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Find the equation of family of circles passing through the point of intersection of the circles $x^2+y^2-2x-4y-4=0$ and $x^2+y^2-10x-12y+40=0$ and whose radius is 4.

Three cards are drawn successively, without replacement from a pack of 52 well-shuffled cards. What is the probability that the third card drawn is a queen given that first card is king of spades while second card is an ace?

Two squares are chosen at random on a chessboard (see figure). The probability that they have a side in common is :

The point on $y-$ axis which is equidistant from the points $(12,3)$ and $(-5,10)$ is

The $x$ and $y$ coordinates of the particle at any time are $x=5t-2t^2$ and $y=10t$ respectively, where $x$ and $y$ are in meters and $t$ in seconds. The acceleration of the particle at $t=2s$ is

Which of the following is correct for $0<x_1<x_2<\frac{\pi }{2}$

A bag contains $(2n+1)$ coins. It is known that $n$ of these coins have a head on both sides, whereas the remaining $(n+1)$ coins are fair. A coin is selected at random from the bag and tossed once. If the probability that the toss results in a head is $\frac{31}{42},$ then $n$ is equal to

Let $C_1$ and $C_2$ be two curves which satisfy the differential equation $2{\left(\frac{dy}{dx}\right)}^2$ + $x\left(\frac{dy}{dx}\right)=y$ and passes through (1, 1). If the area enclosed by the curves $C_1$, $C_2$ and the axes is $\frac{m}{R}$ then m + n = _______________

A group contains $10$ people. A rumour is spread from one person to another. The recipient of the rumour is chosen at random at each stage. However, the person who receives a rumour cannot transmit it back to the person from whom he/she received it. A rumour is passed by one person and spread to a total of five people. What is the number of ways in which it will not be repeated to the first recipient ?

In A.M.'s are introduced between 3 and 17 such that the ratio of the last mean to the first mean is 3 : 1, then the value of n is

The slope of tangent to the curve $y=(x-1)e^x$ at point, where curve intersects the $x-$ axis, is

A particle is in a linear SHM. If the acceleration and the velocity of this particle are $a$ and $v$ respectively, then the graph relating to these values is

Principal solution and general solution

1. 1572

If $\omega$ is a cube root of unity, then $\sin \frac{\pi }{900}\left\{\sum _{r=1}^{10}(r-\omega )(r-{\omega }^2)\right\}=$

The line $x=my+c$ is a tangent to $x^2=4my$, then the distance of this tangent from the parallel normal (in units) is

The area bounded by the
curve,
x-axis and the ordinates x = –1 and x = 1
is given by

[Hint: y
= x² if x > 0 and y = –x²
if x < 0]

A. 0

B.

C.

D.

If $f\left(x\right)=\sqrt{2x+3}$, Then $f^{\prime }\left(x\right)=$___

If ${\log }_{1/2}(4-x)\ge {\log }_{1/2}2-{\log }_{1/2}(x-1)$, then $x\in$

The general solution of differential equation $(x-y)dy+(x+y)dx=dx+dy$ is

(where $C$ is constant of integration )

The locus of the orthocentre of the triangle formed by the lines

$(1+p)x-py+p(1+p)=0,$ $(1+q)x-qy+q(1+q)=0$ and $4y=0,$ where $p\ne q$ is

If X follows binomial distribution with parameters n,p with $n=5,p(x=2)=9=p(x=3),$ then p=

If $\cos (\alpha -\beta )=1$ and $\cos (\alpha +\beta )=\frac{1}{2},$ where $\alpha ,\beta \in (-\pi ,\pi ),$ then the number of ordered pairs $(\alpha ,\beta )$ satisfying both the equations is

If $I=\underset{1}{\overset{2}{\int }}\frac{dx}{\sqrt{2x^3-9x^2+12x+4}}$, then

A standing wave set up in a medium is $y=4\cos \left(\frac{\pi x}{3}\right)\sin 40\pi t$ where $x,y$ are in $\text{cm}$ and $t$ in $\text{sec}$. The velocity of particle at $x=6\text{cm}$ at $t=\frac{1}{8}\text{sec}$ is

The value of the integral $\underset{-\pi /2}{\overset{\pi /2}{\int }}(x^2+\ln \left(\frac{\pi +x}{\pi -x}\right))\cos xdx$ is