A T20 cricket match consists of two teams playing twenty overs each, numbered 1 to 20. The runs scored in any over is a non-negative integer. The run rate at the end of any over is the average runs scored up to and including that over, i.e., the run rate at the end of the kth over is the average number of runs scored in overs numbered \(1, 2, …, k,\) where \(1 \leq k \leq 20, k\) a positive integer.
The following table indicates the run rate of a team at the end of some of the overs during a T20 cricket match (correct up to 2 decimal places), where \(1 \leq N – 2 < N + 6 \leq 20\), N a positive integer. It is also known that the team did not score less than 6 runs and more than 15 runs in any over.
| Over Number | Run Rate |
|---|---|
| N-2 | 8.00 |
| N | 7.43 |
| N+2 | 8.11 |
| N+4 | 8.45 |
| N+6 | 8.08 |
What is the value of N?
- 7
- 13
- 14
- 9
- 12
The Correct Option is A
Solution and Explanation
Step 1: Understand the run rate definition. The run rate at the end of over k is given by:
Run Rate at k = $\frac{\text{Total runs scored in overs 1 to } k}{k}$
From the table:
- At N − 2, the run rate is 8.00:
- At N, the run rate is 7.43:
The additional runs scored in the (N − 1)-th and N-th overs are:
Runs in (N − 1) and N = 7.43N − 8(N − 2).
Simplify:
Runs in (N − 1) and N = 7.43N − 8N + 16 = −0.57N + 16.
Step 2: Verify conditions for N. Since the team did not score less than 6 runs or more than 15 runs in any over, the runs in (N − 1)-th and N-th overs must satisfy:
6 ≤ −0.57N + 16 ≤ 15.
Solve the inequalities:
- −0.57N + 16 ≥ 6:
- −0.57N + 16 ≤ 15:
Thus, N must be an integer between 7 and 13. Testing N = 13 satisfies all conditions.
Final Answer: 13.
In which of these pairs of over numbers, the team could have scored 22 runs in total?
- 6 and 7
- 7 and 8
- 8 and 9
- 9 and 10
- 10 and 11
The Correct Option is D
Solution and Explanation
Step 1: Understand the given conditions. The runs scored in any over are between 6 and 15. For a total of 22 runs in two overs, the possible pair of scores must add to 22.
Step 2: Identify the valid pair of scores. Possible pairs of scores satisfying x + y = 22 are: (7, 15),(8, 14),(9, 13),(10, 12),(11, 11).
Step 3: Match the pairs to the over numbers. Since the valid pairs of scores are within the given range of runs per over, 8 and 14 can be scored in overs 8 and 9.
Final Answer: 8 and 9.
In which of the following over numbers, the team MUST have scored the least number of runs?
- 7
- 8
- 9
- 10
- 11
The Correct Option is A
Solution and Explanation
Step 1: Analyze the run rates. The run rate decreases at N (from the table in Question 23).
This suggests that a relatively low number of runs was scored in over N − 1 or N.
Step 2: Identify the least runs scored. If N = 7 (from Question 23), the team must have scored the least number of runs in over 7, as the run rate drops at this point, indicating a minimum addition to the total.
Final Answer: 7.
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