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Find the equation for the ellipse that satisfies the given conditions,
$\text{Vertices}(\pm 6,0),\text{foci}(\pm 4,0)$

Let $A$ be a $3\times 3$ square matrix. If $B=adj\left(A\right),$ $C=adj\left(adj\right(A\left)\right)$ and $D=adj\left(adj\right(adj\left(A\right)\left)\right),$ then $|adj\left(adj\right(adj\left(adj\right(ABCD\left)\right)\left)\right)|,$ in terms of $|A|$ is

The maximum possible domain $D_f$ and the corresponding range $R_f$ of $f\left(x\right)=(-1{)}^x$ are

If $\prod _{p=1}^r{e}^{ip\theta }=1$ where $\prod$ denotes the continued product, then the most general value of $\theta$ is

Assuming vector X & Y to be of equal magnitude, which of the following approximately represents $\overrightarrow{Y}-\overrightarrow{X}$

The centre of sphere passes through four points (0, 0, 0), (0, 2, 0), (1, 0, 0) and (0, 0, 4) is

1)What is the next number in the series 4743,4137,3129,2319, ?

2)What is the next number in the series 3,7,13,19,29, ?

For any two statements $p$ and $q,$ the expression $\sim (p\vee q)\vee (\sim p\wedge q)$ is logically equivalent to

${lim}_{x\to 0}\frac{e^{bx}-e^{ax}}{x}$ where 0 < a < b

Prove that

(i) $P\left(A\right)=P(A\cap B)+P(A\cap \overline{B})$

(ii) $P(A\cup B)=P(A\cap B)+P(A\cap \overline{B})+P(\overline{A}\cap B)$

Determine the domain and range of the following relations :

(i) $R=\left\{\right(a,b):aϵN,a<5,b=4\}$

(ii) S = \{(a, b) : b = |a - 1|, a \epsilon z\) and $|a|\le 3\}$

Both the roots of x² - 63x + k = 0 are prime numbers. Then the sum of the digits of k is

Find the points at which the function f given by $f\left(x\right)=(x-2{)}^4(x+1{)}^3$ has
(a) Local maxima
(b) Local minima
(c) Point of inflection

If the difference between two complementary angles is ${52}^{\circ }$, then the smaller angle is

The value of $\underset{0}{\overset{2\pi }{\int }}\frac{x{\sin }^{8}x}{{\sin }^{8}x+{\cos }^{8}x}dx$ is equal to :

Let $z_1$ and $z_2$ be any two non-zero complex numbers such that $3|z_1|=4|z_2|$. If $z=\frac{3z_1}{2z_2}+\frac{2z_2}{3z_1}$ then :

let * be a binary operation on z defined by a*b=a+b-4, for all a,b&z, find the identity element

$\frac{d}{dx}\left\{\frac{2x-3}{3x+1}\right\}=$

sin(60° + A) cos(30° - B) + cos(60° + A) sin(30° - B) is equal to

If $P\left(\frac{A}{B}\right)>P\left(A\right)$, then which of the following is correct?

(a)$P\left(\frac{B}{A}\right)>P\left(B\right)\left(b\right)P(A\cap B)<P\left(A\right)P\left(B\right)\left(c\right)P\left(\frac{B}{A}\right)>P\left(B\right)\left(d\right)P\left(\frac{B}{A}\right)=P\left(B\right)$

A team of 8 couples attend a lucky draw in which 4 persons are picked for a prize. Then the probability that there is at least one couple is

A. 11/39 B. 12/39 C.14/39 D. 15/39

In the expansion of $(1+x{)}^2(1+y{)}^3(1+z{)}^4(1+w{)}^5,$ the sum of coefficients of the term of degree $12$ is $k.$ Then the value of $\frac{k}{13}$ is

If the tangent at (1, 1) on \(y^2 = x(2 - x)^2\) meets the curve again at P, then P is

$sin(si{n}^{-1}\left(\frac{1}{2}\right)+co{s}^{-1}\left(\frac{1}{2}\right))=$

The area of the region enclosed by the curves y = x log x and y = 2x-2$x^2$ is

The solution of the differential equation is

Let $PQR$ be a right angled isosceles triangle right angled at $P(2,1)$. If the equation of the line $QR$ is $2x+y=3$, then the equation representing the pair of lines $PQ$ and $PR$ is

If $a_1,a_2,a_3$ .... a_{n} are in A. P., then the common difference is--

Physically transpose of a matrix can be thought of as the mirror image of the original matrix. Similarly what is the physical meaning of the inverse of a matrix?

If f(x) = $$\{\begin{array}{cc}x,& \text{when 0≤ x ≥ 1}\\ 2-x,& \text{2-x when 1 ≤ x ≥ 2}\end{array}$$
then $li{m}_{x\to 1}$ f(x) =

If $f\left(x\right)=xsin\left(\frac{1}{x}\right),x\ne 0,$ then ${lim}_{x\to 0}f\left(x\right)=$

Three numbers are chosen at random from numbers 1 to 30. Write the probability that the chosen numbers are consecutive.

The solution of the differential equation $\frac{dy}{dx}=1+x+y+xy$ is
[AISSE 1985; AI CBSE 1990; MP PET 2003]

$\int \frac{cos\sqrt{x}}{\sqrt{x}}dx=$

10 different books and 2 different pens are given to 3 boys so that each gets equal number of things. The probability that the same boy does not receive both the pens is

Show that the origin is equidistant from the lines 4x + 3y + 10 = 0; 5x - 12y + 26 = 0 and 7x + 24y = 50.

2²⁸ - 1 is exactly divisible by two numbers between 120 and 130. The sum of these two numbers is

Find a particular solution of the differential equation $(x+1)\frac{dy}{dx}=2e^{-y}-1$, given that y = 0 when x = 0

In any $\Delta$ABC, $4(\frac{s}{a}-1)(\frac{s}{b}-1)(\frac{s}{c}-1)$ is equal to

If $lo{g}_6^{30}=a,lo{g}_{15}^{24}=bthenlo{g}_{12}^{60}=$

Examine the following functions for continuity :

$f\left(x\right)=\frac{1}{x-5},x\ne 5$

(sin^{​​​​4}​​​A - cos^{​​​​4}​​​A) (sin⁴A+cos⁴A) = (sin²A-cos²A)(sin²A+cos²A)

How?

The area bounded by the curves f(x) = x² + 1 and g(x) = x - 1 on the interval [1,3] is:

The equation $\sin x(\sin x+\cos x)=k$ has real solutions, where $k$ is a real number. Then

Given that p $\ge$ 1. What is the minimum value of $p^2+\frac{625}{p^2}$?