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A bag contains 3 white, 3 black and 2 red balls. One by one, three balls are drawn without replacing them. Then the probability that the third ball is red , is given by

Solution set of $\frac{x^2-4x+3}{x^2-8x+15}\le 0$ is

Find the principal solutions of $2si{n}^{2}x-sinx-1=0$

The sum ${\sum }_{i=0}^m(\frac{10}{i})(\frac{20}{m-i}),where(\frac{p}{q})=0ifp>q,$ is maximum when m is equal to

Find the principal solution of $2co{s}^{2}x-3cosx-2=0$

Find the middle term(s) in the expansion of $(a+b{)}^{21}$

The mid -points of the sides of a triangle are$(1,5-1),(0,4,-2)and(2,3,4).$ Find its vertices.

If a triangle has its orthocenter at (1, 1) and circumcenter at $(\frac{3}{2},\frac{3}{4})$ , then the coordinates of the centroid of the triangle are

Length of the straight line $x-3y=1$ intercepted by the hyperbola $x^2-4y^2=1$ is

The value of $\cos \frac{\pi }{2^2}\cdot \cos \frac{\pi }{2^3}\cdot ...\cdot \cos \frac{\pi }{2^{10}}\cdot \sin \frac{\pi }{2^{10}}$ is :

Let $a,b,c,d\in R^{+}$ and $256abcd\ge (a+b+c+d{)}^4$ and $3a+b+2c+5d=11$ then $a^3+b+c^2+5d$ is

The sum up to $23$ terms of the A.P. $5,9,13,17,...$ is

If one root of the determinant then the other two roots are

Let $x_n,y_n,z_n,w_n$ denote $n^{\text{th}}$ term of four different arithmetic progressions with positive terms. If $x_4+y_4+z_4+w_4=8$ and $x_{10}+y_{10}+z_{10}+w_{10}=20,$ then maximum possible value of $x_{20}\cdot y_{20}\cdot z_{20}\cdot w_{20}$ is

The differential equation whose solution is $(x-h{)}^2+(y-k{)}^2=a^2$ is (a is a constant)

The first term of the G.P is 1 . If ${(p+q)}^{th}$ term of G.P. is `a` and it's ${(p-q)}^{th}$ term is `b` where a, b ∈ $R^{+}$ then it's $p^{th}$ term is:

A sector $OABO$ of central angle $\theta$ is constructed in a circle with centre $O$ and radius $6.$ The radius of the circle that is circumscribed about the triangle $OAB,$ is

x and y are the sides of two squares such that $y=x-x^2$. The rate of change of area of the second square with respect to that of the first square is

Let $z_1,z_2,\&z_3$ be the vertices and $z_0$ be the circumcentre of an equilateral triangle. then $z_1^2+z_2^2+z_3^2$ =

If $(5+2\sqrt{6}{)}^n=m+f$, where n and m are positive integers and 0 $\le$ f < 1, then $\frac{1}{1-f}-f$ is equal to:

If the relation f is defined by $f\left(x\right)=\{\begin{array}{cc}{x}^2,& 0\le x\le 3\\ 3x,& 3\le x\le 10\end{array}$

and the relation g is defined by $g\left(x\right)=\{\begin{array}{cc}{x}^2,& 0\le x\le 2\\ 3x,& 2\le x\le 10\end{array}$

then,

The locus of a point, from where pair of tangents to the rectangular hyperbola $x^2-y^2=a^2$ contain an angle of ${45}^{\circ }$, is :

Which of the following collections is not a set?

If ${\log }_{a}3=2;{\log }_{b}8=3$, then ${\log }_{a}b=$

A watermelon seed has the following coordinates: x=-5.0 m, y=9.0 m and z=0 m. Find its position vector

(a) In unit-vector notation and as
(b) A magnitude and
(c) An angle relative to the positive direction of the x-axis

$\underset{x\to 0}{lim}\frac{logcosx}{x}=$

$\frac{1}{4}$$[\sqrt{3}cos{23}^{\circ }-sin{23}^{\circ }]$ =

In the quadratic equation $a{x}^2+bx+c=0,\Delta =b^2-4acand\alpha +\beta ,{\alpha }^2+{\beta }^2,{\alpha }^3+{\beta }^3,$ are in G.P. where $\alpha ,\beta$ are the root of $a{x}^2+bx+c=0,$ then

In a composite function f [g(x)], the following condition must be true

Given $\overrightarrow{\alpha }=3\hat{i}+\hat{j}+2\hat{k}$ and $\overrightarrow{\beta }=\hat{i}-2\hat{j}-4\hat{k}$ are the position vectors of the points $A$ and $B.$ Then the distance of the point $-\hat{i}+\hat{j}+\hat{k}$ from the plane passing through $B$ and perpendicular to $AB$ is

Which of the following is in reduced row Echelon form

No. of real roots of the equation $x^3+x^2$ + 10x + sin x = 0, is :

For $\alpha ϵR$ (the set of all real numbers), $a\ne -1$,

${lim}_{n\to \infty }\frac{(1^a+2^a+\dots +n^a)}{(n+1{)}^{a-1}[(na+1)+(na+2)+\dots +(na+n)]}=\frac{1}{60}$

$\int \frac{2sinx}{(3+sin2x)}dx$ is equal to

Some trigonometric ratios and the interval in which $\theta$ lies is given. Match the intervals with the ratios which are positive in those intervals.

$\theta$ gives positive values

p. (0, $\frac{\pi }{2}$) 1. Only sin $\theta$, cosec$\theta$

q. ($\frac{\pi }{2}$, $\pi$) 2. Only cos$\theta$, sec$\theta$

r. $(\pi ,\frac{3\pi }{2})$ 3. Only tan$\theta$, cot$\theta$

s. $(\frac{3\pi }{2},2\pi )$ 4. All sin$\theta$, cos$\theta$, tan$\theta$, cot$\theta$, sec$\theta$, cosec$\theta$

A car will hold two persons in the front seat and 1 in the rear seat. If among six persons only two can drive, the number of ways, in which the car can be filled is :

If the circles $x^2+y^2-16x-20y+164=r^2$ and $(x-4{)}^2+(y-7{)}^2=36$ intersect at two distinct points, then :

A line L passes through the points (1, 1) and (2, 0) and another line L’ passes through $[\frac{1}{2},0]$ and perpendicular to L. Then the area of the triangle formed by the lines L, L’ and y –axis, is