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# Mathematics Delhi Set 1 2013-2014 CBSE Class 10 Question Paper Solution

Course
Mathematics [Delhi Set 1]
Date: March 2014

1

The first three terms of an AP respectively are 3y – 1, 3y + 5 and 5y + 1. Then y equals:

(A) –3
(B) 4
(C) 5
(D) 2

Concept: Arithmetic Progression
Chapter: [2.02] Arithmetic Progressions
2

In Fig. 1, QR is a common tangent to the given circles, touching externally at the point T. The tangent at T meets QR at P. If PT = 3.8 cm, then the length of QR (in cm) is : (A) 3.8
(B) 7.6
(C) 5.7
(D) 1.9

Concept: Circles Examples and Solutions
Chapter: [3.01] Circles
3

n Fig. 2, PQ and PR are two tangents to a circle with centre O. If ∠QPR = 46°, then ∠QOR equals: (A) 67°
(B) 134°
(C) 44°
(D) 46°

Concept: Circles Examples and Solutions
Chapter: [3.01] Circles
4

A ladder makes an angle of 60° with the ground when placed against a wall. If the foot of the ladder is 2 m away from the wall, then the length of the ladder (in metres) is:

(A) 4/sqrt3

(B) 4sqrt3

(C) 2sqrt2

(D)4

Concept: Heights and Distances
Chapter: [4.01] Heights and Distances
5

If two different dice are rolled together, the probability of getting an even number on both dice, is:

(A) 1/36
(B) 1/2
(C) 1/6
(D) 1/4

Concept: Probability Examples and Solutions
Chapter: [5.01] Probability
6

A number is selected at random from the numbers 1 to 30. The probability that it is a prime number is:

(A) 2/3

(B) 1/6

(C) 1/3

(D) 11/30

Concept: Probability Examples and Solutions
Chapter: [5.01] Probability
7

If the points A(x, 2), B(−3, −4) and C(7, − 5) are collinear, then the value of x is:

(A) −63
(B) 63
(C) 60
(D) −60

Concept: Area of a Triangle
Chapter: [6.01] Lines (In Two-dimensions)
8

The number of solid spheres, each of diameter 6 cm that can be made by melting a solid metal cylinder of height 45 cm and diameter 4 cm, is:

3

5

4

6

Concept: Surface Area of a Combination of Solids
Chapter: [7.02] Surface Areas and Volumes
9

Solve the quadratic equation 2x2 + ax − a2 = 0 for x.

Concept: Nature of Roots
10

The first and the last terms of an AP are 5 and 45 respectively. If the sum of all its terms is 400, find its common difference.

Concept: Sum of First n Terms of an AP
Chapter: [2.02] Arithmetic Progressions
11

Prove that the line segment joining the points of contact of two parallel tangents of a circle, passes through its centre.

Concept: Circles Examples and Solutions
Chapter: [3.01] Circles
12

If from an external point P of a circle with centre O, two tangents PQ and PR are drawn such that ∠QPR = 120°, prove that 2PQ = PO.

Concept: Number of Tangents from a Point on a Circle
Chapter: [3.01] Circles
13

Rahim tosses two different coins simultaneously. Find the probability of getting at least one tail.

Concept: Probability Examples and Solutions
Chapter: [5.01] Probability
14

In fig. 3, a square OABC is inscribed in a quadrant OPBQ of a circle. If OA = 20 cm, find the area of the shaded region. (Use π = 3.14) Concept: Areas of Combinations of Plane Figures
Chapter: [7.01] Areas Related to Circles
15

Solve the equation 4/x-3=5/(2x+3); xne0,-3/2 for x .

Concept: Solutions of Quadratic Equations by Factorization
16

If the seventh term of an AP is 1/9 and its ninth term is 1/7, find its 63rd term.

Concept: Arithmetic Progression
Chapter: [2.02] Arithmetic Progressions
17

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

Concept: Construction of Tangents to a Circle
Chapter: [3.03] Constructions
18

If the point A(0, 2) is equidistant from the points B(3, p) and C(p, 5), find p. Also, find the length of AB.

Concept: Distance Formula
Chapter: [6.01] Lines (In Two-dimensions)
19

Two ships are there in the sea on either side of a light house in such a way that the ships and the light house are in the same straight line. The angles of depression of two ships as observed from the top of the light house are 60° and 45°. If the height of the light house is 200 m, find the distance between the two ships. [use √3=1.73]

Concept: Heights and Distances
Chapter: [4.01] Heights and Distances
20

If the points A(−2, 1), B(a, b) and C(4, −1) are collinear and a − b = 1, find the values of a and b.

