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# Board Paper of Class 10 2015 Maths (SET 1) - 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
If $x=-\frac{1}{2},$ is a solution of the quadratic equation $3{x}^{2}+2kx-3=0,$ find the value of k. VIEW SOLUTION
• Question 2
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. VIEW SOLUTION
• Question 3
A letter of English alphabet is chosen at random. Determine the probability that the chosen letter is consonant. VIEW SOLUTION
• Question 4
In Fig. 1, PA and PB are tangents to the circle with centre O such that ∠APB = 50°. Write the measure of ∠OAB.
 Figure 1
VIEW SOLUTION
• Question 5
In Fig. 2, AB is the diameter of a circle with centre O and AT is a tangent. If ∠AOQ = 58°, find ∠ATQ.
 Figure 2

VIEW SOLUTION
• Question 6
Solve the following quadratic equation for x :

VIEW SOLUTION
• Question 7
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. VIEW SOLUTION
• Question 8
Find the middle term of the A.P. 6, 13, 20, ... , 216. VIEW SOLUTION
• Question 9
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. VIEW SOLUTION
• Question 10
Find the ratio in which the point divides the line segment joining the points and B(2, −5). VIEW SOLUTION
• Question 11
Find the area of the triangle ABC with A(1, −4) and mid-points of sides through A being (2, −1) and (0, −1). VIEW SOLUTION
• Question 12
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. VIEW SOLUTION
• Question 13
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. VIEW SOLUTION
• Question 14
Two different dice are rolled together. Find the probability of getting :

(i) the sum of numbers on two dice to be 5.
(ii) even numbers on both dice. VIEW SOLUTION
• Question 15
If ${\mathrm{S}}_{{n}^{1}}$ denotes the sum of first n terms of an A.P., prove that S12 = 3(S8 − S4). VIEW SOLUTION
• Question 16

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.

 Figure 3

VIEW SOLUTION
• Question 17
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.  Figure 4
VIEW SOLUTION
• Question 18
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 . 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. VIEW SOLUTION
• Question 19
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  Figure 5
VIEW SOLUTION
• Question 20
In Fig. 6, find the area of the shaded region [Use π = 3.14]
 Figure 6
VIEW SOLUTION
• Question 21
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 $\frac{29}{20}$. Find the original fraction. VIEW SOLUTION
• Question 22
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?

VIEW SOLUTION
• Question 24
Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. VIEW SOLUTION
• Question 25
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.
 Figure 7
VIEW SOLUTION
• Question 26
Construct a triangle ABC with BC = 7 cm, ∠B = 60° and AB = 6 cm. Construct another triangle whose sides are $\frac{3}{4}$ times the corresponding sides of ∆ABC. VIEW SOLUTION
• Question 27
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. VIEW SOLUTION
• Question 28
A box contains 20 cards numbered from 1 to 20. A card is drawn at random from the box. Find the probability that the number on the drawn card is

(i) divisible by 2 or 3
(ii) a prime number VIEW SOLUTION
• Question 29
If A(−4, 8), B(−3, −4), C(0, −5) and D(5, 6) are the vertices of a quadrilateral ABCD, find its area. VIEW SOLUTION
• Question 30
A well of diameter 4 m is dug 14 m deep. The earth taken out is spread evenly all around the well to form a 40 cm high embankment. Find the width of the embankment. VIEW SOLUTION
• Question 31
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. VIEW SOLUTION
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