Quant Boosters  Kamal Lohia  Set 5

Q26) Given that 2^2010 is a 606digit number whose first digit is 1, how many elements of the set S = {20, 21, 22, …, 2^2009} have a first digit of 4?

The smallest power of 2 with a given number of digits has a first digit of 1, and there are elements of S with n digits for each positive integer n ≤ 605, so there are 605 elements of S whose first digit is 1. Furthermore, if the first digit of 2^k is 1, then the first digit of 2^(k+1) is either 2 or 3, and the first digit of 2^(k+2) is either 4, 5, 6, or 7. Therefore there are 605 elements of S whose first digit is 2 or 3, 605 elements whose first digit is 4, 5, 6, or 7, and 2010  3(605) = 195 whose first digit is 8 or 9. Finally, note that the first digit of 2^k is 8 or 9 if and only if the first digit of 2^(k  1) is 4, so there are 195 elements of S whose first digit is 4.

Q27) Avatar is hosting a party to celebrate the completion of the ritual, but he needs you help to figure out how many guests he can fit in his home. Avatar has five rooms in his home, and he gives you these five facts:
a. The first room can fit half as many guests as the second room
b. The first room and the last room can together fit 11 guests.
c. The third room can fit three less guests than the fourth room
d. The fourth room can fit three times as many guests as the first room
e. The third room can fit two more guests than the second room.
So if he uses all five rooms how many guests can avatar fit in his room

Minimum number of guests that can be in the home is zero. So I am assuming you want to know the maximum capacity of the home.
Let's say Ist room can accomodate x number of guests.
Then IInd room's capacity = 2x, also Vth = 11  x
IVth = 3x. Now IIIrd = 3x  3 = 2x + 2, => x = 5.
So Ist + IInd + IIIrd + IVth + Vth = (x) + (2x) + (2x + 2) + (3x) + (11  x) = 7x + 13 = 7(5) + 13 = 48

Q28) A hollow cube of size 5 cm is taken, with a thickness of 1 cm. It is made of smaller cubes of size 1 cm. If one face of the outer surface of the cube are painted, totally how many faces of the smaller cubes remain unpainted?
a) 900
b) 488
c) 563
d) 800

There are 5^3  3^3 = 98 smaller cubes being used with a total of 6 * 98 = 588 faces and out of which 5^2 = 25 faces are painted which leaves 588  25 = 563 unpainted faces of the smaller cubes

Q29) Rohit drew a grid of 529 cells, arranged in 23 rows and 23 columns. n filled each cell with a no. the no. with which he filled each cell were such that the no. of each row taken from left to right formed an AP and the no. of each column taken from top to bottom formed an AP. the 7th and 17th no of 5th row were 47 and 63, while the 7th and 17th no of 15th rows were 53 and 77. what is the sum of all the no. in the grid.
a. 32798
b. 65596
c. 52900
d. none

As every row and every column is in AP. So sum of all the columns follow an AP and likewise sum of rows. Given sum of 5th row as 23 * (47+63)/2 = 23 * 55 and sum of 15th row as 23 * (53+77)/2 = 23 * 65.
So sum of 12th row is = 23 * 62 and sum of sum of all rows
i.e. sum of all the numbers in grid = 23 * 23 * 62 = 32798

Q30) The integers 1, 2, 3, .... , 40 are written on a blackboard. The following operation is then repeated 39 times. In each operation two numbers say 'a' , 'b' currently on the blackboard are erased and a new number 'ab + a + b' is written. What will be the number left on the board at the end ?
a) 40^3  40 + 1
b) 40^3  40  1
c) No unique value
d) None of these
e) 41!  1

If two numbers are a, b, then the new number you are putting in the place of these two numbers is = ab + a + b = (a + 1)(b + 1)  1 = c (say)
Now when you take c and d next, then the replacement number becomes = cd + c + d
= (c + 1)(d + 1)  1
= (a + 1)(b + 1)(d + 1)  1.
So final number remaining will be (1 + 1)(2 + 1)(3 + 1)....(39 + 1)(40 + 1)  1 = 41!  1.