Directions for Questions 1 to 10: Answer the questions independently of each other.
Sub–Section II-A: Number of Question = 20
Note: Q. 1 to 20 carry one mark each.
1) The total number of integer pairs (x, y) satisfying the equation x + y = xy is
d. None of the above
2) Two boats, traveling at 5 and 10 kms per hour, head directly towards each other. They begin at a distance of 20 kms from each other. How far apart are they (in kms) one minute before they collide?
3) Each family in a locality has at most two adults, and no family’ has fewer than 3 children. Considering all the families together, there are more adults than boys, more boys than girls, and more girls than families. Then the minimum possible number of families in the locality is
4) Suppose 11 is an integer such that the sum of the digits of n is 2, and 1010 < n < 1011. The number of different values for n is
5) In Nuts And Bolts factory, one machine produces only nuts at the rate of 100 nuts per minute and needs to be cleaned for 5 minutes after production of every 1000 nuts. Another machine produces only bolts at the rate of 75 bolts per minute and needs to be cleaned for 10 minutes after production of every 1500 bolts. If both the machines start production at the same time, what is the minimum duration required for producing 9000 pairs of nuts and bolts’?
a. 130 minutes
b. 135 minutes
c. 170 minutes
d. 180 minutes
6) On January 1, 2004 two new societies, S1 and S2, are formed, each with n members. On the first day of each subsequent month, SI adds b members while S2 multiplies its current number of members by a constant factor r. Both the societies have the same number of members on July 2, 2004. If b 10.5n, what is the value of r?
7) Karan and Arjun run a 100-metre race, where Karan beats Arjun by 10 metres, To do a favour to Arjun, Karan starts 10 metres behind the starting line in a second 100-metre race. They both run at their earlier speeds. Which of the following is true in connection with the second race?
a. Karan and Arjun reach the finishing line simultaneously.
b. Arjun heats Karan by 1 metre.
c. Arjun beats Karan by 11 metres.
d. Karan beats Arjun by 1 metre.
8) A father and his son are waiting at a bus stop in the evening. There is a lamp post behind them. The lamp post, the father and his son stand on the same straight line. The father observes that the shadows of his head and his son’s head are incident at the same point on the ground. If the heights of the lamp post, the father and his son are 6 metres, 1.8 metres and 0.9 metres respectively, and the father is standing 2.1 metres away from the post, then how far (in metres) is the son standing from his father?
9) If the sum of the first 11 terms of an arithmetic progression equals that of the first 19 terms, then what is the sum of the first 30 terms?
d. Not unique
10) If then r cannot take any value except:
c. 1/2 or -1
d. –1/2 or -1
Directions for Questions 11 to 13: Answer the questions on the basis of the information given below.
In the adjoining figure, I and II are circles with centres P and Q respectively. The two circles touch each other and have a common tangent that touches them at points Rand S respectively. This common tangent meets the line joining P and Q at 0. The diameters of I and II are in the ratio 4 : 3. It is also known that the length of PO is 28 cm.
9) What is the ratio of the length of PQ to that of QO?
a. 1 : 4
b. 1 : 3
c. 3 : 8
d. 3 : 4
10) What is the radius of the circle II?
a. 2 cm
b. 3 cm
c. 4 cm
d. 5 cm
11) The length of SO is
a. 8√3 cm
b. 10√3 cm
c. 12√3 cm
d. 14√3 cm
Directions for Questions 14 to 20: Answer the questions independently of each other.
14) A milkman mixes 20 litres of water with 80 litres of milk. After selling one-fourth of this mixture, he adds water to replenish the quantity that he has sold. What is the current proportion of water to milk?
a. 2 : 3
b. 1 : 2
c. 1 : 3
d. 3 : 4
15) Let f(x) = ax2 – bx, where a and b are constants. Then at x = 0, f(x) is:
a. maximized whenever a > 0, b > 0
b. maximized whenever a > 0, b < 0
c. minimized whenever a > 0, b > 0
d. minimized whenever a > 0, b < 0
16) If f(x) = x3 – 4x + p, and f(0) and f(1) are of opposite signs, then which of the following is necessarily true?
a. –1 < p < 2
b. 0 < p < 3
c. –2 < p < 1
d. – 3 < p < 0
17) N persons stand on the circumference of a circle at distinct points. Each possible pair of persons, not standing next to each other, sings a two-minute song one pair after the other. If the total time taken for singing is 28 minutes, what is N?
d. None of the above
18) If a man cycles at 10 km/hr, then he arrives at a certain place at 1 p.m. If he cycles at 15 km/hr, he will arrive at the same place at 11 a.m. At what speed must he cycle to get there at noon?
a. 11 km/hr
b. 12 km/hr
c. 13 km/hr
d. 14 km/hr
What is the value of y?
a. (√13 + 3)/2
b. (√13 – 2)/2
c. (√15 + 3)/2
d. (√15 – 3)/2
20) A rectangular sheet of paper, when halved by folding it at the mid point of its longer side, results in a rectangle, whose longer and shorter sides are in the same proportion as the longer and shorter sides of the original rectangle. If the shorter side of the original rectangle is 2, what is the area of the smaller rectangle?
d. None of the above
Sub–Section II-B: Number of Question = 15
Note: Q. 21 to 35 carry two marks each.
Directions for Questions 21 to 29: Answer the questions independently of each other.
21) In the adjoining figure, the lines represent one-way roads allowing travel only northwards or only westwards. Along how many distinct routes can a car reach point B from point A?