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1\3 x-5\2=6

In a parallelogram OABC, vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are respectively the position vectors of vertices A, B, C with reference to O is origin. A point $\overrightarrow{E}$ is taken on the side BC which divides it in the ratio of 2 : 1. Also, the line segment AE intersects the line bisecting the angel O internally in point P. If CP, when extended meets AB in point F. Then

The position vector of point p is

If A = $\left[\begin{array}{cc}\alpha & 2\\ 2& \alpha \end{array}\right]$ and $|A^3|$ = 27, then $\alpha$ =

If f (x) is differentiable in the interval [2, 5], where $f\left(2\right)=\frac{1}{5}$ and $f\left(5\right)=\frac{1}{2}$, then there exists a number c, 2 < c < 5 for which f ' (c) is equal to

In a box containing 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs, none is defective is

(a) ${10}^{-1}$

(b) ${\left(\frac{1}{2}\right)}^5$
(c) ${\left(\frac{9}{10}\right)}^5$
(d) $\frac{9}{10}$

What is the length of tangent from a point $P(x_1,y_1)$ to the circle $x^2+y^2+2gx+2fy+c=0.$

Compute :
(i) $\frac{30!}{28!}$ (ii) $\frac{11!-10!}{9!}$
(iii) L.C.M. (6!,7!,8!)

Derivative of tan^-1 x is

$A\left(z_1\right)$ and $B\left(z_2\right)$ are the points on the circles |z| = 1 and |z| = 3 respectively, then:

The maximum value of $({\sec }^{-1}x{)}^2+(cose{c}^{-1}x{)}^2$ is equal to

Using the method of integration, find the area of the region bounded by lines
2x + y = 4, 3x - 2y = 6 and x - 3y + 5 = 0

If $\left[x\right]$ denotes the greatest integer function, then the solution set of the inequation $\frac{\left[x\right]-1}{2-\left[x\right]}>0,$ is

$InQ>No.1,\text{Write the distance between the circumcentre and orthocentre of}\Delta OAB.$

If $si{n}^{-1}\frac{3}{5}+co{s}^{-1}\left(\frac{12}{13}\right)=si{n}^{-1}C,thenC=$

Let $ABC$ be a triangle with $\angle C={90}^{\circ }$. Draw $CD$ perpendicular to $AB$. Choose points $M$ and $N$ on sides $AC$ and $BC$ respectively such that $DM$ is parallel to $BC$ and $DN$ is parallel to $AC$. If $DM=5,DN=4$, then $AC$ and $BC$ are respectively equal to

If $2y={\left({\cot }^{-1}\left(\frac{\sqrt{3}\cos x+\sin x}{\cos x-\sqrt{3}\sin x}\right)\right)}^2,x\in (0,\frac{\pi }{2})$ then $\frac{dy}{dx}$ is equal to:

$\text{Find the axes,eccentricity, latus-rectum and the co-ordinates of the foci of the hyperbola.}25x^2-36y^2=225$

Find k such that $k+9,k-6$ and 4 form three consecutive terms of a G.P.

What is wrong in the sequence?

The unruly young two brothers

If two distinct chords of a parabola $y^2=ax$ passing through the point $(a,a)$ are bisected by the line $x+y=1,$ then the length of the latus rectum cannot be :

Prove that:
(i) $sin{65}^{\circ }+cos{65}^{\circ }=\sqrt{2}cos{20}^{\circ }$
(ii) $sin{47}^{\circ }+cos{77}^{\circ }=cos{170}^{\circ }$

If $x+\frac{1}{x}=8,$ then the value of $x^3+\frac{1}{x^3}$ is

In how many ways can three jobs, I, II and III be assigned to three persons A,B and C, if oone person is assigned only one job and all are capable of doing each job?

The least value of $\sec A+\sec B+\sec C$ in an acute angle triangle is

If the sum of the roots of the equation $a{x}^2+bx+c=0$ is equal to the sum of the squares of their reciprocals, then $\frac{b^2}{ac}+\frac{bc}{a^2}=$

If m and n are positive integers greater than or equal to 2, m > n, then (mn)! is divisible by

Let f(x) be a function satisfying f ’(x) = f(x) with f(0) = 1 and g(x) be a function that satisfies f(x) + g(x) = $x^2$. Then, the value of the integral ${\int }_0^{1}f\left(x\right)g\left(x\right)dx,$ is

A king comes from a family of two children. What is the probability that his sibling is a sister?

If theta is the angle between $\frac{x+1}{1}=\frac{y-1}{2}=\frac{z-2}{2}$ and the plane $2x-y+\sqrt{\lambda z}+4=0$ and is such that sin(theta) $=\frac{1}{3}$, the value of $\lambda =$

Match the functions given in the first column with their first derivatives.

If $si{n}^{-1}x+si{n}^{-1}y=\frac{\pi }{2}$,then $co{s}^{-1}x+co{s}^{-1}y$ is equal to

Find the particular solution of the differential equation $x^{2}dy=(2xy+y^2)dx,$ given that y= 1 when x = 1.

OR

Find the particular solution of the differential equation $(1+x^2)\frac{dy}{dx}=(e^{mta{n}^{-1}x}-y),$ given that y =1 when x = 0.

If the radius of the circle $x^2+y^2$+ 8x + 10y + k = 0 is 7, Then k =

The value of ${lim}_{x\to 0}\frac{sinax+bx}{ax+sinbx},a,b,a+b\ne 0$, is

If the function $f$ given by
$f\left(x\right)=x^3-3(a-2)x^2+3ax+7$, for some $a\in R$ is increasing in $(0,1]$ and decreasing in $[1,5)$, then a root of the equation,
$\frac{f\left(x\right)-14}{(x-1{)}^2}=0(x\ne 1)$ is :

Answer each of the following questions in one word or one sentence or as per exact requirement of for question:

$Ifina\Delta ABC,\frac{cosA}{a}=\frac{cosB}{b}=\frac{cosC}{c},\text{then find the measures of angles A, B, C.}$

Find the number of combinations and permutations of $4$ letters taken from the word 'EXAMINATION'.

Let $x_1,x_2,x_3$ be three positive numbers such that $x_1+x_2+x_3=z$.
Which of the following is/are correct?

Solve the following system of equations in R.

$2(x-6)<3x-7,11-2x<6-x$

Show that the solution set of the following system of linear equations has no solution:

$2x+y\ge 8,x+2y\ge 10,x\ge o,y\ge 0$

If $sinA=\frac{1}{2},$ $cosB=\frac{12}{13},$ where $\frac{\pi }{2}<A<\pi$ and $\frac{3\pi }{2}<B<2\pi ,$ find tan(A-B)