# Mathematics Foreign Set 3 2013-2014 CBSE (English Medium) Class 10 Question Paper Solution

Mathematics [Foreign Set 3]
Date: March 2014

[1] 1

Two different coins are tossed simultaneously. The probability of getting at least one head is

(A)1/2

(B)1/8

(C)3/4

(D)7/8

Concept: Basic Ideas of Probability
Chapter: [5.01] Probability [5.01] Probability
[1] 2

A bag contains cards numbered from 1 to 25. A card is drawn at random from the bag. The probability that the number on this card is divisible by both 2 and 3 is

(A)1/5

(B)3/25

(C)4/25

(D)2/25

Concept: Basic Ideas of Probability
Chapter: [5.01] Probability [5.01] Probability
[1] 3

If the height of a vertical pole is 3–√3 times the length of its shadow on the ground, then the angle of elevation of the Sun at that time is
(A) 30°
(B) 60°
(C) 45°
(D) 75°

Concept: Heights and Distances
Chapter: [4.01] Heights and Distances
[1] 4

The perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0) is

(A)$7 + \sqrt{5}$
(B) 5
(C) 10
(D) 12

Concept: Right-angled Triangles and Pythagoras Property
Chapter: [3.02] Triangles
[1] 5

A rectangular sheet of paper 40 cm × 22 cm, is rolled to form a hollow cylinder of height 40 cm. The radius of the cylinder (in cm) is

3.5

7

$\frac{80}{7}$

5

Concept: Concept of Surface Area, Volume, and Capacity
Chapter: [7.02] Surface Areas and Volumes [7.02] Surface Areas and Volumes
[1] 6

In Figure 1, a quadrilateral ABCD is drawn to circumscribe a circle such that its sides AB, BC, CD and AD touch the circle at P, Q, R and S respectively. If AB = x cm, BC = 7 cm, CR = 3 cm and AS = 5 cm, find x.

(A) 10
(B) 9
(C) 8
(D) 7

Concept: Number of Tangents from a Point on a Circle
Chapter: [3.01] Circles
[1] 7

The next term of the A.P.$\sqrt{7}, \sqrt{28}, \sqrt{63},$

(A)$\sqrt{70}$

(B)$\sqrt{84}$

(C)$\sqrt{97}$

(D)$\sqrt{112}$

Concept: Sum of First n Terms of an A.P.
Chapter: [2.02] Arithmetic Progressions
[1] 8

Two concentric circles are of radii 5 cm and 3 cm. Length of the chord of the larger circle, (in cm), which touches the smaller circle is
(A) 4
(B) 5
(C) 8
(D) 10

Concept: Number of Tangents from a Point on a Circle
Chapter: [3.01] Circles
[2] 9

In Figure 2, XP and XQ are two tangents to the circle with centre O, drawn from an external point X. ARB is another tangent, touching the circle at R. Prove that XA + AR = XB + BR ?

Concept: Tangent to a Circle
Chapter: [3.01] Circles
[2] 10

In Figure 3, OABC is a quadrant of a circle of radius 7 cm. If OD = 4 cm, find the area of the shaded region  ?$[Use\pi = \frac{22}{7}]$

Concept: Concept of Surface Area, Volume, and Capacity
Chapter: [7.02] Surface Areas and Volumes [7.02] Surface Areas and Volumes
[2] 11

Solve for x:

$\sqrt{3} x^2 - 2\sqrt{2}x - 2\sqrt{3} = 0$
Concept: Pair of Linear Equations in Two Variables
Chapter: [2.01] Pair of Linear Equations in Two Variables
[2] 12

Two different dice are rolled simultaneously. Find the probability that the sum of numbers appearing on the two dice is 10 ?

Concept: Basic Ideas of Probability
Chapter: [5.01] Probability [5.01] Probability
[2] 13

Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

Concept: Number of Tangents from a Point on a Circle
Chapter: [3.01] Circles
[2] 14

The sum of the first n terms of an A.P. is 4n2 + 2n. Find the nth term of this A.P ?

Concept: nth Term of an AP
Chapter: [2.02] Arithmetic Progressions
[3] 15

In Figure 4, from a rectangular region ABCD with AB = 20 cm, a right triangle AED with AE = 9 cm and DE = 12 cm, is cut off. On the other end, taking BC as diameter, a semicircle is added on outside the region. Find the area of the shaded region.$[Use\pi = 3 . 14]$

Concept: Surface Area of a Combination of Solids
Chapter: [7.02] Surface Areas and Volumes
[3] 16

In Figure 5, ABCD is a quadrant of a circle of radius 28 cm and a semi circle BEC is drawn with BC as diameter. Find the area of the shaded region ?$[Use\pi = \frac{22}{7}]$

Concept: Areas of Combinations of Plane Figures
Chapter: [7.01] Areas Related to Circles
[3] 17

Points P, Q, R and S divide the line segment joining the points A(1,2) and B(6,7) in five equal parts. Find the coordinates of the points P,Q and R

Concept: Coordinate Geometry
Chapter: [6.01] Lines (In Two-dimensions) [6.01] Lines (In Two-dimensions)
[3] 18

The sum of the first 7 terms of an A.P. is 63 and the sum of its next 7 terms is 161. Find the 28th term of this A.P ?

