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If A(4, -3), B(3, -2) and C(2, 8) are the vertices of a triangle, then its centroid will be

For any two complex numbers $z_1,z_2$ we have $|z_1+z_2{|}^2=|z_1{|}^2+|z_2{|}^2.$ Then

Let $\alpha ,\beta$ be the roots of $a{x}^2+bx+c=0,a\ne 0$ and ${\alpha }_1,-\beta$ be the roots of $a_1{x}^2+b_{1}x+c_1=0,a_1\ne 0$. Then the quadratic equation whose roots are $\alpha ,{\alpha }_1$ is

${\sum }_{r=0}^{7}ta{n}^2\left(\frac{\pi r}{16}\right)=$___

An ellipse, with foci at $(0,2)$ and $(0,–2)$ and minor axis of length $4$, passes through which of the following points ?

What are the conditions for limit$\underset{x\to a}{lim}f\left(x\right)$ to exist?

The graph of f(x) is given below. The limit of the function f(x) as x approaches 'a' is

If f(x+2y, x-2y)=xy, then f(x, y) equals

For any sets A and B, prove that:
$\left(i\right)A\cup (A\cap B)=A$

$\left(ii\right)A\cap (A\cup B)=A$

Find the term independent of x in the expansion of $x^2{(x+\frac{1}{x})}^{32}$

The value of $(127{)}^{1/3}$ to four decimal places is

There are $10$ points in a plane of which no $3$ points are collinear and $4$ points are concyclic. Number of different circles that can be drawn through at least $3$ points is

The expression ${}^n{C}_r+2{}^n{C}_{r-1}+{}^n{C}_{r-2}$ is equal to

Two systems of rectangular axes have the same origin. If a plane cuts them at distance a, b, c and a', b', c' from the origin, then

Let p and q be real numbers such that p$\ne 0,p^3\ne q$ and $p^3\ne -q$ . If $\alpha$ and $\beta$ are nonzero complex numbers satisfying $\alpha +\beta =-p$ and ${\alpha }^3+{\beta }^3=q$ , then a quadratic equation having $\frac{\alpha }{\beta }$ and $\frac{\beta }{\alpha }$ as its roots is -

If cos$\alpha$ is a root of $25x^2$ + 5x - 12 = 0, (-1 $<$ x $<$ 0). Then a possible value of sin (2α) is

If $19$th term of a non-zero A.P. is zero, then its ($49$th term) : ($29$th term) is :

If $\sin \theta =\frac{24}{25}$ and $\theta$ lies in the second quadrant, then $\sec \theta +\tan \theta =$ .

If ${\left[\frac{1+i}{1-i}\right]}^m=1$, then find the least positive integral value of m.

In how many ways can 5 different balls be distributed among three boxes?

$Therangeofthefunctionf\left(x\right)=\frac{x+3}{|x+3|},x\ne -3is$

What is the value of [x] + [-x] , where [x] is the greatest integer function

The locus point of intersection of tangents to the parabola $y^2=4ax$, the angle between them being always ${45}^{\circ }$

is

If a triangle is inscribed in a rectangular hyperbola, its orthocentre lies

Chord of contact of the point (3, 2) w.r.t. the circle $x^2+y^2=25$ meets the coordinate axes in A and B. The

circumcentre of triangle OAB is

For $2\le r\le n,\left(\begin{array}{c}n\\ r\end{array}\right)+2\left(\begin{array}{c}n\\ r-1\end{array}\right)+\left(\begin{array}{c}n\\ r-2\end{array}\right)$ is equal to

The equation $z^{10}+(13z-1{)}^{10}=0$ has $5$ pairs of complex roots $a_1,b_1,a_2,b_2,a_3,b_3,a_4,b_4,a_5,b_5$. If each pair $a_i,b_i$ are complex conjugates, then

If $\alpha$ and $\beta$ are the roots of the equation $x^2-2x+2=0$, then least value of $n$ for which ${\left(\frac{\alpha }{\beta }\right)}^n=1$ is:

Eight coins are tossed at a time, the probability of getting atleast 6 heads up, is

The normal at $P(2,4)$ to $y^2=8x$ meets the parabola at $Q.$ Then the equation of the circle having normal chord $PQ$ as diameter is

If $|{\log }_{2}x+1|+|1-({\log }_{2}x{)}^2|=|{\log }_{2}x+({\log }_{2}x{)}^2|,$ then the true set of values of $x$ is $\left\{\lambda \right\}\cup [\mu ,\infty ).$ Then

If $x=e^{\theta }(\sin \theta +\cos \theta )$ and $y=e^{\theta }(\sin \theta –\cos \theta ),$ where $\theta$ is a real parameter, then $\frac{d^{2}y}{d{x}^2}$ at $\theta =\frac{\pi }{6}$ is

If $C_r$ represents ${}^{100}{C}_r,$ then $5C_0+8C_1+11C_2+\dots$ upto $101$ terms is equal to

Find the equation of an ellipse whose vertices are at $(\pm 5,0)$ and foci at $(\pm 4,0).$

The equation of the line, passing through the centre and bisecting the chord $7x+y-1=0$ of the ellipse $\frac{x^2}{1}+\frac{y^2}{7}=1,$ is

1. 4.32

If ${\Delta }_1=\left|\begin{array}{cc}1& 0\\ a& b\end{array}\right|$ and ${\Delta }_2=\left|\begin{array}{cc}1& 0\\ c& d\end{array}\right|$, then ${\Delta }_2{\Delta }_1$ is equal to

Find the general solution for the following equation:

cos 4x = cos 2x

The centre of the conic represented by the equation $14x^2-4xy+11y^2-44x-58y+71=0$ is

The A.M. of a set of 50 numbers is 38. If two numbers of the set, namely 55 and 45 are discarded, the A.M. of the remaining set of numbers is

Let $X$ be a set containing $10$ elements and $P\left(X\right)$ be its power set. If $A$ and $B$ are picked up at random from $P\left(X\right)$, with replacement, then the probability that $A$ and $B$ have equal number of elements, is :

The frequency distribution of discrete data is given below, the frequency $x$ against value $0$ is missing

If the mean is $2.5$, then the missing frequency $x$ will be

$\begin{array}{ccccccc}\text{Variable}& 0& 1& 2& 3& 4& 5\\ \text{Frequency}& x& 20& 40& 40& 20& 4\end{array}$

If x, y, and z are in G.P. and $x+3,y+3,Z+3$ are in H.P., then

$37-(3x+5)\ge 9x-8(x-3)$

Reduce the following equations into slope-intercept form and find their slopes and the y-intercepts.

(i) x + 7y = 0, (ii) 6x + 3y -5 = 0, (iii) y = 0