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Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse 4x² + 9y² = 36

The sum of the series ${\tan }^{-1}\frac{1}{3}+{\tan }^{-1}\frac{2}{9}+{\tan }^{-1}\frac{4}{33}+\cdots \text{upto}\infty$ terms is

f(x) = [x], the greatest integer less then or equal to x and k is an integer. Then, ${lim}_{x\to k}f\left(x\right)=$___

If $\{\begin{array}{cc}xsin\left(\frac{1}{x}\right),& x\ne 0\\ 0& x=0\end{array},then{lim}_{x\to 0^{-}}f\left(x\right)$ equal

For the matrix $P=\left[\begin{array}{ccc}3& -2& 2\\ 0& -2& 1\\ 0& 0& 1\end{array}\right]$ one of the eige nvalues is equal to $-2$. Which of the followign is an eigen vector?

Let $X=\{x\in N:1\le x\le 17\}$ and $Y=\{ax+b:x\in X\text{and}a,b\in R,a>0\}$. If mean and variance of elements of $Y$ are $17$ and $216$ respectively then $a+b$ is equal to:

The domain of the function $f\left(x\right)=2^{{\log }_{2}x}+\frac{2x+3}{x^2-4x+3}$ is

If for some $x\in R$, the frequrency distribution of the marks obtained by $20$ students in a test is :

Arrange the given words in the sequence in which they occur in the dictionary and then choose the correct sequence from the options (1) cloth (2) Cinema (3) Chronic (4) Christmas (5) Create

1. 1

Solve the following system of inequalities graphically: 3x + 4y ≤ 60, x + 3y ≤ 30, x ≥ 0, y ≥ 0

The equation of the image of the circle $x^2+y^2+16x-24y+183=0$ by the mirror $4x+7y+13=0$ is

Let $x_1{x}_2{x}_3{x}_4{x}_5{x}_6$ be a six digit number.The number of such numbers if
$x_1<x_2<x_3<x_4<x_5<x_6$ is

The magnitude of component of a vector must be

1- less than the magnitude of vector always

2-equal to magnitude of vector always

3- always greater than magnitude of vector

4-None of the above

If $x=111\dots \dots 1\left(20\text{digits}\right),$ $y=333\dots \dots 3\left(10\text{digits}\right)$ and $z=222\dots \dots 2\left(10\text{digits}\right)$, then $x-z$ is equal to

If $|z_1-1|<2,|z_2-2|<1,$ then maximum possible integral value of $|z_1+z_2|$ is

The solution set of $2lo{g}_{2}lo{g}_{2}x+lo{g}_{1/2}lo{g}_2\left(2\sqrt{2x}\right)=1$ is

If $\underset{x\to 2}{lim}\frac{\sum _{r=1}^n{x}^r-\sum _{r=1}^n{2}^r}{x-2}=f\left(n\right),$ then which of the following is/are CORRECT?

In how many ways can a committee of 5 be made out of 6 men and 4 women containing at least one women ?

If $3cotA=4\text{. Prove that}\frac{1-ta{n}^{2}A}{1+ta{n}^{2}A}=co{s}^{2}A-si{n}^{2}A$.

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

If the normal at P on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ cuts the major and minor axes in Q and R respectively then PQ:PR =

For an isosceles triangle $ABC$ inscribed in a circle of given radius $r$ units, if $h$ is the length of altitude from vertex $A$ to $BC,$ then the value of $\underset{h\to 0}{lim}\frac{\Delta }{P^3}$ is ( where $\Delta ,P$ represent area and perimeter of triangle $ABC$ respectively)

If the term independent of $x$ in the expansion of ${(\sqrt{x}-\frac{m}{x^2})}^{10}$ is $405$, then

Prove the following trigonometric identities.
(i) $\frac{1+\text{cos}\theta +\text{sin}\theta }{1+\text{cos}\theta -\text{sin}\theta }=\frac{1+\text{sin}\theta }{\text{cos}\theta }$

(ii) $\frac{\text{sin}\theta -\text{cos}\theta +1}{\text{sin}\theta +\text{cos}\theta -1}=\frac{1}{\text{sec}\theta -\text{tan}\theta }$

(iii) $\frac{\text{cos}\theta -\text{sin}\theta +1}{\text{cos}\theta +\text{sin}\theta -1}=\text{cosec}\theta +\text{cot}\theta$

(iv) $(\text{sin}\theta +\text{cos}\theta )(\text{tan}\theta +\text{cot}\theta )=\text{sec}\theta +\text{cosec}\theta$

The centre of the conic represented by the equation $x^2-6xy+y^2+6x+14y-2=0$ is

Prove that if a plane
has the intercepts a, b, c and is at a distance
of P units from the origin, then