Q. In Figure, two concentric circles with centre O, have radii $21cm$ and $42cm$. If $\angle AOB={60}^{\circ }$, find the area of the shaded region. [Use $\pi =\frac{22}{7}$]

Q. Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

Q. The angle of elevation of the top of a tower at a distance of 120 m from a point A on the ground is ${45}^{\circ }$. If the angle of elevation of the top of a flagstaff fixed at the top of the tower, at A is ${60}^{\circ }$, then find the height of the flagstaff. [Use $\sqrt{3}=1.73$]

Q. The probability that a number selected at random from the numbers 1, 2, 3, ..., 15 is a multiple of 4, is

1. $\frac{4}{15}$
2. $\frac{2}{15}$
3. $\frac{1}{5}$
4. $\frac{1}{3}$

Q. If the total surface area of a solid hemisphere is $462$ ${cm}^2$, find its volume.
Note: Take $\pi =\frac{22}{7}$

1. $720.87$
2. $716$
3. $718.67$
4. $840$

Q. $150$ spherical marbles, each of diameter $1.4cm$, are dropped in a cylindrical vessel of diameter $7cm$ containing some water, which are completely immersed in water. Find the rise in the level of water in the vessel.

Q. Draw a line segment AB of length 8 cm. Taking A as centre, draw a circle of radius 4 em and taking B as centre, draw another circle of radius 3 cm. Construct tangents to each circle from the centre of the other circle.

Q. Prove that the diagonals of a rectangle ABCD, with vertices A$(2,-1)$, B$(5,-1)$, C $(5,6)$ and D $(2,6)$, are equal and bisect each other.

Q. Find the number of natural numbers between $101$ and $999$ which are divisible by both $2$ and $5$.

Q. A motorboat whose speed in still water is $18$ km/h, takes $1$ hour more to go $24$ km upstream than to return downstream to the same spot. Find the speed of the stream.

Q. $ABCD$ is a rectangle whose three vertices are $B(4,0),C(4,3)$ and $D(0,3)$. The length of one of its diagonals is

1. $5$
2. $4$
3. $3$
4. $25$

Q. Find the ratio in which the line segment joining the points $A(3,-3)$ and $B(-2,7)$ is divided by the x-axis. Also find the coordinates of the point of division.

Q. The incircle of an isosceles triangle $ABC$, in which $AB=AC$, touches the sides $BC,CA$ and $AB$ at $D,E$ and $F$ respectively. Prove that $BD=DC.$

Q. The angle of depression of a car parked on the road from the top of a $150$ m high tower is ${30}^{\circ }$. The distance of the car from the tower (in metres) is

1. $50\sqrt{3}$
2. $75$
3. $150\sqrt{2}$
4. $150\sqrt{3}$

Q. In a school, students decided to plant trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be double of the class in which they are studying. If there are 1 to 12 classes in the school and each class has two sections, find how many trees were planted by the students. Which value is shown in this question?

Q. The angle of elevation of an aeroplane from a point on the ground is ${60}^{\circ }$. After a flight of 30 seconds the angle of elevation becomes ${30}^{\circ }$ If the aeroplane is flying at a constant height of $3000\sqrt{3}$ m, find the speed of the aeroplane.

Q. In a family of 3 children, the probability of having at least one boy is

1. $\frac{7}{8}$
2. $\frac{1}{8}$
3. $\frac{3}{4}$
4. $\frac{5}{8}$

Q. Two different dice are tossed together. Find the probability
(i) that the number on each die is even.
(ii) that the sum of numbers appearing on the two dice is 5.

1. i) $\frac{1}{3}$ and ii) $\frac{1}{8}$
2. i) $\frac{1}{4}$ and ii) $\frac{1}{9}$
3. i) $\frac{1}{2}$ and ii) $\frac{1}{7}$
4. i) $\frac{1}{5}$ and ii) $\frac{1}{6}$

Q. The largest possible sphere is carved out of a wooden solid cube of side $7$ cm. Find the volume of the wood left. $\left[\text{Use}\pi =\frac{22}{7}\right]$

1. $163.3c{m}^3$
2. $164c{m}^3$
3. $170c{m}^3$
4. $165c{m}^3$

Q. In fig, common tangents $AB$ and $CD$ to the two circles with centres $O_1$ and $O_2$ intersect at $E$. Prove that $AB=CD.$

Q. $ABCD$ is a trapezium of area $24.5$ sq. cm. In it, $AD\parallel BC,\angle DAB={90}^{\circ },AD=10cm$ and $BC=4cm$. If $ABE$ is a quadrant of a circle, find the area of the shaded region.

Q. Water in a canal, $6$ m wide and $1.5$ m deep, is flowing at a speed of $4$ km/h. How much area will it irrigate in $10$ minutes, if $8$ cm of standing water is needed for irrigation?

Q. A container open at the top, is in the form of a frustum of a cone of height $24$ cm with radii of its lower and upper circular ends, as $8$ cm and $20$ cm respectively. Find the cost of milk which can completely fill the container at the rate of $21$ per litre. [Use $\pi =\frac{22}{7}$].

Q. The sum of the $2^{nd}$ and the $7^{th}$ terms of an AP is $30$. If its ${15}^{th}$ term is $1$ less than twice its $8^{th}$ term, find the AP.

Q. Solve for $x$ :
$\frac{16}{x}-1=\frac{15}{x+1};x\ne 0,-1$

Q. Two circles touch each other externally at P. AB is a common tangent to the circles touching them at A and B. The value of $\angle APB$ is

1. ${60}^{\circ }$
2. ${30}^{\circ }$
3. ${45}^{\circ }$
4. ${90}^{\circ }$

Q. A chord of a circle of radius $10$ cm subtends a right angle at its centre. The length of the chord (in cm) is

1. $5\sqrt{2}$
2. $10\sqrt{2}$
3. $\frac{5}{\sqrt{2}}$
4. $10\sqrt{3}$

Q. In a right triangle $ABC$, right-angled at $B,BC=12cm$ and $AB=5cm$. The radius of the circle inscribed in the triangle (in cm) is

1. $3$
2. $4$
3. $1$
4. $2$

Q. If $k,2k-1$ and $2k+1$ are three consecutive terms of an A.P., the value of $k$ is

1. $2$
2. $3$
3. $-3$
4. $5$

Q. Find the values of $k$ for which the quadratic equation $9x^2-3kx+k=0$ has equal roots.