Examination Score Calculation

The problem describes a scoring system for an examination with 10 questions. We are given the marks awarded for correct answers, incorrect answers, and unattempted questions. We need to determine which of the given scores is not possible to achieve.

Let's define the variables:

• C: Number of correct answers
• I: Number of incorrect answers
• U: Number of unattempted questions

The total number of questions is 10. So, we have the constraint:

\( C + I + U = 10 \)

Where \( C \), \( I \), and \( U \) must be non-negative integers (\( C \ge 0, I \ge 0, U \ge 0 \)).

The scoring rules are:

• Correct answer: +2 marks
• Incorrect answer: -1 mark
• Unattempted question: 0 marks

The total score (S) is calculated as:

\( S = (2 \times C) + (-1 \times I) + (0 \times U) \)

\( S = 2C - I \)

From the constraint \( C + I + U = 10 \), we know that \( C + I \le 10 \) because \( U \ge 0 \). This also means \( I = 10 - C - U \).

Let's substitute the expression for \( I \) into the score formula:

\( S = 2C - (10 - C - U) \)

\( S = 2C - 10 + C + U \)

\( S = 3C - 10 + U \)

Now, we need to check if the given possible scores can be obtained using this formula, subject to the constraints \( C \ge 0 \), \( U \ge 0 \), and \( C + U \le 10 \) (which comes from \( I = 10 - C - U \ge 0 \)).

Testing Possible Scores

We are asked which score CANNOT be a possible score. We can test each option to see if we can find non-negative integer values for C and U (and consequently I) that satisfy the conditions for that score.

Let's test the score \( S = 19 \).

We need to find if there exist integers \( C \ge 0 \) and \( U \ge 0 \) such that:

\( 3C - 10 + U = 19 \)

Rearranging the equation, we get:

\( 3C + U = 29 \)

Additionally, we must satisfy the constraint \( C + U \le 10 \).

Let's try possible integer values for \( C \) starting from 0 and check if we can find a corresponding non-negative integer \( U = 29 - 3C \) and if the condition \( C+U \le 10 \) is met.

As shown in the table, for all possible non-negative integer values of \( C \) where \( U = 29 - 3C \) is also non-negative, the sum \( C+U \) is always greater than 10. This violates the fundamental constraint that the total number of questions \( C+I+U \) must be 10 (which implies \( C+U \le 10 \)).

Therefore, a score of 19 cannot be achieved in this examination under the given rules.

Verifying Other Options

Let's quickly check if the other options are possible:

• Score -9: Can we find \( C, I, U \) such that \( C+I+U=10 \) and \( 2C-I = -9 \)? Consider \( C=0 \). Then \( -I = -9 \implies I = 9 \). If \( C=0, I=9 \), then \( U = 10 - 0 - 9 = 1 \). Since \( C=0, I=9, U=1 \) are non-negative integers and \( 0+9+1=10 \), a score of -9 is possible.
• Score -7: Can we find \( C, I, U \) such that \( C+I+U=10 \) and \( 2C-I = -7 \)? Consider \( C=1 \). Then \( 2(1) - I = -7 \implies 2 - I = -7 \implies I = 9 \). If \( C=1, I=9 \), then \( U = 10 - 1 - 9 = 0 \). Since \( C=1, I=9, U=0 \) are non-negative integers and \( 1+9+0=10 \), a score of -7 is possible.
• Score 17: Can we find \( C, I, U \) such that \( C+I+U=10 \) and \( 2C-I = 17 \)? Consider \( C=9 \). Then \( 2(9) - I = 17 \implies 18 - I = 17 \implies I = 1 \). If \( C=9, I=1 \), then \( U = 10 - 9 - 1 = 0 \). Since \( C=9, I=1, U=0 \) are non-negative integers and \( 9+1+0=10 \), a score of 17 is possible.

Since -9, -7, and 17 are possible scores, the score that CANNOT be a possible score is 19.