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.
 Q1
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
VIEW SOLUTION  Q2
 Q3
 Q4
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.
VIEW SOLUTION  Q5
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
VIEW SOLUTION  Q6
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.
VIEW SOLUTION  Q7
 Q8
 Q9
Find the value of a so that the point (3, a) lies on the line represented by
VIEW SOLUTION
2x − 3y = 5.  Q10
The decimal expansion of the rational number will terminate after how many places of decimals.
VIEW SOLUTION  Q11
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.
VIEW SOLUTION  Q12
If the polynomial is divided by another polynomial, the remainder comes out to be, find a and b.
VIEW SOLUTION  Q13
If S_{n}, the sum of first n terms of an A.P. is given by S_{n} = , then find its nth term.
VIEW SOLUTION  Q14
 Q15
Find the value(s) of k for which the pair of linear equations kx + 3y = k − 2 and 12x + ky = k has no solution.
VIEW SOLUTION  Q16
 Q17
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.
VIEW SOLUTION  Q18
The sum of 5th and 9th terms of an A.P. is 72 and sum of 7th and 12th terms is 97. Find the A.P.
VIEW SOLUTION  Q19
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
VIEW SOLUTION  Q20
If P (x, y) is any point on the line joining the points A (a, 0) and B (0, b), then show that .
VIEW SOLUTION  Q21
 Q22
Find the point on xaxis which is equidistant from the points (2, − 5) and (− 2, 9)
OR
The line segment joining the points P (3, 3) and Q (6, − 6) is trisected at the points A and B such that A is nearer to P. If A also lies on the line given by 2x + y + k = 0, find the value of k.
VIEW SOLUTION  Q23
Find the value of sin 30° geometrically.
OR
Without using trigonometrical tables, evaluate:
VIEW SOLUTION  Q24
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.
VIEW SOLUTION  Q25
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
VIEW SOLUTION  Q26
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. 7, 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)
VIEW SOLUTION  Q27
Prove that, in a right triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides.
Using the above, prove the following:
Prove that, in a ΔABC, if AD is perpendicular to BC, then AB^{2} + CD^{2} = AC^{2} + BD^{2}.
VIEW SOLUTION  Q28
The angles of depression of the top and bottom of an 8 m tall building from the top of a multistoreyed building are 30° and 45°, respectively. Find the height of the multistoreyed building and the distance between the two buildings.
VIEW SOLUTION  Q29
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.
VIEW SOLUTION 
 Q30
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?
VIEW SOLUTION
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Class X: Math, Board Paper 2009, Set2