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If $\alpha ,\beta$ are the roots of the equation $a{x}^2+bx+c=0$ such that $\beta <\alpha <0,$ then the quadratic equation whose roots are $|\alpha |,|\beta |$ is given by

Assume that each born child is equally likely to be a boy or a girl. If two families have two children each, then the conditional probability that all children are girls given that at least two are girls is :

Two tangents are drawn from the point $P(-1,1)$ to the circle $x^2+y^2-2x-6y+6=0.$ If these tangents touch the circle at points $A$ and $B$, and if $D$ is a point on the circle such that length of the segments $AB$ and $AD$ are equal, then the area of the triangle $ABD$ is equal to

If $a,b,c$ and $d$ are real numbers such that $a^2+b^2+c^2+d^2=1$ and if $A=\left[\begin{array}{cc}a+ib& c+id\\ -c+id& a-ib\end{array}\right],$where $i^2=-1,$ then $A^{-1}=$

Find the equation of a line passing through the point $(2,3)$ and parallel to the line $3x-4y+5=0.$

If $f\left(x\right)=\frac{t+3x-x^2}{x-4},$ where $t$ is a parameter and $f\left(x\right)$ has exactly one minimum and one maximum, then the range of values of $t$ is

For which of the following values of $x$, $7^{\text{th}}$ term is the numerically greatest term in the expansion of ${(1+\frac{2x}{5})}^{12}$

The coordinates of a point on the line $\frac{x-1}{2}=\frac{y+1}{-3}$ at a distance $4\sqrt{14}$ from the point (1, -1, 0)nearer the origin are

The angle at which the circles $(x-1{)}^2+y^2=10and{x}^2+(y-2{)}^2=5$ intersect is

$\int \frac{2x^3-1}{x^4+x}dx$ is equal to

(where $C$ is constant of integration)

$\frac{sin5\theta }{sin\theta }$ is equal to

Value of ${\int }_{\frac{\pi }{6}}^{\frac{\pi }{3}}\frac{\sin x}{x}dx$ lies between

The number of $4$ digit numbers that can be formed using only $0,1,2,3,4,5$ is

(repetition is not allowed)

If a curve $y=f\left(x\right)$ passes through point $(1,-1)$ and satisfy the differential equation $y(1+xy)dx=xdy,$ then $f(-\frac{1}{2})$ equals

Let $A$ is a square matrix of order $3$, such that $det\left(A\right)=a(M_{11}+M_{12}+M_{13})$ and $det\left(5A\right)=b(M_{11}+M_{12}+M_{13})$. Then which of the following is correct. (where $M_{ij}$ is minor of $i^{th}$ row and $j^{th}$ column element)

If the relation R:A$\to$B where A={1,2,3,4} and B={1,3,5} is defined by R={(x,y):x<y,x$\in$A,y$\in$B}, then $RO{R}^{-1}$ is

Find the angle between the X-axis and the line joining the points (3, -1) and (4, -2).

If $x<2$, then $\frac{1}{x}$ lies in the interval

Let $f\left(x\right)={\cos }^{-1}x-\frac{\pi }{2}$ and $g\left(x\right)=e^x+{\cos }^{-1}x$ be two function such that $|f\left(x\right)+g\left(x\right)|=|f\left(x\right)|+|g\left(x\right)|$, then the set of value(s) of $x$ is

If $\underset{0}{\overset{\frac{\pi }{2}}{\int }}(x+e^x)dx=\frac{{\pi }^2-8}{a}+e^b,$ then which of the following is/are true ?

Suppose $A_1,A_2,A_3,\cdots ,A_{30}$ are thirty sets each having 5 elements and $B_1,B_2,\cdots ,B_n$ are n sets each with 3 elements. Let ${\bigcup }_{i-1}^{30}{A}_i={\bigcup }_{j-1}^{n}Bj=S$ and each element of S belongs to exactly 10 of the $A_i^{\prime }s$ and exactly 9 of the $B_j^{\prime }s$. Then n is equal to

The locus of feet of perpendiculars drawn from the origin to the straight lines passing through $(2,1)$ is

I.Q. of a person is given by $I=\frac{M}{C}\times 100$, where $M$ is mental age and $C$ is chronological age. If $80\le I\le 140$ for a group of $12$ years old children, then their mental age can be

f(x) $\{\begin{array}{c}\frac{kcosx}{\pi -2x},ifx\ne \frac{\pi }{2}\\ 3,ifx=\frac{\pi }{2}.\end{array}$

The range of the function $f\left(x\right)=\frac{x}{|x|}$ is

Find the domain and range of 1/(4-cos3x)

The ordinates of the foot of normal(s) to the parabola $y^2=4ax$ from the point $(6a,0)$ is/are

If $\sqrt{\tan y}=e^{\cos 2x}\sin x$, then $\frac{dy}{dx}$ is equal to

The value of $\underset{x\to 1}{lim}(1-x)\tan \left(\frac{\pi x}{2}\right)$ is

Evaluate $\underset{x\to 2}{lim}\left(\frac{x^5-32}{x^3-8}\right)$

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

If $f\left(x\right)=\{\begin{array}{cc}3(1+|\tan x|{)}^{\frac{\alpha }{|\tan x|}},& -\frac{1}{2}<x<0\\ \beta ,& x=0\\ 3{(1+\left|\frac{\sin x}{3}\right|)}^{\frac{6}{|\sin x|}},& 0<x<\frac{2}{3}\end{array}$ is continuous at $x=0,$ then the ordered pair $(\alpha ,\beta )$ is equal to