Q1. Three teams P, Q, R participated in a tournament in which the teams play with one another exactly once. A win fetches a team 2 points and a draw 1 point. A team gets no point for a loss. Each team scored exactly one goal in the tournament. The team P got 3 points, Q got 2 points and R got 1 point. Which of the following statements is/are correct?
I. The result of the match between P and Q is a draw with the score 0 -0.
II. The number of goals scored by R against Q is 1.
Which of the statements given above is/are correct? (2025)
(a) I only
(b) II only
(c) Both I and II
(d) Neither I nor II
Ans: (c)
Teams play each other exactly once, so there are 3 matches in total.
Team P (3 points): P must have won 1 match and drawn 1 match.
Team Q (2 points): Q must have drawn both the matches.
Team R (1 point): R must have lost 1 match and drawn 1 match.
As P has won one match and R has lost one match, it’s obvious that P must have defeated R.
As a team must score at least one goal to win, P must have scored its lone goal in this match.
So, scoreline of P-R match would have been 1-0.
Now, P cannot have scored any other goal in the tournament. Which means that scoreline of PQ match would have been 0-0.
So, the scoreline of Q-R match would have been 1-1.
So, both statements are correct.
Hence, option (c) is the correct answer.
Q2. Let p, q, r and s be distinct positive integers. Let p, q be odd and r, s be even. Consider the following statements:
1. (p-r)2 (qs) is even.
2. (q-s)q2 s is even.
3. (q + r)2 (p + s) is odd.
Which of the statements given above are correct? (2024)
(a) 1 and 2 only
(b) 2 and 3 only
(c) 1 and 3 only
(d) 1, 2 and 3
Ans: (d)
p and q are odd numbers.
Similarly, r and s are even numbers.
Statement 1: (p - r)2 (qs) is even, as s is an even number. This statement is correct.
Statement 2: (q - s)q2 s is even, as s is an even number. This statement is correct.
Statement 3: (q + r)2 (p + s) is odd, as both (q + r) and (p + s) are odd. This statement is correct.
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