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Form the differential equation of the family of ellipse having foci on Y-axis and centre at origin.

The value of $\frac{1-{\tan }^2(\frac{\pi }{4}-A)}{1+{\tan }^2(\frac{\pi }{4}-A)}$ is

where $A\in [0,\frac{\pi }{2}]$

The coordinates of the point on the curve $x^3=y(x-a{)}^2,a>0$ where the ordinate is minimum

For two data sets $X$ and $Y$, each of size $5$, the means are given to be $2$ and $4$ and the corresponding variances are $4$ and $5$, respectively. The variance of the combined data set is

Which of the following function is a monotonically increasing function?

Suppose $a,c$ are the roots of the equation $p{x}^2-3x+2=0$ and $b,d$ are the roots of the equation $q{x}^2-4x+2=0$. Find the value of $p$ and $q$ such that $\frac{1}{a},\frac{1}{b},\frac{1}{c}$ and $\frac{1}{d}$ are in A.P.

Consider two families $A$ and $B.$ Suppose there are $4$ men, $4$ women and $4$ children in family $A$ and $2$ men, $2$ women and $2$ children in family $B.$ The recommended daily amount of calories is $2400$ for a man, $1900$ for a woman, $1800$ for a child and $45$ grams of proteins for a man, $55$ grams for a woman and $33$ grams for a child.

The requirement of calories and proteins for each person is given by matrix $R$ and the number of family members in each family is given by matrix $F.$



Requirement of calories of family $A$ is

If f and g are two functions over real numbers defined as f(x) = 3x + 1, g(x) = x² + 2, then find f-g

If $\underset{0}{\overset{\infty }{\int }}\frac{\sin x}{x}dx=\frac{\pi }{2},$ then $\underset{0}{\overset{\infty }{\int }}\frac{{\sin }^{3}x}{x}dx$ is equal to

If the mappings f and g are given by $f=\left\{\right(1,2),(3,5),(4,1\left)\right\}$ and $g=\left\{\right(2,3),(5,1),(1,3\left)\right\}$, write $fog$.

In the following diagram, the shaded part represents

If the plane $2x-y+2z+3=0$ has the distances $\frac{1}{3}$ and $\frac{2}{3}$ units from the planes $4x-2y+4z+\lambda =0$ and $2x-y+2z+\mu =0,$ respectively, then the maximum value of $\lambda +\mu$ is equal to :

Box $1$ contains three cards bearing numbers $1,2,3;$ box $2$ contains five cards bearing numbers $1,2,3,4,5;$ and box $3$ contains seven cards bearing numbers $1,2,3,4,5,6,7.$ A card is drawn from each of the boxes. Let $x_i$ be the number on the card drawn from the $i^{th}$ box, $i=1,2,3.$



The probability that $x_1,x_2,x_3$ are in an arithmetic progression, is

If the circles $x^2+y^2+2x+2ky+6=0,x^2+y^2+2ky+k=0$ intersect orthogonally, then k is

If $O$ is the circumcentre of the $△ABC$ and $R_1,R_2,R_3$ are the radii of the circumcircles of the triangles $OBC,OCA$ and $OAB$ respectively, then $\frac{a}{R_1}+\frac{b}{R_2}+\frac{c}{R_3}$ is equal to

Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola 9y² – 4x² = 36

Show that the plane $ax+by+cz+d=0$divides the line joining the points

$(x_1,y_1,z_1)and(x^2,y^2,z^2)$ in the ratio

$-\frac{a{x}_1+b{y}_1+c{z}_1+d}{a{x}_2+b{y}_2+c{z}_2+d}$.

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are two vectors such that $|\overrightarrow{a}|=3,|\overrightarrow{b}|=2$ and angle between $\overrightarrow{a}$ and $\overrightarrow{b}\text{is}\frac{\pi }{3},$ then the area of the triangle with adjacent sides $\overrightarrow{a}+2\overrightarrow{b}$ and $2\overrightarrow{a}+\overrightarrow{b}$ in sq. units is

A body cools down from $50{}^{\circ }\text{C}$ to $45{}^{\circ }\text{C}$ in $5\text{minutes}$ and to $40{}^{\circ }\text{C}$ in another $8\text{minutes}$. Find the temperature of the surrounding.

Let the normals at all the points on a given curve pass through a fixed point $(a,b)$. If the curve passes through $(3,-3)$ and $4,-2\sqrt{2}$, and given that $a-2\sqrt{2}b=3$, then $a^2+b^2+ab$ is equal to

If each of the points $(x_1,4),(-2,y_1)$ lies on the line joining the points $(2,-1)$ and $(5,-3)$, then the point $P(x_1,y_1)$ lies on the line

The sum to (n + 1) terms of the series $\frac{c_0}{2}-\frac{c_1}{3}+\frac{c_2}{4}-\frac{c_3}{5}+....is$

If $\theta =ta{n}^{-1}\frac{d}{1+a_1{a}_2}+ta{n}^{-1}\frac{d}{1+a_2{a}_3}+\cdots +ta{n}^{-1}\frac{d}{1+a_{n-1}{a}_n}$, where $a_1,a_2,a_3,\cdots a_n$ are in A.P. with common difference d, then $tan\theta =$

If $I\left(m\right)={\int }_0^1{x}^m.\ln xdx$ and $J\left(m\right)={\int }_0^1{x}^m.(\ln x{)}^{2}dx$ where $m\in N,$ then

$1+i^2+i^4+i^6+..........+i^{2n}$ is

If y = $ta{n}^{-1}\left(\frac{1}{1+x+x^2}\right)+ta{n}^{-1}\left(\frac{1}{x^2+3x+3}\right)+ta{n}^{-1}\left(\frac{1}{x^2+5x+7}\right)$ + ........+

upto n terms they y' (o) is equal to

If f is a continuous function in the interval [a, b], then the value of ${\int }_0^{1}f\left(\left(b-a\right)x+a\right)\mathrm{dx}$ is equal to