Math Mock Test

For the complex number z = (2 - i)² / i (i is the imaginary unit), what is |z|?

Given 0 < θ < π/4, for the hyperbolas C₁: x²/sin²θ - y²/cos²θ = 1 and C₂: y²/cos²θ - x²/sin²θ = 1, which of the following is true?

Given A(-1,1), B(1,2), C(-2,-1), D(3,4), the projection of vector AB onto direction CD is:

For ellipse C: x²/a² + y²/b² = 1 (a>b>0) with left focus F, the line through the origin meets C at A and B. If |AB|=10, |BF|=8, and cos∠ABF = 4/5, the eccentricity e equals:

From integers 1, 2, ..., n pick two distinct numbers at random. If the probability that their sum equals 5 is 1/14, then n equals:

In right triangular prism ABC-A₁B₁C₁, all 6 vertices lie on sphere O. Given AB=3, AC=4 with AB⊥AC, and AA₁=12, the radius of sphere O is:

Let M(x, y) be on ellipse C: x²/16 + y²/9 = 1. The maximum value of x + y is:

Two adjacent rectangular rooms share a wall; the total fence length is 30 m. To maximize total area, the optimal width x is:

For equation x² + px + p = 0 with real roots a and b, if a² + b² = 3, then p equals:

Given log₁₈9 = a and 18ᵇ = 5, compute log₃₆45: