20 carefully crafted problems across all major units. Choose your answer — instant feedback.
Which of the following represents a one-to-one function?
Hint: Think about the Horizontal Line Test
f(x) = x³ is one-to-one because every horizontal line crosses the graph exactly once.
A function is one-to-one (injective) when distinct inputs always produce distinct outputs.
If f(x) = 2x − 3, what is f⁻¹(x)?
To find the inverse, replace f(x) with y, swap x and y, then solve for y:
Trick: Option A is worded confusingly — the correct arithmetic is (x + 3) ÷ 2, which matches C. Option D uses subtraction (wrong sign).
The polynomial p(x) = x³ − 6x² + 11x − 6 has roots at x = 1 and x = 2. What is the third root?
By Vieta's formulas for x³ − 6x² + 11x − 6 = 0:
Verify: (x−1)(x−2)(x−3) = x³ − 6x² + 11x − 6 ✓
What is the horizontal asymptote of f(x) = (3x² + 1) / (x² − 4)?
Compare degrees of numerator and denominator:
Rules to memorize: top < bottom → y = 0; top = bottom → y = ratio; top > bottom → no HA (oblique instead)
Solve for x: log₂(x + 3) + log₂(x − 1) = 5
Domain check: x + 3 > 0 and x − 1 > 0 requires x > 1. So x = −7 is rejected. Only x = 5 is valid. Option D is a trap!
If 4ˣ = 8, what is the exact value of x?
Note: Option C is also mathematically valid (change of base), but B is the exact simplified answer. Options A and D flip the fraction — common mistake!
In which quadrant is an angle θ where sin θ < 0 and cos θ > 0?
ASTC mnemonic: All (Q1), Sin (Q2), Tan (Q3), Cos (Q4) — tells you which trig functions are positive in each quadrant.
What is the exact value of sin(5π/6)?
Common trap: students confuse 5π/6 with 7π/6 (Q3, where sin is negative).
Simplify: (1 − sin²θ) / cos θ
This is one of the most tested identity simplifications. Always look for the Pythagorean identity first!
Solve on [0, 2π): 2cos²x − 1 = 0
Common mistake: forgetting the ± and only finding 2 solutions. The equation has four solutions on [0, 2π).
The equation x² / 9 + y² / 4 = 1 describes which conic section? What are the lengths of its semi-axes?
Common trap: C reads 9 and 4 directly — those are a² and b², not a and b. Always take the square root!
Which equation represents a parabola opening to the right with vertex at the origin?
Memory shortcut: the variable that's squared tells you the axis of symmetry. y² → axis is x-axis → opens left/right.
A geometric sequence has first term a₁ = 2 and common ratio r = 3. What is the sum of the first 4 terms?
What is the coefficient of x³ in the expansion of (x + 2)⁵?
Hmm wait — let me recheck: C = 80? No, 10 × 4 = 40. Option D = 40 is correct. Common mistake: students forget to raise the constant (2) to the power k.
Evaluate: lim(x→2) (x² − 4) / (x − 2)
Direct substitution gives 0/0 (indeterminate form). Always factor and cancel before substituting. The function is defined everywhere except x = 2, but the limit still exists!
What is lim(x→∞) (5x³ − 2x) / (3x³ + x²)?
Same degree on top and bottom → limit = ratio of leading coefficients = 5/3. This is exactly the horizontal asymptote rule!
What is the determinant of the matrix: [[3, 1], [2, 4]]?
Common trap: A is 3×4 + 1×2 = 14 (addition instead of subtraction). Always subtract cross product!
How many solutions does this system have?
2x + 4y = 6
x + 2y = 3
If lines are identical → dependent system (infinite solutions). If lines are parallel but distinct → inconsistent system (no solution). These are commonly confused!
Simplify: i⁴⁷
The key: always divide the exponent by 4 and use the remainder. Remainder 0 → 1, Remainder 1 → i, Remainder 2 → −1, Remainder 3 → −i.
What is the modulus (absolute value) of the complex number z = 3 − 4i?
This is the classic 3-4-5 right triangle. The complex number 3 − 4i lies at point (3, −4) in the complex plane, exactly 5 units from the origin.