Let (*) be a binary operation on a natural number defined by a * b = a - b + (ab)$^2$, then find 3 * 5

Analysis

The question asks us to evaluate a binary operation defined as $a * b = a - b + (ab)^2$ for the specific values $a = 3$ and $b = 5$.

Concept

A binary operation is simply a rule that combines two elements of a set to produce a third element. In this case, the rule is:

1. Subtract the second number from the first ($a - b$).
2. Multiply the two numbers together ($ab$).
3. Square the result of that multiplication ($(ab)^2$).
4. Add the results of step 1 and step 3 together.

Step-by-Step Solution

Step 1: Identify the variables Given $a * b = a - b + (ab)^2$ We are to find $3 * 5$, so: $a = 3$ $b = 5$

Step 2: Substitute the values into the formula $3 * 5 = 3 - 5 + (3 \times 5)^2$

Step 3: Solve the expression inside the parentheses $3 \times 5 = 15$

Step 4: Square the result $15^2 = 15 \times 15 = 225$

Step 5: Complete the calculation $3 - 5 + 225$ $-2 + 225 = 223$

Conclusion

Following the defined operation, $3 * 5$ equals $223$. This matches option A.

Exam Tip

When dealing with binary operations in JAMB, always pay close attention to the order of the variables. If the operation were not commutative (meaning $a * b$ is not the same as $b * a$), swapping the numbers would give you a completely different answer. Always substitute carefully!