Full Grade 11 test

Calculate differences between several consecutive terms and, when the preceding term is nonzero, their ratios. A constant difference indicates an arithmetic progression; a constant ratio indicates a geometric progression. Check every pair.

Use the supplied formula pH = −log₁₀[H⁺]. If concentration is 10⁻ᵏ, its logarithm is −k, so pH = k. For example, 0.001 = 10⁻³ gives pH = 3.

Find α = arcsin s. General solutions of sin x = s are x = α + 360°k or x = 180° − α + 360°k, with integer k. Keep only angles in [0°, 360°] and remove duplicates.

Every logarithm argument must be > 0, with base a > 0 and a ≠ 1. logₐ u = r means u = aʳ. Equal logarithms of the same base have equal positive arguments. Check all arguments and the original equation after finding x.

arcsin s returns the principal angle in [−90°, 90°], while arccos s returns one in [0°, 180°]. The input must be between −1 and 1. Find the angle in the appropriate interval whose sine or cosine equals s.

Use the identity √(u²) = |u|. If −a < x < a, then x − a < 0 and x + a > 0. Therefore |x − a| = a − x and |x + a| = x + a. Add the expressions to obtain the final numerical answer.

logₐ b = c means aᶜ = b. A logarithm is the exponent required on the base. Require a > 0, a ≠ 1 and b > 0. Thus a^(logₐ b) = b; the common logarithm has base 10.

Find α = arctan s. All solutions are x = α + 180°k, with integer k. Keep those in [0°, 360°]. For s = 0, include 0°, 180° and 360°; tangent is undefined at 90° and 270°.

For a valid shared base and positive arguments, logₐ m + logₐ n = logₐ(mn), logₐ m − logₐ n = logₐ(m/n), and k logₐ v = logₐ(vᵏ). A sum of logarithms is not the logarithm of the sum.

A geometric progression has ratio q = b₂/b₁. Its nth term is bₙ = b₁qⁿ⁻¹. For q ≠ 1, the first n terms sum to Sₙ = b₁(qⁿ − 1)/(q − 1); for q = 1, Sₙ = nb₁.