# Mathematics Delhi Set 3 2014-2015 CBSE (English Medium) Class 10 Question Paper Solution

Mathematics [Delhi Set 3]
Date: March 2015

[1] 1

In Fig. 1, PA and PB are tangents to the circle with centre O such that ∠APB = 50°. Write the measure of ∠OAB.

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
[1] 2

If x=1/2, is a solution of the quadratic equation 3x2+2kx3=0, find the value of k

Concept: Nature of Roots
[1] 3

The tops of two towers of height x and y, standing on level ground, subtend angles of 30° and 60° respectively at the centre of the line joining their feet, then find x, y.

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

A letter of English alphabet is chosen at random. Determine the probability that the chosen letter is consonant.

Concept: Basic Ideas of Probability
Chapter: [5.01] Probability [5.01] Probability
[2] 5

From a point T outside a circle of centre O, tangents TP and TQ are drawn to the circle. Prove that OT is the right bisector of line segment PQ.

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
[2] 6

Find the middle term of the A.P. 6, 13, 20, ... , 216.

Concept: Arithmetic Progression
Chapter: [2.02] Arithmetic Progressions
[2] 7

In Fig. 2, AB is the diameter of a circle with centre O and AT is a tangent. If ∠AOQ = 58°, find ∠ATQ.

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
[2] 8

If A(5, 2), B(2, −2) and C(−2, t) are the vertices of a right angled triangle with ∠B = 90°, then find the value of t.

Concept: Distance Formula
Chapter: [6.01] Lines (In Two-dimensions)
[2] 9

Find the ratio in which the point P(3/4,5/12) divides the line segment joining the points A(1/2,3/2) and B(2,-5)

Concept: Section Formula
Chapter: [6.01] Lines (In Two-dimensions)
[2] 10

Solve the following quadratic equation for x :

9x2 − 6b2x − (a4b4) = 0

Concept: Nature of Roots
[3] 11

In Fig. 3, APB and AQO are semicircles, and AO = OB. If the perimeter of the figure is 40 cm, find the area of the shaded region [Use pi=22/7]

Concept: Circumference of a Circle
Chapter: [7.01] Areas Related to Circles [7.01] Areas Related to Circles
[3] 12

A solid wooden toy is in the form of a hemisphere surrounded by a cone of same radius. The radius of hemisphere is 3.5 cm and the total wood used in the making of toy is 166 5/6  cm3. Find the height of the toy. Also, find the cost of painting the hemispherical part of the toy at the rate of Rs 10 per cm2 .[Usepi=22/7]

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

In Fig. 4, from the top of a solid cone of height 12 cm and base radius 6 cm, a cone of height 4 cm is removed by a plane parallel to the base. Find the total surface area of the remaining solid. (Use pi=22/7 and sqrt5=2.236)

Concept: Surface Area of a Combination of Solids
Chapter: [7.02] Surface Areas and Volumes
[3] 14

In Fig. 5, from a cuboidal solid metallic block, of dimensions 15cm ✕ 10cm ✕ 5cm, a cylindrical hole of diameter 7 cm is drilled out. Find the surface area of the remaining block [Use

pi=22/7]

Concept: Surface Area of a Combination of Solids
Chapter: [7.02] Surface Areas and Volumes
[3] 15

The angle of elevation of the top of a building from the foot of the tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 45°. If the tower is 30 m high, find the height of the building.

Concept: Heights and Distances
Chapter: [4.01] Heights and Distances
[3] 16

In Figure, find the area of the shaded region [Use π = 3.14]

Concept: Circumference of a Circle
Chapter: [7.01] Areas Related to Circles [7.01] Areas Related to Circles
[3] 17

Find that non-zero value of k, for which the quadratic equation kx2 + 1 − 2(k − 1)x + x2 = 0 has equal roots. Hence find the roots of the equation.

Concept: Nature of Roots
[3] 18

All red face cards are removed from a pack of playing cards. The remaining cards were well shuffled and then a card is drawn at random from them. Find the probability that the drawn card is

(i) a red card

(ii) a face card

(iii) a card of clubs

Concept: Basic Ideas of Probability
Chapter: [5.01] Probability [5.01] Probability
[3] 19

Find the area of the triangle PQR with Q(3,2) and the mid-points of the sides through Q being (2,−1) and (1,2).

Concept: Area of a Triangle
Chapter: [6.01] Lines (In Two-dimensions)
[3] 20

If Sn denotes the sum of first n terms of an A.P., prove that S30 = 3[S20S10]

Concept: Sum of First n Terms of an AP
Chapter: [2.02] Arithmetic Progressions
[4] 21

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] 22

In Fig. 7, tangents PQ and PR are drawn from an external point P to a circle with centre O, such that ∠RPQ = 30°. A chord RS is drawn parallel to the tangent PQ. Find ∠RQS.

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] 23

From a point P on the ground the angle of elevation of the top of a tower is 30° and that of the top of a flag staff fixed on the top of the tower, is 60°. If the length of the flag staff is 5 m, find the height of the tower.

Concept: Heights and Distances
Chapter: [4.01] Heights and Distances
[4] 24

Ramkali required Rs 2,500 after 12 weeks to send her daughter to school. She saved Rs 100 in the first week and increased her weekly saving by Rs 20 every week. Find whether she will be able to send her daughter to school after 12 weeks.

What value is generated in the above situation?

Concept: Sum of First n Terms of an AP
Chapter: [2.02] Arithmetic Progressions
[4] 25

The numerator of a fraction is 3 less than its denominator. If 2 is added to both the numerator and the denominator, then the sum of the new fraction and original fraction is 29/20. Find the original fraction.

Concept: Solutions of Quadratic Equations by Factorization
[4] 26

Water is flowing at the rate of 2.52 km/h through a cylindrical pipe into a cylindrical tank, the radius of whose base is 40 cm. If the increase in the level of water in the tank, in half an hour is 3.15 m, find the internal diameter of the pipe.

Concept: Concept of Surface Area, Volume, and Capacity
Chapter: [7.02] Surface Areas and Volumes [7.02] Surface Areas and Volumes
[4] 27

If A(−4, 8), B(−3, −4), C(0, −5) and D(5, 6) are the vertices of a quadrilateral ABCD, find its area.

Concept: Area of a Triangle
Chapter: [6.01] Lines (In Two-dimensions)
[4] 28

A 21 m deep well with diameter 6 m is dug and the earth from digging is evenly spread to form a platform 27 m ✕ 11 m. Find the height of the platform.[Use pi=22/7]

Concept: Heights and Distances
Chapter: [4.01] Heights and Distances
[4] 29

A bag contains 25 cards numbered from 1 to 25. A card is drawn at random from the bag. Find the probability that the number on the drawn card is:

(i) divisible by 3 or 5

(ii) a perfect square number

Concept: Basic Ideas of Probability
Chapter: [5.01] Probability [5.01] Probability
[4] 30

Draw a line segment AB of length 7 cm. Taking A as centre, draw a circle of radius 3 cm and taking B as centre, draw another circle of radius 2 cm. Construct tangents to each circle from the centre of the other circle.Steps

Concept: Construction of Tangents to a Circle
Chapter: [3.03] Constructions
[4] 31

Solve for x :

3/(x+1)+4/(x-1)=29/(4x-1);x!=1,-1,1/4

Concept: Solutions of Quadratic Equations by Factorization

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