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Mathematics Delhi Set 3 2015-2016 CBSE Class 10 Question Paper Solution

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Mathematics [Delhi Set 3]
Marks: 90Academic Year: 2015-2016
Date & Time: 19th March 2016, 10:30 am
Duration: 3h

[1]1

Find the 9th term from the end (towards the first term) of the A.P. 5, 9, 13, ...., 185

Concept: Sum of First n Terms of an AP
Chapter: [2.02] Arithmetic Progressions
[1]2

Cards marked with number 3, 4, 5, ...., 50 are placed in a box and mixed thoroughly. A card is drawn at random form the box. Find the probability that the selected card bears a perfect square number.

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

From an external point P, tangents PA and PB are drawn to a circle with centre O. If ∠PAB = 50°, then find ∠AOB.

Concept: Number of Tangents from a Point on a Circle
Chapter: [3.01] Circles
[1]4

In Fig. 1, AB is a 6 m high pole and CD is a ladder inclined at an angle of 60° to the horizontal and reaches up to a point D of pole. If AD = 2.54 m, find the length of the ladder. (use3=1.73)

 

Concept: Heights and Distances
Chapter: [4.01] Heights and Distances
[2]5

Find the ratio in which y-axis divides the line segment joining the points A(5, –6) and B(–1, –4). Also find the coordinates of the point of division.

Concept: Section Formula
Chapter: [6.01] Lines (In Two-dimensions)
[2]6

If `x=2/3` and x =3 are roots of the quadratic equation ax2 + 7x + b = 0, find the values of a and b.

Concept: Nature of Roots
Chapter: [2.03] Quadratic Equations
[2]7

The x-coordinate of a point P is twice its y-coordinate. If P is equidistant from Q(2, –5) and R(–3, 6), find the coordinates of P.

 

Concept: Distance Formula
Chapter: [6.01] Lines (In Two-dimensions)
[2]8

In Fig. 2, a circle is inscribed in a ΔABC, such that it touches the sides AB, BC and CA at points D, E and F respectively. If the lengths of sides AB, BC and CA and 12 cm, 8 cm and 10 cm respectively, find the lengths of AD, BE and CF.

Concept: Circumference of a Circle
Chapter: [7.01] Areas Related to Circles [7.01] Areas Related to Circles
[2]9

In Fig. 3, AP and BP are tangents to a circle with centre O, such that AP = 5 cm and ∠APB = 60°. Find the length of chord AB.

Concept: Areas of Sector and Segment of a Circle
Chapter: [7.01] Areas Related to Circles
[2]10

How many terms of the A.P. 65, 60, 55, .... be taken so that their sum is zero?

Concept: Sum of First n Terms of an AP
Chapter: [2.02] Arithmetic Progressions
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[3]11

A well of diameter 4 m is dug 21 m deep. The earth taken out of it has been spread evenly all around it in the shape of a circular ring of width 3 m to form an embankment. Find the height of the embankment.

Concept: Concept of Surface Area, Volume, and Capacity
Chapter: [7.02] Surface Areas and Volumes [7.02] Surface Areas and Volumes
[3]12

If the sum of first 7 terms of an A.P. is 49 and that of its first 17 terms is 289, find the sum of first n terms of the A.P.

Concept: Sum of First n Terms of an AP
Chapter: [2.02] Arithmetic Progressions
[3]13

The angles of depression of the top and bottom of a 50 m high building from the top of a tower are 45° and 60° respectively. Find the height of the tower and the horizontal distance between the tower and the building (use `sqrt3`=1.73)

Concept: Heights and Distances
Chapter: [4.01] Heights and Distances
[3]14
 

In Fig. 4, ABCD is a square of side 14 cm. Semi-circles are drawn with each side of square as diameter. 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]15
 

In Fig. 5, is a decorative block, made up two solids – a cube and a hemisphere. The base of the block is a cube of side 6 cm and the hemisphere fixed on the top has diameter of 3.5 cm. Find the total surface area of the bock `(Use pi=22/7)`

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

In Fig. 6, ABC is a triangle coordinates of whose vertex A are (0, −1). D and E respectively are the mid-points of the sides AB and AC and their coordinates are (1, 0) and (0, 1) respectively. If F is the mid-point of BC, find the areas of ∆ABC and ∆DEF.

Concept: Area of a Triangle
Chapter: [6.01] Lines (In Two-dimensions)
[3]17

In Fig. 7, are shown two arcs PAQ and PBQ. Arc PAQ is a part of circle with centre O and radius OP while arc PBQ is a semi-circle drawn on PQ ad diameter with centre M. If OP = PQ = 10 cm show that area of shaded region is `25(sqrt3-pi/6)cm^2`.

