- Simplify the product \(i(2+3i)\) to \(-3 + 2i\).
- Expand the full expression \((-3+2i)(c+di)\) to find the real and imaginary parts: \((-3c - 2d) + i(2c - 3d)\).
- Set the imaginary part \(2c - 3d\) to zero to ensure the result is a real number.
- Verify which option satisfies the condition \(2c - 3d = 0\), identifying that \(c=-3, d=-2\) works.
- The final answer is \(\boxed{d}\).
Explanation
- Simplifying the Expression
To determine which values of \(c\) and \(d\) make the expression \(i(2+3i)(c+di)\) a real number, we first simplify the constant part of the expression. Let \(Z_1 = i(2+3i)\). Using the distributive property and the fact that \(i^2 = -1\), we have:
\[Z_1 = 2i + 3i^2 = 2i + 3(-1) = -3 + 2i.\]
Now, we multiply this result by the complex number \((c+di)\):
\[(-3 + 2i)(c + di) = -3c - 3di + 2ci + 2di^2.\]
Recalling that \(i^2 = -1\), we substitute to obtain:
\[(-3 + 2i)(c + di) = -3c - 3di + 2ci - 2d.\]
Grouping the real and imaginary parts, we get:
\[(-3c - 2d) + i(2c - 3d).\]
For this expression to be a real number, the imaginary part must be equal to zero:
\[2c - 3d = 0.\]
- Testing the Options
We need to find which of the given options satisfies the condition \(2c - 3d = 0\):
A\(c=3, d=-2 \implies 2(3) - 3(-2) = 6 + 6 = 12
eq 0\)
B\(c=2, d=3 \implies 2(2) - 3(3) = 4 - 9 = -5
eq 0\)
C\(c=-2, d=3 \implies 2(-2) - 3(3) = -4 - 9 = -13
eq 0\)
D\(c=-3, d=-2 \implies 2(-3) - 3(-2) = -6 + 6 = 0\)
Since option (d) results in an imaginary part of zero, the expression \(-3(-3) - 2(-2) + i(2(-3) - 3(-2)) = 9 + 4 + i(0) = 13\) is a real number. Thus, the correct choice is \(c=-3, d=-2\).
Examples
In electronics engineering, complex numbers are used to represent impedance in AC circuits. The real part of impedance corresponds to resistance, while the imaginary part corresponds to reactance. Being able to determine values that make a circuit's total impedance purely real (zero reactance) is essential for resonance, where a circuit's impedance is at its minimum or maximum, allowing for efficient power transfer in communication systems and radio tuning. This calculation ensures that the system operates at the desired frequency without unwanted phase shifts.