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# Mathematics All India Set 2 2014-2015 CBSE Class 10 Question Paper Solution

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Mathematics [All India Set 2]
Marks: 90Academic Year: 2014-2015
Date: March 2015

General Instructions:
(i) All questions are compulsory.
(ii) The question paper consists of 31 questions divided into four    sections – A, B, C and D.
(iii) Section A contains 4 questions of 1 mark each. Section B contains  6 questions of 2 marks each, Section C contains 10 questions of 3 marks each and Section D contains 11 questions of 4 marks each.
(iv) Use of calculators is not permitted.

1

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: [3.01] Circles
2

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

Concept: Nature of Roots
Chapter: [2.03] Quadratic Equations
3

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: [4.03] Introduction to Trigonometry [4.03] Introduction to Trigonometry
4

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: [5.01] Probability [5.01] Probability
5

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: [6.01] Lines (In Two-dimensions)
6

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 AP
Chapter: [2.02] Arithmetic Progressions
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: [3.01] Circles
8

In the below 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 Δ ABC is 54 cm2, then find the lengths of sides AB and AC. Concept: Triangles Examples and Solutions
Chapter: [3.02] Triangles
9

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

Concept: Solutions of Quadratic Equations by Completing the Square
Chapter: [2.03] Quadratic Equations
10

If A(4, 3), B(-1, y) and C(3, 4) are the vertices of a right triangle ABC, right-angled at A, then find the value of y.

Concept: Distance Formula
Chapter: [6.01] Lines (In Two-dimensions)
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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: [7.02] Surface Areas and Volumes
12

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: [4.01] Heights and Distances
13

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: [7.02] Surface Areas and Volumes
14

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: Basic Ideas of Probability
Chapter: [5.01] Probability [5.01] Probability
15

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: [7.02] Surface Areas and Volumes
16

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
Chapter: [3.03] Constructions
17

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: [7.02] Surface Areas and Volumes
18

All the vertices of a rhombus lie on a circle. Find the area of the rhombus, if the area of the circle is 1256 cm2. [Use p = 3.14]

Concept: Circumference of a Circle
Chapter: [7.01] Areas Related to Circles [7.01] Areas Related to Circles
19

Solve for x :  2x^2+6sqrt3x-60=0

Concept: Nature of Roots
Chapter: [2.03] Quadratic Equations
20

The 16th term of an A.P. is five times its third term. If its 10th term is 41, then find the sum of its first fifteen terms.

Concept: Arithmetic Progression
Chapter: [2.02] Arithmetic Progressions
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21

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: [3.01] Circles [3.01] Circles
22

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: [4.01] Heights and Distances
23

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: [5.01] Probability [5.01] 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: [5.01] Probability [5.01] 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: [5.01] Probability [5.01] 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: [5.01] Probability [5.01] Probability
24

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: [7.01] Areas Related to Circles [7.01] Areas Related to Circles
25

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: [7.02] Surface Areas and Volumes
26

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: [3.02] Triangles
27

Find the 60th 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
Chapter: [2.02] Arithmetic Progressions
28

A bus travels at a certain average speed for a distance of 75 km and then travels a distance of 90 km at an average speed of 10 km/h more than the first speed. If it takes 3 hours to complete the total journey, find its first speed?

Concept: Heights and Distances
Chapter: [4.01] Heights and Distances
29

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

Concept: Tangent to a Circle
Chapter: [3.01] Circles
30

Construct a right triangle ABC with AB = 6 cm, BC = 8 cm and ∠B = 90°. Draw BD, the perpendicular from B on AC. Draw the circle through B, C and D and construct the tangents from A to this circle.

Concept: Constructions Examples and Solutions
Chapter: [3.03] Constructions
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31

Find the values of k so that the area of the triangle with vertices (k + 1, 1), (4, -3) and (7, -k) is 6 sq. units.

Concept: Area of a Triangle
Chapter: [6.01] Lines (In Two-dimensions)

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