Quant Boosters - Shashank Prabhu, CAT 100 Percentiler - Set 4
Let the length be x. A covers x while B covers x-24 and C covers x-36. Similarly, when B covers x, C will cover x-16. So, we can get ratios of distances covered by B and C
(x-24)/(x-36) = x/(x-16)
Q15) When a two–digit number N is divided by the sum of its digits, the result is Q. Find the minimum possible value of Q.
For Q to be minimum, the denominator has to be maximum. So, b/a has to be maximum. The maximum value of b/a can be 9 and so, b=9 and a=1 and Q=1.9
Q16) Suresh and Ramesh are playing a game with a fair die marked 1, 2, 3, 4, 5 and 6 on its six faces. On his turn, a player rolls the die and notes the number s obtained. They roll the die turn by turn. The player, who gets 6 first wins the game. If Suresh gets the first turn to roll the die, then the probability of Ramesh winning the game is ?
Suresh rolls and he doesnt get a 6, probability of 5/6. Then Ramesh rolls and gets a 6, probability of 1/6. The probability of the game ending this way is 5/36
Suresh and Ramesh don't get a 6 in the first go, then Suresh again misses a 6 and Ramesh gets a 6. The probability of the game ending this way is (5/6)^3*1/6
With each addition to this game, the probability will keep on getting multiplied by (5/6)^2 and so, it is an infinite GP with a common ratio (5/6)^2 and the first term as (5/36). Applying the formula a/(1-r), we get 5/11 as the answer.
Q17) Yamini and Zora are standing 25 km apart. Zora starts moving towards Yamini. After 40 minutes Yamini also starts moving towards Zora. By the time Yamini covers 5 km, Zora has covered 15 km. They meet at a point 7 km from the starting point of Yamini. What is the speed of Yamini?
(a) 7.5 km/h
(b) 10.5 km/h
(c) 17.5 km/h
(d) 6 km/h
Yamini covered 5 km in say time t hours. So, Zora has covered 15 km in time (t+2/3) hours. So we get the ratios of their speeds. Also, the time in which Yamini covers 7 km will be 7t/5 and the time taken by Zora to cover 18 km will be 7t/5+2/3.
Equating the two speeds of Zora,
Speed of Yamini is 5/(2/3)=7.5 kmph
Q18) If the product of all the factors of a number is equal to the square of the number and the sum of all the factors of the number, other than the number itself is 13 ,then find the sum of all the values possible for the number.
If the number of factors of a number N is given by f, product of all the factors is given by N^(f/2). As is given, N^(f/2)=N^2. So, number of factors is 4. So, the number is either in the form of a^3 or a*b where a and b are the prime factors.
Case I: a^3... sum of all the factors except the number - 1+a+a^2=13. So, a=-4 or a=3. So a=3 and 27 is one of the numbers
Case II: a * b... sum of all the factors except the number - 1+a+b=13. So, a+b=12 where a and b are prime numbers. The only possible solution is (5, 7). So, 35 is another such number.
Sum of all values = 27+35=62
Q19) In how many ways you can climb up 8 steps if minimum and maximum numbers of steps you can take at a time are 1 and 6 respectively?
Let's say we represent n(x) as the number of ways of climbing x steps.
We have to remove the cases involving 7 and 8 from these. So, n(7) will have 1 case (7) and n(8) will have 2 cases (1, 7) and (7, 1). So, 3 removed. Remaining 125 ways.
Q20) “I am eight times as old as you were when I was as old as you are”, said a man to his son.
Find out their present ages if the sum of their ages is 75 years.
(a) 40 years and 35 years
(b) 56 years and 19 years
(c) 48 years and 27 years
(d) None of these
Let man's present age be m, son's age be s
When the man's age was equal to s, son's age would have been equal to s-(m-s) = 2s-m
Alternatively, use the options. When father was 27, son would have been 6. So, 8 times 6 is 48.
Q21) A jar contains a mixture of two liquids A and B in the ratio 4:1. When 10 litre of the mixture is replaced with liquid B,the ratio becomes 2:3. The volume of liquid A present in the earlier was?
Let total solution be 10x liters
A -> 8x, B -> 2x
(8x - 8)/(2x - 2 + 10) = 2/3
24x - 24 = 4x + 16
x = 2 and so, A -> 16 liters.
Q22) A number x is chosen randomly from first 50 natural numbers. What is the probability that (x + 336/x) is less than or equal to 50.
x^2 + 336 < = 50x
x^2 - 50x + 336 < = 0
(x-42)(x-8) < = 0
8 < = x < = 42
Total of 35 values satisfy this inequality
Total values possible are 50
Probability is 35/50 = 7/10
Q23) Distance b/w x& y is 220km. trains P& Q leave station x at 8 am & 9:51 am resptvly at speeds 25 & 20kmph towards station y. train R leaves station y at 11:30am at 30kmph towards x. when will P be at equal distance from Q and R?
d) 11:48 pm
When R starts moving, the diagram would look like this:
x...(33 km)...Q...(54.5 km)...P...(132.5 km)...R(y)
The midpoint of Q and R will keep on moving towards x at a speed of 25 kmph. The midpoint of Q and R at 1130 is 93.75-54.5 = 39.25 km from P. So, relative speed is 50 kmph and time taken to meet is 39.25/50 = 78.5 minutes ~12:48 pm
Q24) If x, y, z are positive numbers and ax + by + cz = bx + cy + az = cx + ay + bz = 0, then which of the following is/are definitely true?
i. a + b + c = 0
ii. a = b = c
iii. a^2 + b^2 + c^2 = 1
a. Only i
b. Only ii
c. Only i and iii
d. Only i and ii