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If $x_1,x_{2}and{x}_3$ as well as $y_1,y_{2}and{y}_3$ are in GP with same common ratio, the points
$P(x_1,y_1),Q(x_2,y_2)andR(x_3,y_3)$

Which of the following line(s) is nearest to origin

If the line, $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-2}{4}$ meets the plane, $x+2y+3z=15$ at a point P, then the distance of P from the origin is:

The value of $\underset{0}{\overset{\pi }{\int }}x\sin x{\cos }^{2}xdx$ is

The general solution of the differential equation $\frac{dy}{dx}=e^{x+y}$ is
(a)$e^x+e^{-y}=C$
(b)$e^x+e^y=C$
(c)$e^{-x}+e^y=C$
(d)$e^{-x}+e^{-y}=C$

The triangle joining the points P(2, 7), Q(4, -1), R(-2, 6) is

A bag contains 4 white and 5 black balls. Another bag contains 9 white and 7 black balls. A ball is transferred from the first bag to the second and then a ball is drawn at random from the second bag. Find the probability that the ball drawn is white.

A box is constructed from a rectangular metal sheet of $21\text{cm}$ by $16\text{cm},$ by cutting equal squares of sides $x$ from the corners of the sheet and then turning up the projected portions. The value of $x$ for which volume of the box is maximum, is

The value of integral $\int \frac{dx}{(x+7{)}^{3/4}(x-4{)}^{5/4}}$ is

(where $C$ is integration constant)

Let $O$ be the origin. Let $\overrightarrow{OP}=x\hat{i}+y\hat{j}-\hat{k}$ and $\overrightarrow{OQ}=-\hat{i}+2\hat{j}+3x\hat{k},x,y\in R,x>0,$ be

such that $\left|\overrightarrow{PQ}\right|=\sqrt{20}$ and the vector $\overrightarrow{OP}$ is perpendicular to $\overrightarrow{OQ}$. If $\overrightarrow{OR}=3\hat{i}+z\hat{j}-7\hat{k},z\in R$ is coplanar with $\overrightarrow{OP}$ and $\overrightarrow{OQ},$ then the value of $x^2+y^2+z^2$ is equal to :

If PQ is a double ordinate of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ such that OPQ is an equilateral triangle, O being the centre of the hyperbola. Then the eccentricity e of the hyperbola, satisfies

$\int \sec \left(2x\right)\tan \left(2x\right)dx=k\sec \left(mx\right)+C$

Find the value of 2k+m

The length of a rectangle is three times the breadth. If the minimum perimeter of the rectangle is 160 cm, then .

The value of $\frac{3\times 1^3}{1^2}+\frac{5\times (1^3+2^3)}{1^2+2^2}+\frac{7\times (1^3+2^3+3^3)}{1^2+2^2+3^2}+\cdots \text{upto}10\text{terms}$, is

The value of $\sum _{n=1}^{13}(i^n+(i{)}^{n+1}),i=\sqrt{-1},$ is

If tan x tan y = a and x + y = $\frac{\pi }{6}$ , then tan x and tan y satisfy the equation

Cos 60degree-tan square 45degree+3/4 tan square 30 degree+ cos square 30 degree - sin 30 degree

The value of 2$(si{n}^6{735}^{\circ }+co{s}^6{735}^{\circ })$ - 3 $(si{n}^4{735}^{\circ }+co{s}^4{735}^{\circ })$ + 1 is __.

The portion of the tangent intercepted between the point of contact and the directrix of the parabola \( y^2 = 4ax\) subtends at the focus an angle of

Let $A=\{a,e,i,o,u\}$ and $B=\{m,y,g,h,n,s\}$. If $f:A\to B$ is a function, then the maximum number of such functions possible are

The most general valuie that satisfies the equation cosecθ = 2 and cotθ = -√3 is

If $\frac{x+y-8}{2}=\frac{x+2y-14}{3}=\frac{3x-y}{4}$,then the value of $x^2+y^2$ is equal to:

The value(s) of $\left[c\right]$ for which the line $y=4x+c$ touches the curve $x^2+16y^2=16$ is/are (where $[.]$ represents greatest integer function)

If $n\ge 2$ is a positive integer, then the sum of the series ${}^{n+1}{C}_2+2({}^2{C}_2+{}^3{C}_2+{}^4{C}_2+\cdots +{}^n{C}_2)$ is

If $A=\left[\begin{array}{cc}5& 2\\ 2& 1\end{array}\right]$ and matrix $B$ is such that $A{B}^{T}A=B$ and $B^{T}AB=I$ where $I$ is unit matrix of order $2$, then matrix $B^2$ is

If $lo{g}_{a}x,lo{g}_{b}x,lo{g}_{c}x$ be in H.P., then a,b,c are in

Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes.

If $\sum _{k=1}^{n}k(k+1)(k-1)=p{n}^4+q{n}^3+t{n}^2+sn$, where $p,q,t,s$ are constant, then the correct option(s) is/are

Find the equation of the circle with
Centre (0,2) and radius 2

The vertex of the parabola $x^2+2y=8x-7$ is