# Mathematics All India Set 3 2014-2015 English Medium Class 10 Question Paper Solution

Mathematics [All India Set 3]
Date: March 2015

[1] 1

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: Trigonometry
Chapter: [0.043] Introduction to Trigonometry [0.043] Introduction to Trigonometry
[1] 2

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: Basic Ideas of Probability
Chapter: [0.051] Probability [0.051] Probability
[1] 3

If the quadratic equation px2 − 2√5px + 15 = 0 has two equal roots then find the value of p.

Concept: Nature of Roots of a Quadratic Equation
[1] 4

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
Chapter: [0.031] Circles
[2] 5

In an A.P., if S5 + S7 = 167 and S10=235, then find the A.P., where Sn denotes the sum of its first n terms.

Concept: Sum of First ‘n’ Terms of an Arithmetic Progressions
Chapter: [0.022000000000000002] Arithmetic Progressions
[2] 6

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: Right-angled Triangles and Pythagoras Property
Chapter: [0.032] Triangles
[2] 7

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
Chapter: [0.031] Circles
[2] 8

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 cm2, then find the lengths of sides AB and AC.

Concept: Triangles Examples and Solutions
Chapter: [0.032] Triangles
[2] 9

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
Chapter: [0.061] Lines (In Two-dimensions)
[2] 10

Solve the following quadratic equation for x: x2 – 2ax – (4b2 – a2) = 0

Concept: Solutions of Quadratic Equations by Factorization
[3] 11

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
Chapter: [0.07200000000000001] Surface Areas and Volumes
[3] 12

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
Chapter: [0.07200000000000001] Surface Areas and Volumes
[3] 13

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
Chapter: [0.07200000000000001] Surface Areas and Volumes
[3] 14

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
Chapter: [0.07200000000000001] Surface Areas and Volumes
[3] 15

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

Concept: Nature of Roots of a Quadratic Equation
[3] 16

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
Chapter: [0.040999999999999995] Heights and Distances
[3] 17

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
Chapter: [0.071] Areas Related to Circles
[3] 18

The 13th term of an A.P. is four times its 3rd term. If its 5th term is 16, then find the sum of its first ten terms.

Concept: Arithmetic Progression
Chapter: [0.022000000000000002] Arithmetic Progressions
[3] 19

Find the coordinates of a point P on the line segment joining A(1, 2) and B(6, 7) such that AP =(2/5)AB.

Concept: Section Formula
Chapter: [0.061] Lines (In Two-dimensions)
[3] 20

A bag contains, white, black and red balls only. A ball is drawn at random from the bag.If the probability of getting a white ball is 3/10 and that of a black ball is 2/5 then find the
probability of getting a red ball. If the bag contains 20 black balls, then find the total number of balls in the bag.

Concept: Basic Ideas of Probability
Chapter: [0.051] Probability [0.051] Probability
[4] 21

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
Chapter: [0.040999999999999995] Heights and Distances
[4] 22

A card is drawn at random from a well-shuffled deck of playing cards. Find the probability that the card drawn is a card of spade or an ace.

Concept: Basic Ideas of Probability
Chapter: [0.051] Probability [0.051] Probability

A card is drawn at random from a well-shuffled deck of playing cards. Find the probability that the card drawn is a black king.

Concept: Basic Ideas of Probability
Chapter: [0.051] Probability [0.051] Probability

A card is drawn at random from a well-shuffled deck of playing cards. Find the probability that the card drawn is neither a jack nor a king.

Concept: Basic Ideas of Probability
Chapter: [0.051] Probability [0.051] Probability

A card is drawn at random from a well-shuffled deck of playing cards. Find the probability that the card drawn is either a king or a queen.

Concept: Basic Ideas of Probability
Chapter: [0.051] Probability [0.051] Probability
[4] 23

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: Circumference of a Circle
Chapter: [0.071] Areas Related to Circles
[4] 24

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
Chapter: [0.07200000000000001] Surface Areas and Volumes
[4] 25

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: Right-angled Triangles and Pythagoras Property
Chapter: [0.032] Triangles
[4] 26

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
Chapter: [0.031] Circles
[4] 27

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: Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles
Chapter: [0.031] Circles [0.031] Circles
[4] 28

A truck covers a distance of 150 km at a certain average speed and then covers another 200 km at an average speed which is 20 km per hour more than the first speed. If the truck covers the total distance in 5 hours, find the first speed of the truck.

Concept: Heights and Distances
Chapter: [0.040999999999999995] Heights and Distances
[4] 29

An arithmetic progression 5, 12, 19, …. has 50 terms. Find its last term. Hence find the sum of its last 15 terms.

Concept: Arithmetic Progression
Chapter: [0.022000000000000002] Arithmetic Progressions
[4] 30

Construct a triangle ABC in which AB = 5 cm, BC = 6 cm and ∠ABC = 60˚. Now construct another triangle whose sides are 5/7 times the corresponding sides of ΔABC.

Concept: Division of a Line Segment
Chapter: [0.033] Constructions
[4] 31

Find the values of k for which the points A(k + 1, 2k), B(3k, 2k + 3) and (5k – 1, 5k) are collinear.

Concept: Area of a Triangle
Chapter: [0.061] Lines (In Two-dimensions)

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