# Mathematics All India Set 2 2013-2014 English Medium Class 10 Question Paper Solution

Mathematics [All India Set 2]
Date: March 2014

(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.

 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

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

Concept: Heights and Distances
Chapter: [0.040999999999999995] Heights and Distances
 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°

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

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

Concept: Arithmetic Progression
Chapter: [0.022000000000000002] Arithmetic Progressions
 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

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
 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

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

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

Concept: Basic Ideas of Probability
Chapter: [0.051] Probability [0.051] Probability
 9

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

Concept: Triangles Examples and Solutions
Chapter: [0.032] Triangles
 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.

Concept: Basic Ideas of Probability
Chapter: [0.051] Probability [0.051] Probability
 12

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

Concept: Surface Area of a Combination of Solids
Chapter: [0.07200000000000001] Surface Areas and Volumes
 13

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

Concept: Sum of First ‘n’ Terms of an Arithmetic Progressions
Chapter: [0.022000000000000002] Arithmetic Progressions
 14

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

Concept: Nature of Roots of a Quadratic Equation
 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.

Concept: Heights and Distances
Chapter: [0.040999999999999995] Heights and Distances
 16

The largest possible sphere is carved out of a wooden solid cube of side 7 em. Find the volume of the wood left. (Use$\pi = \frac{22}{7}$).

Concept: Surface Area of a Combination of Solids
Chapter: [0.07200000000000001] Surface Areas and Volumes
 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?

Concept: Surface Area of a Combination of Solids
Chapter: [0.07200000000000001] Surface Areas and Volumes
 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] Concept: Surface Area of a Combination of Solids
Chapter: [0.07200000000000001] Surface Areas and Volumes
 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.

Concept: Division of a Line Segment
Chapter: [0.033] Constructions
 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] Concept: Circumference of a Circle
Chapter: [0.071] Areas Related to Circles
 21

Solve for x:

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

Concept: Solutions of Quadratic Equations by Completing the Square
 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.

Concept: Arithmetic Progression
Chapter: [0.022000000000000002] Arithmetic Progressions
 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.

Concept: Construction of Tangents to a Circle
Chapter: [0.033] Constructions
 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.

Concept: Right-angled Triangles and Pythagoras Property
Chapter: [0.032] Triangles
 25

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: [0.031] Circles
 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.

Concept: Surface Area of a Combination of Solids
Chapter: [0.07200000000000001] Surface Areas and Volumes
 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]

Concept: Frustum of a Cone
Chapter: [0.07200000000000001] Surface Areas and Volumes
 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]

Concept: Heights and Distances
Chapter: [0.040999999999999995] Heights and Distances
 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.

Concept: Algebraic Methods of Solving a Pair of Linear Equations - Cross - Multiplication Method
Chapter: [0.021] Pair of Linear Equations in Two Variables
 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?

Concept: Basic Ideas of Probability
Chapter: [0.051] Probability [0.051] Probability
 31

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

Concept: Solutions of Quadratic Equations by Factorization
 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 of red colour.

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

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  an ace.

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

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 a queen.

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

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 a face card.

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

Concept: Area of a Triangle
Chapter: [0.061] Lines (In Two-dimensions)
 34

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

Concept: Number of Tangents from a Point on a Circle
Chapter: [0.031] Circles

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