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# Question Paper Solutions - Mathematics 2013 - 2014 CBSE Class 10

SubjectMathematics
Year2013 - 2014 (March)

#### Alternate Sets

(i) All questions are compulsory.
(ii) The question paper consists of 34 questions divided           into four sections -A, B, C and D.
(iii) Section A contains 8 questions of 1 mark each, which       are multiple choice type questions, Section B contains       6 questions of 2 marks each, Section C contains 10
questions of 3 marks each and Section D contains 10         questions of 4 marks each.
(iv) Use of calculators is not permitted.

Marks: 90
[1]1

The probability that a number selected at random from the numbers 1, 2, 3, ..., 15 is a multiple of 4, is

(A)4/15

(B)2/15

(C)1/5

(D)1/3

Chapter: [4] Probability
Concept: Probability Examples and Solutions
[1]2

The angle of depression of a car parked on the road from the top of a 150 m high tower is 30°. The distance of the car from the tower (in metres) is

(A) 50sqrt3

(B) 150sqrt 3

(C) 150sqrt2

(D) 75

Chapter: [5] Trigonometry
Concept: Heights and Distances
[1]3

Two circles touch each other externally at P. AB is a common tangent to the circles touching them at A and B. The value of ∠ L APB is

(A) 30°

(B) 45°

(C) 60°

(D) 90°

Chapter: [3.02] Circles
Concept: Circles Examples and Solutions
[1]4

If k, 2k- 1 and 2k + 1 are three consecutive terms of an A.P., the value of k is

(A) 2

(B) 3

(C) -3

(D) 5

Chapter: [1] Arithmetic Progression
Concept: Arithmetic Progression
[1]5

A chord of a circle of radius 10 em subtends a right angle at its centre. The length of the chord (in em) is

(A) 5sqrt 2

(B) 10 sqrt2

(C)5/sqrt2

(D) 10sqrt 3

Chapter: [3.02] Circles
Concept: Circles Examples and Solutions
[1]6

ABCD is a rectangle whose three vertices are B (4, 0), C(4, 3) and D(0,3). The length of one of its diagonals is
(A) 5
(B) 4
(C) 3
(D) 25

Chapter: [1] Similarity
Concept: Pythagoras Theorem
[1]7

In a right triangle ABC, right-angled at B, BC = 12 cm and AB = 5 cm. The radius of the circle inscribed in the triangle (in cm) is
(A) 4
(B) 3
(C) 2
(D) 1

Chapter: [1] Similarity
Concept: Pythagoras Theorem
[1]8

In a family of 3 children, the probability of having at least one boy is

(A)7/8

(B)1/8

(C)5/8

(D)3/4

Chapter: [4] Probability
Concept: Probability Examples and Solutions
[2]9

In Figure 1, common tangents AB and CD to the two circles with centres 01and 0intersect at E. Prove that AB = CD.

Chapter: [3.02] Circles
Concept: Circles Examples and Solutions
[2]10

The incircle of an isosceles triangle ABC, in which AB = AC, touches the sides BC, CA and AB at D, E and F respectively. Prove that BD = DC.

Chapter: [3.01] Triangles
Concept: Triangles Examples and Solutions
[2]11

Two different dice are tossed together. Find the probability
(i) that the number on each die is even.
(ii) that the sum of numbers appearing on the two dice is 5.

Chapter: [4] Probability
Concept: Probability Examples and Solutions
[2]12

If the total surface area of a solid hemisphere is 462 cm2 , find its volume.[Take π=22/7]

Chapter: [7.02] Surface Areas and Volumes
Concept: Surface Area of a Combination of Solids
[2]13

Find the number of natural numbers between 101 and 999 which are divisible by both 2 and 5.

Chapter: [2.04] Arithmetic Progressions
Concept: Sum of First n Terms of an AP
[2]14

Find the values of k for which the quadratic equation 9x2 - 3kx + k = 0 has equal roots.

Concept: Nature of Roots
[3]15

The angle of elevation of an aeroplane from a point on the ground is 60°. After a flight of 30 seconds the angle of elevation becomes 300 If the aeroplane is flying at a constant height of 3000 3 m, find the speed of the aeroplane.

Chapter: [5] Trigonometry
Concept: Heights and Distances
[3]16

The largest possible sphere is carved out of a wooden solid cube of side 7 em. Find the volume of the wood left

Chapter: [7.02] Surface Areas and Volumes
Concept: Surface Area of a Combination of Solids
[3]17

Water in a canal, 6 m wide and 1.5 m deep, is flowing at a speed of 4 km/h. How much area will it irrigate in 10 minutes, if 8 cm of standing water is needed for irrigation?

