Board Paper of Class 10 2016 Maths (SET 3) - Solutions

General Instructions :
(i) All questions are compulsory.
(ii) The question paper consists of 31 questions divided into four sections – A, B, C and D.
(iii) Section A contains 4 questions of 1 mark each, Section B contains 6 questions of 2 marks each, Section C contains 10 questions of 3 marks each and Section D contains 11 questions of 4 marks each.
(iv) Use of calculated is not permitted.

Question 1
Find the 9^th term from the end (towards the first term) of the A.P. 5, 9, 13, ...., 185. VIEW SOLUTION

Question 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. VIEW SOLUTION

Question 3
From an external point P, tangents PA and PB are drawn to a circle with centre O. If ∠PAB = 50°, then find ∠AOB. VIEW SOLUTION

Question 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. $(use \sqrt{3} = 1.73)$

VIEW SOLUTION

Question 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. VIEW SOLUTION

Question 6
If $x = \frac{2}{3} and x = - 3$ are roots of the quadratic equation ax² + 7x + b = 0, find the values of a and b. VIEW SOLUTION

Question 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. VIEW SOLUTION

Question 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.

VIEW SOLUTION

Question 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.
VIEW SOLUTION

Question 10
How many terms of the A.P. 65, 60, 55, .... be taken so that their sum is zero? VIEW SOLUTION

Question 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. VIEW SOLUTION

Question 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. VIEW SOLUTION

Question 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 $\sqrt{3} = 1.73$ ) VIEW SOLUTION

Question 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. $(use π = \frac{22}{7})$
VIEW SOLUTION

Question 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. $(useπ = \frac{22}{7})$
VIEW SOLUTION

Question 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.

VIEW SOLUTION

Question 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 $25 (\sqrt{3} - \frac{π}{6}) {cm}^{2} .$

VIEW SOLUTION

Question 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? VIEW SOLUTION

Question 19
Solve the following quadratic equation for x:

$x^{2} + (\frac{a}{a + b} + \frac{a + b}{a}) x + 1 = 0$ VIEW SOLUTION

Question 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 toy. $(Use π = \frac{22}{7})$ VIEW SOLUTION

Question 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? VIEW SOLUTION

Question 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.
VIEW SOLUTION

Question 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. $(Take \sqrt{3} = 1.732)$ . VIEW SOLUTION

Question 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.

VIEW SOLUTION

Question 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. VIEW SOLUTION

Question 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) VIEW SOLUTION

Question 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 $\sqrt{3}$ = 1.73)

VIEW SOLUTION

Question 28
The sum of three numbers in A.P. is 12 and sum of their cubes is 288. Find the numbers. VIEW SOLUTION

Question 29
Prove that the lengths of tangents drawn from an external point to a circle are equal. VIEW SOLUTION

Question 30
The time taken by a person to cover 150 km was $2 \frac{1}{2}$ 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. VIEW SOLUTION

Question 31
Draw a triangle ABC with BC = 7 cm, ∠B = 45° and ∠A = 105°. Then construct a triangle whose sides are $\frac{4}{5}$ times the corresponding sides of ΔABC. VIEW SOLUTION