Board Paper of Class 10 2012 Maths All India(SET 1)  Solutions
1. All questions are compulsory.
2. The question paper consists of 34 questions divided into four sections A, B, C and
D.
3. Section A contains 10 questions of 1 mark each, which are multiple choices type
questions, Section B contains 8 questions of 2 marks each, Section C contains 10
questions of 3 marks each, Section D contains 6 questions of 4 marks each.
4. There is no overall choice in the paper. However, internal choice is provided in one
question of 2 marks, 3 questions of 3 marks each and two questions of 4 marks each.
5. Use of calculators is not permitted.
 Question 1
 Question 2
If the n^{th} term of an A.P. is (2n + 1), then the sum of its first three terms is
A. 6n + 3
B. 15
C. 12
D. 21
VIEW SOLUTION
 Question 3
From a point Q, 13 cm away from the centre of a circle, the length of tangent PQ to the circle is 12 cm. The radius of the circle (in cm) is
A. 25
B.
C. 5
D. 1
VIEW SOLUTION
 Question 4
In Figure 1, AP, AQ and BC are tangents to the circle. If AB = 5 cm, AC = 6 cm and BC
= 4 cm, then the length of AP (in cm) is
A. 7.5
B. 15
C. 10
D. 9
VIEW SOLUTION
 Question 5
The circumference of a circle is 22 cm. The area of its quadrant (in cm^{2}) is
A.
B.
C.
D.
VIEW SOLUTION
 Question 6
A solid right circular cone is cut into two parts at the middle of its height by a plane parallel to its base. The ratio of the volume of the smaller cone to the whole cone is
A. 1 : 2
B. 1 : 4
C. 1 : 6
D. 1 : 8
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 Question 7
A kite is flying at a height of 30 m from the ground. The length of string from the kite to the ground is 60 m. Assuming that there is no slack in the string, the angle of elevation of the kite at the ground is
A. 45°
B. 30°
C. 60°
D. 90°
VIEW SOLUTION
 Question 8
 Question 9
In Figure 2, P (5, −3) and Q (3, y) are the points of trisection of the line segment joining A (7, −2) and B (1, −5). Then y equals
A. 2
B. 4
C. −4
D.
VIEW SOLUTION
 Question 10
Cards bearing numbers 2, 3, 4, ..., 11 are kept in a bag. A card is drawn at random from the bag. The probability of getting a card with a prime number is
A.
B.
C.
D.
VIEW SOLUTION
 Question 11
Find the value of p for which the roots of the equation px (x − 2) + 6 = 0, are equal.
VIEW SOLUTION
 Question 12
How many twodigit numbers are divisible by 3?
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 Question 13
In Figure 3, a right triangle ABC, circumscribes a circle of radius r. If AB and BC are of lengths of 8 cm and 6 cm respectively, find the value of r.
VIEW SOLUTION
 Question 14
Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
VIEW SOLUTION
 Question 15
In Figure 4, ABCD is a square of side 4 cm. A quadrant of a circle of radius 1 cm is drawn at each vertex of the square and a circle of diameter 2 cm is also drawn. Find the area of the shaded region. (Use π = 3.14)
VIEW SOLUTION
 Question 16
A solid sphere of radius 10.5 cm is melted and recast into smaller solid cones, each of radius 3.5 cm and height 3 cm.Find the number of cones so formed.
VIEW SOLUTION
 Question 17
Find the value of k, if the point P (2, 4) is equidistant from the points A(5, k) and B (k, 7).
VIEW SOLUTION
 Question 18
A card is drawn at random from a wellshuffled pack of 52 cards. Find the probability of getting
(i) a red king.
(ii) a queen or a jack.
VIEW SOLUTION
 Question 19
 Question 20
 Question 21
Draw a triangle ABC with BC = 7 cm, ∠ B = 45° and ∠C = 60°. Then construct another triangle, whose sides are times the corresponding sides of ΔABC.
 Question 22
In Figure 5, a circle is inscribed in a triangle PQR with PQ = 10 cm, QR = 8 cm and PR =
12 cm. Find the lengths of QM, RN and PL.
 Question 23
In Figure 6, O is the centre of the circle with AC = 24 cm, AB = 7 cm and ∠BOD = 90°.
Find the area of the shaded region. [Use π = 3.14]
 Question 24
A hemispherical bowl of internal radius 9 cm is full of water. Its contents are emptied in a cylindrical vessel of internal radius 6 cm. Find the height of water in the cylindrical vessel.
 Question 25
The angles of depression of the top and bottom of a tower as seen from the top of a m high cliff are 45° and 60° respectively. Find the height of the tower.
 Question 26
Find the coordinates of a point P, which lies on the line segment joining the points A (−2, −2), and B (2, −4), such that .
 Question 27
If the points A (x, y), B (3, 6) and C (−3, 4) are collinear, show that x − 3y + 15 = 0.
 Question 28
All kings, queens and aces are removed from a pack of 52 cards. The remaining cards are well shuffled and then a card is drawn from it. Find the probability that the drawn card is
(i) a black face card.
(ii) a red card.
 Question 29
The numerator of a fraction is 3 less than its denominator. If 1 is added to the denominator, the fraction is decreased by .Find the fraction.
OR
In a flight of 2800 km, an aircraft was slowed down due to bad weather. Its average speed is reduced by 100 km/h and time increased by 30 minutes. Find the original duration of the flight.
 Question 30
Find the common difference of an A. P. whose first term is 5 and the sum of its first four terms is half the sum of the next four terms.
 Question 31
Prove that the lengths of tangents drawn from an external point to a circle are equal.
 Question 32
A hemispherical tank, full of water, is emptied by a pipe at the rate of litres per sec.
How much time will it take to empty half the tank if the diameter of the base of the tank is 3 m?
OR
A drinking glass is in the shape of the frustum of a cone of height 14 cm. The diameters of its two circular ends are 4 cm and 2 cm. Find the capacity of the glass.
VIEW SOLUTION
 Question 33
A military tent of height 8.25 m is in the form of a right circular cylinder of base diameter
30 m and height 5.5 m surmounted by a right circular cone of same base radius. Find the
length of the canvas used in making the tent, if the breadth of the canvas is 1.5 m.
 Question 34
The angles of elevation and depression of the top and bottom of a lighthouse from the top of a 60 m high building are 30° and 60° respectively. Find
(i) the difference between the heights of the lighthouse and the building.
(ii) the distance between the lighthouse and the building.

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