Quant Boosters  Soumil Jain, CAT Quant 100 percentiler, IIM Calcutta  Set 4

Number of Questions : 30
Topic : Theory of Equations
Solved ? : Yes
Source : Quant100 Prep Forum

Q1) A sphere of radius r is cut by a plane at a distance of h from its center, thereby breaking this sphere into two different pieces. The cumulative surface area of these two pieces is 25% more than that of the sphere. What is the value of h?

Q2) If a^2 + 1 = a, then find the value of a^12 + a^6 + 1.

Q3) How many integrals angled triangles are possible whose angles are in AP

Q4) If f(x) is cubic polynomial with coefficient of x^3 as 1. It is given that it has nonnegative real roots and f(0) = 64. Find the largest possible value of f(1).
a) 125
b) 65
c) 64
d) 120
e) None of these

Q5) Classroom window was broken. The principal had four students in his office. He knew that only one of them did it, and he also knew that only one of the students told the truth, but not sure which one.
Alex said: Bob did;
Bob said: Dean did;
Cam said: not me;
Dean said: Bob lied.
Who broke the window?

Q6) A man covered a distance of 200 kms partly by train and partly by bus. Had he covered the entire distance by bus, he would have taken 10 hours more and had he covered the entire distance by train, he would have taken 6 hours less. What was the difference between the distances (in kms) covered by him by bus and train?
a) 60
b) 75
c) 50
d) 90

Q7) Mr. Seth distributed 100 gold coins amongst his four children such that the average of the number of coins received by his three daughters is same as the the number of coins received by his only son. The ratio of the number of coins received by two of his daughters is 1 : 7. If the son received the second highest number of gold coins and if each child received more than one gold coin , then what could be the maximum possible number of coins received by any of the four children?
a) 67
b) 65
c) 63
d) 61

Q8) Two men A and B play a match of two games on the condition: if A wins, B pays him one rupee more than half the money B has and if B wins, A pays him one rupee less than half the money A has. If A wins the first game and B the second, then the difference in the amounts with A and B becomes Rs. 16. The initial amount with ‘A’ and ‘B’ in that order could be
a.Rs.35,Rs.38
b.Rs.20,Rs.35
c.Rs.20,Rs.38
d.Either(a)or(c)

Q9) In how many ways can eight directors, the vice chairman and chairman of a firm be seated at a round table, if the chairman has to sit between the vice chairman and a director?

Q10) The length of a ladder is exactly equal to the height of the wall it is learning against. If lower end of the ladder is kept on a stool of height 3 m and the stool is kept 9 m away from the wall, the upper end of the ladder coincides with the top of the wall. Then the height of the wall is

Q11) Euclid has a triangle in mind. Its longest side has length 20 and another of its sides has length 10. Its area is 80. What is the exact length of its third side?

Q12) I live X floors above the ground floor of a highrise building. It takes me 30 s per floor to walk down the steps and 2 s per floor to ride the lift. What is X, if the time taken to walk down the steps to the ground floor is the same as to wait for the lift for 7 min and then ride down?
a) 4
b) 7
c) 14
d) 15

Q13) A, B, C and D are four towns, any three of which are noncollinear. Then the number of ways to construct three roads each joining a pair of towns so that the roads do not form a triangle is

Q14) A monkey climbs a tree. He climbs 10 metres every second but fall back 5 metres in the next second. In how much time will he cover a distance of 100 metres for the first time.

Q15) Largest value of min(2 + x^2, 6–3x), when x > 0, is

Q16) Amar has 5 butternaans and 10 rotis, Bharat has 9 butter naans and 8 rotis. Suman joined them and they shared the butternaans and rotis equally based on their calorific value. The cost of the butternaan as well as the roti is proportional to its calorific value. Suman paid Rs.10 and Rs.30 to Amar and Bharat respectively. If the calorific value of one butter naan is 150 calories, then what is the calorific value (in calories) of one roti?

Q17) A can complete a piece of work in 4 days. B takes double the time taken by A, C takes double that of B, and D takes double that of C to complete the same task. They are paired in groups of two each. One pair takes twothirds the time needed by the second pair to complete the work. Which is the first pair?
a) A and B
b) B and C
c) A and C
d) A and D

Q18) In a fourdigit number, the sum of the first 2 digits is equal to that of the last 2 digits. The sum of the first and last digits is equal to the third digit. Finally, the sum of the second and fourth digits is twice the sum of the other 2 digits. What is the third digit of the number?

Q19) Train X departs from station A at 11 a.m. for station B, which is 180 km so far. Train Y departs from station B at 11 a.m. for station A. Train X travels at an average speed of 70 km/hr and does not stop anywhere until it arrives at station B. Train Y travels at an average speed of 50 km/hr, but has to stop for 15 min at station C, which is 60 km away from station B enroute to station A. Ignoring the lengths of the trains, what is the distance, to the nearest kilometre, from station A to the point where the trains cross each other?