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Mathematics All India Set 2 2013-2014 CBSE Class 10 Question Paper Solution

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Mathematics
All India Set 2
2013-2014 March
Marks: 90

 (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]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: Probability Examples and Solutions
Chapter: [5.01] Probability
[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`

Concept: Heights and Distances
Chapter: [4.01] 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°

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

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

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

Concept: Pythagoras Theorem
Chapter: [3.02] Triangles
[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

Concept: Pythagoras Theorem
Chapter: [3.02] Triangles
[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`

 
Concept: Probability Examples and Solutions
Chapter: [5.01] Probability
[2]9

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

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

Concept: Triangles Examples and Solutions
Chapter: [3.02] Triangles
[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.

Concept: Probability Examples and Solutions
Chapter: [5.01] Probability
[2]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: [7.02] Surface Areas and Volumes
[2]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 AP
Chapter: [2.02] Arithmetic Progressions
[2]14

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

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

Concept: Heights and Distances
Chapter: [4.01] 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. (Use\[\pi = \frac{22}{7}\]).

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

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

 

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

Concept: Division of a Line Segment
Chapter: [3.03] Constructions
[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]

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

Solve for x:

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

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

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

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

Concept: Pythagoras Theorem
Chapter: [3.02] Triangles
[4]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: [3.01] Circles
[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.

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

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

Concept: Heights and Distances
Chapter: [4.01] 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.

Concept: Algebraic Methods of Solving a Pair of Linear Equations - Cross - Multiplication Method
Chapter: [2.01] Pair of Linear Equations in Two Variables
[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?

Concept: Probability Examples and Solutions
Chapter: [5.01] Probability
[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
Chapter: [2.03] Quadratic Equations
[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 of red colour.

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

Concept: Area of a Triangle
Chapter: [6.01] Lines (In Two-dimensions)
[4]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: [3.01] Circles

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