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Normal at $(5,3)$ of rectangular hyperbola $xy-y-2x-2=0$ intersects it again at a point

Name the octants in which the following points lie (1, 2, 3), (4, -2, 3), (4, -2, -5), (-4, 2, -5), (-4, 2, 5), (-4, 2, 5), (-3, -1, 6), (2, -4, -7)

The sum of the squares of perpendicuars on any tangents of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(a>b)$ from two points on minor axis each one at a distance of $\sqrt{a^2-b^2}$ unit from the centre is

If the pair of straight lines given by $A{x}^2+2Hxy+B{y}^2=0,(H^2>AB)$ forms an equilateral triangle with line ax + by + c = 0, then (A + 3B)(3A + B) is

The coordinates of a point on the hyperbola, $\frac{x^2}{24}-\frac{y^2}{18}=1,$ which is nearest to the line $3x+2y+1=0$ are

A parabola passing through the point (-4,-2) has its vertex at the origin and y-axis as its axis. The latus rectum of

the parabola is

The inequality $\left|x^2\sin x+{\cos }^{2}x{e}^x+{\ln }^{2}x\right|<x^2\left|\sin x\right|+{\cos }^{2}x{e}^x+{\ln }^{2}x$ is true for $x\in$

The solution set of $\frac{1}{x^2-9}\ge 0$ is

Find the derivative of $x^2+x\left(sinx\right)$ ?

Number of ways of dividing $80$ cards into $5$ equal groups of $16$ each is :

If the minimum value of $f\left(x\right)=a{x}^2+2x+5,a>0$ is equal to the maximum value of $g\left(x\right)=3+2x-x^2$, then the value of ${}^{\prime }{a}^{\prime }$ is

A particle is oscillating according to the equation $X=7cos0.5\pi t$, where t is in second. The point moves from the position of equilibrium to maximum displacement in time

The set of all real values of $a$ for which the function $f\left(x\right)=(a+2)x^3-3a{x}^2+9ax-1$ decreases monotonically throughout for all real $x,$ is

Determine the number of 4 card combinations out of a deck of 52 cards if there is no ace in each combination.

The following table expresses the age of eight students. Find the median age.

Orthocentre of the triangle formed by the lines x + y = 1 and xy = 0 is

If $e_1$ and $e_2$ are the eccentricities of the ellipse , $\frac{x^2}{18}+\frac{y^2}{4}=1$ and the hyperbola, $\frac{x^2}{9}-\frac{y^2}{4}=1$ respectively and $(e_1,e_2)$ is a point on th ellipse , $15x^2+3y^2=k$. Then $k$ is equal to :

There are $15$ two bed room flats in a building and $10$ two bed room flats in second building and $8$ two bed room flats in third building. The number of choices a customer will have for buying a flat is

If $|z-z_1|+|z+z_2|$ = $\lambda ,\lambda \in R^{+},z_1,z_2$ are fixed complex numbers, represents an ellipse, then:

If centroid of the tetrahedron OABC, where coordinates of $A,B,C$ are $(a,2,3),(1,b,3)$ and $(2,1,c)$ respectively be $(1,2,3),$ then find the distance of a point $(a,b,c)$ from the origin, where O is the origin.

The complete set of values of $k$, for which the quadratic equation $x^2-kx+k+2=0$ has equal roots, consists of

If $asi{n}^{2}x+bco{s}^{2}x=c,$ $bsi{n}^{2}y+aco{s}^{2}y=d,$ and $atanx=btany,$ then $\frac{a^2}{b^2}$ is equal to

If $sin\theta$ and $cos\theta$ are the roots of $a{x}^2+bx+c=0,$ then $co{s}^{-1}(a^2-b^2+2ac)=$

The integral value(s) of $x$ which satisfies the inequality $4x+5\le 2x+17$ is/are

Which one of the following well formed formulae is a tautology?

How many of below given statements are correct?

