QCM : Advanced Complex Numbers and Roots — 20 questions

Explication

The modulus is the distance of the complex number from the origin, so it is always non-negative. The angle with the positive real axis is the argument, not the modulus.

Explication

The complex conjugate is formed by changing the sign of the imaginary part only. So -2 + 3i becomes -2 - 3i.

Explication

The four fourth roots of unity cancel in pairs, so their sum is 0. Their product, by contrast, is 1.

Explication

For a primitive fourth root of unity, squaring gives -1 and the fourth power returns 1. This matches the fact that a fourth root satisfies z^4=1.

Explication

In polar form, r is the modulus and θ is the argument. The argument is the angle, not the distance from the origin.

Explication

The angle is given by θ = tan^{-1}(y/x) in the source material. A common mistake is to reverse the ratio.

Explication

A function assigns each input in its domain exactly one output. This is why its graph must pass the vertical line test.

Explication

The smallest positive shift that leaves the function unchanged is the period. Any multiple of that period is also a period.

Explication

An arithmetic progression is defined by a constant difference between consecutive terms. A constant ratio would describe a geometric progression.

Explication

The standard sum formula for an arithmetic progression is n/2{2a + (n-1)d}. It adds the first and last terms in a structured way across all n terms.

Explication

Permutations count ordered selections, so nP_r is about arrangement. Combinations, by contrast, ignore order.

Explication

By convention, 0! = 1. This makes factorial formulas consistent in combinatorics and algebra.

Explication

The factor theorem directly links a zero of the polynomial to the factor (x - a). The remainder theorem instead says the remainder upon division by (x - a) is f(a).

Explication

Using the factor theorem with x = 2 gives f(2) = 0, which leads to k = 10. This is a standard application of checking a factor by substitution.

Explication

The source gives cos(α + β) + cos(α - β) = 2cos α cos β. The difference of the cosine terms gives a sine-product identity instead.

Explication

Allied angles are angle pairs that add up to 180°. The pair θ and 180° - θ is the standard example.

Explication

The derivative at a point is the limit of the difference quotient as x approaches a. This is the standard limit definition of the derivative.

Explication

The source states that for f(x) = x^{1/3}, f'(8) = 1/12. This is a direct derivative evaluation at the given point.

Explication

An angle of π/2 means the vectors are perpendicular. Parallel vectors have angle 0 or π instead.

Explication

If the cross product is zero, the vectors are parallel, provided they are non-zero. The cross product measures the sine of the angle between them.