Paper Solution:. How many positive integers less than 000 have the property that the sum of the digits of each such number is divisible by 7 and the number itself is divisible by 3? Sum of Digits Drivable by 7 & 3 so, LCM of (7,3) = 2 Sum of digits should be 2 minimum single digit and two digit cannot be no Three digit no < 000 xyz < 000 x + y + z = 2 Case-() take Z = 9 Pairs of (x,y) 7 x + y + z 3 9 9 9 3 9 Similarly y = 9 Pairs of (x,2) 7 x = 9 Pair of ( y,2) 7 But 399, 993, 939 have 2 9 s Total = 7 + 7+ 7-3 = 8 Numbers 399 489 498 579 594 669 696 759 795 849 894 993 939 975 957 966 984 948 2. Suppose a, b are positive real numbers such that a a b b 83. a b b a 83Find 9 ( ) 5 a b a a b b 83..(i)
a b b a 83 (ii) Equation (i) + 3.(ii) a b 9 9 a b 365 73 5 5..(iii) using (ii) and (iii) 3. A contractor has two teams of workers: team A and team B. Team A can complete a job in 2 days and team B can do the same job in 36 days. Team A starts working the job and team B joins team A withdraws after two more days. For how many more days should team B work to complete the job. 4 2 2 2 36 36 x Answer: x =6 4. Let a, b be integers such that all the roots of the equation (x 2 + a + 20) (x 2 + 7x + b) =0 are negative integers. What is the smallest possible value of a + b? roots of quadratic for which a is minimum are -4, -5. for the minimum value of b roots of quadratic are -6, - then hence, value of a + b is 25. 5. Let u, v, w be real numbers in geometric progression such that u > v > w. Suppose u 40 = u n = w 60. Find the value of n. u 40 = u n = w 60 = k 20 these u = k 3 20 n v k n=48 (since, u, v, w are in G.P.) w = k 2 6. Let the sum n n ( n )( n Written in its slowest terms be p 2) q 9. Find the value of q p. Applying V n method T n 2 n n 2 n n 27 Sn 0
so, q p = 83. 7. Find the number of positive integers n, such that n n Solving the given inequation n 60 total value of n = 29. 8. A pen costs Rs. and a notebook costs Rs. 3. Find the number of ways in which a person can spend exactly Rs. 000 to buy pens and notebooks. Let the numbers of pens x & number of notebooks y. x + 3y = 000 x (00 y) 2y y = {5, 6, 27, 38, 49, 60, 7} Total seven ways. Answer: 7 9. There are five cities A, B, C, D, E on a certain island. Each city is connected to every other city by road. In how many ways can a person starting from city A come back to A after visiting some cities without visiting a city more than once and without taking the same road more than once? (The order in which he visits the cities also meters: e.g., the routs A B C A and A C B A are different.) C 2! C 3! C 4! 60 4 4 4 2 3 4 0. There are eight rooms on the first floor of a hotel, with four rooms on each side of the corridor, symmetrically situated (that is each room is exactly opposite to one other room). Four guests have to be accommodated in four of the eight rooms (that is, one in each) such that no two guests are in adjacent rooms or in opposite rooms. In how many ways can the guests be accommodated? There are two possibilities for the accommodation pattern. each possibility has two guests on both sides rooms in alternate rooms positions.
Possibility - Possibility -2 Answer 2. 4! = 48 x 3x. Let f( x) sin cos for all real x. End the least natural number n such that 3 0 f ( n X ) f ( x) for all real x. n n 2 2 6 3 2 20 3 3 0 LCM of n and n 2 = 30 so, answer n=30. 2. In a class, the total numbers of boys and girls are in the ratio 4:3. On one day it was found that 8 boys and 4 girls were absent from the class, and that the number of boys was the square of the number of girls. What is the total number of students in the class? Total students 7k. given (4k - 8) = (3k 4) 2 k = 6 Answer : 42 3. In a rectangle ABCD, E is the midpoint of AB; F is a point on AC such that BF is perpendicular to AC; and FE perpendicular to BD. Suppose BC 8 3. Find AB. Let the BAC then In In a ABC tan 90...( i) 8 3 a BFC cos...( ii) 6 3 (i) and (ii) gives 30 o, now using (ii) a=24 (length of side AB).
4. Suppose x is a positive real number such that {x}, [x] and x are in a geometric progression. Find the least positive integer u such that x>00. (Here [x] denotes the integer part for x and { x } = x [x].) Given [x] 2 = x{x} solving the above equation x 5, [x] =. 2 Using the above information (.6) n > 00, least value satisfying n = 0. 5. Integers, 2, 3..,n, where n > 2, are written on a board. Two numbers m, k such that < m < n, < k < n are removed and the average of the remaining numbers is found to be 7. What is the maximum sum of the two removed numbers? 6. Five distinct 2-digit numbers are in a geometric progression. Find the middle term. Answer: 36
7. Suppose the altitudes of a triangle are 0, 2 and 5. What is its semi-perimeter? 2 2 a, b, 0 2 c 2 5 now using herons formula semi perimeter is 22.67 8. If the real numbers x, y, z are such that x 2 + 4y 2 + 6z 2 = 48 and xy + 4yz + 2zx = 24, what is the value of x 2 + y 2 = z 2? Solving given equation x=4, y=2, z= Using above information answer is 2. 9. Suppose, 2, 3 are the roots of the equation x 4 + ax 2 + bx = c. Find the value of c. Sum of all the roots = 0 6 (Let 4 th root is ). Product of all the roots = - c.2.3.(-6) = - c c = 36 20. What is the number of triples (a, b, c) of positive integers such that (i) a < b < c < 0 and (ii) a, b, c, 0 form the sides of a quadrilateral?
2. Find the number of ordered triplets (a, b, c) of positive integers such that abc = 08. 08 = 2 2 3 3 distribution of index 2 among 3 variables, so that each variable can take none one or more. 2 3 2 3 2 3 total number of ordered triplets (a, b, c) = 2 2 C 3 2 C 2 2 60 22. Suppose in the plane 0 pair wise nonparallel lines intersect one another. What is the maximum possible number of polygons (with finite areas) that can be formed? Number of non-overlapping polygons = 56 20 = 46 line divide plane into 2 regions 2 lines divide plane into 4 regions 3 lines divide plane into 7 regions 4 lines divide plane into regions 5 lines divide plane into 6 regions 6 lines divide plane into 22 regions 7 lines divide plane into 29 regions 8 lines divide plane into 37 regions 9 lines divide plane into 46 regions 0 lines divide plane into 56 regions Now open regions for 3 lines are 6 23. Suppose an integer x, a natural number n and a prime number p satisfy the equation 7 x 2 44 x + 2 = p n. Find the largest value of p.
Sol 24. Let P be can interior point of a triangle ABC whose side lengths are 26, 65, 78. The line through P parallel to BC meets AB in K and AC in L. The line through P parallel to AB meets CA in S and CB in T. If KL, MN, ST are of equal lengths, find this common length. Sol 25. Let ABCD be a rectangle and let E and F be points CD and BC respectively such that area (ADE) = 6, area (CEF) =9 and area (ABF) =25. What is the area of triangle AEF?
26. Let AB and CD be two parallel chords in a circle with radius 5 such that the centre O lies between these chords. Suppose AB = 6, CD =8. Suppose further that the area of the part of the circle lying between the chords AB and CD is ( m n) / k, where m, n, k are positive integers with gcd(m, n, k)=. What is the value of m + n + k? BD arc subtend 90 o at O. area = 2 2 5 6 4 8 3 4 2 2 25 48 2 Ans m + n + k =75 27. Let be a circle with centre O and let AB be a diameter of. Let P be a point on the segment OB different from O. Suppose another circle with centre P lies in the interior of.tangents are drawn from A and B to the circle intersecting again at A and B respectively such that A and B are on the opposite sides of Ab. Given that A B = 5. AB = 5 and OP=0, find the radius of. Let radius of = R Radius of 2 = R 2 5 R 2R...( i) R 0 2
5 R 2R...( ii) R 0 2 From (i) and (ii) R = 20 28. Let p.q be prime numbers such that n 3pq n is a multiple of 3pq for all positive integers n. Find the least possible value of p + q. 29. For each positive integer n, consider the highest common factor h n of the two numbers n!+ and (n + )!. For n < 00, find the largest value of h n. n =2 then highest common factor is 3. n =4 then highest common factor is 5. n =6 then highest common factor is 7. Similarly n = 96 then highest common factor is 97. 30. Consider the areas of the four triangles obtained by drawing the diagonals AC and BD of a trapezium ABCD. The products of these areas, taken two at time, are computed. If among the six products so obtained, two products are 296 and 576, determine the square roots of the maximum possible area of the trapezium to the nearest integer.