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The number of ways of choosing $2$ cards from a pack of $52$ where atleast one face card is choosen

Let $P(3,3)$ be a point on the hyperbola, $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. If the normal to it at $P$ intersects the $x-$axis at $(9,0)$ and $e$ is its eccentricity, then the ordered pair $(a^2,e^2)$ is equal to

P speaks truth in 75% cases while Q in 90%.in what % r they likely to contradict each other in stating the same fact .? Do u think B is true

Do

Find the angle between two vectors a and b with magnitudes $\sqrt{3}$ and 2 respectively, having $a.b=\sqrt{6}$

If $y=\{x{\}}^{\left[x\right]},$ then $\underset{0}{\overset{3}{\int }}ydx$ is equal to,where {.} and [.] are fractional part function and greatest integer function respectively.

The graph of $f\left(x\right)=-|\log (x-3)|+3$ will be

Solve the following system of equations in R.
$|x-1|+|x-2|+|x-3|\ge 6$

If $C_0,C_1,C_2,\cdots C_n$, are the binomial coefficients of the expansion $(1+x{)}^n$, where $n$ is even, then $C_0+(C_0+C_1)+(C_0+C_1+C_2)+\cdots \cdots +(C_0+C_1+C_2+\cdots +C_{n-1})=$

A five digit number is chosen at random.The probability that all the digits are distinct and digits at odd place are odd and digits at even places are even is

The principal value of $si{n}^{-1}(-\frac{1}{2})$ is

Integrate the following functions.
$\int x\sqrt{x+2}dx.$

The number of non-zero integral solutions of the equation $|1-i{|}^x$=$2^x$ is

Let $f_n\left(\theta \right)=\sum _{r=0}^n\frac{1}{4^r}{\sin }^4\left(2^r\theta \right)$, then which of the following alternative (s) is/are correct?

$cos{35}^{\circ }+cos{85}^{\circ }+cos{155}^{\circ }=$

The value of the integral $\underset{-\frac{3\pi }{4}}{\overset{\frac{5\pi }{4}}{\int }}\frac{(\sin x+\cos x)}{e^{x-\frac{\pi }{4}}+1}dx$ is

The remainder when $2^{2003}$ is divided by $17$ is

The principal solution(s) for $\tan x=-1$ is/are

If the pair of lines $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ intersect on the y axis, then

The smallest positive term of the sequence $25,22\frac{3}{4},20\frac{1}{2},18\frac{1}{4},\cdots$ is

$\int (px+q{)}^{4}dx$

n cadets have to stand in a row. If all possible permutations are equally likely, then the probability that two particular cadets stand side by side, is

If $P$ is a point $(x,y)$ on the line $y=-3x$ such that $P$ and the point $(3,4)$ are on the opposite sides of the line $3x-4y=8$, then

Find the mean and
variance for the data

The domain of the function $y=\frac{1}{\sqrt{{81}^{1/(x-1)}-3}}$ is

$Iff:(0,\infty )\to (0,\infty )$ satisfy

$f\left(xf\right(y\left)\right)=x^2{y}^2(a\in R)$,then

${\sum }_{r=1}^n-f\left(r\right){}^n{C}_{r}is$

Let $y=y\left(x\right)$ be the solution of the differential equation $\frac{dy}{dx}=2(y+2\sin x-5)x-2\cos x$ such that $y\left(0\right)=7$. Then $y\left(\pi \right)$ is equal to

Consider a circle $C_1$ with equation $(x-1{)}^2+y^2=1$ and a shrinking circle $C_2$ with radius $r$ and centre at the origin. $P$ is the point $(0,r),Q$ is the point of intersection of the two circles(above x-axis) and $R$ is the point of intersection of the line $PQ$ with the x-axis. Find the x-coordinate of the point $R,$ as $C_2$ shrinks, that is $r\to 0$.