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If $\left|\begin{array}{ccc}{x}^n& x^{x+2}& x^{x+4}\\ y^n& y^{n+2}& y^{n+4}\\ z^n& z^{n+2}& z^{n+4}\end{array}\right|=(\frac{1}{y^2}-\frac{1}{x^2})(\frac{1}{z^2}-\frac{1}{y^2})(\frac{1}{x^2}-\frac{1}{z^2})$ then n=

A circle of radius $2\sqrt{10}$ cm is divided by a chord of length $12$ cm into two segments. Then maximum area of a rectangle that can be inscribed into the smaller circular segment is

The coefficient of $x^5$ in the expansion of $(1+x^2{)}^5$ $(1+x{)}^4$ is

The number of ways $10$ boys can be divided into $2$ groups of $5$, such that two particular boys are in the different groups, is

If $A$ is a $4\times 4$ matrix such that $|A|=1,$ then the value of $|\text{adj}\left(2A\right)|$ is

The number of complex numbers z such that is:

The differentiation of ${\cos }^{-1}\left(\frac{1-x^2}{1+x^2}\right)$ w.r.t. x is

Distance between two parallel lines is unity. A point $P$ lies between the lines at a distance ${}^{\prime }{a}^{\prime }$ from one of them. Find the length of a side of an equilateral $△PQR$ such that $Q$ lies on one of the parallel lines while $R$ lies on the other.

The line x + y = 1 meets x - axis at A and y - axis at B. P is the mid - point of AB $P_1$ is the foot of the perpendicular from P to OA; $M_1$ is that from $P_1$ to OP; $P_2$ is that from $M_1$ to OA; $M_2$ is that from $P_2$ to OP; $P_3$ is that from $M_2$ to OA and so on. If $P_n$ denotes the nth foot of the perpendicular on OA from $M_{n-1}$, then $O{P}_n$ =

The system of equations

X+2y+3z=4

2x+3y+4z=5

For every integer $n,$ let $a_n$ and $b_n$ be real numbers. Let function $f:R\to R$ be given by

$f\left(x\right)=\{\begin{array}{c}{a}_n+\sin \pi x,\text{for}x\in [2n,2n+1]\\ b_n+\cos \pi x,\text{for}x\in (2n-1,2n)\end{array}$ , for all integers $n.$

If $f$ is continuous, then which of the following hold(s) for all $n?$

If $Z=5x+2y$ subject to the following constraints : $x-2y\le 2,$ $3x+2y\le 12,$ $-3x+2y\le 3,$ $x\ge 0,y\ge 0,$ then the absolute value of the difference of the maximum and minimum value of $Z$ is



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On which of the following lines lies the point of intersection of the line, $\frac{x-4}{2}=\frac{y-5}{2}=\frac{z-3}{1}$ and the plane, $x+y+z=2$?

Construct a 3 $\times$ 2 matrix whose elements are given by $a_{ij}=e^{i.x}$ sin jx.

Find the principal values of the following questions:

$co{s}^{-1}(-\frac{1}{2})$

Let $A_1,A_2,....A_n$ be the vertices of an $n-$sided regular polygon such that $\frac{1}{A_1{A}_2}=\frac{1}{A_1{A}_3}+\frac{1}{A_1{A}_4}$. Then the value of $n$ is:

If $1+\sin \theta +{\sin }^2\theta +....upto\infty =2\sqrt{3}+4$, then $\theta =$

The only statement among the following that is a tautology is

The sum of each of two sets of three terms in A.P. is 15. The common difference of the first set is greater than that of the second by 1 and the ratio of the products of the terms in the first set and that of the second set is 7:8 the ratio of the smallest terms in two sets of terms is

The range of $y=\frac{3x-5}{5x-1},x\ne \frac{1}{5}$ is

Find the mean deviation
about median for the following data:

For natural number $m$, $n$, if $(1-y{)}^m(1+y{)}^n=1+a_{1}y+a_2{y}^2+...,$ and $a_1=a_2=10$, then

A school has 8 teachers. One of them is the headmaster (a) How many committees of 5 can be formed? (b) How many of them have the headmaster as a member?

The value of ${\int }_0^{\pi }\frac{cos3\theta }{cos\theta +sin\theta }d\theta$ is

Integrate the function.
$\int \sqrt{x^2+4x-5}dx.$

Three vectors, $\overrightarrow{A},\overrightarrow{B}$ and $\overrightarrow{C}$ are such that $\overrightarrow{A}\cdot \overrightarrow{B}=0$ and $\overrightarrow{A}\cdot \overrightarrow{C}=0$, then $\overrightarrow{A}$ is collinear to -

Find the coefficient of $(a^5+b^4+c^7)$ in the expansion of $(bc+ca+ab{)}^8$

Show that the height of the cylinder of
maximum volume that can be inscribed in a sphere of radius R
is.
Also find the maximum volume.