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Points P, Q and the centre of the circle O lies on a straight line. P, Q are both outside the circle. When chords of contact of P and Q are drawn to the circle they are found to coincide. Whats the distance between P and Q?

The value of for which the projection of on is 4 units is:

A group of $10$ persons including ''Manager'', ''Assistant manager'' and ''Secretory'' are to be seated around a circular table.
The total number of possible arrangements, if ''Manager'', ''Assistant manager'' and ''Secretory'' had to sit together is

The number of ways in which we can get a score of $11$ by throwing three dice is :

If the equation $x^2+px+q=0$ and $x^2+qx+p=0$ have exactly one root in common, then equation with other roots is

A matrix which is both symmetric as well as skew-symmetric is a null matrix. Prove.

If a differentiable function f(x) has a relative minimum at x = 0, then the function y = f(x) + ax + b has a relative minimum at x = 0 for

If $lo{g}_{0.04}(x-1)\ge lo{g}_{0.2}(x-1)$ then x belongs to the interval

Which of the following relation in x and y is general solution of the differential equation $\frac{dy}{dx}=y$

If P is a point on the rectangular hyperbola $x^2-y^2=a^2,$ C is its centre and $S,S^{\prime }$ are the two foci, the $SP\cdot S^{\prime }P=$

The chord of contact of tangents from a point 'P' to a circle passes through Q. If $l_{1}and{l}_2$ are the lengths of the tangents from P and Q to the circle, then PQ is equal to

What is the value of cos 25⁰ - cos35⁰?

If origin is shifted to the point $(2,3)$ without rotation of axes then the coordinates of the point $P$ which divides the join of $A(4,8)$ and $B(7,14)$ in the ratio $1:2$ or $2:1$ with respect to the new system of coordinates can be

Let A, B and C are the angles of a plane triangle and $tan\frac{A}{2}=\frac{1}{3},tan\frac{B}{2}=\frac{2}{3}$.Then $tan\frac{C}{2}$ is equal to

The greatest value of $x^2{y}^3$ where $x>0,y>0$ and $3x+4y=5$ is

The equation of second degree $x^2+2\sqrt{2}xy+2y^2+4x+4\sqrt{2}y+1=0$ represents a pair of straight lines. The distance between them is

If $P=\{x\in N:\frac{14x}{x+1}-\left(\frac{9x-30}{x-4}\right)\le 0\},$
$Q=\{x\in Z:|x-1|\le 5\text{and}|x-1|\ge 2\}$
and $R=\{x\in R:{\log }_{6}x+\frac{2}{{\log }_{6}x}\}=3,$ then which of the following options is (are) CORRECT?

If $\theta ={\sin }^{-1}x+{\cos }^{-1}x-{\tan }^{-1}x,x\ge 0$ then the smallest interval in which $\theta$ lies is

A square is inscribed in the circle $x^2+y^2-2x+4y+3=0$, whose sides sre parallel to the coordinate axes. One vertex of the square is

If $m$ is chosen in the quadratic equation $(m^2+1)x^2–3x+(m^2+1{)}^2=0$ such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is -

There is a narrow rectangular plot reserved for a school in Mahuli village the length and breadth of the plot are in the ratio 11 : 24 at the rate hundred per metre it will cost the village panchayat rupees 75000 to fence the plot what are the dimensions of the plot

For the parabola $y^2=4ax$, the ratio of the sub-tangent to the abcissa is

Write the value of $li{m}_{x\to a}\frac{xf\left(a\right)-a\left(x\right)}{x-a}.$

The direction cosines of the line joining the points (2, -1, 8) and (-4, -3, 5) are:

If 5 of your friends ate one slice each from a medium pizza containing 6 equal slices, what will be the fraction of the pizza left over?

If $x+by+c=0$ is normal to the parabola $y^2=12x,$ tben complete set of all values of $c$ is

Let $a=min\{x^2+2x+3:x\in R\}$ and $b=\underset{\theta \to 0}{lim}\frac{1-\cos \theta }{{\theta }^2}.$ Then $\sum _{r=0}^n{a}^r{b}^{n-r}$ is

Distance between two parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 is

In the expansion of $(1+x{)}^m(1-x{)}^n$, the coefficients of x and $x^2$ are respectively 3 and -6. Then m equals :

Value of ${\int }_0^1(1+x+x^2+x^3)(1+3x+5x^2+7x^3)dx$ is

If $y_1\left(x\right)$ is a solution of the differential equation $\frac{dy}{dx}+f\left(x\right)y=0$, then a solution of differential equation $\frac{dy}{dx}+f\left(x\right)y=r\left(x\right)$ is

If $n_{P-4}=360$, find the value of n.

The equation(s) of normals(s) to the curve 3x² - y² = 8 which is parallel to the line x + 3y = 4 is.

If A = $A=\left\{\right(x,y):y=\frac{1}{x},0\ne xϵR\}$ and B = {$(x,y):y=-x,xϵR$}, then write $A\cap B$.

The $x$-coordinate of the vertices of a square of unit area are the roots of the equation $x^2-3|x|+2=0$ and the $y$-coordinates of the vertices are the roots of the equation $y^2-5y+6=0$. The number of such square is

Find the roots of $(4+4i{)}^{\frac{p}{q}}$ . Where p and q are integers and q $\ne$ 0

If z = 4 + i $\sqrt{7}$ , then value of $z^3-4z^2-9z+91$ equals

If $f\left(x\right)=(1+x{)}^n$, then the value of $f\left(0\right)+f^{\prime }\left(0\right)+\frac{f^{\prime \prime }\left(0\right)}{2!}+.....+\frac{f^n\left(0\right)}{n!}$ is

The set of real values of x satisfying $lo{g}_{\frac{1}{2}}(x^2-6x+12)\ge -2$ is

Equation of the parabola whose axis is $y=x$, distance from origin to vertex is $\sqrt{2}$ and distance from origin to focus is $2\sqrt{2},$ is (Focus and vertex lie in 1st quadrant) :

A fair coin is tossed n times. If the probability that head occurs 6 times is equal to the probability that head occurs 8 times, then n is equal to

[Kurukshetra CEE 1998; AMU 2000]

$\frac{C_0}{1}$ + $\frac{C_1}{2}$ + $\frac{C_2}{3}$ +...........+ $\frac{C_n}{n+1}$ =

Let P be the image of the point $(3,1,7)$ with respect to the plane $x-y+z=3$. Then the equation of the plane passing through P and containing the straight line
$\frac{x}{1}=\frac{y}{2}=\frac{z}{1}$ is