Five OMPT questions, solved slowly

Students hunting for sample questions usually want reassurance about difficulty. Fair enough, but a sample question read passively teaches almost nothing. So treat this post as a mini-session: cover the solution, attempt each question on paper, then read my working and compare it with yours line by line. Where your path differed matters more than whether your answer matched. These are my own questions, written in the official style; the real test randomises its numbers anyway, so the method is the only thing worth memorising.

Question 1: an exponential equation (A territory)

Divide both sides by 5 first: 2^(x-1) = 8. Now both sides can live in base 2, since 8 = 2³, so the exponents must match: x - 1 = 3, giving x = 4. Two moves, no calculator, and that is deliberate. OMPT exponential equations are built so the bases align once you isolate the power. If you find yourself reaching for logs of ugly decimals on a question like this, you have missed the isolation step, not the theory. More of these live in the exponential functions drills.

Question 2: a quadratic inequality

Factor: (x - 2)(x - 3) ≤ 0. The roots are 2 and 3, and the parabola opens upward, so it dips below zero exactly between its roots. Answer: 2 ≤ x ≤ 3. The classic error here is solving the equation, writing "x = 2 or x = 3", and stopping, which answers a question nobody asked. An inequality wants a region. Sketch the parabola every single time, even badly; a three-second sketch has saved more OMPT points than any identity I teach. The inequalities drills push this until it is automatic.

Question 3: a logarithm equation

The difference of logs is the log of the quotient: log₃((x+6)/(x-2)) = 2, so (x+6)/(x-2) = 3² = 9. Multiply out: x + 6 = 9x - 18, so 24 = 8x and x = 3. Now the step that separates a pass from a near-miss: check the domain. Both x + 6 and x - 2 must be positive, so x must exceed 2, and x = 3 survives. On log equations the algebra frequently produces a candidate the domain then executes, and the OMPT's exact-answer interface gives no partial credit for a rejected root you failed to reject. Say "domain?" out loud at the end of every log solve during your logarithm practice and the habit will hold on test day.

Question 4: a trig equation (B and D territory)

One solution comes from the unit circle directly: x = π/3. The second comes from symmetry: sine is positive in the second quadrant too, at x = π - π/3 = 2π/3. Answer: x = π/3 or x = 2π/3. Finding one solution and moving on is the single most common trig error I mark; sine hits every value in (−1, 1) twice per cycle, so a lone answer should feel suspicious by reflex. Notice also that the interval was given in radians and the answer stays in radians. If your instinct was 60 degrees, the angles and radians drills are where to spend an evening, because the OMPT lives in radians.

Question 5: a multi-step calculus problem (D flavour)

Differentiate: f′(x) = 3x² - 12x + 9 = 3(x² - 4x + 3) = 3(x - 1)(x - 3). Stationary points where f′(x) = 0: x = 1 and x = 3. Classify by the sign of f′: positive before 1, negative between 1 and 3, positive after 3. So x = 1 is a local maximum and x = 3 a local minimum. Finally evaluate: f(1) = 1 - 6 + 9 = 4 and f(3) = 27 - 54 + 27 = 0. The stationary points are (1, 4), a local maximum, and (3, 0), a local minimum.

This is what a long-form OMPT-D or F question feels like: four dependent stages, each easy alone, where an error in the derivative poisons everything after it. The defence is checking at the joints. After factoring f′, expand it back in your head; after evaluating f(3), notice that 27 - 54 + 27 collapses neatly, which is the kind of clean landing OMPT authors design on purpose. Ugly intermediate numbers are usually a signal to recheck the previous line rather than push on. The differentiation drills run this full routine repeatedly.

What these five have in common

Every one resolves through structure into clean exact values: integers, simple fractions, multiples of π. That is the OMPT house style, and it doubles as a self-check. It also means answers get entered exactly, not as rounded decimals; type 0.33 where the answer is 1/3 and a correct solution scores zero. Practice in an environment that enforces exactness, which the timed mocks here do, and read the calculator post for why arithmetic by hand is the habit underneath all five solutions.