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Find the equation for the ellipse that satisfies the given conditions,
Vertices $(\pm 5,0),Foci(\pm 4,0)$

The general solution of the differential equation

$\frac{d^{4}y}{d{x}^4}-2\frac{d^{3}y}{d{x}^3}+2\frac{d^{2}y}{d{x}^2}-2\frac{dy}{dx}+y=0$

Let $z_k=\cos \left(\frac{2k\pi }{10}\right)+i\sin \left(\frac{2k\pi }{10}\right);k=1,2,\cdots ,9.$



$\begin{array}{cccc}& List\left(I\right)& & List\left(II\right)\\ \text{P}.& \text{For each}{z}_k\text{there exist a}{z}_j& \left(1\right)& \text{True}\\ & \text{such that}{z}_k\cdot z_j=1& & \\ \text{Q}.& \text{There exists a}k\in \{1,2,\cdots ,9\}& & \\ & \text{such that}{z}_1\cdot z=z_k\text{has no solution}& \left(2\right)& \text{False}\\ & z\text{in the set of complex numbers.}& & \\ \text{R}.& \frac{|1-z_1||1-z_2|\cdots |1-z_9|}{10}\text{equals}& \left(3\right)& 1\\ \text{S}.& 1-\sum _{k=1}^9\cos \left(\frac{2k\pi }{10}\right)\text{equals}& \left(4\right)& 2\end{array}$



Which of the following option is correct?

If x,y, z are in A.P. and $A_1$ is the A.M. of x and y and $A_2$ is the A.M. of y and z, then prove that the A.M. of $A_1$ and $A_2$ is y.

Let the tangent to the parabola $S:y^2=2x$ at the point $P(2,2)$ meet the $x-$axis at $Q$ and normal at it meet the parabola $S$ at the point $R.$ Then the area (in sq. units) of the triangle $PQR$ is equal to

If $A_i$ is the area bounded by $|x-a_i|+|y|=b_i,$ where $a_{i+1}=a_i+\frac{3}{2}{b}_i$ and $b_{i+1}=\frac{b_i}{2};$ $a_1=0,b_1=32,$ then

${\int }_0^{\pi /2}\log \left(\sin x\right)dx=$

Let $l_1={\int }_0^1(2007{)}^{x^2}dx,l_2={\int }_0^1(2007{)}^{x^3}dx,l_3={\int }_1^2(2007{)}^{x^x}dx$ and $l_4={\int }_0^1(2007{)}^{x^4}dx$. Then

In the given frequency distribution where the marks are ordered, which 2 observations (students) are the median observations?

$$\begin{array}{cc}Marksobtained& Numberofstudents\left(Frequency\right)\\ 20& 6\\ 25& 20\\ 28& 24\\ 29& 28\\ 33& 15\\ 38& 4\\ 42& 2\\ 43& 1\end{array}$$

The vector equation of the plane through the point $\hat{i}+2\hat{j}-\hat{k}$ and perpendicular to the line of intersection of the plane $\overline{r}.(3\hat{i}-9\hat{j}+\hat{k})=1$ and $\overline{r}.(\hat{i}+4\hat{j}-2\hat{k})=2$
is:

$P$ is a point on the line $y+2x=1$ and $Q$, and $R$ are two points on the line $3y+6x=6$ such that triangle $PQR$ is an equilateral triangle. The length of the side of the triangle is

$f\left(x\right)=\{\begin{array}{cc}1+x,& ifx\le 2\\ 5-x,& ifx>2\end{array}$

If $|z-1-i|=1$, then the locus of a point represented by the complex number $5(z-i)-6$ is

Using elementary transformations, find the inverse of the followng matrix.

$\left[\begin{array}{cc}3& 1\\ 5& 2\end{array}\right]$

If a,b are roots of equation

x²+px-q=0 and

c,d are roots of equation

x²+px+r=0

Prove that

(a-c)(a-d)=(b-c)(b-d)=q+r

The general solution(s) of $4\sin \theta \sin 2\theta \sin 4\theta =\sin 3\theta$ can be

Find the sum of integers divisible from 1 to 100 that are divisible by both 3 and 4 is

The values of $m$ such that exactly one root of $x^2+2(m-3)x+9=0$ lies between $1$ and $3$, is

If the line $ax+by=2$ is a normal to the circle $x^2+y^2-4x-4y=0$ and a tangent to the circle $x^2+y^2=1,$ then

In a triangle ABC, a : b : c = 4 : 5 : 6. The ratio of the radius of the circumcircle to that of the incircle is

If A and B are subsets of X such that $A\subseteq B.,then(X-B)\subseteq (X-A).$

The equation of an ellipse, centred at origin and passing through the points $(4,3)$ and $(-1,4),$ is

The lats rectum of a parabola whose directrix is x + y - 2 = 0 and focus is (3,-4), is

If $\alpha$ is one of the real roots of $a{x}^2+x+b=0$ where $ab>0$, then the value(s) of ${\tan }^{-1}\alpha +{\tan }^{-1}\frac{1}{\alpha }$ is/are

Tangents are drawn from the points on the parabola $y^2=-8(x+4)$ to the parabola $y^2=4x.$ Then the locus of mid-point of chord of contact of $y^2=4x$ is

If $C_1$ and $C_2$ are circles whose equations are $x^2+y^2-20x+64=0$ and $x^2+y^2+30x+144=0$, then the length of the shortest line segment $PQ$ that is tangent to $C_1$ at $P$ and to $C_2$ at $Q$ is

One hundred identical coins, each with probability p of showing up heads are tossed once. If 0 < p < 1 and the probability of heads showing on 50 coins is equal to that of heads showing on 51 coins, then the value of p is

Find the equation of
the line which passes through the point (1, 2, 3) and is parallel to
the vector.