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Line $\frac{x}{a}+\frac{y}{b}=1$ cuts the coordinate axes at A (a, 0) and B(0, b) and the line $\frac{x}{a^{\prime }}+\frac{y}{b^{\prime }}=-1$ at A' (-a', 0) and B' (0, -b'). If the point A, B, A', B' are concyclic then the orthocenter of the triangle ABA' is

Integrate the following functions.
$\int \frac{cos\sqrt{x}}{\sqrt{x}}dx.$

$Iff\left(x\right)=co{s}^{2}x+se{c}^{2}x,then$

The plane x + 2y - z = 4 cuts the sphere $x^2+y^2+z^2-x+z-2=0$ in a circle of radius
[AIEEE 2005]

Find the value of (1001)^⅓ .

The value of integral $\frac{\underset{0}{\overset{10}{\int }}\left[x\right]dx}{\underset{0}{\overset{10}{\int }}\left\{x\right\}dx}$

( [.] and {.} represents the greatest integer function and the fractional part funtion respectively ) is

A man takes a step forward with probability 0.4 and one step backward with probability 0.6. then the probability that at the end of eleven steps he is one step away from the starting point is

The term independent of $x(x>0,x\ne 1)$ in the expansion of ${[\frac{(x+1)}{(x^{2/3}-x^{1/3}+1)}-\frac{(x-1)}{(x-\sqrt{x})}]}^{10}is$

Find the coordinates of the centre and radius of each of the following circles :

(i) $x^2+y^2+6x-8y-24=0$
(ii) $2x^2+2y^2-3x+5y=7$
(iii) $\frac{1}{2}(x^2+y^2)+xcos\theta +ysin\theta -4=0.$
(iv) $x^2+y^2-ax-by=0$

If the tangent to the curve $2y^3=a{x}^2+x^3$ at the point $(a,a)$ cuts off intercepts $\alpha$ and $\beta$ on the co-ordinate axes, where ${\alpha }^2+{\beta }^2=61$, then the possible value(s) of $a$ is/are

Let $A=\{1,2,3,....,10\}$ and $f:A\to A$ be defined as $f\left(k\right)=\{\begin{array}{cc}k+1& \text{if}k\text{is odd}\\ k& \text{if}k\text{is even}\end{array}.$ Then the number of possible functions $g:A\to A$ such that $gof=f$ is:

1. 1

If U = {x: x is a natural number and x ≤ 10},

A = {a: a is an even natural number and a < 10}, and B = {b: b is a prime number and b ≤ 9}, then find A' ∩ B'.

The value of $\sum _{r=0}^6({}^6{C}_r\cdot {}^6{C}_{6-r})$ is equal to:

A cricket club has $15$ members, of whom only $5$ can bowl. If the names of these $15$ members are put into a box and $11$ are drawn at random, then the probability of getting an eleven containing at least $3$ bowlers is

Let $z_1$ and $z_2$ be two complex numbers such that $\overline{z_1}+i\overline{z_2}=0,\arg \left(z_1{z}_2\right)=\pi .\text{Then}$ which of the following is/are ture

Write the coordinates of the centre of the circle inscribed in the square formed by the lines x = 2, x = 6, y 5 and y = 9.

Find the equation of the circle which passes through the points (3, 7), (5, 5) and has its centre on the line x - 4y = 1.

Complete set of values of '$m$' for which function $f\left(x\right)=e^{\cos x}+2m\cos x+1$ is increasing $\forall xϵ(0,\frac{\pi }{2})$ is

Suppose a machine produces metal parts that contain some defective parts with probability $0.05$. How many parts should be produced in order that the probability of at least one part being defective is $\frac{1}{2}$ or more?
(Given ${\log }_{10}95=1.977$ and ${\log }_{10}2=0.3)$

Which among the following statements is not a contradiction

The value of $\underset{\ln 3}{\overset{\ln 27}{\int }}\frac{x^{1/4}}{x^{1/4}+(4\ln 3-x{)}^{1/4}}dx$ is

The value of ${\sin }^{-1}\left(\sin 5\right)-{\cos }^{-1}\left(\cos 5\right)$ is

The area bounded by the tangent on the curve

$y=4x^2+2x\text{at (0, 0)},y-10=-x\text{and}y=0$ is