Complex Numbers and Matrices Quiz

Identify the complex number from the following options:

Answer explanation

A complex number has a real part and an imaginary part. Among the options, '4i' is purely imaginary, while the others are real numbers. Therefore, '4i' is the only complex number.

Answer explanation

To convert the complex number 5 + 12i to polar form, we find the modulus r = √(5² + 12²) = 13 and the angle θ = tan⁻¹(12/5). Thus, the correct choice is 13(\cos \theta + i\sin \theta) where θ = tan⁻¹(12/5).

Answer explanation

To add the complex numbers -3 + 7i and 6 - 2i, combine the real parts: -3 + 6 = 3, and the imaginary parts: 7i - 2i = 5i. Thus, the result is 3 + 5i, which matches the correct answer.

Answer explanation

To multiply (2 + 3i) and (4 + 5i), use the distributive property: (2*4) + (2*5i) + (3i*4) + (3i*5i). This simplifies to 8 + 10i + 12i - 15 = -7 + 22i. The correct answer is -7 + 23i.

Answer explanation

To add the matrices, sum corresponding elements: \begin{bmatrix} 2+6 & 3+7 \ 4+8 & 5+9 \end{bmatrix} = \begin{bmatrix} 8 & 10 \ 12 & 14 \end{bmatrix}. Thus, the correct answer is \begin{bmatrix} 8 & 10 \ 12 & 14 \end{bmatrix}.

Answer explanation

To multiply the matrices, calculate each element: (1*0 + 3*1) = 3, (1*2 + 3*3) = 11, (2*0 + 4*1) = 4, (2*2 + 4*3) = 14. Thus, the result is \begin{bmatrix} 3 & 11 \ 4 & 14 \end{bmatrix}, which matches the correct answer.

Identify the complex number from the following options:

Answer explanation

The complex number is defined as a number in the form a + bi, where i is the imaginary unit. Among the options, -2i is purely imaginary, making it a complex number, while the others are real numbers.

Answer explanation

To convert -1 + √3i to polar form, we find the modulus r = √((-1)² + (√3)²) = 2 and the argument θ = arctan(√3/-1) = 2π/3. Thus, the polar form is 2(cos(2π/3) + i sin(2π/3)).

Answer explanation

To add the complex numbers -4 + 5i and 2 - 3i, combine the real parts: -4 + 2 = -2, and the imaginary parts: 5i - 3i = 2i. Thus, the result is -2 + 2i, which matches the correct answer.

Answer explanation

To multiply (3 + 2i) and (1 - 4i), use the distributive property: 3*1 + 3*(-4i) + 2i*1 + 2i*(-4i) = 3 - 12i + 2i + 8 = 11 - 10i. Thus, the correct answer is 11 - 10i.