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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

Write whether the rational number will have a terminating decimal expansion or a non-terminating repeating decimal expansion.

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• Q2

Write the polynomial, the product and sum of whose zeroes are and respectively.

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• Q3

Write whether the following pair of linear equations is consistent or not:

x + y = 14

xy = 4

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• Q4

Write the nature of roots of quadratic equation

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• Q5

For what value of k, are the numbers x, 2x + k and 3x + 6 three consecutive terms of an A.P.

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• Q6

In a ΔABC, DE||BC. IF DE = BC and area of ΔABC = 81 cm2, find the area of ΔADE.

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• Q7

If sec A = and A + B = 90°, find the value of cosec B.

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• Q8

If the mid-point of the line segment joining the points P (6, b − 2) and Q (−2, 4) is (2, −3), find the value of b.

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• Q9

The length of the minute hand of a wall clock is 7 cm. How much area does it sweep in 20 minutes?

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• Q10

What is the lower limit of the modal class of the following frequency distribution?

 Age in (years) 0 − 10 10 − 20 20 − 30 30 − 40 40 − 50 50 − 60 Number of patients 16 13 6 11 27 18

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• Q11

Without drawing the graph, find out whether the lines representing the following pair of linear equations intersect at a point, are parallel or coincident:

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• Q12

The 17th term of an A.P. exceeds its 10th term by 7. Find the common difference.

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• Q13

Without using trigonometric tables, evaluate:

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• Q14

Show that the points (−2, 5); (3, −4) and (7, 10) are the vertices of a right angled isosceles triangle.

OR

The centre of a circle is (2α − 1, 7) and it passes through the point (−3, −1). If the diameter of the circle is 20 units, then find the values(s) of α.

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• Q15

If C is a point lying on the line segment AB joining A (1, 1) and B (2, −3) such that 3 AC = CB, then find the coordinates of C.

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• Q16

Show that the square of any positive odd integer is of the form 8m + 1, for some integer m.

OR

Prove that is not a rational number.

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• Q17

If the polynomial 6x4 + 8x3 − 5x2 + ax + b is exactly divisible by the polynomial 2x2 − 5, the find the values of a and b.

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• Q18

If 9th term of an A.P. is zero, prove that its 29th term is double of its 19th term.

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• Q19

Draw a circle of radius 3 cm. From a point P, 6 cm away from its centre, construct a pair of tangents to the circle. Measure the lengths of the tangents.

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• Q20

In figure 1, two triangles ABC and DBC lie on the same side of base BC. P is a point on BC such that PQ || BA and PR || BD. Prove that QR || AD.

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• Q21

In figure 2, a triangle ABC is right angled at B. Side BC is trisected at points D and E. Prove that 8 AE2 + 5 AD2.

OR

In figure 3, a circle is inscribed in a triangle ABC having side BC = 8 cm, AC = 10 cm and AB = 12 cm. Find AD, BE and CF.

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• Q22

Prove that

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• Q23

Find a relation between x and y if the points (x, y), (1, 2) and (7, 0) are collinear.

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• Q24

In figure 4, the shape of the top of a table in a restaurant is that of a sector of a circle with centre O and ∠BOD = 90°. If BO = OD = 60 cm, find

(i) the area of the top of the table.

(ii) the perimeter of the table top.

(Take π = 3.14)

OR

In figure 5, ABCD is a square of side 14 cm and APD and BPC are semicircles. Find the area of shaded region. (Take)

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• Q25

A box has cards numbered 14 to 99. Cards are mixed thoroughly and a card is drawn from the bag at random. Find the probability that the number on the card, drawn from the box is

(i) an odd number,

(ii) a perfect square number,

(iii) a number divisible by 7.

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• Q26

A trader bought a number of articles for Rs 900. Five articles were found damaged. He sold each of the remaining articles at Rs. 2 more than what he paid for it. He got a profit of Rs. 80 on the whole transaction. Find the number of articles he bought.

OR

Two years ago the man’s age was three times the square of his son’s age. Three years hence his age will be four times his son’s age. Find their present ages.

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• Q27

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

Using the above theorem prove the following:

The area of the equilateral triangle described on the side of a square is half the area of the equilateral triangle described on its diagonal.

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• Q28

The angle of elevation of the top of a building from the foot of a tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 50 m high, find the height of the building.

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• Q29

A spherical copper shell, of external diameter 18 cm, is melted and recast into a solid cone of base radius 14 cm and height cm. Find the inner diameter of the shell.

OR

A bucket is in the form of a frustum of a cone with a capacity of 12308.8 cm3. The radii of the top and bottom circular ends of the bucket are 20 cm and 12 cm respectively. Find the height of the bucket and also the area of metal sheet used in making it.

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• Q30

Find the mode, median and mean for the following data:

 Marks obtained 25 − 35 35 − 45 45 − 55 55 − 65 65 − 75 75 − 85 Number of students 7 31 33 17 11 1

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