Quant Boosters  Hemant Yadav  Set 4

Q14) How many ordered triplets of natural numbers (a, b, c) are there such that GCD(a, b, c) = 1 and all a, b, c are factors of 2310 ?

Q15) How many positive integral solutions does the following equation has : x + y + z + w = xyzw

Q16) If x and y are integers with (y − 1)^(x + y) = 4^3, then the number of possible values for x is

Q17) Dolly, Molly and Polly each can walk at 6 km/h. They have one motorcycle, which travels at 90 km/h, which can accommodate at most two of them at once (and cannot drive by itself!). Let t hours be the time taken for all three of them to reach a point 135 km away. Ignoring the time required to start, stop or change directions, what is true about the smallest possible value of t ?

Q18) A and B together can complete a task in 20 days. B and C together can complete the same task in 30 days. A and C together can complete the same task in 30 days. What is the respective ratio of the number of days taken by A when completing the same task alone to the number of days taken by C when completing the same task alone ?

Q19) A cube is divided into eight smaller cubes each of which has an edge of length equal to half the length of the edge of the bigger cube. Also each edge of the smaller cubes represents a path. Find the no. of ways of reaching the diagonally opposite point of the bigger cube from one any one corner, so that one has to travel the shortest distance.

Q20) Set S is formed by selecting some of the numbers from the first 110 natural numbers such that the HCF of any two numbers in the set is the same. If every pair of numbers of set S has to be relatively prime and set S has the maximum number of elements possible, then in how ways can the set S be selected?

Q21) Five points A,B,C,D,E lie on a line L1 and points P,Q,R,S,T lie on a line L2. Each of the five points on L1 is connected to each of the points on L2, by means of straight lines terminated by the points.Then excluding the given points the maximum number of points at which the lines can intersect is ?

Q22) Amit has two congruent sheets of paper in triangular shape. He built a parallelogram with them by three different methods. It is known that the perimeters of the three parallelograms are 24, 33 and 35. Find the perimeter of the triangular piece of paper.

Q23) If a and b are chosen at randomly from the set containing first nine natural numbers with replacement and let f(x)= a(x^4) + b(x^3) + (a+1)(x^2) + bx + 1. Then in how many ways f(x) > 0 for all x belongs to R ?

Q24) One writes 268 numbers around a circle such that the sum of 20 consecutive numbers is always equal to 75. The numbers 3, 4 and 9 are written in positions 17, 83 and 144 respectively. Find the number in position 210.

Q25) A rope 20m long is randomly cut into two segments, each of which is used to form the perimeter of a square.
(a) Find the probability that the larger square has an area greater than 9 m^2
(b) Find the probability that the total area of the two squares is greater than 20.5 m^2

Q26) How many threedigit numbers are there such that no two adjacent digits of the number are consecutive?

Q27) In a science competition, participants are required to form teams. Each team is of 6 or 7 students from the same school. n students has been sent each by schools A and B for the competition. The students from School A are divided into 31 teams while the students from School B are divided into 36 teams. Find sum of all possible values of n.

Q28) Raman saved some money in January, 2011. In each morning of Febuary, 2011, his mother gave him 150 rupees as pocket money. He then spent 10% of the total amount he had for lunch every day. Given that the amount he had was 1350 rupees at the end of Febuary, how many dollars did he spend on lunch in February?

Q29) 8100 square boxes were aligned as a square with a length of 90 boxes. Alan put a coin in one of the boxes. Betty wanted to find the coin. Every time she could choose one of the boxes, open it and the (at most eight) boxes surrounding it. At least how many boxes must Betty choose to guarantee that she must find the coin?

Q30) There are 10 consecutive positive integers written on a blackboard. One number is erased. The sum of remaining nine integers is 2011. Which number was erased?