(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.
 Q1
 Q2
 Q3
 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.
VIEW SOLUTION  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
 Q6
A wristwatch 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.
VIEW SOLUTION  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
VIEW SOLUTION  Q8
 Q9
 Q10
If the sum to the first n terms of an AP is given by S_{n} = n(n + 1), find the 20^{th} term of the AP.
VIEW SOLUTION  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
AF^{2} + BD^{2} + CE^{2} = OA^{2} + OB^{2} + OC^{2} − OD^{2} − OE^{2} − OF^{2}
VIEW SOLUTION  Q12
Construct a ΔABC in which base BC = 6 cm, ∠B = 45° and ∠C = 60°. Draw a
circumcircle of ΔABC.
VIEW SOLUTION  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.
VIEW SOLUTION  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.
VIEW SOLUTION  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
(iii) a spade card
VIEW SOLUTION  Q16
 Q17
The coordinates of the midpoints of the sides of a triangle are (4, 3), (6, 0) and (7, −2). Find the coordinates of the centroid of the triangle.
VIEW SOLUTION  Q18
If the distance of P(x, y) from two points with coordinates (5, 1) and (−1, 5) is equal, prove that 3x = 2y.
VIEW SOLUTION  Q19
A loan of Rs. 24600 is to be paid back in two equal semiannual instalments. If the interest is charged at 10% per annum, compounded semiannually, find the instalment.
VIEW SOLUTION  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.
VIEW SOLUTION  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 AB^{2}^{ }= 2AC^{2}.
VIEW SOLUTION  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.
VIEW SOLUTION  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.
VIEW SOLUTION  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.
VIEW SOLUTION  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 12^{th} 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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Class X: Math, Board Paper 2007, Set1