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If a and b denote respectively the coefficients of $x^m$ and $x^n$ in the expansion of $(1+x{)}^{m+n}$, then write the relation between a and b.

Find the equation of the hyperbola whose

(i) focus is at (5,2),vertex at (4,2) and centre at(3,2)

(ii) focus is at (4,2), centre at (6,2) and e=2.

The distance between the origin and the tangent to the curve $y=e^{2x}+x^2$ drawn at the point $x=0$ is

If A = {1, 2, 3}, B = {4, 5, 6} which of the following are relations from A to B ?

Give reasons in support of your answer.

(i) {(1, 6), (3, 4), (5, 2)}

(ii) {(1, 5), (2, 6), (3, 4), (3, 6)}

(iii) {(4, 2), (4, 3), (5, 1)}

(iv) $A\times B$.

If $z=e^{\frac{i\pi }{13}}$ then $\frac{1}{1-z}$ is equal to $(i=\sqrt{-1})$

If the equation of hyperpola is $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. Then which of following statements are correct

Solution of the equation $cosy=cosx$

If set A has 4 elements and B = {5, 6}, then the number of elements in A x B are

If $\alpha co{s}^{2}3\theta +\beta co{s}^4\theta =16co{s}^6\theta +9co{s}^2\theta$ is an identity then

Find the sum of the possible values of $\theta ϵ(0,\pi )$ such that $\sin \left(\theta \right)+\sin \left(4\theta \right)+\sin \left(7\theta \right)=0$ are

If the mean of 1, 2, 3,..........,n is $\frac{6n}{11}$, then n is

Tangents are drawn from the origin to the curve y = sin x. Their points of contact lie on the curve

The mean of 100 observations is 50 and their standard deviation is 5. The sum of all squares of all the observations is

If the slope of the curve $y=\frac{ax}{b-x}$ at the point (1, 1) is 2 then values of a and b respectively

If $cos(si{n}^{-1}\frac{2}{5}+co{s}^{-1}x)=0,$ then x is equal to

(a) $\frac{1}{5}$ (b) $\frac{2}{5}$ (c) 0 (d) 1

For the following question verify that the given function (explicit or implicit) is a solution of the corresponding differential equation.

y=cosy=x and (ysiny+cosy+x)y'=y.

Let $f:R\to R$ be defined by $f\left(x\right)=\frac{x}{1+x^2},x\in R.$ Then the range of $f$ is

What is the sign of the sinA and tanA in third quadrant respectively

${lim}_{x\to 3}(\frac{1}{x-3}-\frac{2}{x^2-4x+3})$

The derivative of the function $f\left(x\right)=x^x$ is equal to

The value of ${\sum }_{n=0}^{1947}\frac{1}{2^n+\sqrt{2^{1947}}}$ is equal to

If the length of the major axis of a vertical ellipse is three times length of the minor axis, then its eccentricity is equal to

The value(s) of $\lambda$ for which the line $y=x+\lambda$ touches the ellipse

$9x^2+16y^2=144$ is/are:

A line passing through the points (a, 2a) and (-2, 3) is perpendicular to the line $4x+3y+5=0$, find the value of a.

$\frac{a-b}{a+b}=\frac{tan\left(\frac{A-B}{2}\right)}{tan\left(\frac{A+B}{2}\right)}$

Let $a$ and $b$ be real numbers such that $\sin a+\sin b=\frac{1}{\sqrt{2}}$ and $\cos a+\cos b=\frac{\sqrt{6}}{2}$ then the value of $\sin (a+b)$ is :

The slope(s) of the common tangent(s) to the two hyperbolas $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$ is/are

Write the domain and range of function $f\left(x\right)=\frac{1}{\sqrt{x-|x|}}$.

Let f, g be two real functions defined by $f\left(x\right)=\sqrt{x+1}$ and $g\left(x\right)=\sqrt{9-x^2}$. Then, describe each of the following functions:

(i) $f+g$
(ii) $g-f$
(iii) $fg$
(iv) $\frac{f}{g}$
(v) $\frac{g}{f}$
(vi) $2f-\sqrt{5}g$
(vii) $f^2+7f$
(viii) $\frac{5}{8}$

If $\sec x+\tan x=p,$ then the value of $\frac{\sec x-\tan x}{2\sec x}$ is

If the position vector of a point P is $\overrightarrow{r}=x\hat{i}=y\hat{j}+z\hat{k},$ where x, y, z$ϵ$ N and projection of $\overrightarrow{r}$ on $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}is\frac{10}{\sqrt{3}}$ then number of possible of P is also equal to

Find the area of the region lying in the first quadrant and bounded by $y=4x^2,x=0,y=1$ and y = 4.

A what angle do the two forces (P+q)and (p-q) actso that the resultant is √3p²+√q_²

If $z=(-\sqrt{3}+\sqrt{-2})(2\sqrt{3}-i)$, then

The straight lines $3x+4y=5$ and $4x-3y=15$ intersect at the point $A$. On these lines points $B$ and $C$ are chosen so that $AB=AC$.Possible equation(s) of the line $BC$ passing through $(1,2)$ is/are

let $I_1=\underset{0}{\overset{n}{\int }}\left[x\right]dx$ and $I_2=\underset{0}{\overset{n}{\int }}\left\{x\right\}dx,$ where $\left[x\right]$ and $\left\{x\right\}$ are integral and fractional parts of $x$ and $n\in N-\left\{1\right\}$. Then $\frac{I_1}{I_2}$ is equal to

If $|z_1+z_2|=|z_1|+|z_2|$ where $z_1$ and $z_2$ are different non-zero complex numbers, then :

If $a,b,c,d$ are distinct positive numbers which are in A.P. with positive common difference, then which of the following is/are correct?

Which of the following is roster form of $\{x:x$ is even prime number greater than $2\}$?

The coefficient of $x^4$ in the expansion of the ${(1+x+x^2+x^3)}^{10}$

Statements: G $ H, J # K, H * K

Conclusions:

I. H $ J

II. J * H