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Let $f\left(n\right)=\left|\begin{array}{ccc}n& n+1& n+2\\ {}^n{P}_n& {}^{n+1}{P}_{n+1}& {}^{n+1}{P}_{n+2}\\ {}^n{C}_n& {}^{n+1}{C}_{n+1}& {}^{n+1}{C}_{n+2}\end{array}\right|$

Then, f(n) is divisible by

Suppose the probability for A to win a game against B is 0.4. If A has an option of playing either a "best of 3 games" or a "best of 5 games" match against B, which option should A choose so that the probability of his winning the match is higher ? (No game ends in a draw)

Find the local maxima of the function $f\left(x\right)=-x^2+7x-12$

The condition that the straight line $lx+my+n=0$ touches the parabola $x^2=4ay$ is

Let two points be $A(1,-1)$ and $B(0,2)$. If a point $P(x^{\prime },y^{\prime })$ be such that the area of $\Delta PAB=5$ sq. units and it lies on the line, $3x+y-4\lambda =0$, then the value of $\lambda$ is:

A box contains $6$ red and $4$ white marbles. A marble is drawn and replaced back in the box for three times. The probability that two white marbles are drawn is:

Christine wears her thermal coat when the temperature is colder than $-4$ ${(}^{\circ }C)$. If $T$ is the temperature in ${(}^{\circ }C)$ at which Christine wears her winter coat, which of the following inequalities best models the situation?

Consider the lines $L_1:\frac{x-1}{2}=\frac{y}{-1}=\frac{z+3}{1},$ $L_2:\frac{x-4}{1}=\frac{y+3}{1}=\frac{z+3}{2}$ and the planes $P_1:7x+y+2z=3,$ $P_2:3x+5y-6z=4.$ Let $ax+by+cz=d$ be the equation of plane passing through the point of intersection of the lines $L_1$ and $L_2$, and perpendicular to the planes $P_1$ and $P_2$.



Match $\text{List - I}$ with the $\text{List - II}$ and select the correct answer using the code given below the lists :

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{P}& a=& 1.& 13\\ \text{Q}& b=& 2.& -3\\ \text{R}& c=& 3.& 1\\ \text{S}& d=& 4.& -2\end{array}$

Let $a_i=i+\frac{1}{i}$ for $i=1,2,...,20.$ Put
$p=\frac{1}{20}(a_1+a_2+\cdots +a_{20})\text{and}q=\frac{1}{20}(\frac{1}{a_1}+\frac{1}{a_2}+\cdots +\frac{1}{a_{20}}).$
Then

Let a matrix $A=\left[\begin{array}{cc}\cos x& \sin x\\ \tan x& \cot x\end{array}\right],$ then which of the following statement(s) is(are) true for atleast one value of $x\in [0,\frac{\pi }{2}]$ ?

(where $C_{ij}$ is co-factor of element $\left[a_{ij}\right]$)

Locus of the centre of the circle which touches the two circle $x^2+y^2+8x–9=0and{x}^2+y^2–8x+7=0$ externally, is

A parallelogram is cut by two sets of m lines parallel to its sides. Find the number of parallelograms thus formed.

Differentiate the following functions with respect to x :

$\frac{x}{si{n}^{n}x}$

If A and B are two sets such that n (A) = 20, n(B) = 25 and $n(A\cup B)$= 40, then write $n(A\cap B)$.

Find the points on the curve y =
x³ at which the slope of the tangent is equal to
the y-coordinate of the point.

Find the
unit vector in the direction of vector,
where P and Q are the points

(1, 2, 3)
and (4, 5, 6), respectively.

The general solution of the equation ${\sin }^3\frac{x}{2}-{\cos }^3\frac{x}{2}=\frac{\cos x\times (2+\mathrm{sinx})}{3}$ is

If $|x^2-3x+2|=-(x-1)(x-2)$, then $x\in$

A sweet shop was placing an order for making cardboard boxes of packing their sweet two size of boxes one $20cm\times 15cm\times 5cm$ and the other $12cm\times 10cm\times 5cm$ were required for all the overlaps $5\%$ of the total surface area is required extra. If the cost of the cardboards is $Rs.4.50$ per $1000c{m}^2$, find the cost of cardboard required for supplying $100$ boxes of each kind.

If $4\hat{i}+7\hat{j}+8\hat{k},2\hat{i}+3\hat{j}+4\hat{k}$ and $2\hat{i}+5\hat{j}+7\hat{k}$ are the position vectors of the vertices $A,B$ and $C$ respectively of triangle $ABC$. The position vector of the point where the bisector of $\angle A$ meets $BC$

The domain of $f\left(x\right)={\left(\frac{2x+1}{x^2-10x-11}\right)}^{2020}$ is

If vectors $\overrightarrow{a},\overrightarrow{b}$and $\overrightarrow{c}$ are unit vectors such that $\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=0$ then the value of $(\overrightarrow{a}.\overrightarrow{b}+\overrightarrow{b}.\overrightarrow{c}+\overrightarrow{c}.\overrightarrow{a})$ is

The integral $\underset{0}{\overset{\pi }{\int }}\sqrt{1+4{\sin }^2\frac{x}{2}-4\sin \frac{x}{2}}dx$ equals to: