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# Class X: Math, Board Paper 2009, Set-3

General Instructions:
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
• Q1

If, then find the value of (2cot2θ + 2).

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• Q2

For what value of p, (−4) is a zero of the polynomial x2 − 2x − (7p + 3)?

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• Q3

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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• Q4

Find the value of a so that the point (3, a) lies on the line represented by
2x − 3y = 5.

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• Q5

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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• Q6

Find the distance between the points and.

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• Q7

The decimal expansion of the rational number will terminate after how many places of decimals?

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• Q8

For what value of p, are 2p − 1, 7 and 3p three consecutives terms of an A.P.?

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• Q9

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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• Q10

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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• Q11

If the polynomial is divided by another polynomial, the remainder comes out to be, find p and q.

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• Q12

Simplify:

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• Q13

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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• Q14

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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• Q15

If Sn, the sum of first n terms of an A.P. is given by Sn = , then find its nth term.

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• Q16

Prove that is an irrational number.

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• Q17

Solve the following pair of equations:

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• Q18

Find the value of sin 30° geometrically.

OR

Without using trigonometrical tables, evaluate:

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• Q19

Construct a ΔABC in which BC = 6.5 cm, AB = 4.5 cm and ∠ACB = 60°. Construct another triangle similar to ΔABC such that each side of new triangle is of the corresponding sides of ΔABC.

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• Q20

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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• Q21

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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• Q22

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 DE2 = BD × EC.

Fig. 5

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• Q23

Find the point on y-axis 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
2xy + k = 0, find the value of k.

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• Q24

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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• 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

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• 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. 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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• Q27

Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

In a trapezium ABCD, AC and BD are intersecting at O. AB || DC and AB = 2 CD. If area of ΔAOB = 84 cm2, find the area of ΔCOD.

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• Q28

A pole of height 5 m is fixed on the top of a tower. The angle of elevation of the top of the pole as observed from a point A on the ground is 60° and the angle of depression of the point A from the top of the tower is 45°. Find the height of the tower. [Take]

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• 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.

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• 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?

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