2}{3x}\right)+{\cos }^{-1}\left(\frac{3}{4x}\right)=\frac{\pi }{2}(x>\frac{3}{4}),$ then $x$ is equal to:

The value of $(-i{)}^{\frac{1}{3}}$ is

In $\Delta ABC,\angle C=\frac{2\pi }{3}$, then the value of $co{s}^{2}A+co{s}^{2}B-cosA.cosB=$___

Solution of the equation $xdy=(y+x\frac{f\left(\frac{y}{x}\right)}{f^{\prime }\left(\frac{y}{x}\right)})dx$

The mean of the values $0,1,2,\dots \dots \dots ,n$ having corresponding weight ${}^n{C}_0{,}^n{C}_1{,}^n{C}_2,\dots \dots \dots {\dots }^n{C}_n,$ respectively is

If f(x) = $\{\begin{array}{cc}\sin x& x\ne n\pi ,n=0,\pm 1,\pm \mathrm{2...}\\ 2,& otherwise\end{array}$ and g(x) = $\{\begin{array}{cc}{x}^2+1,& x\ne 0,2\\ 4,& x=0\\ 5,& x=2\end{array}$, then $\underset{x\to 0}{lim}g\left\{f\right(x\left)\right\}$ is

If $10$ different balls has to placed in $4$ distinct boxes at random, then the probability that two of these boxes contain exactly $2$ and $3$ balls is :

If $\alpha ,\beta$ and $\gamma$ are real numbers such that ${\alpha }^2+{\beta }^2+{\gamma }^2=1$ and $\alpha +\beta +\gamma =\sqrt{3},$ then $\beta =$

(correct answer + 1, wrong answer - 0.25)

The coordinates of a point on the parabola $y^2=8x$, whose focal distance is $4$ units, is/are

If one root of the quadratic equation $a{x}^2$ + bx + c = 0 is equal to the n^th power of the other root, then the value of $(a{c}^n{)}^{\frac{1}{n+1}}$ + $(a^{n}c{)}^{\frac{1}{n+1}}$

The range of $f\left(x\right)=\frac{x+2}{x-3}$ is

Find the real numbers x and y such that: (x + iy)(3 + 2i) = 1 + i

Are the following pairs of statements negations of each other:

(i) The number x is not a rational number. The number x is not an irrational number.

(ii) The number x is a rational number. The number x is an irrational number.

The value of $\sum _{r=1}^{19}\frac{{}^{20}{C}_{r+1}\cdot (-1{)}^r}{2^{2r+1}}$ is

The number of terms in the expansion of $(x+y+z{)}^{10}$ is

Total number of ways of selecting $3$ smallest squares on a normal chess board, so that they don't belong to the same row, same column or same diagonal line, is

Three positive numbers form an increasing G.P. If the middle term in this G.P is doubled, then new numbers are in A.P. Then, the common ratio of the G.P. is

Three lines

$L_1$ : x-y+6 = 0

$L_2$ _: 2x+y-3 = 0

$L_3$ : x-2y+m = 0 are given. Which of the following can be the value of m if $L_1$_, $L_2$ and $L_3$ form a triangle.

The number of goals scored by two teams in a football session were as under:

$\begin{array}{ccccccc}\text{No of goals scored}& 0& 1& 2& 3& 4& 5\\ \text{No of matches (Team A)}& 15& 10& 7& 5& 3& 2\\ \text{No of matches (Team B)}& 20& 10& 5& 4& 2& 1\end{array}$

Which team is more consistent?

By LMVT, which of the following is true for x>1

Find the equation of plane passing through the points P (1,1,1) , Q (3, -1, 2) and R (-3, 5 , -4).

The sum of first $20$ common terms between the series $3+7+11+15+\cdots$ and $1+6+11+\cdots$ is

The principal solution of the equation $\sin x=\frac{1}{2}$ that is less than $\frac{\pi }{2}$ is

What are the coordinates of a point on the hyperbola $16x^2-{25}^2=1$ given by the parameter ${30}^{\circ }$?

Given the following frequency distribution with some missing frequencies

$\begin{array}{cccccccc}\text{Class}& 10-20& 20-30& 30-40& 40-50& 50-60& 60-70& 70-80\\ \text{Frequency}& 180& & 34& 180& 136& & 50\end{array}$

If the total frequency is $685$ and approximate value of median is $42.6$, then the approximate values for missing frequencies are

The sum of all those terms which are rational numbers in the expansion of ${(2^{1/3}+3^{1/4})}^{12}$ is

The value of Z satisfying the equation $logZ+log{z}^2+log{z}^3+.....+log{z}^n=0$ is

The value(s) of $m$, for which the line $y=mx+\frac{25\sqrt{3}}{3}$ , is a normal to the conic $\frac{x^2}{16}-\frac{y^2}{9}=1$ is/are

The value of $co{s}^{-1}\left(cos10\right)$=

Let $A=\left(\begin{array}{ccc}1& 0& 0\\ 2& 1& 0\\ 3& 2& 1\end{array}\right)$ , If ${\mu }_1$ and ${\mu }_2$ are column matrices such that $A{\mu }_1=\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)$ and $A{\mu }_2=\left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)$, then ${\mu }_1+{\mu }_2$ is equal to: