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P and Q are the centres of two circles whose radii are 5 cm and 11 cm, respectively. If the direct common tangent to the circles meets PQ at M, then, M divides PQ in the ratio ___

The value of $\tan \left(\frac{11\pi }{12}\right)$ is

Consider the hyperbola $H:x^2-y^2=1$ and a circle $S$ with center $N(x_2,0).$ Suppose that $H$ and $S$ touch each other at a point $P(x_1,y_1)\text{with}{x}_1>1\text{and}{y}_1>0.$ The common tangent to $H$ and $S$ at $P$ intersects the $x-$axis at point $M$. If $(l,m)$ is the centroid of the triangle $△PMN,$ then the correct expression(s) is (are)

If the complex number $z_1,z_2$ the origin form an equilateral triangle then $z_1^2+z_2^2$ =

The approximate value of $(1.0002{)}^{3000}$ is

If $\left[\begin{array}{cc}1/25& 0\\ x& 1/25\end{array}\right]={\left[\begin{array}{cc}5& 0\\ -a& 5\end{array}\right]}^{-2}$, then the value of $x$ is:

The zeroes of the quadratic polynomial $f\left(x\right)=x^2+7x+10$ are

The total number of $4$ digit numbers that can be formed by using the digits $1,2,3,4,5,6$ and $7$ if at least one digit is repeated is

If ${\cos }^{4}x-(\lambda +2){\cos }^{2}x-(\lambda +3)=0$ has a solution, then

The area of an acute triangle $ABC$ is $\Delta$, the area of its pedal triangle is '$p$', where $\cos B=\frac{2p}{\Delta }$ and $\sin B=\frac{2\sqrt{3}p}{\Delta }$. The value of $8({\cos }^{2}A\cos B+{\cos }^{2}C)$ is

If $T_n=3n-1$ of an A.P., then

The curves x²+y²=16 and y²=6x intersects at

If $\left\{y\right\}$ denotes the fractional part of $y,$ then the value of the definite integral $\underset{-\infty }{\overset{\ln 3}{\int }}\left\{e^x\right\}dx$ equals

The possible values of $\frac{1}{x^2+3}$ lie in the interval

The equation of line(s) joining the vertex of $y^2=6x$ to the point on it whose abscissa is $24,$ is (are)

Let P(n) denote the statement that $n^2$ + n is odd. It

is seem that P(n) ⇒ P(n + 1), $P_n$ is true for all

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are non coplanar non zero vectors such that $\overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{a},\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{c}$ and $\overrightarrow{c}\times \overrightarrow{a}=\overrightarrow{b},$ then which of the following is not correct?

The point (2, 3) is a limiting point of a coaxial system of circles of which $x^2+y^2=9$ is a member. The co-ordinates of the other limiting point is given by

The lateral edge of a regular hexagonal pyramid is 1 cm. If the volume is maximum, then its height is

The value of $\underset{x\to 0}{lim}\frac{\sin x^n}{(\sin x{)}^m},(m\le n)$, can be

If $\alpha ,\beta \in C$ are the distinct roots of the equation $x^2-x+1=0,$ then ${\alpha }^{101}+{\beta }^{107}$ is equal to

Two sides of a parallelogram are along the lines $4x+5y=0$ and $7x+2y=0$. If the equation of one of the diagonals of the parallelogram is $11x+7y=9$, then other diagonal passes through the point

$△ABC$ is an equilateral triangle with side 4 units and a circle inscribed in it. A line segment goes from a vertex C to the midpoint of AB, M. What is the ratio of the length outside the circle to the length inside it of this line segment CM?

Match the following prefixes with their multiples:

Using properties of sets, show that for any two sets A and B, $(A\cup B)\cap (A\cap B^{\prime })=A.$

In how many ways can 7 letters posted in 4 letter boxes?

A straight line $L$ with negative slope passes through the point $(8,4)$ and cuts the positive co-ordinate axes at points $A$ and $B$. The minimum value of $OA+OB$, as $L$ varies, is (Here $O$ is the origin)

Write the range of the real function f(x) = |x|

A person
buys a lottery ticket in 50 lotteries, in each of which his chance of
winning a prize is.
What is the probability that he will in a prize (a) at least once (b)
exactly once (c) at least twice?

Consider the following 2 x 2 matrix A where two elements are unknown and are marked by a and b. The eigen values of this matrix are - 1 and 7. What are the values of a and b? $A=\left[\begin{array}{cc}1& 4\\ b& a\end{array}\right]$

Let an ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ $a^2>b^2,$ passes through $(\sqrt{\frac{3}{2}},1)$, and has eccentricity$\frac{1}{\sqrt{3}}$. If a circle, centered at focus $F(\alpha ,0),$ $(\alpha >0),$ of $E$ and radius $\frac{2}{\sqrt{3}}$, intersects $E$ at two points $P$ and $Q,$ then $P{Q}^2$ is equal to