#### Alternate Sets

If the quadratic equation px^{2} − 2√5px + 15 = 0 has two equal roots then find the value of p.

Concept: Nature of Roots

In the below given figure, a tower AB is 20 m high and BC, its shadow on the ground, is 20√3 m long. Find the sun’s altitude.

Concept: Introduction to Trigonometry Examples and Solutions

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

Concept: Probability Examples and Solutions

In the figure given below, PQ is a chord of a circle with centre O and PT is a tangent. If ∠QPT = 60°, find ∠PRQ.

Concept: Tangent to a Circle

In the below given figure, two tangents RQ and RP are drawn from an external point R to the circle with centre O. If∠PRQ = 120°, then prove that OR = PR + RQ.

Concept: Number of Tangents from a Point on a Circle

the below given figure, a triangle ABC is drawn to circumscribe a circle of radius 3 cm, such that the segments BD and DC are respectively of lengths 6 cm and 9 cm. If the

area of ΔABC is 54 cm^{2}, then find the lengths of sides AB and AC.

Concept: Triangles Examples and Solutions

Solve the following quadratic equation for x: `4x^2 + 4bx – (a^2 – b^2) = 0`

Concept: Solutions of Quadratic Equations by Completing the Square

In an A.P., if S5 + S7 = 167 and S_{10=}235, then find the A.P., where Sn denotes the sum of its first n terms.

Concept: Sum of First n Terms of an AP

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

Concept: Pythagoras Theorem

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

Concept: Area of a Triangle

The 14^{th} 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.

Concept: Arithmetic Progression

Solve for x: `sqrt(3x^2)-2sqrt(2)x-2sqrt3=0`

Concept: Nature of Roots

The angle of elevation of an aeroplane from point A on the ground is 60˚. After flight of 15 seconds, the angle of elevation changes to 30˚. If the aeroplane is flying at a constant height of 1500√3 m, find the speed of the plane in km/hr.

Concept: Heights and Distances

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

Concept: Division of a Line Segment

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

Concept: Probability Examples and Solutions

Find the area of the minor segment of a circle of radius 14 cm, when its central angle is 60˚. Also find the area of the corresponding major segment.[use π=22/7]

Concept: Areas of Sector and Segment of a Circle

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 same diameter but of 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 π=22/7]

Concept: Surface Area of a Combination of Solids

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

Concept: Surface Area of a Combination of Solids

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 100 sq. cm. [Use π = 3.14]

Concept: Surface Area of a Combination of Solids

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 and hence find its surface area.

[Use π=22/7]

Concept: Surface Area of a Combination of Solids

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.

Concept: Pythagoras Theorem

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

Concept: Arithmetic Progression

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?

Concept: Algebraic Methods of Solving a Pair of Linear Equations - Cross - Multiplication Method

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

Concept: Number of Tangents from a Point on a Circle

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.

Concept: Circles Examples and Solutions

Construct a Δ ABC in which AB = 6 cm, ∠A = 30° and ∠B = 60°, Construct another ΔAB’C’ similar to ΔABC with base AB’ = 8 cm.

Concept: Division of a Line Segment

At a point A, 20 metres above the level of water in a lake, the angle of elevation of a cloud is 30˚. The angle of depression of the reflection of the cloud in the lake, at A is 60˚.

Find the distance of the cloud from A.

Concept: Heights and Distances

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.

Concept: Probability Examples and Solutions

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.

Concept: Area of a Triangle

In the figure given below, PQRS is square lawn with side PQ = 42 metres. Two circular flower beds are there on the sides PS and QR with centre at O, the intersections of its

diagonals. Find the total area of the two flower beds (shaded parts).

Concept: Problems Based on Areas and Perimeter Or Circumference of Circle, Sector and Segment of a Circle

From each end of a solid metal cylinder, metal was scooped out in hemispherical from of same diameter. The height of the cylinder is 10 cm and its base is of radius 4.2 cm.

The rest of the cylinder is melted and converted into a cylindrical wire of 1.4 cm thickness. Find the length of the wire [Use π=22/7]

Concept: Conversion of Solid from One Shape to Another