# Quant Capsules by Shashank Prabhu - Set 15

• Using only 2, 5, 10, 25, and 50 paisa coins, what will be the minimum number of coins required to pay exactly 78 paise, 69 paise and Rs. 1.01 to three different persons?

78 -> 2+2+2+2+10+10+50 -> 7 coins
69 -> 2+2+5+10+50 -> 5 coins
101 -> 2+2+2+10+10+25+50 -> 7 coins
Total = 19

Appurv and Vikram play a game in which they roll a 6-faced die alternately starting with Appurv. Each of them keeps on adding the numbers rolled by him and the first one to get to a sum of at least 3 wins the game. What is the probability of Vikram winning the game?
(1) 2/9
(2) 293/1296
(3) 299/1296
(4) None of these

Let the game continue as A -> V -> A -> V .... The possible outcomes are:
1 -> 3/4/5/6 -> 1/6 * 4/6
1 -> 1 -> 1 -> 2/3/4/5/6 -> 1/6 * 1/6 * 1/6 * 5/6
1 -> 2 -> 1 -> 1/2/3/4/5/6 -> 1/6 * 1/6 * 1/6 * 1
2 -> 3/4/5/6 -> 1/6 * 4/6
Total: 144/6^4 + 5/6^4 + 6/6^4 + 144/6^4 = 299/1296

A sequence is formed from the set of first n natural numbers by deleting all the perfect squares and all the perfect cubes. If the elements are arranged in an ascending order then, what is the 240th term of the sequence?

240 numbers as it is, there will be 15 squares and 6 cubes in these 240 numbers. But 1 and 64 will be counted under both squares and cubes. So, total eliminations will be 15+6-2=19. But, when we are advancing 240 by 19, we are crossing 256 which is a square and so, we will advance by 20 positions to take care of it. So, it would be 260.

The sum of the coefficients of the polynomial (x - 1)^7 * (x - 2)^2 * (x - 4) is

In any expression ax^n + bx^(n-1) + cx^(n-2)... kx+m, the sum of coefficients is nothing but the sum of a, b, c, ... k, m. This can be found out by putting x=1. That's the concept.

What is the remainder obtained when the sum of the squares of any thirty consecutive natural numbers is divided by 12?
(a) 0
(b) 3
(c) 11
(d) Not unique

11 is correct. The best way is to check for the most basic case and then understand that the difference in the squares will be divisible by 12. The technically correct way of doing it is as follows:
n+1, n+2 ... n+30 are the numbers
30n^2 + 2n(1+2...+30) + (1^2+2^2+...30^2)
30n^2 + 930n + 9455
30n(n+31) + 9455
9455 mod 12 is 11
Also, either n or n+31 will be an even number. So, the first part is a multiple of 60 and hence, divisible by 12.
Remainder of the expression will always be 11 irrespective of the value of n

Tony Stark, Wade Wilson and Scott Lang can destroy the Tesseract alone in 15, 20 and 25 days respectively. However, while teaming up along with somebody the efficiency of Tony Stark, Wade Wilson and Scott Lang reduces by 30%, 20% and 50% respectively. If none of them is allowed to work for three consecutive days, then what is the maximum possible fraction of the Tesseract that they can destroy in four days?
a) 21/50
b) 17/50
c) 8/25
d) None of these

Let the total work be 300 units. Tony does 20 units, Wade does 15 units and Scott does 12 units while working alone. While working with someone else, Tony does 14 units, Wade does 12 units and Scott does 6 units. The best case would be to make sure that all three work together for the maximum amount of time i.e. for 2 days.
Day 1: All 3 -> 32 units
Day 2: Tony+Wade -> 26 units
Day 3: Scott alone -> 12 units
Day 4: All 3 -> 32 units
Total 102/300 or 17/50
PS: You can also go for all 3, Tony alone, Wade+Scott, all 3 which will also give you 102.

A man, starting from a point P, takes six equal steps. Each step is in one of the four directions – East, West, North and South. What is the total number of ways in which the man ends up at point P after the six steps?
(a) 200
(b) 256
(c) 400
(d) 512

EEEWWW =6!/(3!)^2 = 20
SSSNNN = 6!/(3!)^2 = 20
NNSSEW = 6!/(2!)^2 = 180
EEWWNS=6!/(2!)^2 = 180
Total = 400

The function f(x) is defined for all positive values of x and y as f(xy)=f(x)+f(y). Also, f(2)=2 and f(3)=3. What is the value of f(32/27)?

f(4)=f(2)+f(2)
f(4)=4
f(16)=f(4)+f(4)=8
f(32)=f(16)+f(2)=10
f(9)=f(3)+f(3)=6
f(3)=f(9)+f(1/3)
f(1/3)=-3
f(1/9)=f(1/3)+f(1/3)=-6
f(1/27)=f(1/9)+f(1/3)=-9
f(32/27)=f(32)+f(1/27)=10-9=1

Short cut: here f(x) = log(x) => log(32/27) = log 32 – log 27 = 5log2 – 3log3 = 5f(2) – 3f(3) = 10 – 9 = 1

A dealer offers a discount of 20% and still makes a profit of 20%, even when he further allows 16 articles to a dozen to a particular sticky customer. How much % is his items marked up?

Let MP be 1 Rs per article. For a dozen, MP will be 12 Rs but a discount of 20% on this will make it 9.6 Rs. But, he has sold 16 articles. That makes SP of one article as 9.6/16 = 0.6. As he is making 20% profit, 1.2CP = 0.6 which will make CP = 0.5 hence, 100% mark-up on Cost Price

The area of triangle ABE is 2sq. cm and that of triangle BEF is3sqcm.What is the area of blue shaded figure ACDE?

AE:FE=2:3
AB:DF=2:3
A(ABE)/A(DEF)=(AB/DF)^2=4/9
A(DEF)=4.5
A(BDF)=7.5
A(FBCD)=15
A(AECD)=5.5

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