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If the equation of the locus of a point equidistant from the points $(a_1,b_1)and(a_2,b_2)is(a_1-a_2)x+(b_1-b_2)y+c=0$

then the value of c is

The set $X=\{5,10,15,20,25\}$ can be written in Set-builder form as:

The sixth term in the expansion of ${[\sqrt{\left\{2^{\log (10-3^x)}\right\}}+\sqrt[5]{5\left\{2^{(x-2)\log 3}\right\}}]}^m$is equal to $21$. If it is known that the binomial coefficient of the $2^{\text{nd}},3^{\text{rd}}$ and $4^{\text{th}}$ terms in the expansion represents respectively the first, third and fifth terms of an A.P. (the symbol log stands for logarithm to the base $10$) then the value of $m$ is

An equation of the curve satisfying $xdy-ydx=\sqrt{x^2-y^2}$ dx and $y\left(1\right)=0$ is

If in a parallelogram $ABDC$, the coordinates of $A,B$ and $C$ are respectively $(1,2),(3,4)$ and $(2,5)$, then the equation of the diagonal $AD$ is :

If $U=\{1,3,5,7,9\}$ and $A=\{3,5,7\}$, then $(A^{\prime }{)}^{\prime }$ is:

Solution of the differential equation

$cosxdy=y(sinx-y)dx,0<x<\frac{\pi }{2}$ is

If $\theta =\frac{\pi }{2^{100}+1},$ then $\cos \theta \cos 2\theta \cos 2^2\theta \cdots \cos 2^{99}\theta$ is

The point(s) which lies inside the region bounded by the curves $y^2=3x$ and $(x-2{)}^2=-4(y-4)$ is/are

If $f$ and $g$ are differentiable functions in $[0,1]$ satisfying $f\left(0\right)=2=g\left(1\right),g\left(0\right)=0$ and $f\left(1\right)=6$, then for some $c\in [0,1]$:

The number of solution of the equation $\cos 3x+\sin (2x-\frac{7\pi }{6})=-2$ for $[0,2\pi ]$ is

Considering only the principal values of inverse functions, the set $A=\{x\ge 0:{\tan }^{-1}(2x)+{\tan }^{-1}(3x)=\frac{\pi }{4}\}$

Let $PS$ be the median of a triangle with vertices $P(2,2),$ $Q(6,-1)$ and $R(7,3).$ The equation of the line passing through $(1,-1)$ and parallel to $PS$ is

There are $2n$ guests at a dinner party. Supposing that the master and mistress of the house have fixed seats opposite one another, and that there are two specified guests who must not be placed next to one another, the number of ways in which the company can be placed is

Which among the following relations on $Z$ is an equivalence relation

Which of the following statements is/are correct?

1. The radical axis of two circles is the locus of points whose power with respect to the two circles is equal.

2. The common point of intersection of the radical axes of three circles taken two at a time called the radical center of three circles.

Let $a,r,s,t$ be nonzero real numbers. Let $P(a{t}^2,2at),Q,R(a{r}^2,2ar)\text{and}S(a{s}^2,2as)$ be distinct points on the parabola $y^2=4ax$. Suppose that $PQ$ is the focal chord and lines $QR$ and $PK$ are parallel, where $K$ is the point $(2a,0).$



The value of $r$ is

If $\alpha ,\beta \in R$ are such that $1-2i$ (here $i^2=-1$) is a root of $z^2+\alpha z+\beta =0,$ then $(\alpha -\beta )$ is equal to

Papu takes a test and he is very 'determined' about passing the test. He will not give up until he passes the test. How would the sample space of results look like if p stands for pass and F stands for fail, given that he can take the test atmost five times and no test after passing?

If $0\le x<2\pi$ , then the number of real values of x, which satisfy the equation $\cos x+\cos 2x+\cos 3x+\cos 4x=0$, is

The 4 term of a H.P is $\frac{3}{5}$ and 8th term is $\frac{1}{3}$, then its 6th term is

If the equation $x^4-4x^3+a{x}^2+bx+1=0$ has four roots and all of them are positive real roots then the value of a and b are

The value of $y^{\prime \prime }\left(1\right)$ if $x^3-2x^2{y}^2+5x+y-5=0$ when $y\left(1\right)=1$, is equal to

In $△ABC$, if $A,B$ and $C$ represent the angles of a triangle, then the maximum value of $\sin \frac{A}{2}+\sin \frac{B}{2}+\sin \frac{C}{2}$ is

If $(1+x)(1+x+x^2).....(1+x+.....+x^n)=a_0+a_{1}x+a_2{x}^2+a_3{x}^3+.....$, then the value of $a_0+a_2+a_4+.....$ is

If the inclination of the line joining the points $(x,-3)$ and $(2,5)$ is ${135}^{\circ }$, then the value of $x$ is

The vertices $B$ and $C$ of a $△ABC$ lie on the line, $\frac{x+2}{3}=\frac{y-1}{0}=\frac{z}{4}$ such that $BC=5$ units. Then the area (in sq. units) of this triangle, given that the point $A(1,-1,2)$, is :

If the roots of $a(b-c)x^2+b(c-a)x+c(a-b)$ = 0 be equal then a,b,c are in

If $\sin (x+{20}^{\circ })=2\sin x\cos {40}^{\circ }$ where $x\in (0,\frac{\pi }{2})$, then which of the following is/are correct ?

If $(1+x{)}^n=C_0+C_{1}x+C_2{x}^2+........+C_n{x}^2$, then

$C_0^2+C_1^2+C_2^2+C_3^2+..........+C_n^2$ =

Let $f\left(x\right)=\sum _{r=0}^{20}{a}_r{x}^r$ and $g\left(x\right)=\sum _{r=0}^9{b}_r{x}^r+\sum _{r=10}^{20}{x}^r.$ If $f\left(x\right)=g(x+1),$ then the value of $a_{10}$ is

If $n\left(A\right)=m,m>0$, then number of reflexive relations from $A$ to $A$ is

The angle between any two diagonals of a cube is:

If $tan\theta$ = $\frac{sin\alpha -cos\alpha }{sin\alpha +cos\alpha }$, then $sin\alpha +cos\alpha$ and

$sin\alpha -cos\alpha$ must be equal to

A car has a velocity $\overrightarrow{v}=80km/hr$ towards east. Another car on the road has a velocity = $\overrightarrow{-v}$. The magnitude and direction of velocity of second car is

If $\sin x=\frac{12}{13}$ and $x\in [0,\pi ]$, then the possible value(s) of $\sec x+\tan x$ is/are

The maximum value of $f\left(x\right)=(x+3)(4-x)+3$ is

The coordinates of a point common to a directrix and an asymptote of the hyperbola $\frac{x^2}{25}-\frac{y^2}{16}=1$ are

Ahyperbola $\frac{x^2}{25}-\frac{y^2}{16}=1$ is given and a normal is drawn at the point $(5\sqrt{3},4\sqrt{2}).$

What is the abscissa of the point at which it meets the x-axis.

If $A=\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$ then AA' =

The locus of midpoint of chord of the circle $x^2+y^2-2x-2y-2=0$, which makes an angle of ${120}^{\circ }$ at the centre, is

If $\sin x=\frac{12}{13}$ and $x\in [0,\pi ]$, then the value(s) of $\sec x+\tan x$ is/are

Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2x – 3y + 4z – 6 = 0.