Concept: Arithmetic Progressions Examples and Solutions
Chapter: [2.02] Arithmetic Progressions
[3] 19

A 5 m wide cloth is used to make a conical tent of base diameter 14 m and height 24 m. Find the cost of cloth used at the rate of Rs 25 per metre ?$[Use \pi = \frac{22}{7}]$

Concept: Frustum of a Cone
Chapter: [7.02] Surface Areas and Volumes
[3] 20

A girl empties a cylindrical bucket, full of sand, of base radius 18 cm and height 32 cm, on the floor to form a conical heap of sand. If the height of this conical heap is 24 cm, then find its slant height correct upto one place of decimal?

Concept: Concept of Surface Area, Volume, and Capacity
Chapter: [7.02] Surface Areas and Volumes [7.02] Surface Areas and Volumes
[3] 21

Two ships are approaching a light-house from opposite directions. The angles of depression of the two ships from the top of the light-house are 30° and 45°. If the distance between the two ships is 100 m, find the height of the light-house. $[Use \sqrt{3} = 1 . 732]$

Concept: Heights and Distances
Chapter: [4.01] Heights and Distances
[3] 22

If 1 is a root of the quadratic equation 3x2 + ax – 2 = 0 and the quadratic equation a(x2 + 6x) – b = 0 has equal roots, find the value of b ?

Concept: Nature of Roots of a Quadratic Equation
[3] 23

Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other an angle of 60º.

Concept: Construction of Tangents to a Circle
Chapter: [3.03] Constructions
[3] 24

Find the value(s) of p for which the points (3p + 1, p), (p + 2, p – 5) and (p + 1, –p) are collinear ?

Concept: Area of a Triangle
Chapter: [6.01] Lines (In Two-dimensions)
[4] 25

The angle of elevation of the top of a chimney form the foot of a tower is 60° and the angle of depression of the foot of the chimney from the top of the tower is 30° . If the height of the tower is 40 meters. Find the height of the chimney.

Concept: Heights and Distances
Chapter: [4.01] Heights and Distances
[4] 26

A quadrilateral is drawn to circumscribe a circle. Prove that the sums of opposite sides are equal ?

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
[4] 27

The midpoint P of the line segment joining points A(-10, 4) and B(-2, 0) lies on the line segment joining the points C(-9, -4) and D(-4, y). Find the ratio in which P divides CD. Also, find the value of y.

Concept: Coordinate Geometry
Chapter: [6.01] Lines (In Two-dimensions) [6.01] Lines (In Two-dimensions)
[4] 28

A hemispherical depression is cut out from one face of a cubical block of side 7 cm, such that the diameter of the hemisphere is equal to the edge of the cube. Find the surface area of the remaining solid. [Use π = $\frac{22}{7}$]

Concept: Volume of a Combination of Solids
Chapter: [7.02] Surface Areas and Volumes
[4] 29

If Sn denotes the sum of the first n terms of an A.P., prove that S30 = 3 (S20 − S10) ?

Concept: nth Term of an AP
Chapter: [2.02] Arithmetic Progressions
[4] 30

A metallic bucket, open at the top, of height 24 cm is in the form of the frustum of a cone, the radii of whose lower and upper circular ends are 7 cm and 14 cm respectively. Find :
(i) the volume of water which can completely fill the bucket.
(ii) the area of the metal sheet used to make the bucket.
[Use π =$\frac{22}{7}$

Concept: Frustum of a Cone
Chapter: [7.02] Surface Areas and Volumes
[4] 31

The sum of the squares of two consecutive multiples of 7 is 637. Find the multiples ?

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

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] 33

A piggy bank contains hundred 50 p coins, fifty Rs 1 coins, twenty Rs 2 coins and ten Rs 5 coins. If it is equally likely that one of the coins will fall out when the bank is turned upside down, what is the probability that the coin

(i) Will be a 50 p coin?

(ii) Will not be a Rs.5 coin?

Concept: Probability - A Theoretical Approach
Chapter: [5.01] Probability
[4] 34

Solve for x:

$3\left( \frac{7x + 1}{5x - 3} \right) - 4\left( \frac{5x - 3}{7x + 1} \right) = 11; x \neq \frac{3}{5}, - \frac{1}{7}$
Concept: Pair of Linear Equations in Two Variables
Chapter: [2.01] Pair of Linear Equations in Two Variables

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