Q. Prove that the lengths of the tangents drawn from an external point to a circle are equal.

Q. A card is drawn at random from a well-shuffled deck of playing cards. Find the probability that the card drawn is
i. a card of spade or an ace.
ii. a black king.
iii. neither a jack nor a king
iv. either a king or a queen.

Q. A hemispherical bowl of internal diameter $36cm$ contains liquid. This liquid is filled into $72$ cylindrical bottles of diameter $6cm$. Find the height of each bottle, if $10\%$ liquid is wasted in this transfer.

1. $5.2$ cm
2. $6.0$ cm
3. $5.4$ cm
4. $50$ cm

Q. $504$ cones, each of diameter $3.5$ cm and height $3$ cm, are melted and recast into a metallic sphere. Find the diameter of the sphere . Use $\pi =\frac{22}{7}$.

Q. Find the values of $k$ so that the area of the triangle with vertices $(1,-1),(-4,2k)$ and $(-k,-5)$ is $24$ sq. units.

Q. Two different dice are tossed together. Find the probability that the product of the two numbers on the top of the dice is $6$.

Q. If the quadratic equation $p{x}^2-2\sqrt{5}px+15=0$ has two equal roots then find the value of $p$.

Q. A cubical block of side $10$cm is surmounted by a hemisphere. What is the largest diameter that the hemisphere can have? Find the cost of painting the total surface area of the solid so formed, at the rate of Rs. $5$ per sq. cm. $[Use\pi =22/7]$

Q. Construct a $△ABC$ in which $AB=6cm$, $\angle A={30}^{\circ }$ and $\angle B={60}^{\circ }$. Construct another $△A{B}^{\prime }{C}^{\prime }$ similar to $△ABC$ with base $A{B}^{\prime }=8cm$.

Q. Find the area of the minor segment of a circle of radius 14 cm, when its central angle is ${60}^o$. Also find the area of the corresponding major segment. $[Use\pi =\frac{22}{7}]$

Q. Due to sudden floods, some welfare associations jointly requested the government to get $100$ tents fixed immediately and offered to contribute $50\%$ of the cost. If the lower part of each tent is of the form of a cylinder of diameter $4.2$ m and height $4$ m with the conical upper part of the same diameter but height $2.8$ m, and the canvas to be used costs Rs. $100$ per sq. m, find the amount, the associations will have to pay. What values are shown by these associations $[Use\pi =\frac{22}{7}]$

Q. If the coordinates of points $A$ and $B$ are $(-2,-2)$ and $(2,-4)$ respectively, find the coordinates of $P$ such that $AP=\frac{3}{7}AB$, where $P$ lies on the line segment $AB$.

Q. In the following figure, a tower AB is $20$ m high and its shadow BC on the ground is $20\sqrt{3}$ m long. Find the distance of the tip of the tower from point $C$.

Q. Prove that the tangent drawn at the mid-point of an arc of a circle is parallel to the chord joining the end points of the arc.

Q. The probability of selecting a red ball at random from a jar that contains only red, blue and orange balls is $\frac{1}{4}$. The probability of selecting a blue ball at random from the same jar $\frac{1}{3}$. If the jar contains $10$ orange balls, find the total number of balls in the jar.

Q. A train travels at a certain average speed for a distance of 54 km and then travels a distance of 63 km at an average speed of 6 km/h more than the first speed. If it takes 3 hours to complete the total journey, what is its first speed?

Q. The diagonal of a rectangular field is 16 metres more than the shorter side. If the longer side is 14 metres more than the shorter side, then find the lengths of the sides of the field.

Q. In the following figure, PQRS is square lawn with side $PQ=42$ meters. Two semi-circular flower beds are there on the sides PS and QR with center at O, the intersections of its diagonals. Find the total area of the two flower beds (shaded parts).

Q. In the following figure, two tangents $RQ$ and $RP$ are drawn from an external point $R$ to the circle with centre $O$, If $\angle PRQ={120}^o$, then prove that $OR=PR+RQ.$

Q. In the following figure, PQ is a chord of a circle with centre O and PT is a tangent. If $\angle QPT={60}^o$, find $\angle PRQ$.

Q. Solve the following quadratic equation for $x$:
$4x^2+4bx-(a^2-b^2)=0$

Q. The angle of elevation of an aeroplane from point A on the ground is ${60}^o$. After flight of $15$ seconds, the angle of elevation changes to ${30}^o$. If the aeroplane is flying at a constant height of $1500\sqrt{3}$ m, find the speed of the plane in km/hr.

Q. Solve for $x$:
$\sqrt{3}{x}^2-2\sqrt{2}x-2\sqrt{3}=0$

Q. In an AP, if $S_5+S_7=167$ and $S_{10}=235$, then find the A.P., where $S_n$ denotes the sum of its first $n$ terms.

Q. Find the ${60}^{th}$ term of the AP 8, 10, 12, ., if it has a total of 60 terms and hence find the sum of its last 10 terms.

Q. Find the relation between $x$ and $y$ if the points $A(x,y),B(-5,7)$ and $C(-4,5)$ are collinear.

Q. The 14th term of an A.P. is twice its $8^{th}$ term. If its $6^{th}$ term is $-8$, then find the sum of its first $20$ terms.

Q. The points $A(4,7),B(p,3)$ and $C(7,3)$ are the vertices of a right triangle, right-angled at $B$, Find the values of $P$.

Q. In Figure $5$, a $△ABC$ is drawn to circumscribe a circle of radius $3cm$, such that the segments $BD$ and $DC$ are respectively of lengths $6cm$ and $9cm$. If the area of $△ABC$ is $54c{m}^2$, then find the lengths of sides $AB$ and $AC$.