Q. Find the locus of the complex number $z=x+iy$, satisfying relations $\arg (z-1)=\frac{\pi }{4}$ and $|z-2-3i|=2$. Illustrate the locus on the Argand plane.

Q. The following table shows the mean and standard deviation of the marks of Mathematics and Physics scored by the students in a school

	Mathematics	Physics
Mean	$84$	$81$
Standard Deviation	$7$	$4$

The correlation coefficient between the given marks is $0.86$. Estimate the likely marks in Physics if the marks in Mathematics are $92$.

Q. Find the equation of planes parallel to the plane $2x-4y+4z=7$ and which are at a distance of five units from the point $(3,-1,2).$

Q. Solve the following differential equation
$y{e}^{y}dx=(y^3+2x{e}^y)dy$, given that $x=0,y=1.$

Q.

Q. Using L' Hospital's Rule, evaluate
$\underset{x\to 0}{lim}\left(\frac{e^x-e^{-x}-2x}{x-\sin x}\right)$

Q. Using properties of determinants, prove that
$\omega \left|\begin{array}{ccc}x& y& z\\ x^2& y^2& z^2\\ y+z& z+x& x+y\end{array}\right|=(x-y)(y-z)(z-x)(x+y+z)$

Q. Evaluate : ${\int }_0^1\log (\frac{1}{x}-1)dx$

Q. Verify Lagrange's Mean Value Theorem for the function $f\left(x\right)=\sqrt{x^2-x}$ in the interval $[1,4].$

Q. Find the equation of a line passing through the point $(-1,3,-2)$ and perpendicular to the lines: $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ and $\frac{x+2}{-3}=\frac{y-1}{2}=\frac{z+1}{5}$.

Q. From the following information, find the equation of Hyperbola and the equation of its Transverse Axis:

Q. Prove that : ${\tan }^{-1}\frac{1}{4}+{\tan }^{-1}\frac{2}{9}=\frac{1}{2}{\sin }^{-1}\frac{4}{5}$

Q. Evaluate : $\int \frac{{\cos }^{-1}x}{x^2}dx$

Q. Three persons, Aman, Bipin and Mohan attempt a Mathematics problem independently. The odds in favour of Aman and Mohan solving the problem are $3:2$ and $4:1$ respectively and the odds against Bipin solving the problem are $2:1$. Find
(i) The probability that all the three will solve the problem.
(ii) The probability that problem will be solved.

Q. Find the area bounded by the curve $y=2x-x^2$, and the line $y=x.$

Q. Find the value of $\lambda$ for which the four points $A,B,C,D$ with position vectors $-\hat{j}-\hat{k};4\hat{i}+5\hat{j}+\lambda \hat{k};3\hat{i}+9\hat{j}+4\hat{k}$ and $-4\hat{i}+4\hat{j}+4\hat{k}$ are coplanar.

Q. Bag $A$ contains three red and four white Balls; bag $B$ contains two red and three white balls. If one ball is drawn from bag $A$ and two balls from bag $B$, find the probability that:
(i) One ball is red and two balls are white;
(ii) All the three balls are of the same colour.

1. i) $\frac{33}{70}$ and ii) $\frac{3}{14}$
2. i) $\frac{22}{31}$ and ii) $\frac{1}{3}$
3. i) $\frac{1}{2}$ and ii) $\frac{1}{2}$
4. i) $\frac{8}{25}$ and ii) $\frac{2}{7}$

Q. Find $A^{-1}$, where $A=\omega \left[\begin{array}{ccc}4& 2& 3\\ 1& 1& 1\\ 3& 1& -2\end{array}\right]$
Hence, solve the following system of linear equations
$4x+2y+3z=2$
$x+y+z=1$
$3x+y-2z=5$

Q. If two balls are drawn from a bag containing three red balls and four blue balls, find the probability that:
(a) They are of the same colour.
(b) They are of different colours.

Q. If 1, $\omega$ and ${\omega }^2$ are the cube roots of unity, evaluate $(1-{\omega }^4+{\omega }^8)(1-{\omega }^8+{\omega }^{16})$

Q. Solve the differential equation:
$\log \left(\frac{dy}{dx}\right)=2x-3y$

Q. Solve for $x:{\sin }^{-1}x+{\sin }^{-1}(1-x)={\cos }^{-1}x$.

Q. If $\overrightarrow{a}$ and $\overrightarrow{b}$ are unit vectors and $\theta$ is the angle between them, then show that $|\overrightarrow{a}-\overrightarrow{b}|=2\sin \frac{\theta }{2}$.

Q. Find the Karl Pearson's co-efficient of correlation between $x$ and $y$ for the following data:

$x$	$16$	$18$	$21$	$20$	$22$	$26$	$27$	$15$
$y$	$22$	$25$	$24$	$26$	$25$	$30$	$33$	$14$

Q. Find the maximum volume of the cylinder which can be inscribed in a sphere of radius $3\sqrt{3}$cm. (Leave the answer in terms of $\pi$).

Q. If $(A-2I)(A-3I)=0$, where $A=\left[\begin{array}{cc}4& 2\\ -1& x\end{array}\right]$ and $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ , find the value of $x$.

Q. Find the value(s) of $k$ so that the line $2x+y+k=0$ may touch the hyperbola $3x^2-y^2=3$

Q. If $y=({\cot }^{-1}x{)}^2$, show that $(1+x^2{)}^2\frac{d^{2}y}{d{x}^2}+2x(1+x^2)\frac{dy}{dx}=2$.

Q. Construct a circuit diagram for the following Boolean Function
$(BC+A)(A^{\prime }{B}^{\prime }+C^{\prime })+A^{\prime }{B}^{\prime }{C}^{\prime }$
Using laws of Boolean Algebra, simplify the function and draw the simplified circuit.

Q. Two regression lines are represented by $4x+10y=9$ and $6x+3y=4$. Find the line of regression of $y$ on $x$.