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A(-1, 8), B(4, -2) and C(-5, -3) are the vertices of a triangle, find the equation of the median through (-1, 8).

T is the region of the plane x + y + z = 1 with x, y, z >0. S is the set of points (a, b, c) in T such that just two of the following three inequalities hold: $a<\frac{1}{2},b\le \frac{1}{3},c\le \frac{1}{6}$

Area of the region S is (in sq. units)

If $cos(\theta -\alpha )$ = a, $sin(\theta -\beta )$ = b,

then $co{s}^2(\alpha -\beta )$ + 2ab $sin(\alpha -\beta )$ is equal to

Divide $4x^3+12x^2+11x+3byx+1$ and then find the quotient.

Find the multiplicative inverse of the following complex numbers :

$\left(i\right)1-i\left(ii\right)(1+i\sqrt{3}{)}^2(iii)4-3i(iv)\sqrt{5}+3i$

RS is the chord of contact of the point P on a circle center at O. if $m_1=slopeof\overline{RS}and{m}_2=slopeof\overline{OP}$

1. 1.1667

The standard equation of the circle whose parametric equation are $x=5-5\sin t$ and $y=4+5\cos t$ is

If the coefficients of $x^3$ and $x^4$ in the expansion of $(1+ax+b{x}^2$) $(1–2x{)}^{18}$ in powers of $x$ are both zero, then $(a,b)$ is equal to

The number of solution(s) of the equation ${\tan }^{-1}x-{\cot }^{-1}x={\cos }^{-1}(2-x)$ is

In a town of $10000$ families, it was found that $40\%$ families buy newspaper $A$, $20\%$ buy newspaper $B$ and $10\%$ buy newspaper $C$, $5\%$ families buy $A$ and $B$, $3\%$ buy $B$ and $C$ and $4\%$ buy $A$ and $C$. If $2\%$ families buy all the three newspapers, then number of families which buy newspaper $A$ only

The equation of circle passing through the origin and cutting off equal intercepts of $2$ units on the lines $\sqrt{3}{y}^2-\sqrt{3}{x}^2-2xy=0$ is/are

The coefficient of $x^5$ in the expansion of $(1+x{)}^{21}+(1+x{)}^{22}+...+(1+x{)}^{30}$ is

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are non-zero, non-coplanar vectors then $\{\overrightarrow{a}\times (\overrightarrow{b}+\overrightarrow{c}\left)\right\}\times \{\overrightarrow{b}\times (\overrightarrow{c}-\overrightarrow{a}\left)\right\}$ is collinear with the vector(s)

Let $y=y\left(x\right)$ be the solution of the differential equation $\cos x\frac{dy}{dx}+2y\sin x=\sin 2x,x\in (0,\frac{\pi }{2})$. If $y\left(\frac{\pi }{3}\right)=0,$ then $y\left(\frac{\pi }{4}\right)$ is equal to:

If the point A is symmetric to the point B(4, –1) with respect to the bisector of the first quadrant, then the length of AB is

If $\underset{n\to \infty }{lim}\sum _{r=1}^n\frac{a(1+3+5+\dots +(2r-1\left)\right)+(a-2)rn}{n\sum _{r=1}^n(2r-1)}=4$, then the value of $a$ is

If $|2x+1|+|x-2|+|x+3|<8$, then $x\in$

Let $f:N\to R$ be a function satisfying the following conditions:
$f\left(1\right)=1$ and $f\left(1\right)+2f\left(2\right)+\dots +nf\left(n\right)=n(n+1)f\left(n\right)$ for $n\ge 2$.
If $f\left(1003\right)=\frac{1}{K}$, then $K$ equals

If ${lim}_{x\to a}\frac{x^9-a^9}{x-a}=9$, find all possible values of a.

Box $1$ contains three cards bearing numbers $1,2,3;$ box $2$ contains five cards bearing numbers $1,2,3,4,5;$ and box $3$ contains seven cards bearing numbers $1,2,3,4,5,6,7.$ A card is drawn from each of the boxes. Let $x_i$ be the number on the card drawn from the $i^{th}$ box, $i=1,2,3.$



The probability that $x_1+x_2+x_3$ is odd, is

If $f\left(x\right)=\frac{(4x+3)}{(6x-4)},x\ne \frac{2}{3}$ show that (fof)(x)=x, for all $x\ne \frac{2}{3}$.What is the inverse of f?

If A + B + C= pi, Cos A= cos B. cos c, show that 2cot B 2cot c=1

The value of the determinant

$\Delta =\left|\begin{array}{ccc}logx& logy& logz\\ log2x& log2y& log2z\\ log3x& log3y& log3z\end{array}\right|$ is

Find the equation of
the line in vector and in Cartesian form that passes through the
point with position vector

and is in the direction
.

Can APS be greater than one? Give reasons in support of your answer.

$ta{n}^{-1}\left(\frac{1}{3}\right)+ta{n}^{-1}\left(\frac{1}{7}\right)+........+ta{n}^{-1}\left(\frac{1}{n^2+n+1}\right)=$