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The value of the expresssion $C_0^2+2C_1^2+3C_2^2+...+(n+1)C_n^2$ is equal to

Sixteen men compete with one another in running, swimming and riding. How many prize lists could be made if there were altogether $6$ prizes of different values, one for running, $2$ for swimming and $3$ for riding.

1. 2

If 3 – 2 cos x – 4 sin x – cos 2x + sin 2x = 0, then x =( n $ϵ$ Z)

The only elastic modulus that applies to fluids is

If $y=y\left(x\right)$ is the solution of the differential equation, $x\frac{dy}{dx}+2y=x^2$ satisfying $y\left(1\right)=1$, then $y\left(\frac{1}{2}\right)$ is equal to:

How many real numbers satisfy the relation $\left[x\right]$ = $\frac{3}{2}$ {x}?

For circles $x^2+y^2+2x-8y+13=0$ and $x^2+y^2-12x-14y+76=0$ equation of all the common tangents are:

Locus of the point equidistant from (0,-1) and the line y=1 is

Prove that $\frac{sinx}{1+\text{cos}x}=\text{tan}\frac{x}{2}$

There are m-stations on a railway line. A train has to stop at 3 intermediate stations. Then probability that no two stopping stations are adjacent is

Find the indicated terms in each of the sequences where $n^{th}$ term is :

$a_n=(-1{)}^{n-1}{n}^3;a_9$

If both the roots of the quadratic equation $x^2-(2n+18)x-n-11=0,n\in Z$ are rational, then the value(s) of $n$ is/are

Let $f\left(\alpha \right)=\left[\begin{array}{ccc}cos\alpha & -sin\alpha & 0\\ sin\alpha & cos\alpha & 0\\ 0& 0& 1\end{array}\right]$, then $\left(f\right(\alpha ){)}^{-1}$ is equal to

$\text{The value of}\underset{x\to 0}{lim}\frac{\underset{0}{\overset{x^2}{\int }}cos{t}^{2}dt}{xsinx}\text{is}$

Solve the inequalities: $-15<\frac{3(x-2)}{5}\le 0$

Express the following in standard form: i²⁰ + (1-2i)³

Let $f:(-\infty ,+1]\to R,g:[-1,\infty )\to R$ be such that $f\left(x\right)=\sqrt{1-x}$ and $g\left(x\right)=\sqrt{1+x}$, then $f\left(x\right)+\frac{1}{g\left(x\right)}$ exist if $x\in$

The value of $e^{2+i}$ is

Express the following in standard form: (2 – 3i)²

The mean of $100$ items is $49$. It was found that three items which should have been $60,70,80$, were wrongly read as $40,20,50$ respectively. The corrected mean is

If $A=\{x\in R:|x|<2\}$ and $B=\{x\in R:|x-2|\ge 3\}$ then :

The logically equivalent proposition of $p\iff q$ is

The value of the expression $\frac{(1+\tan \frac{\pi }{6})(1-\cot \frac{\pi }{6})}{(1+\cos \frac{\pi }{3})(1-\sec \frac{\pi }{3})}$ is

An ellipse with major and minor axis length as $2a$ and $2b$ units touches coordinate axis in first quadrant. If foci are $(x_1,y_1)$ and $(x_2,y_2),$ then the value of $x_1{x}_2+y_1{y}_2$ is

Let $T_n$ be the $n^{\text{th}}$ term and $S_n$ be the sum of $n$ terms of the series $\frac{1^3}{1}+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}+\cdots n\text{terms}$. Then which of the following is/are true?

If $2^{4n+4}-15n-16,n\in N$ is divided by $225$, then the remainder is

Which of the following Venn-diagram best represents the sets of males, females and mothers?

If $5({\tan }^{2}x-{\cos }^{2}x)=2\cos 2x+9,$ then the value of $\cos 4x$ is:

The number of distinct normals that can be drawn from $(-2,1)$ to the parabola $y^2-4x-2y-3=0$, is

Three numbers are chosen at random without replacement from {1, 2, ......, 15}. Let $E_1$ be the event that minimum of the chosen numbers is 5 and $E_2$ be that their maximum is 10 then

The locus of mid point of the chords of $x^2-y^2=4,$ that also touches the parabola $y^2=8x$ is

The negation of $\sim s\vee (\sim r\wedge s)$ is equivalent to

The polar form of $\frac{-\sqrt{3}}{2}-\frac{i}{2}$ (where i = $\sqrt{-1}$) is

The conjugate of complex number $\frac{2-3i}{4-i}$ is_______.

The triangle whose vertices are (0,7,-10), (1,6,-6) and (4,9,-6) is a ___.

The sum 1(1!) + 2(2!) + 3(3!) + ........ + n(n!) equals

The area of triangle formed by the lines x = 0, y = 0 and $\frac{x}{a}+\frac{y}{b}=1$, a and b are positive , is

If $x\in R$ and $x+\frac{x}{2}+\frac{x}{4}<7$, then $x$ lies in

Tweleve balls are distributed among three boxes. The probability that the first box contains three balls is

Find the domain of f(2x - 1) if the domain of f(x) is [-1,1]

The last three digits in 10! are ______.

The locus of z satisfying the inequality $lo{g}_{\frac{1}{3}}$|z+1| > $lo{g}_{\frac{1}{3}}$|z-1| is

If the ratio of sum of $p$ terms to $q$ terms of an A.P. is $\frac{p^2+p}{q^2+q}$, then the ratio of $p^{th}$ term to $q^{th}$ term is equal to

Points A$(1,2,3),B(-1,-2,-1),C(2,3,2)andD(4,7,6)$ are the vertices of _____