## QBM042

Q1. Find the smallest natural number greater than 1 billion (109) that has exactly 1000 positive divisors. (The term divisor includes 1 and the number itself. So, for example, 9 has three positive divisors.) Q2. Describe as explicitly as you can all cubic polynomials with integer coefficients having (a)three distinct real roots,
(b) local maximum and minimum values at integers, and
(c) point of inflection at an integer. Q3. In a 16-team basketball league, each team plays every other team exactly 4 times. Find the total number of games played in the league. a.480
b.960
c.900
d.450
e. None of these Q4. If (x + 2) 2 = 9 and (y + 3) 2 = 25, then the maximum value of x/y is. A.1/2
B. 5/2
C.5/8
D.1/8 Q5. The rate of inflation was 1000%. What will be the cost of an article, which costs 6 units of currency now, two years from now? (1)666
(2)660
(3)720
(4)726 Q6. You are being evaluated in three different categories. In category I you can receive either a 0, 1, or 2. In categories II and III you can receive either a 0, 1, 2, 3, or 4. Your ultimate rating is the sum of the points you receive in each category. In how many different ways could you end up with a rating of 5? (a)10
(b)11
(c)12
(d)13
(e) None of the above Q7. Of 100 students, 42 took mathematics, 38 took chemistry, and 20 took both mathematics and chemistry. How many took neither mathematics nor chemistry? a.20
b.25
c.30
d.35
e. None of these Q8. What is the angle between the minute hand and the hour hand when the time is 1540 hours? A.150
B.160
C.140
D.130 Q9. In a game, a basket is kept in the centre, with 16 potatoes placed on either side of the basket in a single line at equal intervals of 6 feet. How long will a competitor take to bring the potatoes one by one into the basket, if he starts from the basket and runs at an average speed of 12 feet a second? a)144s
b)72s
c)84s
d) 272s Q10. The income of A is twice that of B. A spends 90% of his income while B spends 80% of his income. What is the ratio of their savings? 1)2 : 1
2) 1 : 2
3) 1 : 1
4) 8 : 9 Q11. Question deleted Q12. Take any two positive integers N and a. Show that Na is the sum of N consecutive odd integers. As an easy example, note that 713 = 96889010407 = 13841287195 + 13841287197 + 13841287199 + 13841287201 + 13841287203 + 13841287205 + 13841287207 Most solvers noted that the result is not true if a = 1 and N is even, but is otherwise correct. Since Na is the sum of N times the integer Na-1 it is also the sum of N integers whose average is Na-1. Take these integers to be the N odd integers centered at the odd integer Na-1 - N + 1. Q13. Suppose n fair 6-sided dice are rolled simultaneously. What is the expected value of the score on the highest valued die? Q14. Let a, m, and n be positive integers, with a > 1, and m odd. What is the greatest common divisor of am âˆ’ 1 and an + 1? Q15. A man invests Rs. 3000 at a rate of 5% per annum. How much more should he invest at a rate of 8%, so that he can earn a total of 6% per annum? (1)Rs. 1200
(2)Rs.1300
(3)Rs.1500
(4)Rs. 2000 END

## Question 3

If every team play with others only once

Total number of matches played = 15 + 14 + . . . . 1

= (16 x 15) /2 = 8 x 15 = 120

So when all teams paly 4 mathces each other.

Total number of matches = 120 x 4 = 480

## HUMRAJ!!!!!!

Q15. A man invests Rs. 3000 at a rate of 5% per annum. How much more should he invest at a rate of 8%, so that he can earn a total of 6% per annum?
(1)Rs. 1200
(2)Rs.1300
(3)Rs.1500
(4)Rs. 2000

Sol.

ratio of amount at a rate of 5% to amount at a rate of 8% = 2:1

Ans. Rs. 1500

## HUMRAJ!!!!

Q8. What is the angle between the minute hand and the hour hand when the time is 1540 hours?
A.150
B.160
C.140
D.130

Sol.

Minute hand moves 60 in 1 minute.

Minute hand moves b/w 3hr and 8hr is 1500.

hour hand moves (1/2)0 in 1minute.

hour hand moves 200 in 40minutes.

the angle between the minute hand and the hour hand when the time is 15:40 hours is (1500 - 200) = 1300

## HUMRAJ!!!!!

Q9. In a game, a basket is kept in the centre, with 16 potatoes placed on either side of the basket in a single line at equal intervals of 6 feet. How long will a competitor take to bring the potatoes one by one into the basket, if he starts from the basket and runs at an average speed of 12 feet a second?
a)144s
b)72s
c)84s
d) 272s

Sol. total distance = 2(12 + 24 + 36.......... + 96) = 864 feet

time = 864/12 = 72sec

## HUMRAJ!!!!

Q6. You are being evaluated in three different categories. In category I you can receive either a 0, 1, or 2. In categories II and III you can receive either a 0, 1, 2, 3, or 4. Your ultimate rating is the sum of the points you receive in each category. In how many different ways could you end up with a rating of 5?
(a)10
(b)11
(c)12
(d)13
(e) None of the above

Sol.

Sum of the points is 5 in 3 different categories.

as (0, 1,4) 4 ways

(0, 2, 3) 4 ways

(1,2, 2) 3 ways

(1,1, 3) 2 ways

total 13 ways