Quant Boosters  Soumil Jain, CAT Quant 100 percentiler, IIM Calcutta  Set 3

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

Q1) A point P(a,b) lies on the line 3x=2y, where a and b are integers. What is the value of a^2 + b^2 + ab if the distance between P and another point Q (7, 5) is minimum possible?

Q2) A language school has 2001 students. The percentage of students of this school who study French is between 80 and 85 and the percentage of the students of the school who study Spanish is between 30 and 40. Each student of this school studies at least one of the two languages. What is the absolute difference between the minimum and maximum possible numbers of students who study both French and Spanish?
(1) 201
(2) 299
(3) 499
(4) 298

Q3) Mr. Ganguly, the captain, distributed toffees between players of his team. The first player received 50 toffees and then one tenth of the remaining toffees available with Mr. Ganguly. The second player received 100 toffees and then one tenth of the remaining toffees available with the captain. The third player received 150 toffees and then one tenth of the remaining toffees available with the captain. The sequence continued like this till Mr.Ganguly was left with no toffee. After distributing the toffees Mr. Ganguly found that every player of his team had got the same number of toffees. How many players are there in the team including the captain?
(1) 9
(2) 10
(3) 11
(4) 12

Q4) Sanjay bought a basket of oranges, a basket of apples, a basket of mangoes and a basket of peaches. He paid a total of Rs.790 for the baskets of fruits bought by him. While driving back to home, he realized that if he decreased the price of the basket of mangoes by Rs.4, increased the price of basket of peaches by Rs.7, multiplied the price of the basket of oranges by 3 and divided the price of the basket of apples by 2, then all the baskets would have the same price. Find the price of the basket of peaches.
(1) Rs. 160
(2) Rs. 180
(3) Rs. 95
(4) Rs. 176
(5) Rs. 170

Q5) My current age is two years more than thrice the age of one of my two sisters named Richa. After p years, My age will be two years more than thrice the age of my other sister named Namita. What is the minimum possible integral difference (in years) between the age of my two mentioned sisters? (p is a natural number.)
(1) 10
(2) 6
(3) 4
(4) 2

Q6) There are two alloys of gold and copper. In the first alloy, there is twice as much gold as copper and in the second alloy, the quantity of copper is five times the quantity of gold. In what ratio we must mix the first and the second alloy such that the amount of copper is twice the amount of the gold in this new alloy?
(1) 1 : 1
(2) 1 : 2
(3) 1 : 1.5
(4) 1 : 2.5
(5) 2 : 1

Q7) A and B work for m and n hours per day respectively. They complete a work independently in ‘a’ and ‘b’ days respectively. If each of them reduces their working hours per day by 4 hrs, each would take 5 days more to finish the work independently. If they work together at their original work hours and if they start at the same time everyday, the job will get done in (a, b, m, n are distinct natural numbers and m, n < 16):
(1) 3 days 3 hours
(2) 4 days
(3) 5 days
(4) 5 days 6 hours
(5) Cannot be determined

Q8) The total number of terms in an arithmetic progression (A.P.) is odd. The sum of all the even placed terms is 7 and the sum of all the odd placed terms is 8.75 in this A.P. It is also given that the last term of this A.P. exceeds the first term by 2. Find the total number of terms in the A.P..
(1) 5
(2) 9
(3) 11
(4) 21
(5) Cannot be determined

Q9) ABCD is a cyclic quadrilateral with AB = 8 units, BC = 5 units and CD = 6 units. If the diagonals of this quadrilateral intersect at right angles, then AC is approximately equal to
(1) 10 units
(2) 11 units
(3) 12 units
(4) 9 units
(5) None of these

Q10) Two boats start towards each other, from the two points exactly opposite of each other on the opposite banks of a river, simultaneously. They meet at a distance of 410 m from one of the banks and continue sailing further till they reach the opposite banks. They take rest for 1 hr each and start off the return journey taking the same route. Now they meet at a distance of 230 m from the other bank. Find the distance between the two banks. (Assume that river water is still.)
(1) 750 m
(2) 840 m
(3) 1100 m
(4) 1200 m
(5) 1000 m

Q11) An escalator is moving downwards with the speed of 4 steps per minute. A person takes 6 minutes less to get down if he is coming down on the moving escalator as compared to when he comes down via the stationary escalator. Another person takes 6 minutes more to get up if he is going up on the moving escalator as compared to when he goes up via the stationary escalator. If they start together from the top and the bottom respectively, they meet after 4 minutes on the moving escalator. How many steps are there in the escalator?
(1) 60
(2) 56
(3) 48
(4) Cannot be determined

Q12) In the month of January, what is the ratio of probability of someone’s birthday falling on a date that is a prime number to the probability of the birthday falling on a date that is perfect square of a prime number? (It is known that the birthday falls in the month of January.)

Q13) ABCDEFGH is a regular octagon. The line segments AE and CF intersect at the point K. If AB = 1 unit, then what is the length of the line segment CK?

Q14) A dealer marks up the cost price of an article by ‘p%’ and then gives a discount of ‘p%’ on the marked price. Now the price of the article is Rs. 21 less than the cost price of the article. He again marks up the decreased price by ‘p%’ and then gives a discount of ‘p%’. If the price of article now is Rs. 2058, then find the approximate value of p.
(1) 8
(2) 9
(3) 10
(4) 12

Q15) Pawan was designing a Mock test. He had 8 questions for this Mock test and he had to assign a total of 30 marks to these 8 questions. If the minimum marks assigned to a question was 2 and each question carried integral marks, then in how many ways was it possible for Pawan to distribute 30 marks in that Mock test?

Q16) Given that a^5 + b^5 + c^5 = 91849, where a, b, and c are distinct digits. What is the remainder when a sixdigit number 2a5b1c is divided by 11?
a) 4
b) 6
c) 2
d) 1
e) 0

Q17) How many fourdigit numbers with distinct digits are there such that the sum of the digits is even?
(1) 2376
(2) 2256
(3) 2208
(4) 2152
(5) 2156

Q18) Find the number of positive integral solutions of the equation a(a^2 – b) = (b^3 + 61).
a) 0
b) 1
c) 2
d) None of these

Q19) A box contains 25 red balls and 20 blue balls. An unbiased die is rolled, if the result is an even number, then 3 red balls and 1 blue ball are taken out from the box without replacement and if the result is an odd number, then 2 blue balls and 1 red ball are taken out from the box without replacement. If the die is rolled 4 times, then what is the probability that the total number of red balls, which are taken out, is more than the number of blue balls, which are taken out ?