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1. There are 312, 260 and 156 students in class X, XI and XII respectively. Buses are to be hired to take these students to a picnic. Find the maximum number of students who can sit in a bus if each bus takes equal number of students. (3)
2. Show that, if ‘n’ is odd, then x +1 is a factor of x n +1 . (3)
3. Form a quadratic equation whose one root is 3 + √2. (3)
4. If x − y = 1 and 2x + y = 6 have solution x = α, y = β, find α² + β². (3)
5. Let x = √12 + 3√27 + 2√18 + 4√8. If x = a√2 + b√3, find a + b. (3)
6. The first and last terms of an arithmetic progression are 1 and 11. If the sum of its terms is 36, then find its number of terms. (3)
7. If the cost price of 6 pencils is equal to the selling price of 5 pencils, then find the gain percent.
15 x 10 z (3)
8. If = = = , then find the values of x , y and z. 18 6 y 30 (3)
9. Aman’s income is 20% less than that of Anil. How much percent is Anil’s income more than Aman’s income? (3)
10. Arjun borrowed 16,000 from a finance company at 10% interest per annum, compounded half-yearly. What amount of money will discharge his debt after 1½ years? (3)
11. The arithmetic mean of a set of ‘n’ numbers is x . If the sum of first (n −1) terms is ‘k’, then find the nth number. (3)
12. Jaison buys shares of face value 50 of a company that pays 10% of dividend. At what price did he buy each share from the market if his profit is 16% of his investment? (3)
13. ABCD is a cyclic quadrilateral in which AC and BD are diagonals. If ∠DBC = 55° and ∠BAC = 45°, find ∠BCD. (3)
14. A tree is broken at a height of 5 metre from the ground and its top touches the ground at a distance of 12 metre from the base of the tree. Find the original height of the tree. (3)
15. A cone of height 24 cm and radius of base 6 cm is made up of modeling clay. A child reshapes it in the form of a sphere. Find the radius of the sphere. (3)
16. In triangle PQR, AB is parallel to QR, PA = 4 cm, PQ = 12 cm and PR = 24 cm. Find the value of PB. (3)
17. In a triangle ABC, right angled at B, if tan A = , then find the value of sin A cos C + cos A sin C . (3)
18. Find a point on the y-axis which is equidistant from the points A (6, 5) and B ( − 4, 3 ) . (3)
19. A man decided to cover a distance of 6 km in 84 minutes. He decided to cover two-thirds of the distance at 4 km/hr and the remaining at some different speed. Find the speed after the two-third distance has been covered. (3)
20. Let X and Y be the sets of all positive divisors of 400 and 1000 respectively (including 1 and the number). Find the number of elements in X ∩ Y. (3)
21. Prove that there is no rational number r such that r² = 2. (5)
22. Show that any positive odd integer is of the form 4q +1 or 4 q + 3 , where ‘q’ is some integer. (5)
23. Find the quotient and the remainder when x 4 − 3x 2 + 4 x + 5 is divided by x 2 +1 − x . (5)
24. Find two consecutive odd positive integers, sum of whose squares is 290. (5)
25. For which values of ‘a’ and ‘b’ does the following pair of linear equations have an infinite number of solutions?
2x + 3 y = 7 (a − b) x + (a + b) y = 3a + b − 2 (5)
26. Consider the number 53240. (a) Is it a perfect cube? (b) If not, than by which smallest natural number should 53240 be divided so that the quotient is a perfect cube? Also find the perfect cube in that case. (5)
27. A polygon has 25 sides, the lengths of which starting from the smallest side are in arithmetic progression. If the perimeter of the polygon is 2100 cm and the length of the largest side is 20 times that of the smallest, then find the length of the smallest side. (5)
28. On selling a pen for 48, a shopkeeper loses 20%. In order to gain 20%, what should be the selling price? (5)
29. Arun bought 400 eggs at 54 a dozen. At what price per hundred must he sell them so as to earn a profit of 15%? (5)
30. The present age of the father is 42 years and that of his son is 14 years. Find the ratio of
(a) Present age of the father to the present age of the son.
