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Find the equation of a circle whose centre is $z_1$ and radius r.

Four distinct integers are picked at random from $\{0,1,2,3,4,5,6\}.$ If the probability that among those selected, the second smallest is $3,$ is $p,$ then $p$ is equal to

Evaluate the definite integrals.
${\int }_0^{\frac{\pi }{4}}sin2xdx.$

Let $N=2^6{.5}^5{.7}^6{.10}^7$, then the total number of even factors of $N$ is

The number of all numbers having $5$ digits, with distinct digits is

Let $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k},\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{c}=\hat{i}-\hat{j}-\hat{k}$ be three vectors. A vector $\overrightarrow{v}$ in the plane of

$\overrightarrow{a}$ and $\overrightarrow{b}$, whose projection on $\overrightarrow{c}$ is $\frac{1}{\sqrt{3}}$, is given by

If $[x{]}^2-5[x]+6=0$, where [.] denotes the greatest integer function, then

If $A$ is a non-singular skew-symmetric matrix and $B$ is a square matrix such that $B={\left(\right(A^T{B}^T\left)A^{-1}\right)}^T,$ then $(A+B{)}^2$ is equal to

If the real part of the complex number ${(1-\cos \theta +2i\sin \theta )}^{-1}$ is $\frac{1}{5}$ for $\theta \in (0,\pi ),$ then the value of the integral $\underset{0}{\overset{\theta }{\int }}\sin xdx$ is equal to :

An experiment has $10$ equally likely outcomes. Let $A$ and $B$ are two non-empty events of the experiment. If $A$ consists of $4$ outcomes, then number of possible outcomes that $B$ must have so that $A$ and $B$ are independent is/are ?

If $f\left(x\right)=\{\begin{array}{cc}x+3& ;x<3\\ 3x^2+1& ;x\ge 3\end{array}$, then the value of $\underset{2}{\overset{5}{\int }}f\left(x\right)dx$ is

In the following case, find the coordinates of the foot of the perpendicular drawn from the origin:
x +y +z =1

If $ta{n}^{-1}(x+1)+ta{n}^{-1}(x-1)=ta{n}^{-1}\left(\frac{8}{31}\right)$, then x is equal to

If $n\left(A\right)=52$, $n(A\cup B)=80$, $n(A\cap B)=31,$ then $n(A\cap B^{\prime })=$

The solution set of the inequality $||x|-1<1-x,2\forall xϵR$ is equal to

The range of the function $f\left(x\right)=\sqrt{x^2-3x+5}$ is

The equation of the curve that passes through the point (1,2) and satisfies the differential equation $\frac{dy}{dx}=\frac{-2xy}{(x^2+1)}$ is

If the ${10}^{th}$ term of a GP is 9 and $4^{th}$ term is 4, then its $7^{th}$ term is

Find the equation of the stright line passing through the point of intersection of $2x+3y+1=0$ and $3x-5y-5=0$and equally inclined to the axes.

The sum of $n$ terms of the following series; $1^3+3^3+5^3+7^3+...$ is

If $\overrightarrow{a}=10,\overrightarrow{b}=2and\overrightarrow{a}.\overrightarrow{b}=12,thenthevalueof|\overrightarrow{a}\times \overrightarrow{b}|$ is

(a) 5 (b) 10 (c) 14 (d) 16

A metal crystallises in a face centred cubic structure. If the edge length of its unit cell is $‘a^{\prime }$, the closest approach between two atoms in a metallic crystal will be:

If $x+y+z=1$, $x,y,z>0$. Then greatest value of $x^2{y}^3{z}^4$ is

If cos$\alpha$+cos$\beta$+cos$\gamma$=0=sin$\alpha$+sin$\beta$+sin$\gamma$ then

cos2$\alpha$+cos2$\beta$+cos2$\gamma$ equals

If $\cot x-\tan x=2,$ then generalized solution is (here $n$ is integer)

The least value of expression $x^2+4y^2+9z^2$ - 2x + 8y + 27z + 15 is :

Of all the closed cylindrical cans (right circular), which enclosed a given volume 100 cubic centimeters, find the dimensions of the minimum surface area.

A monkey seated before a type writer with $26$ keys on the key board denoting the English alphabet. Then the probability for that monkey to type the word $\text{SIR}$ is

A wire of length 2 units is cut into two parts which are bent respectively to form a square of side = x units and a circle of radius = r units. If the sum of the areas of the square and the circle so formed is minimum, then:

The locus of middle point of the portion of the normal to $y^2=4ax$ intercepted between curve and axis of parabola is

If $\alpha ,\beta ,\gamma$ are roots of equation $x^3-x-1=0$, then the equation whose roots are $\frac{1}{\beta +\gamma },\frac{1}{\gamma +\alpha },\frac{1}{\alpha +\beta }$ is -

Find the value of $\underset{x\to \infty }{lim}\frac{2x^2-3x+11}{7x^2+8x+15}$

If $U_n=(1+\frac{1}{n^2}){(1+\frac{2^2}{n^2})}^2\dots {(1+\frac{n^2}{n^2})}^n$, then $\underset{n\to \infty }{lim}(U_n{)}^{\frac{-4}{n^2}}$ is equal to

Words with or without meaning are to be formed using all the letters of the word $\text{EXAMINATION}$. The probability that the letter $\text{M}$ appears at the fourth position in any such word is