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# Class X: Math, Board Paper 2007, Set-1

General Instructions:
(i) All questions are compulsory.
(ii) The question paper consists of 25 questions divided into three sections A, B and C. Section A comprises of 7 questions of two marks each, Section B comprises of 12 questions of three marks each, and Section C comprises of 6 questions of five marks each.
(iii) Use of calculators is not permitted.
Question 1
• Q1

Find the GCD of the following polynomials.

12x4 + 324x; 36x3 + 90x2 − 54x

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• Q2

Solve for x and y.

OR

Solve for x and y.

31x + 29y = 33, 29x + 31y = 27

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• Q3

Find the sum of all three-digit whole numbers which are multiples of 7.

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• Q4

In figure 1, PQ || AB and PR || AC. Prove that QR || BC.

OR

In figure 2, incircle of ABC touches its sides AB, BC and CA at D, E and F respectively. If AB = AC, prove that BE = EC.

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• Q5

If the mean of the following frequency distribution is 49, find the missing frequency p.

 Class Frequency 0 − 20 2 20 − 40 6 40 − 60 p 60 − 80 5 80 − 100 2

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• Q6

A wrist-watch is available for Rs 1000 cash or Rs 500 as cash down payment followed by three equal monthly instalments of Rs 180. Calculate the rate of interest charged under the instalment plan.

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• Q7

An unbiased die is tossed once. Find the probability of getting

(i) a multiple of 2 or 3

(ii) a prime number greater than 2

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• Q8

Solve the following system of equations graphically.

2x + y = 8

x + 1 = 2y

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• Q9

Simplify the following rational expression in the lowest terms.

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• Q10

If the sum to the first n terms of an AP is given by Sn = n(n + 1), find the 20th term of the AP.

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• Q11

In a cyclic quadrilateral ABCD, diagonal AC bisects C. Prove that the tangent to the circle at A is parallel to the diagonal BD.

OR

In figure 3, O is any point in the interior of ΔABC. OD, OE and OF are drawn perpendiculars to the sides BC, CA and AB respectively. Prove that

AF2 + BD2 + CE2 = OA2 + OB2 + OC2 − OD2 − OE2 − OF2

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• Q12

Construct a ΔABC in which base BC = 6 cm, B = 45° and C = 60°. Draw a

circumcircle of ΔABC.

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• Q13

The diameter of a solid copper sphere is 18 cm. It is melted and drawn into a wire of uniform cross section. If the length of the wire is 108 m, find its diameter.

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• Q14

The expenditure (in rupees) of a family for a month is as follows:

 Item Rent Food Education Electricity and water Others Expenditure 800 3000 1200 400 1800

Represent the above data by a pie chart.

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• Q15

From a pack of 52 cards, red face cards are removed. After that a card is drawn at random from the pack. Find the probability that the card drawn is

(i) a queen

(ii) a red card

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• Q16

Prove that:

OR

If A, B and C are the interior angles of a triangle ABC, show that

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• Q17

The coordinates of the mid-points of the sides of a triangle are (4, 3), (6, 0) and (7, −2). Find the coordinates of the centroid of the triangle.

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• Q18

If the distance of P(x, y) from two points with coordinates (5, 1) and (−1, 5) is equal, prove that 3x = 2y.

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• Q19

A loan of Rs. 24600 is to be paid back in two equal semi-annual instalments. If the interest is charged at 10% per annum, compounded semi-annually, find the instalment.

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• Q20

Prove that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

Using the above, prove the following.

In figure 4, O is the centre of the circle. If BAO = 30° and BCO = 40°, find the value of AOC.

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• Q21

State and prove Pythagoras theorem.

Use the above to prove the following.

ABC is an isosceles right triangle, right angled at C. Prove that AB2 = 2AC2.

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• Q22

The side of a square exceeds the side of another square by 4 cm and the sum of the areas of the two squares is 400 sq. cm. Find the dimensions of the squares.

OR

A fast train takes 3 hours less than a slow train for a journey of 600 km. If the speed of the slow train is 10 km/hour less than that of the fast train, find the speeds of the two trains.

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• Q23

A hollow copper sphere of external and internal diameters 8 cm and 4 cm respectively is melted into a solid cone of base diameter 8 cm. Find the height of the cone.

OR

If the radii of the circular ends of a bucket 45 cm high are 28 cm and 7 cm, find the capacity and surface area of the bucket.

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• Q24

An observer in a lighthouse observes two ships on the same side of the lighthouse, and in the same straight line with the base of the lighthouse. The angles of depression of the ships approaching it are 30° and 60°. If the height of the lighthouse is 150 m, find the distance between the ships.

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• Q25

Satish (aged 67 years) has monthly income of Rs 30000 (excluding HRA). He donates Rs 80000 to a charitable orphanage (50% exemption). He contributes Rs 30000 towards Public Provident Fund and purchases NSCs worth Rs 20000. He pays Rs 1500 as income tax per month for 11 months. Calculate the income tax to be paid by him in the 12th month of the year.

Use the following table to calculate the income tax.

 (a) Savings 100% exemption for permissible savings up to Rs 100000 (b) Rates of Income tax for Senior Citizens (over 65 years) - - Slab Income tax - (i) Up to Rs 185000 No tax - (ii) From Rs 185001 to Rs 250000 20% of the taxable income exceeding Rs. 185000 - (iii) From Rs 250001 and above Rs 13000 + 30% of the taxable income exceeding Rs 250000 (c) Education Cess 2% of Income tax payable

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