# Mathematics All India Set 3 2016-2017 CBSE (English Medium) Class 10 Question Paper Solution

Mathematics [All India Set 3]
Marks: 90 Academic Year: 2016-2017
Date & Time: 2nd April 2017, 12:30 pm
Duration: 3h

[1] 1

The probability of selecting a rotten apple randomly from a heap of 900 apples is 0.18. What is the number of rotten apples in the heap?

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

If a tower 30 m high, casts a shadow 10sqrt3 m long on the ground, then what is the angle of elevation of the sun?

Concept: Heights and Distances
Chapter: [4.01] Heights and Distances
[1] 3

If the angle between two tangents drawn from an external point P to a circle of radius a and centre O, is 60°, then find the length of OP

Concept: Tangent to a Circle
Chapter: [3.01] Circles
[1] 4

What is the common difference of an A.P. in which a21 – a7 = 84?

Concept: Arithmetic Progression
Chapter: [2.02] Arithmetic Progressions
[2] 5

A circle touches all the four sides of a quadrilateral ABCD. Prove that AB + CD = BC + DA

Concept: Tangent to a Circle
Chapter: [3.01] Circles
[2] 6

Prove that the tangents drawn at the end points of a chord of a circle make equal angles with the chord.

Concept: Number of Tangents from a Point on a Circle
Chapter: [3.01] Circles
[2] 7

A line intersects the y-axis and x-axis at the points P and Q respectively. If (2, –5) is the mid-point of PQ, then find the coordinates of P and Q.

Concept: Coordinate Geometry
Chapter: [6.01] Lines (In Two-dimensions) [6.01] Lines (In Two-dimensions)
[2] 8

If the distances of P(x, y) from A(5, 1) and B(–1, 5) are equal, then prove that 3x = 2y

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

Find the value of p, for which one root of the quadratic equation px2 – 14x + 8 = 0 is 6 times the other.

Concept: Nature of Roots
Chapter: [2.03] Quadratic Equations
[2] 10

For what value of n, are the nth terms of two APs 63, 65, 67, and 3, 10, 17, … equal?

Concept: nth Term of an AP
Chapter: [2.02] Arithmetic Progressions
[3] 11

On a straight line passing through the foot of a tower, two points C and D are at distances of 4 m and 16 m from the foot respectively. If the angles of elevation from C and D of the top of the tower are complementary, then find the height of the tower.

Concept: Heights and Distances
Chapter: [4.01] Heights and Distances
[3] 12

A bag contains 15 white and some black balls. If the probability of drawing a black ball from the bag is thrice that of drawing a white ball, find the number of black balls in the bag.

Concept: Simple Problems on Single Events
Chapter: [5.01] Probability
[3] 13

Three semicircles each of diameter 3 cm, a circle of diameter 4.5 cm and a semicircle of radius 4.5 cm are drawn in the given figure. Find the area of the shaded region

Concept: Circumference of a Circle
Chapter: [7.01] Areas Related to Circles [7.01] Areas Related to Circles
[3] 14

In what ratio does the point (24/11, y) divide the line segment joining the points P(2, –2) and Q(3, 7)? Also find the value of y.

Concept: Section Formula
Chapter: [6.01] Lines (In Two-dimensions)
[3] 15

Water in a canal, 5·4 m wide and 1·8 m deep, is flowing with a speed of 25 km/hour. How much area can it irrigate in 40 minutes, if 10 cm of standing water is required for irrigation?

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

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: [7.01] Areas Related to Circles [7.01] Areas Related to Circles
[3] 17

The dimensions of a solid iron cuboid are 4·4 m × 2·6 m × 1·0 m. It is melted and recast into a hollow cylindrical pipe of 30 cm inner radius and thickness 5 cm. Find the length of the pipe

Concept: Concept of Surface Area, Volume, and Capacity
Chapter: [7.02] Surface Areas and Volumes [7.02] Surface Areas and Volumes
[3] 18

A toy is in the form of a cone of radius 3.5 cm mounted on a hemisphere of same radius. The total height of the toy is 15.5 cm. Find the total surface area of the toy [Use π =22/7]

Concept: Surface Area of a Combination of Solids
Chapter: [7.02] Surface Areas and Volumes
[3] 19

How many terms of the AP. 9, 17, 25 … must be taken to give a sum of 636?

Concept: Sum of First n Terms of an AP
Chapter: [2.02] Arithmetic Progressions
[3] 20

If the roots of the equation (a2 + b2) x2 – 2(ac + bd) x + (c2 + d2) = 0 are equal, prove that a/b = c/d

Concept: Relationship Between Discriminant and Nature of Roots
Chapter: [2.03] Quadratic Equations
[4] 21

If the points A(k + 1, 2k), B(3k, 2k + 3) and C(5k − 1, 5k) are collinear, then find the value of k

Concept: Coordinate Geometry
Chapter: [6.01] Lines (In Two-dimensions) [6.01] Lines (In Two-dimensions)
[4] 22

Construct a triangle ABC with sides BC = 7 cm, ∠B = 45° and ∠A = 105°. Then construct a triangle whose sides are 3/4 times the corresponding sides of ∆ABC.

Concept: Application of Pythagoras Theorem in Acute Angle and Obtuse Angle
Chapter: [3.02] Triangles
[4] 23

Two different dice are thrown together. Find the probability that the numbers obtained have

1) even sum, and

2) even product.

Concept: Basic Ideas of Probability
Chapter: [5.01] Probability [5.01] Probability
[4] 24

In the given figure, XY and X’Y’ are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X’Y’ at B. Prove that ∠AOB=90°

Concept: Number of Tangents from a Point on a Circle
Chapter: [3.01] Circles
[4] 25

In a rain-water harvesting system, the rain-water from a roof of 22 m × 20 m drains into a cylindrical tank having diameter of base 2 m and height 3·5 m. If the tank is full, find the rainfall in cm. Write your views on water conservation.

Concept: Concept of Surface Area, Volume, and Capacity
Chapter: [7.02] Surface Areas and Volumes [7.02] Surface Areas and Volumes
[4] 26

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] 27

If the ratio of the sum of the first n terms of two A.Ps is (7n + 1) : (4n + 27), then find the ratio of their 9th terms.

Concept: Sum of First n Terms of an AP
Chapter: [2.02] Arithmetic Progressions
[4] 28

Solve for x

(x - 1)/(2x + 1) + (2x + 1)/(x - 1) = 2, "where x" != -1/2, 1

Concept: Solutions of Quadratic Equations by Factorization
Chapter: [2.03] Quadratic Equations
[4] 29

A takes 6 days less than B to do a work. If both A and B working together can do it in 4 days, how many days will B take to finish it?

Concept: Pair of Linear Equations in Two Variables
Chapter: [2.01] Pair of Linear Equations in Two Variables
[4] 30

From the top of a tower, 100, high, a man observes two cars on the opposite sides of the tower and in same straight line with its base, with angles of depression 30° and 45°. Find the distance between the cars. [Take sqrt3 = 1.732]

Concept: Heights and Distances
Chapter: [4.01] Heights and Distances
[4] 31

In the given figure, O is the centre of the circle with AC = 24 cm, AB = 7 cm and ∠BOD = 90°. Find the area of the shaded region.

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

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