Chapter: [7.02] Surface Areas and Volumes
Concept: Surface Area of a Combination of Solids
[3]18

In Figure 2, ABCD is a trapezium of area 24.5 sq. cm. In it, AD|| BC, ∠ DAB = 900, AD = 10 cm and BC = 4 cm. If ABE is a quadrant of a circle, find the area of the shaded region. [Take π=22/7]

Chapter: [7.02] Surface Areas and Volumes
Concept: Surface Area of a Combination of Solids
[3]19

Find the ratio in which the line segment joining the points A(3,- 3) and B(- 2, 7) is divided by x-axis. Also find the coordinates of the point of division.

Chapter: [4] Geometric Constructions
Concept: Division of a Line Segment
[3]20

In Figure , two concentric circles with centre O, have radii 21cm and 42 cm. If ∠ AOB = 60°, find the area of the shaded region. [use π=22/7]

Chapter: [7.01] Areas Related to Circles
Concept: Problems Based on Areas and Perimeter Or Circumference of Circle, Sector and Segment of a Circle
[3]21

Solve for x:

16/x-1=15/(x+1);x!=0,-1

Concept: Solutions of Quadratic Equations by Completing the Square
[3]22

The sum of the 2nd and the 7th terms of an AP is 30. If its 15th term is 1 less than twice its 8th term, find the AP.

Chapter: [1] Arithmetic Progression
Concept: Arithmetic Progression
[3]23

Draw a line segment AB of length 8 cm. Taking A as centre, draw a circle of radius 4 cm and taking B as centre, draw another circle of radius 3 cm. Construct tangents to each circle from the centre of the other circle.

Chapter: [3.05] Constructions
Concept: Construction of Tangents to a Circle
[3]24

Prove that the diagonals of a rectangle ABCD, with vertices A(2, -1), B(5, -1), C(5, 6) and D(2, 6), are equal and bisect each other.

Chapter: [1] Similarity
Concept: Pythagoras Theorem
[4]25

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

Chapter: [3.04] Circles
Concept: Tangent to a Circle
[4]26

150 spherical marbles, each of diameter 1.4 cm, are dropped in a cylindrical vessel of diameter 7 cm containing some water, which are completely immersed in  water. Find the rise in the level of water in the vessel.

Chapter: [7.02] Surface Areas and Volumes
Concept: Surface Area of a Combination of Solids
[4]27

A container open at the top, is in the form of a frustum of a cone of height 24 cm with radii of its lower and upper circular ends, as 8 cm and 20 cm respectively. Find the cost of milk which can completely fill the container. at the rate of 21 per litre. [use π=22/7]

Chapter: [7.02] Surface Areas and Volumes
Concept: Frustum of a Cone
[4]28

The angle of elevation of the top of a tower at a distance of 120 m from a point A on the ground is 45°. If the angle of elevation of the top of a flagstaff fixed at the top of the tower, at A is 60°, then find the height of the flagstaff. [use √3=1.73]

Chapter: [5] Trigonometry
Concept: Heights and Distances
[4]29

A motorboat whose speed in still water is 18 km/h, takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.

Chapter: [3] Linear equations in two variables
Concept: Algebraic Methods of Solving a Pair of Linear Equations - Cross - Multiplication Method
[4]30

In a school, students decided to plant trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be double of the class in which they are studying. If there are 1to 12 classes in the school and each class has two sections, find how many trees were
planted by the students. Which value is shown in this question?

Chapter: [4] Probability
Concept: Probability Examples and Solutions
[4]31

Solve for x: (x-3)/(x-4)+(x-5)/(x-6)=10/3; x!=4,6

Concept: Solutions of Quadratic Equations by Factorization
[4]32

All the red face cards are removed from a pack of 52 playing cards. A card is drawn at random from the remaining cards, after reshuffling them. Find the probability that the drawn card is
(i) of red colour
(ii) a queen
(iii) an ace
(iv) a face card

Chapter: [4] Probability
Concept: Probability Examples and Solutions
[4]33

A(4, - 6), B(3,- 2) and C(5, 2) are the vertices of a 8 ABC and AD is its median. Prove that the median AD divides Δ ABC into two triangles of equal areas.

Chapter: [6.01] Lines (In Two-dimensions)
Concept: Area of a Triangle
[4]34

Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

Chapter: [2] Circle
Concept: Number of Tangents from a Point on a Circle

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