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A plane passes through the points $A(1,2,3),B(2,3,1)$ and $C(2,4,2).$ If $O$ is the origin and $P$ is $(2,-1,1),$ then the projection of $\overrightarrow{OP}$ on this plane is of length :

If |z - 25i| $\le$ 15, then |max amp(z) - min.amp(z)| =

A line passing through $P(6,4)$ meets the coordinates axes at $A$ and $B$ respectively. If $O$ is the origin, then the locus of the centre of the circumcircle of triangle $OAB$ is

The value of $\frac{1}{2-3i}+7+8i$ is

The line $x+y=1$ touches the parabola $y^2-y+x=0$ at the point

Evaluate $\underset{0}{\overset{10}{\int }}\mathrm{sgn}(1-\cos x)dx$

If $A=\{1,2,4\},B=\{2,4,5\},C=\{2,5\}$ then $(A-C)\times (B-C)$ is

The distributive law from algebra states that for all real numbers, c, $a_{1}and{a}_2$, we have $c(a_1+a_2)=c{a}_1+c{a}_2$.

Use this law and mathematical induction to prove that, for all natural numbers, $n\ge 2$, if c, $a_1,a_2,.....a_n$ are any numbers, then

$c(a_1+a_2+....+a_n)=c{a}_1+c{a}_2+......+c{a}_n$.

Which of the following collections are sets ? Justify your answer :

(i) A collection of all natural numbers less than 50.

(ii) The collection of good hockey players in India.

(iii) The collection of all girls in your class.

(iv) The collection of most talented writers of India.

(v) The collection of difficult topics in Mathematics.

(vi) The collection of all months of a year beginning with the letter J.

(vii) A collection of novels written by Munshi Prem Chand.

(viii) The collection of all questions in this chapter.

(ix) A collection of most dangerous animals of the world.

(x) The collection of prime integers.

The number of integral triples (a, b, c) such that a + b cos 2x + c $si{n}^2$ = 0, for all x, is

If $x_1$ and $x_2$ are two distinct solutions of the equation ${\log }_5({\log }_{64}|x|+(25{)}^x-\frac{1}{2})=2x,$ then

The solution set of $x^3-10x^2+21x>0$ is

A jet of enemy is flying along the path given by y=x^2+7 , Find the nearest distance from where a soldier placed at (3,7) can shoot down the helicopter

Let $f$ be differentiable for all $x$. If $f\left(1\right)=-2$ and $f^{\prime }\left(x\right)\ge 2$ for all $x\in [1,6],$ then

If $P$ be any point on the plane $lx+my+nz=p$ and $Q$ be a point on the line $OP$ such that $OP.OQ=p^2$, then the locus of the point $Q$ is:

$A(1,-1,-3),B(2,1,-2),C(-5,2,-6)$ are the position vectors of the vertices of a triangle ABC. The length of the bisector of its internal angle at A is :

In a triangle ABC with $\angle A={90}^{\circ }$, P is a point on BC such that PA : PB = 3:4. If AB $\sqrt{7}$ and AC = $\sqrt{5}$, then BP:PC is

If $\int \frac{dx}{x^2(x^n+1{)}^{\frac{(n-1)}{n}}}=-\left[f\right(x){]}^{\frac{1}{n}}+C$ then f(x) is

Find
the domain and the range of the real function f
defined by.

If A(1, 2, -1) and B(-1, 0, 1) are given, then the co-ordinates of P which divides AB externally in the ratio 1:2, are [MP PET 1989]

The locus of foot of the perpendiculars drawn from the line $\frac{x-2}{2}=\frac{y-1}{1}=\frac{z}{3}$ to the plane $x+y+z=1$ is

The equation of the common tangent touching the circle $(x-3{)}^2+y^2=9$ and the parabola $y^2=4x$ above the X-axis is

Which of the following plane is equally inclined to the planes 4x + 3y – 5z = 0 and 5x – 12y + 13z = 0 is

Which of the following points lie on the same side of origin with respect to the line

x - 3y + 7 = 0?

If $f:R\to Sdefinedbyf\left(x\right)=sinx-\sqrt{3}cosx+1$ is onto ,then the interval of $S$ is