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Find the equation to the ellipse (referred to its axes as the axes of x and y respectively) which passes through the point(-3,1) and has eccentricity $\sqrt{\frac{2}{5}}$.

Find the domain of each of the following real valued functions of real variable :
(i) $f\left(x\right)=\frac{1}{x}$
(ii) $f\left(x\right)=\frac{1}{x-7}$
(iii) $f\left(x\right)=\frac{3x-2}{x+1}$
(iv) $f\left(x\right)=\frac{2x+1}{x^2-9}$
(v) $f\left(x\right)=\frac{x^2+2x+1}{x^2-8x+12}$

If the lines $x=ay+b,z=cy+d$ and $x=a^{\prime }z+b^{\prime },y=c^{\prime }z+d^{\prime }$ are perpendicular, then :

If the area of the triangle formed by the lines $y=x$, $x+y=2$ and the line through $P(h,k)$ and parallel to $x$-axis is $4h^2$, the locus of $P$ can be

A car will hold two persons in the front seat and 1 in the rear seat. If among six persons only two can drive, the number of ways, in which the car can be filled is :

If the polar coordinates of a point are $(6,\frac{\pi }{3})$, then the Cartesian coordinates are

If $y=y\left(x\right)$ is the solution of the differential equation $\frac{5+e^x}{2+y}\cdot \frac{dy}{dx}+e^x=0$ satisfying $y\left(0\right)=1$, then a value of $y\left({\log }_{e}13\right)$ is:

Solve each of the following system of equation in R.

$|x+\frac{1}{3}|>\frac{8}{3}$

Rearrange the following sentences (A), (B), (C), (D), (E) and (F) to make a meaningful paragraph and answer the questions which follow:

The set of all real numbers x for which $x^2-|x+2|+x>0$, is

If $x\left[\begin{array}{c}2\\ 1\end{array}\right]+y\left[\begin{array}{c}3\\ 5\end{array}\right]+\left[\begin{array}{c}-8\\ -11\end{array}\right]=O,$ then

If the value of x is so small that $x^2$ and greater powers can be neglected, then $\frac{\sqrt{1+x}+\sqrt[3]{3(1-x{)}^2}}{1+x+\sqrt{1+x}}$ is equal to

The focus and vertex of a parabola are (4,5) and (3,6). The equation of axis is

If $S_n={\cot }^{-1}\left(3\right)+{\cot }^{-1}\left(7\right)+{\cot }^{-1}\left(13\right)+{\cot }^{-1}\left(21\right)+\dots n\text{terms}$, then

Let a vector $\overrightarrow{a}$ be coplanar with vectors $\overrightarrow{b}=2\hat{i}+\hat{j}+\hat{k}$ and $\overrightarrow{c}=\hat{i}-\hat{j}+\hat{k}.$ If $\overrightarrow{a}$ is perpendicular to $\overrightarrow{d}=3\hat{i}+2\hat{j}+6\hat{k}$, and $|\overrightarrow{a}|=\sqrt{10}.$ Then a possible value of $\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]+\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{d}\right]+\left[\overrightarrow{a}\overrightarrow{c}\overrightarrow{d}\right]$ is equal to

A semi-circle of diameter 1 unit sits at the top of a semi-circle of diameter 2 units. The shaded region inside the smaller semi-circle but outside the larger semi-circle is called a lune. The area of the lune is.

If $A$ is the area formed by the positive $x$ axis, and the normal and tangent to the circle $x^2+y^2=4$

at $(1,\sqrt{3})$ then the value of $A/\sqrt{3}$ is

Integrate the following functions.
$\int \frac{(logx{)}^2}{x}dx.$

Let $f\left(x\right)=\{\begin{array}{cc}2-x,& x\le 0\\ x+1,& x>0\end{array}$ and $g\left(x\right)=\{\begin{array}{cc}x+3,& x<1\\ x^2-2x-2,& 1\le x<2\\ x-5,& x\ge 2\end{array}$. If $g\left(f\right(x\left)\right)$ is continuous at $x=0$, then the value of $g\left(f\right(0\left)\right)$ is

The complete solution set of the inequality $\frac{1}{1+\ln x}+\frac{1}{1-\ln x}>2$ is

The mean deviation of the variates 40, 62, 68, 76, 54 from their arithmetic mean is

The feasible region of an LPP is shown in the figure. If $Z=8x+3y,$ then the minimum value of $Z$ occurs at





[1 mark]

Consider the function f : R $\to$ R, defined as $\{\begin{array}{c}{x}^2-x+3,xϵ(-\infty ,3)\cap Q\\ x+a,xϵ(-\infty ,2)-Q\\ 2^x+1,xϵ(2,3)-Q\\ 9tan\left(\frac{\pi x}{12}\right),xϵ[3,6]\end{array}$

If f(x) is continuous at x = 2 then the value of a is

If one end point of the focal chord of the parabola $y^2=4ax$ is $(1,2)$, then second end point lies on

If a curve is represented parametrically by the equations $x=f\left(t\right)$ and $y=g\left(t\right),$ then $\frac{\left(\frac{d^{2}y}{d{x}^2}\right)}{\left(\frac{d^{2}x}{d{y}^2}\right)}$ is equal to

$($where $f^{\prime }\left(t\right)\ne 0$ and $g^{\prime }\left(t\right)\ne 0)$

If $\int \sqrt{(x-1)(2-x)}dx=\frac{y}{4}-\frac{1}{16}\sin 4y+C,$ then which of the following(s) can not be the value(s) of $y.$

(Where $C$ is integration constant)

The value of $\int \frac{\sin 7x}{\cos x}dx$ is

(where $C$ is constant of integration)

Let $P$ and $Q$ be $3\times 3$ matrices with $P\ne Q$. If $P^3=Q^3$ and $P^{2}Q=Q^{2}P$, then the determinant of $(P^2+Q^2)$ is equal to