Board Paper of Class 10 2009 Maths (SET 1)  Solutions
1. All questions are compulsory.
2. The question paper consists of 30 questions divided into four sections – A, B, C and D. Section A comprises of ten questions of 1 mark each, Section B comprises of five questions of 2marks each, Section C comprises of ten questions of 3 marks each and Section D comprises of five questions of 6marks each.
3. All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of the question.
There is no overall choice. However, an internal choice has been provided in one question of 2 marks each, three questions of 3 marks each and two questions of 6 marks each. You have to attempt only one of the alternatives in all such questions.
4. In question on construction, the drawing should be neat and as per the given measurements.
5. Use of calculators is not permitted.
 Question 1
The decimal expansion of the rational number will terminate after how many places of decimals?
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 Question 2
For what value of k, (−4) is a zero of the polynomial x^{2} − x − (2k + 2)?
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 Question 3
For what value of p, are 2p − 1, 7 and 3p three consecutives terms of an A.P.?
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 Question 4
In Fig. 1, CP and CQ are tangents to a circle with centre O. ARB is another tangent touching the circle at R. If CP = 11 cm, and BC = 7 cm, then find the length of BR.
Fig. 1
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 Question 5
In Fig. 2, ∠M = ∠N = 46°. Express x in terms of a, b and c where a, b and c are lengths, of LM, MN and NK respectively.
Fig. 2
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 Question 6
If, then find the value of (2cot^{2}θ + 2).
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 Question 7
Find the value of a so that the point (3, a) lies on the line represented by
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2x − 3y = 5.
 Question 8
A cylinder and a cone are of same base radius and of same height. Find the ratio of the volume of cylinder to that of the cone.
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 Question 9
Find the distance between the points and.
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 Question 10
Write the median class of the following distribution:
Classes Frequency
0−10 4
10−20 4
20−30 8
30−40 10
40−50 12
50−60 8
60−70 4
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 Question 11
If the polynomial is divided by another polynomial, the remainder comes out to be, find a and b.
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 Question 12
Find the value(s) of k for which the pair of linear equations kx + 3y = k − 2 and 12x + ky = k has no solution.
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 Question 13
If S_{n}, the sum of first n terms of an A.P. is given by S_{n} = , then find its nth term.
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 Question 14
Two tangents PA and PB are drawn to a circle with centre O from an external point P. Prove that ∠APB = 2 ∠OAB.
Fig. 3
OR
Prove that the parallelogram circumscribing a circle is a rhombus.
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 Question 15
Simplify:
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 Question 16
Prove that is an irrational number.
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 Question 17
Solve the following pair of equations:
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 Question 18
The sum of 4th and 8th terms of an A.P. is 24 and sum of 6th and 10th terms is 44. Find A.P.
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 Question 19
Construct a ΔABC in which BC = 6.5 cm, AB = 4.5 cm and ∠ABC = 60°. Construct a triangle similar to this triangle whose sides are of the corresponding sides of the triangle ABC.
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 Question 20
In Fig. 4, ΔABC is right angled at C and DE ⊥ AB. Prove that ΔABC ∼ ΔADE and hence find the lengths of AE and DE.
Fig. 4
OR
In Fig, 5, DEFG is a square and ∠BAC = 90°. Show that DE^{2} = BD × EC.
Fig. 5
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 Question 21
Find the value of sin 30° geometrically.
OR
Without using trigonometrical tables, evaluate:
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 Question 22
Find the point on yaxis which is equidistant from the points (5, −2) and (−3, 2)
OR
The line segment joining the points A (2, 1) and B (5, −8) is trisected at the points P and Q such that P is nearer to A. If P also lies on the line given by
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2x − y + k = 0, find the value of k.
 Question 23
If P (x, y) is any point on the line joining the points A (a, 0) and B (0, b), then show that .
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 Question 24
In Fig. 6, PQ = 24 cm, PR = 7 cm and O is the centre of the circle. Find the area of shaded region (take π = 3.14)
Fig. 6
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 Question 25
The king, queen and jack of clubs are removed from a deck of 52 playing cards and the remaining cards are shuffled. A card is drawn from the remaining cards. Find the probability of getting a card of (i) heart (ii) queen (iii) clubs.
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 Question 26
The sum of the squares of two consecutive odd numbers is 394. Find the numbers.
OR
Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?
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 Question 27
Prove that, if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.
Using the above result, do the following:
In Fig. 7, DEBC and BD = CE. Prove that ΔABC is an isosceles triangle.
Fig. 7
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 Question 28
A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30°, which is approaching the foot of the tower with a uniform speed. Six seconds later the angle of depression of the car is found to be 60°. Find the time taken by the car to reach the foot of the tower from this point.
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 Question 29
From a solid cylinder whose height is 8 cm and radius 6 cm, a conical cavity of height 8 cm and of base radius 6 cm, is hollowed out. Find the volume of the remaining solid correct to two places of decimals. Also find the total surface area of the remaining solid. (take π = 3.1416)
OR
In Fig. 8, ABC is a right triangle right angled at A. Find the area of shaded region if AB = 6 cm, BC = 10 cm and O is the centre of the incircle of ΔABC.
(take π = 3.14)
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 Question 30
The following table gives the daily income of 50 workers of a factory:

Daily income (in Rs.)
100−120
120−140
140−160
160−180
180−200
Number of workers
12
14
8
6
10
Find the Mean, Mode and Median of the above data.
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