Q. A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The radius of the hemisphere is 7 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel.

Q. If $K(5,4)$ is the mid-point of the line segment $PQ$ and co-ordinates of $Q$ are $(2,3),$ then find the co-ordinates of point $P$.

Q. A copper rod of radius 1 cm and length 2 cm is drawn into a wire of length 18 m of uniform thickness. Find the thickness of the wire.

Q. Write the length of an arc of a sector of circle with radius r and angle with degree measure $\theta$.

Q. Find the value of $\frac{\tan {67}^o}{\cot {23}^o}$.

Q. If tangents $RA$ and $RB$ from a point $R$ to a circle with centre $O$ are inclined to each other at an angle of $\theta$ and $\angle AOB={40}^o$ then find the value of $\theta$ (in degrees).

Q. If $3\cot A=4$, then evaluate $\frac{1-{\tan }^{2}A}{1+{\tan }^{2}A}$

Q. Show that $\sin {28}^o\cos {62}^o+\cos {28}^o\sin {62}^o=1$

Q. Write the solution of the pair of linear equations $4x+2y=5$ and $x-2y=0$.

Q. Prove that $\sqrt{2}$ is an irrational number.

Q. In the given figure, O is the centre of a circle and two tangents KR, KS are drawn on the circle from a point K lying outside the circle. Prove that KR = KS.

Q. One card is drawn from a well-shuffled deck of 52 cards. Calculate the probability that the card will not be an ace.

1. $\frac{11}{13}$
2. $\frac{12}{13}$
3. $\frac{10}{13}$
4. $\frac{9}{13}$

Q. Find the value of k if the points $A(2,3),B(4,k)$ and $C(6,-3)$ are collinear.

Q. How many tangents can be constructed to any point on the circle of radius 4 cm?

Q. Write the distance of the point (-5, 4) from x-axis.

Q. $O$ is any point inside rectangle $ABCD$. Prove that $O{B}^2+O{D}^2=O{A}^2+O{C}^2$.

Q. In the following distribution, calculate mean $\overline{x}$:

Class-interval	$10-25$	$25-40$	$40-55$	$55-70$	$70-85$	$85-100$
Frequency	$2$	$3$	$7$	$5$	$6$	$7$

If $\overline{x}=ab$, then $a+b$ is :

Q. How many terms of the A.P. 17, 15, 13, ...... must be taken, so that their sum is 81?

Q. Construct a triangle of sides 4 cm, 5 cm and 6 cm and then a triangle similar to it whose sides are $\frac{3}{5}$ time of the corresponding sides of the given triangle.

Q. Divide $3x^3+x^2+2x+5$ by $1+2x+x^2$

Q. The speed of a boat in still water is 18 km/h. It takes $\frac{1}{2}$ an hour extra to going 12 km upstream instead of going the same distance downstream. Find the speed of the steam ( in km/hr ).

Q. Find the circumference of a circle whose diameter is 14 cm ( in cm. )

Q. Write the common difference of the A.P. 7, 5, 3, 1, -1, -3, .........

Q. In the given figure, $\frac{PK}{KS}=\frac{PT}{TR}$ and $\angle PKT=\angle PRS$. Prove that $\Delta PSR$ is an isosceles triangle.

Q. Neeraj and Dheeraj are friends. Find the probability of their birthdays when
(i) birthdays are different.
(ii) birthdays are same.

Q. Find the area of corresponding major sector of a circle with radius 7 cm and angle 120${}^o$.

Q. From a point on a bridge across a river the angles of depression of the banks on opposite sides of the river are 30${}^o$ and 45${}^o$ respectively. If the bridge is at a height of 4 m from the banks, find the width of the river.

Q. In the figure, $\Delta OPR\sim \Delta OSK,\angle POS={125}^o$ and $\angle PRO={70}^o$. Find the values of $\angle OKS$ and $\angle ROP$.

1. $\angle ROP={50}^o$ and $\angle OKS={80}^o$
2. $\angle ROP={55}^o$ and $\angle OKS={70}^o$
3. $\angle ROP={75}^o$ and $\angle OKS={15}^o$
4. $\angle ROP={45}^o$ and $\angle OKS={45}^o$

Q.

Find the HCF of $96$ and $404$.

Q. Prove that ${\left[\frac{1+\tan A}{1+\cot A}\right]}^2={\tan }^{2}A$.