A test has 150 questions. A candidate gets 2 marks for each correct answer and loses 1 mark for each wrong answer and loses `(1)/(2)` mark for leaving the question unattempted. A student score 165 marks. If the student left 18 questions unattempted, find the number of questions he marked wrong.

Answer

To solve the problem, we need to set up equations based on the information given.

### Step 1: Define Variables
Let:
- \( c \) = number of correct answers
- \( w \) = number of wrong answers
- \( u \) = number of unattempted questions

From the problem, we know:
- Total questions = 150
- Unattempted questions = 18
- Therefore, \( u = 18 \)

### Step 2: Calculate Attempted Questions
The number of attempted questions is:
\[ \text{Attempted questions} = 150 - u = 150 - 18 = 132 \]
So, the number of attempted questions is 132.

### Step 3: Set Up the Equation
The total number of attempted questions is the sum of correct and wrong answers:
\[ c + w = 132 \]

### Step 4: Set Up the Score Equation
The score is calculated as follows:
- For each correct answer, the student gets 2 marks.
- For each wrong answer, the student loses 1 mark.
- For each unattempted question, the student loses \( \frac{1}{2} \) mark.

The total score can be expressed as:
\[ \text{Score} = 2c - w - \frac{1}{2}u \]
Substituting \( u = 18 \):
\[ \text{Score} = 2c - w - \frac{1}{2}(18) \]
\[ \text{Score} = 2c - w - 9 \]
Given that the student scored 165 marks, we can set up the equation:
\[ 2c - w - 9 = 165 \]

### Step 5: Simplify the Score Equation
Rearranging the score equation gives:
\[ 2c - w = 165 + 9 \]
\[ 2c - w = 174 \]

### Step 6: Solve the System of Equations
Now we have a system of two equations:
1. \( c + w = 132 \) (Equation 1)
2. \( 2c - w = 174 \) (Equation 2)

We can solve these equations simultaneously.

From Equation 1, we can express \( w \) in terms of \( c \):
\[ w = 132 - c \]

Substituting \( w \) in Equation 2:
\[ 2c - (132 - c) = 174 \]
\[ 2c - 132 + c = 174 \]
\[ 3c - 132 = 174 \]
\[ 3c = 174 + 132 \]
\[ 3c = 306 \]
\[ c = \frac{306}{3} = 102 \]

### Step 7: Find the Number of Wrong Answers
Now substitute \( c \) back into Equation 1 to find \( w \):
\[ c + w = 132 \]
\[ 102 + w = 132 \]
\[ w = 132 - 102 = 30 \]

### Conclusion
The number of questions the student marked wrong is \( w = 30 \).