Concept: Area of a Triangle
Chapter: [6.01] Lines (In Two-dimensions)
21

In  Fig 4, a circle is inscribed in an equilateral triangle ABC of side 12 cm. Find the radius of inscribed circle and the area of the shaded region.[Use π=3.14 and √3=1.73] Concept: Problems Based on Areas and Perimeter Or Circumference of Circle, Sector and Segment of a Circle
Chapter: [7.01] Areas Related to Circles
22

In PSR, RTQ and PAQ are three semicircles of diameters 10 cm, 3 cm and 7 cm respectively. Find the perimeter of the shaded region. [Use π=3.14] Concept: Problems Based on Areas and Perimeter Or Circumference of Circle, Sector and Segment of a Circle
Chapter: [7.01] Areas Related to Circles
23

A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank which is 10 m in diameter and 2 m deep. If the water flows through the pipe at the rate of 4 km per hour, in how much time will the tank be filled completely?

Concept: Surface Area of a Combination of Solids
Chapter: [7.02] Surface Areas and Volumes
24

A solid metallic right circular cone 20 cm high and whose vertical angle is 60°, is cut into two parts at the middle of its height by a plane parallel to its base. If the frustum so obtained be drawn into a wire of diameter 1/12 cm, find the length of the wire.

Concept: Surface Areas and Volumes Examples and Solutions
Chapter: [7.02] Surface Areas and Volumes
25

The difference of two natural numbers is 5 and the difference of their reciprocals is 1/10. Find the numbers

Concept: Algebraic Methods of Solving a Pair of Linear Equations - Substitution Method
Chapter: [2.01] Pair of Linear Equations in Two Variables
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: [3.01] Circles
27

The angles of elevation and depression of the top and the bottom of a tower from the top of a building, 60 m high, are 30° and 60° respectively. Find the difference between the heights of the building and the tower and the distance between them.

Concept: Heights and Distances
Chapter: [4.01] Heights and Distances
28

A bag contains cards numbered from 1 to 49. A card is drawn from the bag at random, after mixing the cards thoroughly. Find the probability that the number on the drawn card is an odd number.

Concept: Probability Examples and Solutions
Chapter: [5.01] Probability

A bag contains cards numbered from 1 to 49. A card is drawn from the bag at random, after mixing the cards thoroughly. Find the probability that the number on the drawn card is multiple of 5.

Concept: Probability Examples and Solutions
Chapter: [5.01] Probability

A bag contains cards numbered from 1 to 49. A card is drawn from the bag at random, after mixing the cards thoroughly. Find the probability that the number on the drawn card is a perfect square.

Concept: Probability Examples and Solutions
Chapter: [5.01] Probability

A bag contains cards numbered from 1 to 49. A card is drawn from the bag at random, after mixing the cards thoroughly. Find the probability that the number on the drawn card is an even prime number.

Concept: Probability Examples and Solutions
Chapter: [5.01] Probability
29

Find the ratio in which the point P(x, 2) divides the line segment joining the points A(12, 5) and B(4, −3). Also, find the value of x.

Concept: Section Formula
Chapter: [6.01] Lines (In Two-dimensions)
30

Find the values of k for which the quadratic equation (k + 4) x2 + (k + 1) x + 1 = 0 has equal roots. Also find these roots.

Concept: Nature of Roots
31

In an AP of 50 terms, the sum of first 10 terms is 210 and the sum of its last 15 terms is 2565. Find the A.P.

Concept: Arithmetic Progression
Chapter: [2.02] Arithmetic Progressions
32

Prove that a parallelogram circumscribing a circle is a rhombus.

Concept: Number of Tangents from a Point on a Circle
Chapter: [3.01] Circles
33

Sushant has a vessel, of the form of an inverted cone, open at the top, of height 11 cm and radius of top as 2.5 cm and is full of water. Metallic spherical balls each of diameter 0.5 cm are put in the vessel due to which 2/th of the water in the vessel flows out. Find how many balls were put in the vessel. Sushant made the arrangement so that the water that flows out irrigates the flower beds. What value has been shown by Sushant?

Concept: Surface Areas and Volumes Examples and Solutions
Chapter: [7.02] Surface Areas and Volumes
34

From a solid cylinder of height 2.8 cm and diameter 4.2 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid [take π=22/7]

Concept: Surface Area of a Combination of Solids
Chapter: [7.02] Surface Areas and Volumes

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