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What is the numerical value of F?

If ${}^{22}{P}_{r+1}{:}^{20}{P}_{r+2}=11:52,$ find r.

Solve the following system of equations in R.

3x-6 >,2x-5>0

Number of ways of dividing $80$ cards into $5$ equal groups of $16$ each is :

Sir in the following question we have to answer true or false..

A={1,2,{3,4},5}

1-- fi € A

2 fi subset of A

3 {fi} is a subset of A

Please help sir

Thank you

Triangle $ABC$ has $AB=90,BC=50$ and $CA=70.$ A circle is drawn with centre $P$ on $AB$ such that $CA$ and $CB$ are tangents to the circle. Find $\frac{2}{3}AP.$
(correct answer + 3, wrong answer 0)

Determine the values of a and b so that the points (a,b,3),(2,0,-1) and (1,-1,-3) are collinear.

If $ax+by=1$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,$ then the value of $a^2–b^2$ is

If ( x-3) and (x-1/3) are both factors of ax²+5x+b, then show that a=-3/2.

How is U related to L ?

Statement I. S's sister K has married U's brother M, who is the only son of his parents.

Statement II. L is the only daughter of M, and K.

If $a+b+c+d=63,$ where $a,b,c,d\in I^{+},$ then the maximum value of $ab+bc+cd$ is

Let $f:R\to R,g:R\to R$, be two functions, such that f(x) =2x – 3, g (x) = $x^3$ + 5.
The function $(fog{)}^{-1}$ (x) is equal to

In a knockout tournament 16 equally skilled players namely $P_1,P_2,------P_{16}$ are participating. In each round players are divided in pairs at random and winner from each pair moves to the next round. If $P_2$ reaches the semifinal, then the probability that $P_1$ will win the tournament is

A tower subtends angles $\theta ,2\theta$ and $3\theta$ at three points $A,B,C$ respectively lying on a horizontal line through the foot of tower. Then the ratio $\frac{AB}{BC}$ equals

Check whether the function f given by $f\left(x\right)=x^{100}+sinx-1$ strictly decreasing for the given interval.

$(0,\frac{\pi }{2})$

The area bounded by the curve $y^2=4a(a-|x-a|);a>0$, is

Let $f\left(x\right)={\int }_1^x\frac{{\tan }^{-1}t}{t}dt$ $(x>0)$ then $f\left(e^2\right)-f\left(\frac{1}{e^2}\right)$ is

If $\alpha ,\beta$ are the roots of $2x^2+3x+1=0$, then the equation whose roots are $\frac{1}{\alpha },\frac{1}{\beta }$ is

The number of four digit numbers that can be formed with the digits $1,2,3,4$ and $5$ in which atleast two digits are identical is

In the expansion of $(1+x{)}^{70}$, the sum of coefficients of odd powers of x is

A ladder 20 ft long has one end on the ground and the other end in contact with a vertical wall. The lower end slips along the ground. If the lower end of the ladder is 16 ft away from the wall, upper end is moving $\lambda$ times as fast as the lower end, then $\lambda$ is

If $\alpha$ and $\beta$ are complex numbers with $|\beta |=1$, find $\left|\begin{array}{c}\frac{\beta -\alpha }{1-\overline{\alpha }\beta }\end{array}\right|$.

A manufacturer has 460 litres of a 9% acid solution. How many litres of a 3% acid solution must be added to it so that the acid content in the resulting mixture be more than 5% but less than 7%?

Compute the indicated product.
$$\left[\begin{array}{ccc}3& -1& 3\\ -1& 0& 2\end{array}\right]$$$$\left[\begin{array}{cc}2& -3\\ 1& 0\\ 3& 1\end{array}\right]$$

${lim}_{x\to 0}\frac{e^{3x}-e^{2x}}{x}$

Find the points on the curve $y=x^3$ at which the slope of the tangent is equal to the y-coordinate of the point.

Complete set of values of $x$, satisfying the in equality $x^2+\frac{x^2}{(x+1{)}^2}<\frac{5}{4},$ is

The solution of the equation is

Find the equation of a sphere, whose end points of a diameter are (0,0,0) and (1,2,3)

If $f\left(x\right)=\{\begin{array}{cc}\frac{x-|x|}{x},& whenx\ne 0\\ 2,& whenx=0\end{array}$ , then

The graph of $f\left(x\right)=2x^2-3x+2$ is

If $\theta$ is an angle between the lines given by the equation $6x^2+5xy-4y^2+7x+13y-3=0$, then equation of the line passing through the point of intersection of these lines and making an angle $\theta$ with the positive x - axis is

The number of different five digits numbers that can be formed from the digits $2,4,9,3,5$ (without repetation) which are not divisible by $5$ are

Five cards are drawn successively with replacement from well- shuffled deck of 52 cards. What is the probability that

(i) all the five cards are spades?

(ii) only 3 cards are spades?

(iii) None is a spade?

Solve the equation $x^4+4x^3+6x^2+4x+5=0$, given one root is $\sqrt{(-1)}$. Find the value of other three roots.

If $y=mx+c$ touches the parabola $y^2=4a(x+a)$, then

Show that the points A(1, 2, 7), B(2, 6, 3) and C(3, 10, -1) are collinear.

Let ABCD be a parallelogram whose diagonals intersect at P and let O be the origin, then $\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}+\overrightarrow{OD}$ equals

$\text{The locus of the point of intersecion of the lines}\sqrt{3}x-y-4\sqrt{3}\gamma =0and\sqrt{3}\gamma x+\gamma y-4\sqrt{3}=0isahyperbolaofeccentricity$

If n arithmetic means are inserted between 1 and 31 such that the ratio of the first mean and nth mean is 3 : 29, then the value of n is