Quant Boosters  Hemant Yadav  Set 4

Q8) If it is known that log2 = 0.3010, then find the smallest possible positive integral value of n such that 2^(10n) does not begin with 1

Q9) The lattice points (points having integral coordinates) lying on a circle centered at origin having radius 5 forms a polygon. Find the area of the polygon.

4 * {area of quad. with vertices (5,0),(3,4),(4,3),(0,5) + triangle with vertices (0,0,), (0,5), (5,0) }
= 4 * (6 + 25/2) = 4 * 37/2 = 74

Q10) A faulty 12 hour digital clock displays 9 whenever it is supposed to display 1. For example when it is 1:21, the clock incorrectly shows 9:29. For what fraction of the day will the clock show the right time?
a) 1/2
b) 2/3
c) 3/4
d) 5/6

Q11) Two sides of a triangle are 12.5 and 15.5. How many different integral values can the third side take?
[OA : 24]

Q12) An acute triangle has sides as 12, 16 and ‘n’. How many different integral values ‘n’ can take?

Case i) when n is the largest side
=> n² < 12² + 16² = 20²
=> n < 20Case ii) When 16 is the largest side
=> 16² < n² + 12²
=> n² > 112
=> n > 10So, 11 to 19
Hence 9 values!!

Q13) If perimeter of a right angle triangle is 12 + 8√3 and sum of squares of the sides is 294, then find the area of the triangle.

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