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The complex number $z$ which simultaneously satisfies the equations $\left|\frac{z-12}{z-8i}\right|=\frac{5}{3}$ and $\left|\frac{z-4}{z-8}\right|=1$ is

A set $X$ is the set of vowels in the word $"CRYPT",$ then

If the data given to construct a triangle ABC is a=5,b=7, $sinA=\frac{3}{4}$ then it is possible to construct

For any $\Delta ABC.$ The value of $\frac{c-b\cos A}{b-c\cos A}$ will be equal to:

(where $\angle C\ne {90}^0$)

1. 5

If the direction ratios of two lines are given by $3lm-4ln+mn=0$ and $l+2m+3n=0,$ then the angle between the lines is

If $f\left(x\right)=\{\begin{array}{cc}3+|x-k|,& \text{for}x\le k\\ a^2-2+\frac{sin(x-k)}{x-k},& \text{for}x>k\end{array}$ has minimum at x = k, then

If $3\sin \theta +4\cos \theta =5$, then the value of $4\sin \theta -3\cos \theta$ is

If the lines $lx+my+n=0,mx+ny+l=0$ and $nx+ly+m=0$ are concurrent then ($l,m,n\ne 0$)

Find the area enclosed between the
parabola y² = 4ax and the line y = mx

Show that
f: [−1, 1] → R, given byis
one-one. Find the inverse of the function f: [−1, 1] →
Range f.

(Hint: For
y ∈Range f, y
=,
for some x in [−1, 1], i.e.,)

Find the value of the expression $\sec \left(\frac{9\pi }{4}\right)\tan (-\frac{9\pi }{4}).$

For each positive integer n, let $s_n=\frac{3}{\mathrm{1.2.4}}+\frac{4}{\mathrm{2.3.5}}+\frac{5}{\mathrm{3.4.6}}+\dots \dots +\frac{n+2}{n(n+1)(n+3)}$. Then $\underset{n\to \infty }{lim}{s}_n$ equals

The common tangents to the circle $x^2+y^2=2$ and the parabola $y^2=8x$ touch the circle at the points $P,Q$ and the parabola at the points $R,S$. Then the area of the quadrilateral $PQRS$ is

If $P(\alpha ,\beta )$ is the mid point of the chord of contact of the point $(3,2)$ with respect to the ellipse $x^2+4y^2=9$, then the slope of $OP$ is $(O$ is origin $)$ equal to

A particle is moving in a straight line and at some moment it occupied the positions $(5,2)$ and $(-1,-2)$. Then the position of the particle when it is on $x-$axis is

If $a+b+c=1$, where $a,b,c$ are positive real numbers, then the minimum value of $\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ca}$ is

Given that
the events A and B are such thatand
P (B) = p. Find p if they are (i) mutually exclusive
(ii) independent.

If the truth value of the statement $p\to (\sim q\vee r)$ is false$\left(F\right),$ then the truth values of the statements $p,q,r$ are respectively :

Write the value of sin ${10}^{\circ }+si{n}^2{0}^{\circ }+....+sin{360}^{\circ }$

$If4si{n}^2\theta =1,thenthevalueof\theta are.$

Find the equation for the ellipse that satisfies the given conditions: Major axis on the x-axis and passes through the points (4, 3) and (6, 2).

The value of $\underset{n\to \infty }{lim}(\frac{\sqrt{n}}{{(3+4\sqrt{n})}^2}+\frac{\sqrt{n}}{\sqrt{2}{(3\sqrt{2}+4\sqrt{n})}^2}+\frac{\sqrt{n}}{\sqrt{3}{(3\sqrt{3}+4\sqrt{n})}^2}+\cdots +\frac{1}{49n})$

is of the form $\frac{1}{p},$ where $p\in N.$ Then possible factors of $p$ is/are

Let $F_1(x_1,0)$ and $F_2(x_2,0),$ where $x_1<0$ and $x_2>0$ be the foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{8}=1$ suppose a parabola having vertex at the origin and focus at $F_2$ intersects the ellipse at point M in the first quadrant and at point N in the fourth quadrant.

The orthocentre of $\Delta F_{1}MN$ is

If the coefficient of $x^2$ and $x^3$ in the expansion of $(3+ax{)}^9$ are the same, then the value of a is