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The line L has intercepts a and b on the coordinate axes. The coordinate axes are rotated through a fixed angle, keeping the origin fixed. If p and q are the intercepts of the line L on the new axes, then $\frac{1}{a^2}-\frac{1}{p^2}+\frac{1}{b^2}-\frac{1}{q^2}$ is equal to

Equation of a line in the plane $\pi :2x-y+z-4=0$ which is perpendicular to the line $l$ whose equation is $\frac{x-2}{1}=\frac{y-2}{-1}=\frac{z-3}{-2}$ and which passes through the point of intersection of $l$ and $\pi$ is

The chord AB of the parabola $y^2=4ax$ cuts the axis of the parabola at C. If $A=(a{t}_1^2,2a{t}_1),B=(a{t}_2^2,2a{t}_2)$ and AC : AB = 1:3 then

Let $f\left(x\right)=5x^3+px+q$, where $p$ and $q$ are real numbers. When $f\left(x\right)$ is divided by $x^2+x+1$, the remainder is $0$. Then the value of $p-q$ is

The angle of elevation of top of tower from a point A due south of the tower is $\alpha$ and from a point B due East of the tower is$\beta$. If AB = d, then the height of tower is

If the circle $x^2+y^2+6x-2y+k=0$ bisects the circumference of the circle $x^2+y^2+2x-6y-15=0$, then k _______

Determine the direction cosines of the normal to plane and the distance from the origin:

2x +3y -z = 5

Let $\overrightarrow{a},\overrightarrow{b}and\overrightarrow{c}$ be three vectors such that $|\overrightarrow{a}|=3,|\overrightarrow{b}|=4,|\overrightarrow{c}|=5$ and mutually perpendicular to each other, then $|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}|=$

If the latus rectum of a hyperbola through one focus subtends ${60}^{\circ }$ angle at the other focus, then its eccentricity e is

The number of positive integral solutions of the equation $ta{n}^{-1}x+co{s}^{-1}\frac{y}{\sqrt{1+y^2}}=si{n}^{-1}\frac{3}{\sqrt{10}}$

Two numbers ‘a’ & ‘b’ are chosen from the set of {1,2,3……3n}. In how many ways can these integers be selected such that $a^2-b^2$ is divisible by 3

$\text{If}{\left(\frac{1+i}{1-i}\right)}^3-{\left(\frac{1-i}{1+i}\right)}^3=x+iy,\text{find (x,y)}$

Prove that following identities:

$sin5A=5co{s}^{4}AsinA-10co{s}^{2}Asi{n}^{3}A+si{n}^{5}A$

A bag contains 3 red, 4 white and 5 blue balls (All balls are different). If two balls are drawn at random, then the probability that they are of different colours is:

If $y=\sqrt{\frac{1-cos2x}{1+cos2x}},find\frac{dy}{dx}$.

The value of $\left|\begin{array}{ccc}cos(\theta +\propto )& -sin(\theta +\propto )& cos2\propto \\ sin\theta & cos\theta & sin\propto \\ -cos\theta & sin\theta & \lambda cos\propto \end{array}\right|$ is

The probability of hitting a target by three men is $\frac{1}{2},\frac{1}{3}$ and $\frac{1}{4}$ respectively. If the probability that exactly two of them will hit the target is $\lambda$ and that atleast two of them hit the target is $\mu$ then $\lambda +\mu$ is -

Which of the following is always true about a function f(x) on the interval [a, b] ?

A sequence $x_1,x_2,x_3,......$ is defined by letting $x_1=2$ and $x_k=\frac{x_{k-1}}{n}$ for all natural numbers k, $k\ge 2$. Show that $x_n=\frac{2}{n!}$ for all $nϵN$.

Using binomial theorem, find the value of $(103{)}^4$

Let $f\left(x\right)=(\begin{array}{c}{x}^2-1,\mathrm{if}0<x<2\\ 2x+3,\mathrm{if}2\le x<3\end{array}$, a quadratic equation whose roots are $\underset{x\to 2^{-}}{\lim }f\left(x\right)\mathrm{and}\underset{x\to 2^{+}}{\lim }f\left(x\right)$ is

The distance between the orthocentre and circumcentre of a triangle whose vertices are $P(3,0),Q(0,0)$ and $R(\frac{3}{2},\frac{-3\sqrt{3}}{2})$ is

The range of $f\left(x\right)=x^3+x$ is

Suppose $A$ is a 3 × 3 matrix consisting of integer entries that are chosen at random from the set $\{–1000,–999,\dots ,999,1000\}$. Let $P$ be the probability that either $A^2=–I$ or $A$ is diagonal matrix, where $I$ is the 3 × 3 identity matrix. Then

A circle of radius $14\text{units}$ touches the coordinate axes in the first quadrant. If the circle makes three complete rolls along positive direction of $Y-\text{axis}$, then the equation of circle in the new position is

Let $\ast$ be a binary operation on the set Q of rational number as follows:
$\left(iv\right)a\ast b=(a-b{)}^2$
Show that none of the operations has an identity.

A vector $\overrightarrow{r}$ is inclined at equal angles to the three axes.If the magnitude or $\overrightarrow{r}is2\sqrt{3}$ units, then find the value of $\overrightarrow{r}$ .

If $\alpha ,\beta ,\gamma$ are the roots of $x^3-x^2-1=0,$ then the value of $\frac{1+\alpha }{1-\alpha }+\frac{1+\beta }{1-\beta }+\frac{1+\gamma }{1-\gamma }=$

If k is a natural number and the roots of the equation x²+11x+6k =0 are rational numbers, then find the smallest value of k .

How to solve this using simplest method?

A. Maria unnecessarily picked up a quarrel with Rani and left the party hurried.
B. Acquisition of certain specific skills can be facilitated by general awareness, education to novel situations.
C. He asked the crowd if they thought he was right and the crowd shouted that they did.
D. Why should the candidates be afraid of English Language is not clear.

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Which of the following graphs represents $f\left(x\right)=|x-2|-|x+6|$?

If $x^m.y^n=(x+y{)}^{m+n},\text{then}\frac{dy}{dx}=$

Solve the following system of equations in R.
$\frac{|x-2|}{x-2}>0$

When should we look for RHL and LHL for a limit given?

The point on the $Y-$axis equidistant from the point $(9,3)$ and $(-5,2),$ is

Find the perpendicular distance from the origin of the perpendicular from the point (1, 2) upon the straight line $x=\sqrt{3}y+4=0$.

Statement $(p\wedge \sim q)\wedge (\sim p\vee q)$ is

If $Z=\left|\begin{array}{ccc}2& 5-i& 7+i\\ 5+i& 2& 3-i\\ 7-i& 3+i& 7\end{array}\right|$ and arg (z) = $\theta$ then $\theta$ =………………………

Evaluate the determinants.

$\left|\begin{array}{cc}cos\theta & -sin\theta \\ sin\theta & cos\theta \end{array}\right|$

$\left|\begin{array}{cc}{x}^2-x{+}_1& x-1\\ x+1& x+1\end{array}\right|$