Q. If the angle between two tangents drawn from an external point $P$ to a circle of radius a and a center $O$, is ${60}^o$ , then find the length of $OP$.

Q. A line intersects the y-axis and x-axis at points P and Q respectively. If (2, -5) is the mid point of PQ, then find the coordinates of P and Q.

Q. On a straight line passing through the foot of the tower, two points $C$ and $D$ are at distances of $4$ m and $16$ m from the foot respectively. If the angles of elevation from $C$ and $D$ of the top of the tower are complementary, then find the height of the tower.

Q. A circle touches all the four sides of a quadrilateral ABCD. Prove that AB + CD = BC + DA.

Q. The probability of selecting a rotten apple randomly from a heap of 900 apples is 0.18. What is the number of rotten apple in the heap?

Q. Two different dice are thrown together, Find the probability that the numbers obtained have
(i) even sum, and
(ii) even product

Q. In the given figure, $△ABC$ is right-angled at $A$. Semi circles are drawn on $AB,AC$ and $BC$ as diameters. It is given that $AB=3cm$ and $AC=4cm$. Find the area of the shaded region in $c{m}^2$.

Q. If the ${10}^{th}$ term of an A.P. is 52 and the ${17}^{th}$ term is 20 more than the ${13}^{th}$ term, find the A.P.

Q. In a rain-water harvesting system, the rain-water from a roof of 22 m x 20 m drains into a cylindrical tank having diameter of base 2 m and height 3.5 m. If the tank is full, find the rainfall in cm. Write your views on conservation.

Q. Prove that the tangents drawn at the end points of a chord of a Circle make equal angles with the chord.

Q. A bag contains 15 white and some black balls. If the probability of drawing a black ball from the bag is thrice that of drawing a white ball, find the number of black balls in the bag.

Q. A train covers a distance of 300 km at a uniform speed. If the speed of the train is increased by 5 km/hour, it takes 2 hours less in the journey. Find the original speed of the train.

Q. Three semicircles each of diameter $3cm$, a circle of diameter $4.5cm$ and a semicircle of radius $4.5cm$ are drawn in the given figure. Find the area of the shaded region.

Q. If the ratio of the sum of the first n terms of two $A.P.s$ is $(7n+1):(4n+27)$, then find the ratio of their $9^{th}$ terms.

Q. Water in a canal, 5.4 m wide and 1.8 m deep, is flowing with a speed of 25 km/hour. How much area can it irrigate in 40 minutes, if 10 cm of standing water is required for irrigation

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

Q. A man observes a car from the top of a tower, which is moving towards the tower with a uniform speed. If the angle of depression of the car changes from ${30}^o$ to ${45}^o$ in $12minutes$, find the time taken by the car now to reach the tower.

Q. If the distance of P(x, y) from A(5, 1) and B(-1, 5) are equal, then prove that 3x = 2y.

Q. Find the value of $\mu$ for which one root of the quadratic equation $\mu x^2-14x+8=0$ in 6 times the other.

Q. From a solid right circular cylinder of a height $2.4$ cm and radius $0.7$ cm, a right circular cone of same height and same radius is cut. Find the total surface area of the remaining solid.

Q. In the given figure, XY and X' Y' are two parallel tangents to a circle with centre O and another tangent AB with point of contact C, is intersecting XY and X'Y' at B. Prove that $\angle AOB={90}^o$.

Q. Which term of the A.P. 8, 14, 20, 26.... will be 72 more than its $41st$ term?

Q. If a tower $30$ m high, cause a shadow $10\sqrt{3}$ long on the ground, then what is the angle of elevation of the sun?

Q. The dimension of a solid iron cuboid arw 4.4 m x 2.6 m x 1.0 m . It is melted and react into a hollow cylindrical pipe of 30cm inner radius and thickness 5 cm. Find the length of the pipe.

Q. If the roots of the equation $(c^2-ab)x^2-2(a^2-bc)x+b^2-ac=0$ in x are equal, then show that either $a=0$ or $a^3+b^3+c^3=3abc$

Q. What is the common difference of an A.P., in which $a_{21}-a_7=644$.