Concept: Circumference of a Circle
Chapter: [7.01] Areas Related to Circles [7.01] Areas Related to Circles
[3]18

A box consists of 100 shirts of which 88 are good, 8 have minor defects and 4 have major defects. Ramesh, a shopkeeper will buy only those shirts which are good but 'Kewal' another shopkeeper will not buy shirts with major defects. A shirt is taken out of the box at random. What is the probability that

(i) Ramesh will buy the selected shirt?

(ii) 'Kewal' will buy the selected shirt?

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

Solve the following quadratic equation for x:

`x^2+(a/(a+b)+(a+b)/a)x+1=0`

Concept: Solutions of Quadratic Equations by Factorization
Chapter: [2.03] Quadratic Equations
[3]20

A toy is in the form of a cone of base radius 3.5 cm mounted on a hemisphere of base diameter 7 cm. If the total height of the toy is 15.5 cm, find the total surface area of the top (Use π = 22/7)

Concept: Surface Area of a Combination of Solids
Chapter: [7.02] Surface Areas and Volumes
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[4]21

A passenger, while boarding the plane, slipped form the stairs and got hurt. The pilot took the passenger in the emergency clinic at the airport for treatment. Due to this, the plane got delayed by half an hour. To reach the destination 1500 km away in time, so that the passengers could catch the connecting flight, the speed of the plane was increased by 250 km/hour than the usual speed. Find the usual speed of the plane

What value is depicted in this question?

Concept: Heights and Distances
Chapter: [4.01] Heights and Distances
[4]22

In Fig. 8, O is the centre of a circle of radius 5 cm. T is a point such that OT = 13 cm and OT intersects circle at E. If AB is a tangent to the circle at E, find the length of AB, where TP and TQ are two tangents to the circle.

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

A bird is sitting on the top of a 80 m high tree. From a point on the ground, the angle of elevation of the bird is 45°. The bird flies away horizontally in such a way that it remained at a constant height from the ground. After 2 seconds, the angle of elevation of the bird from the same point is 30°. Find the speed of flying of the bird. 

`("Take"sqrt3=1.732)`

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

A game of chance consists of spinning an arrow on a circular board, divided into 8 equal parts, which comes to rest pointing at one of the numbers 1, 2, 3, ..., 8 (Fig. 9), which are equally likely outcomes. What is the probability that the arrow will point at (i) an odd number (ii) a number greater than 3 (iii) a number less than 9.

Concept: Basic Ideas of Probability
Chapter: [5.01] Probability [5.01] Probability
[4]25

Prove that the area of a triangle with vertices (t, t −2), (t + 2, t + 2) and (t + 3, t) is independent of t.

Concept: Area of a Triangle
Chapter: [6.01] Lines (In Two-dimensions)
[4]26

A bucket open at the top is in the form of a frustum of a cone with a capacity of 12308.8 cm3. The radii of the top and bottom circular ends are 20 cm and 12 cm, respectively. Find the height of the bucket and the area of metal sheet used in making the bucket. (use π = 3.14)

Concept: Heights and Distances
Chapter: [4.01] Heights and Distances
[4]27

An elastic belt is placed around the rim of a pulley of radius 5 cm. (Fig. 10) From one point C on the belt, the elastic belt is pulled directly away from the centre O of the pulley until it is at P, 10 cm from the point O. Find the length of the belt that is still in contact with the pulley. Also find the shaded area (use π = 3.14 and `sqrt3=1.73)`

Concept: Circumference of a Circle
Chapter: [7.01] Areas Related to Circles [7.01] Areas Related to Circles
[4]28

The sum of three numbers in A.P. is 12 and sum of their cubes is 288. Find the numbers.

Concept: Arithmetic Progression
Chapter: [2.02] Arithmetic Progressions
[4]29

Prove that the lengths of the tangents drawn from an external point to a circle are equal.

Concept: Number of Tangents from a Point on a Circle
Chapter: [3.01] Circles
[4]30

The time taken by a person to cover 150 km was 2 1/2 hours more than the time taken in the return journey. If he returned at a speed of 10 km/hour more than the speed while going, find the speed per hour in each direction.

Concept: Quadratic Equations
Chapter: [2.03] Quadratic Equations
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[4]31

Draw a triangle ABC with BC = 7 cm, ∠B = 45° and ∠A = 105°. Then construct a triangle whose sides are`4/5` times the corresponding sides of ΔABC.

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