1. sin2A = 2sinA.cosA

2.cos2A = $si{n}^{2}A-co{s}^{2}A$

3.tan A = $\frac{2tan\frac{A}{2}}{1-ta{n}^2\frac{A}{2}}$

4. $si{n}^2\frac{A}{2}$ = 1 - cosA

5. $ta{n}^{2}A$ = $\frac{1-cos2A}{1+cos2A}$

6. $si{n}^{2}A$ = $\frac{2tanA}{1-ta{n}^{2}A}$

Let f(x) = x - |x| then f(x) is,

The value of the definite integral $\underset{0}{\overset{\pi /2}{\int }}\sqrt{\tan x}dx$ is

The maximum value of ${\left(\frac{1}{x}\right)}^x$ is

Let M and N be two 3 $\times$ 3 skew - symmetric matrices such that MN = NM. If $P^T$ denotes the transpose of P, then $M^2{N}^2\left(M^{T}N{)}^{-1}\right(M{N}^{-1}{)}^T$ is equal to

Find the equation of the hyperbola satisfying the given conditions.
Foci $(\pm 3\sqrt{5},0)$ the latus rectum is of length 8.

The sum of n terms of three A.P.'s whose first term is 1 and

common differences are 1, 2, 3 are $S_1$, $S_2$, $S_3$ respectively. The true

relation is

The equation of the line passing through the points $(0,-3)$ and $(4,3)$ is

If f(xy) = f(x)+f(y), and f(e) = 1, then find the value of $f\left(e^2\right)$

If $H_1,H_2,\dots \dots ,H_{20}$ be 20 harmonic means between $2$ and $3$, then the value of $\frac{H_1+2}{H_1-2}+\frac{H_{20}+3}{H_{20}-3}$ is

Solve the following inequalities graphically in two dimensional plane: $3y-5x<30$

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

Prove ${10}^{2n-1}+1\text{is divisible by 11.}$

For $2\le r\le n,\left(\begin{array}{c}n\\ r\end{array}\right)+2\left(\begin{array}{c}n\\ r-1\end{array}\right)+\left(\begin{array}{c}n\\ r-2\end{array}\right)$ is equal to

If $a^2+4b^2=12ab$, then log(a + 2b) is

The range of values of $a$ such that the angle $\theta$ between the pair of tangents drawn from $(a,0)$ to the circle $x^2+y^2=1$ satisfies $\frac{\pi }{2}<\theta <\pi$, lies in

For any two sets $A\&B;n(A-B)=$ $\text{and}n(B-A)=$

In the following, state whether A = B or not :

(i) A = {a, b, c, d} B = {d, c, b, a}

(ii) A = {4, 8, 12, 16} B = {8, 4, 16, 18}

(iii) A = {2, 4, 6, 8, 10} B = {x : x is a positive even integer and $x\le 10$

(iv) A = {x : x is a multiple of 10} B = {10, 15, 20, 25, 30, ...........}

Evaluate the following limit:
$\underset{x\to 0}{\text{lim}}\frac{\text{sin ax}}{\text{sin bx}},a,b\ne 0$

The number of solutions of the equation $\cos 3x+\cos 2x=\sin \frac{3x}{2}+\sin \frac{x}{2}$ lying in the interval $[0,2\pi ]$ is

The number of permutations by using all the digits of the number $986754$ which niether begins with $8$ nor ends with $5$ is $\lambda$ , then the value of $\frac{\lambda }{2}$ is

Let $S_n={\sum }_{k=1}^n\frac{n}{n^2+kn+k^2}and{T}_n={\sum }_{k=0}^{n-1}\frac{n}{n^2+kn+k^2}$ for a = 1, 2, 3,...... Then,

Find the coordinates of the point which divides the line segment joining the points $(-2,3,5)$ and $(1,-4,6)$ in the ratio. (i) 2 : 3 internally (ii) 2 : 3 externally.