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The maximum value of the function $f\left(x\right)=2x^3-18x^2+48x-11$ over the set $S=\{x\in R:x^2+42\le 13x\}$ is

Let $f\left(x\right)=\{\begin{array}{cc}\frac{1-si{n}^{3}x}{3co{s}^{2}x}:& x<\frac{\pi }{2}\\ p:& x=\frac{\pi }{2}\\ \frac{q(1-sinx)}{(x-2x{)}^2}:& x>\frac{\pi }{2}\end{array}$ If f(x) is continuous at $x=\frac{\pi }{2},(p,q)=$

Range of the function $f\left(x\right)=|x-1|+|x-2|+|x+1|+|x+2|$ where $xϵ[-2,2]$, is

A rod is hinged at point ‘O’.What part of length should be submerged in water so that it remains in equilibrium, if its specific gravity is 0.5.

Let, $S_n$ denote the sum of first $n$ terms of an arithmetic progression whose first term is $-4$ and common difference is $1$. If $V_n=2S_{n+2}-2S_{n+1}+S_n(n\in N)$, then the minimum value of $V_n$ is

If $A=\left[\begin{array}{cc}a& b\\ b& a\end{array}\right]$ and $A^2=\left[\begin{array}{cc}\alpha & \beta \\ \beta & \alpha \end{array}\right]$, then

If $1+{\log }_5(x^2+1)\ge {\log }_5({ax}^2+4x+a)$ for all $x\in R,$ then $a$ can be equal to

The sum of the series upto infinity
${\tan }^{-1}(\frac{1}{{2.1}^2})+{\tan }^{-1}(\frac{1}{{2.2}^2})+{\tan }^{-1}(\frac{1}{{2.3}^2})+...\infty$ is

An n-digit number is a positive number with exactly n digits. Nine hundered distinct n-digit numbers are to be formed using only three digits 2, 5 and 7.The smallest value of n for which this is possible is ___

If the complex numbers $z_1,z_2,z_3$ represents vertices of a triangle ABC satisfy the equation $z^3$ = 1. If there is a complex number $z_0$ satisfying $\frac{1}{z_0-z_1}+\frac{1}{z_0-z_2}+\frac{1}{z_0-z_3}$ = 0 then $z_0$ is

Let
$\begin{array}{cc}\left|\begin{array}{ccc}x& 2& x\\ x^2& x& 6\\ x& x& 6\end{array}\right|& =A{x}^4+B{x}^3+C{x}^2+Dx+E\end{array}$

Write the distance between the lines $4x+3y-11=0$ and $8x+6y-15=0$.

${lim}_{x\to 10}\frac{\sqrt{a+x}-\sqrt{a}}{x\sqrt{a^2+ax}}$

Let R be a relation on N defined by x + 2y = 8. The domain of R is

The fourth term of a G.P. is 27 and the 7th term is 792, find the G.P.

$C_{1}and{C}_2$ are circles of unit radius with centres at (0,0) and (1,0) respectively. $C_3$ is a circle of unit radius passes through the centres of the circles $C_{1}and{C}_2$ and have its centre above x-axis. Equation of the common tangent to $C_{1}and{C}_3$ which does not pass through $C_2$ is

Complete solution set of $|x^2-5x+7|+|x^2-5x-14|=21$ is -

___________ ÷ 137 = -2.

Numerically greatest term in the expansion of $(2+3x{)}^9,$ where $x=\frac{3}{2}$ is

State the second principle of mathematical induction.

Equation of the ellipse with vertices ($\pm 5,0$) foci ($\pm 4,0$) is

A point P moves so that its distance from the point (a, 0) is always equal to its distance from the line x + a = 0. The

locus of the point is

If S denotes the sum of an infinite G.P. $S_1$ denotes the sum of the squares of its terms, then prove that the first term and common ratio are respectively $\frac{2S{S}_1}{S^2+S_1}$ and $\frac{S^2-S_1}{S^2+S_1}$.

A graph may be defined as a set of points connected by lines called edges. Every edge connects a pair of points. Thus, a triangle is a graph with 3 edges and 3 points. The degree of a point is the number of edges connected to it. For example, a triangle is a graph with three points of degree 2 each. Consider a graph with 12 points. It is possible to reach any point from any point through a sequence of edges. The number of edges, e, in the graph must satisfy the condition

Find $\int \sec \left(x\right)\tan \left(x\right)dx$

In the expansion of $(1+x+x^3+x^4{)}^{10},$ the coefficient of $x^4$ is

$\frac{d}{dx}\left(\frac{1}{x\sqrt{x}}\right)=$

The standard deviation of 4 consecutive numbers which are in A.P is $\sqrt{5}$. The common difference (d) of this A.P is

and atleast one of the coordinate axes as axis of symmetry of the square is

Two points A and B have coordinates (1, 0) and (-1, 0) respectively and Q is a point which satisfies the relation

AQ - BQ = $\pm$ 1. The locus of Q is

There are four machines and it is known that exactly two of them are faulty. They are tested, one by one, is a random order till both the faulty machines are identified. Then the probability that only two tests are needed is

[IIT 1998]

The co-ordinates of the foot of perpendicular drawn from point P(1,0,3) to the line joining the points A(4,7,1) and B(3,5,3) is

A class consists of 80 students, 25 of them are girls and 55 are boys. If 10 of them are rich and the remaining are poor and also 20 of them are intelligent, then the probability of selecting an intelligent rich girl is

Let ${\alpha }_1,{\alpha }_2$ and ${\beta }_1,{\beta }_2$ be the roots of $a{x}^2+bx+c=0$ and $p{x}^2+qx+r=0$ respectively. If the system of equations ${\alpha }_{1}y+{\alpha }_{2}z=0$ and ${\beta }_{1}y+{\beta }_{2}z=0$ has a non-trivial solution, then which of the following options is CORRECT ?

Prove that following identities:

$cotA+cot({60}^{\circ }+A)+cot({120}^{\circ }+A)=3cot3A$

${lim}_{x\to \frac{\pi }{6}}\frac{co{t}^{2}x-3}{cosecx-2}$

If $\sqrt{(1-x^{2n})}+\sqrt{(1-y^{2n})}=a(x^n-y^n)$, then $\sqrt{\left(\frac{1-x^{2n}}{1-y^{2n}}\right)}\frac{dy}{dx}$ is equal to

The equation of the directrix of the parabola whose vertex and focus are (1,4) and (2,6)

If n is a positive integer, prove $3^{3n}-26n-1$ is divisible by 676.

Peter's father is Audrey's ______.

$\int e^x\left(\frac{1+sinx}{1+cosx}\right)$ dx is......................