Q. Two coins are tossed once. Find the probability of getting:
(i) $2$ heads
(ii) at least $1$ tail.

Q. The line through $P(5,3)$ intersects $y$-axis at $Q$.
(i) Write the slope of the line.
(ii) Write the equation of the line.
(iii) Find the co-ordinates of $Q$.

Q. Find the mode and median of the following frequency distribution:

$x$	$10$	$11$	$12$	$13$	$14$	$15$
$f$	$1$	$4$	$7$	$5$	$9$	$8$

Q. Using the factor theorem factorise the following polynomial completely $3x^3+2x^2-19x+6$.

Q. A man invests Rs. $9,600$ on Rs. $100$ share at Rs. $80$. If the company pays him $18\%$ dividend find:
(i) the number of shares he buys.
(ii) his total dividend
(iii) his percentage return on the shares.

Q. A page from the Saving Bank Account of Mr. Prateek is given below:

Date	Particulars	Withdrawal (In Rs.)	Deposit (In Rs.)	Balances (In Rs.)
Jan $1^{st}$ $2006$	B/F	-	-	$1,270$
Jan. $7^{th}$ $2006$	By Cheque	-	$2,310$	$3,580$
March $9^{th}$ $2006$	To Self	$2,000$	-	$1,580$
March ${26}^{th}$ $2006$	By Cash	-	$6,200$	$7,780$
June ${10}^{th}$ $2006$	To Cheque	$4,500$	-	$3,280$
July ${15}^{th}$ $2006$	By Clearing	-	$2,630$	$5,910$
October ${18}^{th}$ $2006$	To Cheque	$530$	-	$5,380$
October ${27}^{th}$ $2006$	To Self	$2,690$	-	$2,690$
November $3^{rd}$ $2006$	By cash	-	$1,500$	$4,190$
December $6^{th}$ $2006$	To Cheque	$950$	-	$3,240$
December ${23}^{rd}$ $2006$	By Transfer	-	$2,920$	$6,160$

If he receives Rs. $198$ as interest on $1^{st}$ January, $2007$, find the rate of interest paid by the bank.

Q. The following distribution represent the height of $160$ students of a school.

Height (in $cm$)	No. of Students
$140-145$	$12$
$145-150$	$20$
$150-155$	$30$
$155-160$	$38$
$160-165$	$24$
$165-170$	$16$
$170-175$	$12$
$175-180$	$8$

Draw an ogive for the given distribution taking $2cm=5cm$ of height on one axis and $2cm=20$ students on the other axis. Using the graph, determine:
(i) The median height
(ii) The inter quartile range
(iii) The number of students whose height is above $172cm$.

Q. On what sum of money will the difference between the compound interest and simple interest for $2$ years be equal to Rs. $25$ if the rate of interest charged for both is $5\%$ p.a?

Q. As observed from the top of a $80m$ tall lighthouse, the angles depression of two ships on the same side of the light house in horizontal line with its base are ${30}^{\circ }$ and ${60}^{\circ }$ respectively. Find the distance between the two ships. Give your answer correct to the nearest metre.

Q. Given $\left[\begin{array}{cc}2& 1\\ -3& 4\end{array}\right]X=\left[\begin{array}{c}7\\ 6\end{array}\right]$. Write:
(i) the order of the matrix $X$
(ii) the matrix $X$.

Q. Kiran deposited Rs. $200$ per month for $36$ months in a banks's recurring deposit account. If the bank pays interest at the rate of $11\%$ per annum, find the amount (in Rs.) she gets on maturity.

Q. Prove that $\frac{{\tan }^2\theta }{(\sec \theta -1{)}^2}=\frac{1+\cos \theta }{1-\cos \theta }$

Q. The line through $A(-2,3)$ and $B(4,b)$ is perpendicular to the line $2x-4y=5$. Find the value of $|b|$.

Q. If $x=\frac{\sqrt{a+1}+\sqrt{a-1}}{\sqrt{a+1}-\sqrt{a-1}}$, then show that $x^2-2ax+1=0$.

Q. $ABC$ is an isosceles right angled triangle with $\angle ABC={90}^o$. A semi-circle is drawn with $AC$ as the diameter. If $AB=BC=7cm$, find the area (in $c{m}^2$) of shaded region. $(\text{Take}\pi =\frac{22}{7})$

Q. The monthly pocket money of Ravi and Sanjeev are in the ratio $5:7$. Their expenditures are in the ratio $3:5$. If each save $Rs80$ every month, find their monthly pocket money.

Q. Without solving the following quadratic equation, find the value of $m$ for which the given equation has real and equal roots.
$x^2+2(m-1)x+(m+5)=0$

Q. Solve the following equation and give your answer correct to 3 significant figures
$5x^2-3x-4=0$

Q. Using graph paper and taking $1cm=1$ unit along both $x$-axis and $y$-axis.
(i) Plot the points $A(-4,4)$ and $B(2,2)$
(ii) Reflect $A$ and $B$ in the origin to get images $A^{\prime }$ and $B^{\prime }$ respectively.
(iii) Write down the co-ordinates of $A^{\prime }$ and $B^{\prime }$.
(iv) Give the geometrical name for the figure $AB{A}^{\prime }{B}^{\prime }$
(v) Draw and name its lines of symmetry.

Q. In the given figure $O$ is the centre of the circle and $AB$ is a tangent at $B$. If $AB=15cm$ and $AC=7.5cm$. Calculate the radius of the circle.

Q. Solve the following inequation and represent the solution set on the number line:
$4x-19<\frac{3x}{5}-2\le -\frac{2}{5}+x,x\in R$

Q. Evaluate without using trigonometric tables
${\cos }^2{26}^{\circ }+\cos {64}^{\circ }\sin {26}^{\circ }+\frac{\tan {36}^{\circ }}{\cot {54}^{\circ }}$

Q. A hollow sphere of internal and external radii $6cm$ and $8cm$ respectively is melted and recast into the small cylinder of base radius $2cm$ and height $8cm$. Find the number of cylinders.

Q. In the given figure $△ABC$ and $△AMP$ are right angled at $B$ and $M$ respectively.
Given $AB=10cm,AP=15cm$ and $PM=12cm$.
(i) Prove $△ABC\sim △AMP$
(ii) Find $AB$ and $BC$.

Q. In $△PQR,PQ=24cm,QR=7cm$ and $\angle PQR={90}^{\circ }$. Find the radius (in $cm$) of the inscribed circle.

Q. If $A=\left[\begin{array}{cc}3& 1\\ -1& 2\end{array}\right]$ and $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, find $A^2-5A+7I$

Q. A car cover a distance of $400km$ at a certain speed. Had the speed $12km/h$ more, the time taken for the journey would have been $1$ hour $40$ minutes less. Find the original speed of the car.

Q. The printed price of an article is Rs. $60,000$. The wholesaler allows a discount of $20\%$ to the shopkeeper. The shopkeeper sells the article to the customer at the printed price. Sales tax (under VAT) is charged at the rate of $6\%$ at every stage.
Find:
(i) the cost to the shopkeeper inclusive of tax.
(ii) VAT paid by the shopkeeper to the Government.
(iii) the cost to the customer inclusive of tax.

Q. Marks obtained by $40$ students in a short assessment is given below, where $a$ and $b$ are two missing data.

Marks	$5$	$6$	$7$	$8$	$9$
No. of students	$6$	$a$	$16$	$13$	$b$

If the mean of the distribution is $7.2$, find $a$ and $b$.