SECTION I

Sub-Section I-A

Number of Questions : 10

Note: Questions 1 to 10 carry one mark each.

DIRECTIONS for Q1 to Q5: Answer the questions independently of each other.

1. If x = (16³ + 17³ + 18³ + 19³), then x divided by 70 leaves a remainder of

2. A chemical plant has four tanks (A, B, C and D), each containing 1000 litres of a chemical. The chemical is being pumped from one tank to another as follows:
From A to B @ 20 litres/minute
From C to A @ 90 litres/minute
From A to D @ 10 litres/minute
From C to D @ 50 litres/minute
From B to C @ 100 litres/minute
From D to B @ 110 litres/minute
Which tank gets emptied first and how long does it take (in minutes) to get empty after pumping starts?

3. Two identical circles intersect so that their centres, and the points at which they intersect, form a square of side 1 cm. The area in sq. cm of the portion that is common to the two circles is

4. A jogging park has two identical circular tracks touching each other, and a rectangular track enclosing the two circles. The edges of the rectangles are tangential to the circles. Two friends, A and B, start jogging simultaneously from the point where one of the circular tracks touches the smaller side of the rectangular track. A jogs along the rectangular track, while B jogs along the two circular tracks in a figure of eight. Approximately, how much faster than A does B have to run, so that they take the same time to return to their starting point?

5. In a chess competition involving some boys and girls of a school, every student had to play exactly one game with every other student. It was found that in 45 games both the players were girls, and in 190 games both were boys. The number of games in which one player was a boy and the other was a girl is

DIRECTIONS for Q.6 & Q.7:
Answer the questions on the basis of the information given below.

Ram and Shyam run a race between points A and B, 5 km apart. Ram starts at 9 a.m. from A at a speed of 5 km/ hr, reaches B, and returns to A at the same speed. Shyam starts at 9:45 a.m. from A at a speed of 10 km/hr, reaches B and comes back to A at the same speed

6. At what time do Ram and Shyam first meet each other?

7. At what time does Shyam overtake Ram?

DIRECTIONS for Q.8 to Q.10:
Answer the questions independently of each other

8. if R=(30⁶⁵ – 29⁶⁵)/(30⁶⁴ + 29⁶⁴)

9. What is the distance in cm between two parallel chords of lengths 32 cm and 24 cm in a circle of radius 20 cm?

10. For which value of k does the following pair of equations yield a unique solution for x such that the solution is positive? x² – y² = 0
(x – k)² + y² = 1

SECTION I

Sub-Section I-B

Number of Questions : 20

Note: Questions 11 to 30 carry two marks each.

Directions for Questions 11 to 30: Answer the questions independently of each other.

11. Let n! = 1 x 2 x 3 x … x n for integer n ? 1. If p = 1! (2 x 2!) + (3 x 3!) + . . . + (10 x 10!), then p + 2 when divided by 11! leaves a remainder of

12. Consider a triangle drawn on the X-Y plane with its three vertices at (41, 0), (0, 41) and (0, 0), each vertex being represented by its (X, Y) coordinates. The number of points with integer coordinates inside the triangle (excluding all the points on the boundary) is

13. The digits of a three-digit number A are written in the reverse order to form another three-digit number B. If B > A and B-A is perfectly divisible by 7, then which of the following is necessarily true?

14. If a₁ = 1 and a_n+1 – 3a_n + 2 = 4n for every positive integer n, then a₁₀₀ equals

15. Let S be the set of five-digit numbers formed by the digits 1, 2, 3, 4 and 5, using each digit exactly once such that exactly two odd positions are occupied by odd digits. What is the sum of the digits in the rightmost position of the numbers in S?

16. The rightmost non-zero digit of the number 30²⁷²⁰ is

17. Four points A, B, C and D lie on a straight line in the X-Y plane, such that AB = BC = CD, and the length of AB is 1 metre. An ant at A wants to reach a sugar particle at D. But there are insect repellents kept at points B and C. The ant would not go within one metre of any insect repellent. The minimum distance in metres the ant must traverse to reach the sugar particle is

18. If x ? y and y > 1, then the value of the expression log_x(x/y)+log_y(y/x) can never be

19. For a positive integer n, let p_n denote the product of the digits of n, and s_n denote the sum of the digits of n. The number of integers between 10 and 1000 for which p_n + s_n = n is

20. Rectangular tiles each of size 70 cm by 30 cm must be laid horizontally on a rectangular floor of size 110 cm by 130 cm, such that the tiles do not overlap. A tile can be placed in any orientation so long as its edges are parallel to the edges of the floor. No tile should overshoot any edge of the floor. The maximum number of tiles that can be accommodated on the floor is

21. In the X-Y plane, the area of the region bounded by the graph of |x + y| + |x – y| = 4 is

22. In the following figure, the diameter of the circle is 3 cm. AB and MN are two diameters such that MN is perpendicular to AB. In addition, CG is perpendicular to AB such that AE:EB = 1:2, and DF is perpendicular to MN such that NL:LM = 1:2. The length of DH in cm is

23. Consider the triangle ABC shown in the following figure where BC = 12 cm, DB = 9 cm, CD = 6 cm and ÐBCD = ÐBAC.

What is the ratio of the perimeter of the triangle ADC to that of the triangle BDC?

24. P, Q, S, and R are points on the circumference of a circle of radius r, such that PQR is an equilateral triangle and PS is a diameter of the circle. What is the perimeter of the quadrilateral PQSR?

25. Let S be a set of positive integers such that every element n of S satisfies the conditions
a) 1000 = n = 1200
b) every digit in n is odd
Then how many elements of S are divisible by 3?

26. Let . Then x equals

27. Let g(x) be a function such that g(x + l) + g(x – l) = g(x) for every real x. Then for what value of p is the relation g(x + p) = g(x) necessarily true for every real x?

28. A telecom service provider engages male and female operators for answering 1000 calls per day. A male operator can handle 40 calls per day whereas a female operator can handle 50 calls per day. The male and the female operators get a fixed wage of Rs.250 and Rs.300 per day respectively. In addition, a male operator gets Rs.15 per call he answers and a female operator gets Rs.10 per call she answers. To minimize the total cost, how many male operators should the service provider employ assuming he has to employ more than 7 of the 12 female operators available for the job?

29. Three Englishmen and three Frenchmen work for the same company. Each of them knows a secret not known to others. They need to exchange these secrets over person-to-person phone calls so that eventually each person knows all six secrets. None of the Frenchmen knows English, and only one Englishman knows French. What is the minimum number of phone calls needed for the above purpose?.

30. A rectangular floor is fully covered with square tiles of identical size. The tiles on the edges are white and the tiles in the interior are red. The number of white tiles is the same as the number of red tiles. A possible value of the number of tiles along one edge of the floor is