Q. 2 For some real numbers a and b, the system of
equations x + y = 4 and (a + 5) x + (b^{2} – 15)y = 8b
has infinitely many solutions for x and y. Then, the
maximum possible value of ab is

Q. 4 Let n and m be two positive integers such that there
are exactly 41 integers greater than 8^{m} and less than
8^{n}, which can be expressed as powers of 2. Then,
the smallest possible value of n + m is

Q. 5 A quadratic equation x^{2} + bx + c = 0 has two real
roots. If the difference between the reciprocals of the
roots is 1/3, and the sum of the reciprocals of the
squares of the roots is 5/9, then the largest possible
value of (b + c) is

Q. 7 Let n be any natural number such that 5^{n – 1} < 3^{n + 1}.
Then, the least integer value of m that satisfies
3^{n + 1} < 2^{n + m} for each such n, is

Q. 8 A boat takes 2 hours to travel downstream a river
from port A to port B, and 3 hours to return to port A.
Another boat takes a total of 6 hours to travel from
port B to port A and return to port B. If the speeds of
the boats and the river are constant, then the time, in hours, taken by the slower boat to travel from port
A to port B is

Q. 9 The population of a town in 2020 was 100000. The
population decreased by y% from the year 2020 to
2021, and increased by x% from the year 2021 to
2022, where x and y are two natural numbers.
If population in 2022 was greater than the population
in 2020 and the difference between x and y is 10,
then the lowest possible population of the town in
2021 was

Q. 10 Rahul, Rakshita and Gurmeet, working together,
would have taken more than 7 days to finish a job.
On the other hand, Rahul and Gurmeet, working
together would have taken less than 15 days to finish
the job. However, they all worked together for 6 days,
followed by Rakshita, who worked alone for 3 more
days to finish the job. If Rakshita had worked alone
on the job, then the number of days she would have
taken to finish the job, cannot be

Q. 11 There are three persons A, B and C in a room. If a
person D joins the room, the average weight of the
persons in the room reduces by x kg. Instead of D, if
person E joins the room, the average weight of the
persons in the room increases by 2x kg. If the weight
of E is 12 kg more than that of D, then the value of x
is

Q. 12 A merchant purchases a cloth at a rate of Rs.100
per meter and receives 5 cm length of cloth free for
every 100 cm length of cloth purchased by him. He
sells the same cloth at a rate of Rs.110 per meter
but cheats his customers by giving 95 cm length of
cloth for every 100 cm length of cloth purchased by
the customers. If the merchant provides a 5%
discount, the resulting profit earned by him is

Q. 13 Anil mixes cocoa with sugar in the ratio 3 : 2 to
prepare mixture A, and coffee with sugar in the ratio
7 : 3 to prepare mixture B. He combines mixtures A
and B in the ratio 2 : 3 to make a new mixture C.
If he mixes C with an equal amount of milk to make a drink, then the percentage of sugar in this drink
will be

Q. 14 A fruit seller has a stock of mangoes, bananas and
apples with at least one fruit of each type. At the
beginning of a day, the number of mangoes make up
40% of his stock. That day, he sells half of the
mangoes, 96 bananas and 40% of the apples.
At the end of the day, he ends up selling 50% of the
fruits. The smallest possible total number of fruits in
the stock at the beginning of the day is

Q. 15 Gautam and Suhani, working together, can finish a
job in 20 days. If Gautam does only 60% of his usual
work on a day, Suhani must do 150% of her usual
work on that day to exactly make up for it. Then, the
number of days required by the faster worker to
complete the job working alone is

Q. 16 The number of coins collected per week by two
coin-collectors A and B are in the ratio 3 : 4. If the
total number of coins collected by A in 5 weeks is a
multiple of 7, and the total number of coins collected
by B in 3 weeks is a multiple of 24, then the minimum
possible number of coins collected by A in one week
is

Q. 17 A rectangle with the largest possible area is drawn
inside a semicircle of radius 2 cm. Then, the ratio of
the lengths of the largest to the smallest side of this
rectangle is

Q. 18 Let ΔABC be an isosceles triangle such that AB and
AC are of equal length. AD is the altitude from A on
BC and BE is the altitude from B on AC. If AD and
BE intersect at O such that ∠AOB = 105°, then
AD/BE equals

Q. 22 Suppose f(x, y) is a real-valued function such that
f(3x + 2y, 2x – 5y) = 19x, for all real numbers
x and y. The value of x for which f(x, 2x) = 27, is