(b) Age of the father to the age of the son, when the son was
12 years old.
(c) Age of the father after 10 years to the age of the son after 10 years.
(d) Age of father to the age of son when father was 30 years old. (5)
31. Divide 14,000 among A, B and C such that A gets 50% of what B gets and B gets 50% of what C gets. (5)
32. In a direct election between two contestants for the post of secretary, 4% of the total votes cast are declared to be illegal. One contestant secures 55% of the valid votes and wins with a majority of 240 votes. Find the total number of votes cast. (5)
33. The simple interest on a sum of money for 2 years at 8% per annum is 2,400. What will be the compound interest on that sum at the same rate and for the same period. (5)
34. The difference between the compound interest and the simple interest on a certain sum for 2 years at 6% per annum is Rs.90. Find the sum. (5)
35. The arithmetic mean of the marks of boys in a class is 52 and that of girls is 42. The arithmetic mean of marks of boys and girls combined together is 50. Find the percentage of boys in the class. (5)
36. If the arithmetic mean of the numbers 27 + x , 31 + x , 89 + x , 107 + x and 156 + x is 82, then find the arithmetic mean of the numbers 130 + x , 126 + x , 50 + x , 68 + x and 1 + x . (5)
37. A shopkeeper sells an article at its marked price 7,500 and charges sales tax at the rate of 12% from the customer. If the shopkeeper pays a VAT of Rs.180, calculate the price inclusive of the tax paid by the shopkeeper. (5)
38. A circular part of radius 20 m is situated in a colony. Three boys Ankur, Syed and David are sitting at an equal distance on its boundary, each having a toy telephone in his hands to talk to each other. Find the length of the string of each phone. (5)
39. The cost of fencing a circular field at the rate of 24 per metre is 5,280. The field is to be ploughed at the rate of 0.50 per square metre. Find the cost of ploughing the field. (Take π = ) (5)
40. ABCD is a parallelogram with ∠A = 80°. The internal bisectors of ∠B and ∠C meet at O. Find the measures of the three angles of triangle BCO. (5)
41. In triangle ABC, PQ is parallel to BC and the area of the quadrilateral PBCQ is 42 m2. If AP : PB = 2:3, then find the area of triangle APQ. (5)
42. A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top, which is open, is 5 cm. It is filled with water upto the brim. When lead shots, each of which is a sphere of radius 0.5 cm are dropped into the vessel, one fourth of the water flows out. Find the number of lead shots dropped in the vessel. (5)
43. A girl of height 90 cm is walking away from the base of a lamp-post at a speed of 1.2 m/s. If the lamp is 3.6 m above the ground, find the length of her shadow after 4 seconds. (5)
44. On one bank of a river, there is a tree. On another bank, an observer makes an angle of elevation of 60° at the top of the tree. The angle of elevation of the top of the tree at a distance 20 m away from the bank is 30°. Find the width of the river.
( ) (5)
45. Find k for which (k − 3)x − (4 − k²)y + k² − 7k + 6 = 0 is (a) parallel to the x-axis, (b) parallel to the y-axis, and (c) passes through the origin. (5)
46. A market research group conducted a survey of 1000 customers and reported that 720 consumers like product A and 450 consumers like product B. What is the least number that must have liked both products? (5)
47. A train can travel 50% faster than a car. Both start from point A at the same time and reach point B, which is 75 km away from A at the same time. On the way, however, the train lost about 12.5 minutes while stopping at the stations. Find the speed of the car. (5)
48. A and B can do a piece of work in 12 days; B and C can do it in 15 days while C and A can finish it in 20 days. If A, B, C work together, in how many days with they finish the work? In how many days will each one of them finish it, working done? (5)
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