Full Grade 11 test

The equation |x − a| = d means that point x is d units away from point a on the number line. Therefore it normally has two solutions: x = a − d and x = a + d.

The intersection quadratic discriminant D determines the number of real intersections: D > 0 gives two, D = 0 one and D < 0 none.

For an angle in radians, arc length is radius times angle. Multiply the coefficients exactly and retain π.

One full turn is 2π radians. Remove full turns, then convert the remainder to degrees.

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°.

Determine monotonicity from the base. Evaluate the interval endpoints, accounting for both shifts. Include the endpoint values because the interval is closed.

Check whether one divisor is a multiple of the other. Multiples of that divisor are then also multiples of the other divisor. Check equality separately; if inclusion holds in neither direction, none of the first statements applies.

Graphical solutions of f(x) = a are the x coordinates where the graph meets y = a. Algebra often gives exact values while a graph generally gives approximations. Check every intersection and the specified interval.

Substitute g(x) for the whole argument of f. Expand and solve the resulting linear equation.

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.

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.

The fractional exponent ^m⁄_n means that you may first take the nth root and then raise the result to the power m.

Rule: a^m/n = (ⁿ√a)^m = ⁿ√(a^m).

1. Look at the denominator n of the exponent: it gives the root index.
2. Take the nth root of the base.
3. Raise the resulting number to the power m shown by the numerator.

In this question, the base is generated as an exact nth power, so the root is an integer.

The vertex is B, so use vectors BA and BC. The sign of their dot product determines whether the angle is less than, equal to or greater than 90°.

The period of the tangent function is 180°. Therefore, adding or subtracting any integer multiple of 180° does not change the tangent value.

Rule: tan(α + 180° · k) = tan α, where k is an integer.

1. Notice that 180° · k has been added to the angle.
2. This is a whole number of tangent periods, so that part may be removed.
3. The expression becomes tan α.

Thus, keep the tangent function and the original angle that appears before the added multiple of 180°.