Board Paper of Class 10 2015 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.
(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
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
 Question 2
If $x=\frac{1}{2},$ is a solution of the quadratic equation $3{x}^{2}+2kx3=0,$ find the value of k. VIEW SOLUTION
 Question 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. VIEW SOLUTION
 Question 4
A letter of English alphabet is chosen at random. Determine the probability that the chosen letter is consonant. VIEW SOLUTION
 Question 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. VIEW SOLUTION
 Question 6
Find the middle term of the A.P. 6, 13, 20, ... , 216. VIEW SOLUTION
 Question 7
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
 Question 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. VIEW SOLUTION
 Question 9
Find the ratio in which the point $\mathrm{P}\left(\frac{3}{4},\frac{5}{12}\right)$ divides the line segment joining the points $\mathrm{A}\left(\frac{1}{2},\frac{3}{2}\right)$ and B(2, −5). VIEW SOLUTION
 Question 10
Solve the following quadratic equation for x :
9x^{2} − 6b^{2}x − (a^{4} − b^{4}) = 0 VIEW SOLUTION
 Question 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. $\left[\mathrm{Use}\mathrm{\pi}=\frac{22}{7}\right]$
Figure 3
VIEW SOLUTION
 Question 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 $166\frac{5}{6}{\mathrm{cm}}^{3}$. 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 cm^{2}. $\left[\mathrm{Use}\mathrm{\pi}=\frac{22}{7}\right]$ VIEW SOLUTION
 Question 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. $\left(\mathrm{Use}\mathrm{\pi}=\frac{22}{7}\mathrm{and}\sqrt{5}=2.236\right)$
Figure 4
 Question 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 $\left[\mathrm{Use}\mathrm{\pi}=\frac{22}{7}\right]$Figure 5
 Question 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. VIEW SOLUTION
 Question 16
 Question 17
Find that nonzero value of k, for which the quadratic equation kx^{2} + 1 − 2(k − 1)x + x^{2} = 0 has equal roots. Hence find the roots of the equation. VIEW SOLUTION
 Question 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 VIEW SOLUTION
 Question 19
Find the area of the triangle PQR with Q(3,2) and the midpoints of the sides through Q being (2,−1) and (1,2). VIEW SOLUTION
 Question 20
If S_{n} denotes the sum of first n terms of an A.P., prove that S_{30} = 3[S_{20} − S_{10}] VIEW SOLUTION
 Question 21
Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. VIEW SOLUTION
 Question 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.
Figure 7
 Question 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. VIEW SOLUTION
 Question 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?
VIEW SOLUTION
 Question 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 $\frac{29}{20}$. Find the original fraction. VIEW SOLUTION
 Question 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. VIEW SOLUTION
 Question 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. VIEW SOLUTION
 Question 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. $\left[\mathrm{Use}\mathrm{\pi}=\frac{22}{7}\right]$ VIEW SOLUTION
 Question 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 VIEW SOLUTION
 Question 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 VIEW SOLUTION
 Question 31
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