Board Paper of Class 10 2014 Maths (SET 2) - Solutions

General Instructions:
(i) All questions are compulsory.
(ii) The question paper consists of 34 questions divided into four sections A, B, C and D.
(iii) Sections A contains 8 questions of one mark each, which are multiple choice type questions, section B contains 6 questions of two marks each, section C contains 10 questions of three marks each, and section D contains 10 questions of four marks each.
(iv) Use of calculators is not permitted.

Question 1
A ladder makes an angle of 60° with the ground when placed against a wall. If the foot of the ladder is 2 m away from the wall, then the length of the ladder (in metres) is:
(A) $\frac{4}{\sqrt{3}}$

(B) $4 \sqrt{3}$

(C) $2 \sqrt{2}$

(D) 4 VIEW SOLUTION

Question 2
If two different dice are rolled together, the probability of getting an even number on both dice, is:
(A) $\frac{1}{36}$
(B) $\frac{1}{2}$
(C) $\frac{1}{6}$
(D) $\frac{1}{4}$ VIEW SOLUTION

Question 3
A number is selected at random from the numbers 1 to 30. The probability that it is a prime number is:
(A) $\frac{2}{3}$

(B) $\frac{1}{6}$

(C) $\frac{1}{3}$

(D) $\frac{11}{30}$ VIEW SOLUTION

Question 4
If the points A(x, 2), B(−3, −4) and C(7, − 5) are collinear, then the value of x is:
(A) −63
(B) 63
(C) 60
(D) −60 VIEW SOLUTION

Question 5
In Fig. 1, QR is a common tangent to the given circles, touching externally at the point T. The tangent at T meets QR at P. If PT = 3.8 cm, then the length of QR (in cm) is :

(A) 3.8
(B) 7.6
(C) 5.7
(D) 1.9 VIEW SOLUTION

Question 6
In Fig. 2, PQ and PR are two tangents to a circle with centre O. If ∠QPR = 46°, then ∠QOR equals:

(A) 67°
(B) 134°
(C) 44°
(D) 46° VIEW SOLUTION

Question 7
The number of solid spheres, each of diameter 6 cm that can be made by melting a solid metal cylinder of height 45 cm and diameter 4 cm, is:
(A) 3
(B) 5
(C) 4
(D) 6 VIEW SOLUTION

Question 8
The first three terms of an AP respectively are 3y – 1, 3y + 5 and 5y + 1. Then y equals:
(A) –3
(B) 4
(C) 5
(D) 2 VIEW SOLUTION

Question 9
If from an external point P of a circle with centre O, two tangents PQ and PR are drawn such that ∠QPR = 120°, prove that 2PQ = PO. VIEW SOLUTION

Question 10
Rahim tosses two different coins simultaneously. Find the probability of getting at least one tail. VIEW SOLUTION

Question 11
In fig. 3, a square OABC is inscribed in a quadrant OPBQ of a circle. If OA = 20 cm, find the area of the shaded region. (Use π = 3.14)
VIEW SOLUTION

Question 12
Solve the quadratic equation 2x² + ax − a² = 0 for x. VIEW SOLUTION

Question 13
Prove that the line segment joining the points of contact of two parallel tangents of a circle, passes through its centre. VIEW SOLUTION

Question 14
The first and the last terms of an AP are 7 and 49 respectively. If sum of all its terms is 420, find its common difference. VIEW SOLUTION

Question 15
In Fig 4, a circle is inscribed in an equilateral triangle ABC of side 12 cm. Find the radius of inscribed circle and the area of the shaded region. [Use π = 3.14 and $\sqrt{3} = 1.73$ ]

VIEW SOLUTION

Question 16
In Fig.5, PSR, RTQ and PAQ are three semicircles of diameters 10 cm, 3 cm and 7 cm respectively. Find the perimeter of the shaded region. [Use π = 3.14]
VIEW SOLUTION

Question 17
A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank which is 10 m in diameter and 2 m deep. If the water flows through the pipe at the rate of 4 km per hour, in how much time will the tank be filled completely? VIEW SOLUTION

Question 18
A solid metallic right circular cone 20 cm high and whose vertical angle is 60°, is cut into two parts at the middle of its height by a plane parallel to its base. If the frustum so obtained be drawn into a wire of diameter $\frac{1}{12}$ cm, find the length of the wire. VIEW SOLUTION

Question 19
If the seventh term of an AP is $\frac{1}{9}$ and its ninth term is $\frac{1}{7}$ , find its 63^rd term. VIEW SOLUTION

Question 20
Draw a right triangle ABC in which AB = 6 cm, BC = 8 cm and ∠B = 90°. Draw BD perpendicular from B on AC and draw a circle passing through the points B, C and D. Construct tangents from A to this circle. VIEW SOLUTION

Question 21
Two ships are there in the sea on either side of a light house in such a way that the ships and the light house are in the same straight line. The angles of depression of two ships as observed from the top of the light house are 60° and 45°. If the height of the light house is 200 m, find the distance between the two ships. [Use $\sqrt{3} = 1.73$ ] VIEW SOLUTION

Question 22
Solve the equation $\frac{3}{x + 1} - \frac{1}{2} = \frac{2}{3 x - 1}; x \neq - 1, x \neq \frac{1}{3}$ , for x. VIEW SOLUTION

Question 23
Points A(–1, y) and B(5, 7) lie on a circle with centre O(2, –3y). Find the values of y. Hence find the radius of the circle. VIEW SOLUTION

Question 24
If the points P(–3, 9), Q(a, b) and R(4, – 5) are collinear and a + b = 1, find the values of a and b. VIEW SOLUTION

Question 25
The angles of elevation and depression of the top and the bottom of a tower from the top of a building, 60 m high, are 30° and 60° respectively. Find the difference between the heights of the building and the tower and the distance between them. VIEW SOLUTION

Question 26
Find the ratio in which the point P(x, 2) divides the line segment joining the points A(12, 5) and B(4, – 3). Also find the value of x. VIEW SOLUTION

Question 27
In an AP of 50 terms, the sum of first 10 terms is 210 and the sum of its last 15 terms is 2565. Find the A.P. VIEW SOLUTION

Question 28
Prove that a parallelogram circumscribing a circle is a rhombus. VIEW SOLUTION

Question 29
Sushant has a vessel, of the form of an inverted cone, open at the top, of height 11 cm and radius of top as 2.5 cm and is full of water. Metallic spherical balls each of diameter 0.5 cm are put in the vessel due to which $\frac{2}{5}$ th of the water in the vessel flows out. Find how many balls were put in the vessel. Sushant made the arrangement so that the water that flows out irrigates the flower beds. What value has been shown by Sushant? VIEW SOLUTION

Question 30
From a solid cylinder of height 2.8 cm and diameter 4.2 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid. $[Take π = \frac{22}{7}]$ VIEW SOLUTION

Question 31
The difference of two natural numbers is 3 and the difference of their reciprocals is $\frac{3}{28}$ . Find the numbers. VIEW SOLUTION

Question 32
Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact. VIEW SOLUTION

Question 33
All the black face cards are removed from a pack of 52 playing cards. The remaining cards are well shuffled and then a card is drawn at random. Find the probability of getting a:
(i) face card
(ii) red card
(iii) black card
(iv) king VIEW SOLUTION

Question 34
Find the values of k for which the quadratic equation (3k + 1) x²+ 2(k + 1) x + 1 = 0 has equal roots. Also, find the roots. VIEW SOLUTION