Q. A box contains 30 bolts and 40 nuts. Half of the bolts and half of the nuts are rusted. If two items are drawn at random from the box, what is the probability that either both are rusted or both are bolts?

Q. The candidates received percentage marks in two subjects as follows

Candidate	$A$	$B$	$C$	$D$	$E$	$F$	$G$	$H$	$I$	$J$
Mathematics	$80$	$88$	$76$	$74$	$68$	$65$	$40$	$43$	$40$	$80$
Statastics Marks	$72$	$84$	$90$	$66$	$54$	$50$	$54$	$38$	$30$	$43$

Calculate Spearman's rank correlation coefficient and interpret your result.

Q. From the equation of the two regression lines, $4x+3y+7=0$ and $3x+4y+8=0$, find: Regression coefficients.

Q. Find the product of the matrices $A$ and $B$ where
$A=\left[\begin{array}{ccc}-5& 1& 3\\ 7& 1& -5\\ 1& -1& 1\end{array}\right],B=\left[\begin{array}{ccc}1& 1& 2\\ 3& 2& 1\\ 2& 1& 3\end{array}\right]$

Q. Solve the differential equation:
$(x{y}^2+x)dx+(x^{2}y+y)dy=0$

Q. Evaluate ${\int }_1^2\frac{\sqrt{x}}{\sqrt{3-x}+\sqrt{x}}dx$.

Q. From the equation of the two regression lines, $4x+3y+7=0$ and $3x+4y+8=0$, find: Coefficient of correlation.

Q. $\sum x=30,\sum y=42,\sum xy=199,\sum x^2=184,\sum y^2=318,n=6.$ Find the correlation coefficient between x and y.

Q. Using properties of determinants, prove that
$\left|\begin{array}{ccc}a& a+b& a+b+c\\ 2a& 3a+2b& 4a+3b+2c\\ 3a& 6a+3b& 10a+6b+3c\end{array}\right|=a^3$

Q. $\sum x=30,\sum y=42,\sum xy=199,\sum x^2=184,\sum y^2=318,n=6.$ The regression coefficients. Find the likely value of y when x = 10.

Q. Evaluate $\int \frac{dx}{x\left\{6\right(\log x{)}^2+7\log x+2\}}$.

Q. Prove that ${\sec }^2\left({\tan }^{-1}2\right)+{\text{cosec}}^2\left({\cot }^{-1}3\right)=15$.

Q. Find the locus of the complex number, Z = x + iy
given $\left|\frac{x+iy-2i}{x+iy+2i}\right|=\sqrt{2}$

Q. Verify Rolle's theorem for the function:
$f\left(x\right)=\log \left\{\frac{x^2+ab}{(a+b)x}\right\}$ in the interval $[a,b]$ where, $0\notin [a,b]$.

Q. A bag contains 8 red and 5 white balls. Two successive draws of 3 balls are made at random from the bag without replacements. Find the probability that the first draw yields 3 white balls and the second draw 3 red balls.

Q. Three persons A, B and C shoot to hit a target. If in trials, A hits the target 4 times in 5 shots, B hits 3 times in 4 shots and C hits 2 times in 3 trials. find the probability that
(a) Exactly two persons hit the target
(b) At least two persons hit the target.

Q. Find the equation of the ellipse with its centre at $(4,-1)$ focus at $(1,-1)$ and given that it passes through $(8,0).$

Q. Find the area of the region bounded by the curves $x=4y-y^2$ and the y-axis.

Q. A printed page is to have a total area of $80$ sq. cm with a margin of $1$ cm at the top and on each side and a margin of $1.5$ cm at the bottom. What should be the dimensions of the page, so that the printed area will be maximum?

Q. Using De Moivre's theorem prove that
${\left(\frac{1+\cos \theta +i\sin \theta }{1+\cos \theta -i\sin \theta }\right)}^n=\cos n\theta +i\sin n\theta ,$ where $i=\sqrt{-1}$

Q.

Q. Solve for x and y if $\left(\begin{array}{c}{y}^2\\ x^2\end{array}\right)+2\left(\begin{array}{c}3y\\ 2x\end{array}\right)=3\left(\begin{array}{c}-3\\ 7\end{array}\right)$

Q. $\sum x=30,\sum y=42,\sum xy=199,\sum x^2=184,\sum y^2=318,n=6.$ The regression coefficients. Find the regression equation of y on x.

Q. Prove that: ${\cos }^{-1}\frac{63}{65}+2{\tan }^{-1}\frac{1}{5}={\sin }^{-1}\frac{3}{5}$

Q. Find the equation of the hyperbola whose Transverse and Conjugate axes are the $x$ and $y$ axes respectively, given that the length of conjugate axis is $5$ and distance between the foci is $13$.

Q. If, $e^y(1+x)=1$, then show that
$\frac{d^{2}y}{d{x}^2}={\left(\frac{dy}{dx}\right)}^2$

Q. From the equation of the two regression lines, $4x+3y+7=0$ and $3x+4y+8=0$, find: Mean of x and y.

Q. Evaluate ${lim}_{x\to \pi /2}(\cos x.\log (\tan x\left)\right)$.

Q. Evaluate: $\int e^x(\tan x+\log (\sec x\left)\right)dx.$

Q. (i) Write the Boolean expression corresponding to the circuit given below.
(ii) Simplify the expression using laws of Boolean Algebra and construct the simplified circuit.