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Pre-Calculus Β· Comprehensive Review

Mastery Quiz

20 Exam-Style Problems Across All Core Units

πŸ“‹ 20 Questions ⏱ 40 Minutes πŸ“Š Multiple Choice 🎯 Exam Level
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UNIT 01 Functions & Graphs

πŸ“Œ Key Concepts to Memorize

β–ΈA function maps each input x to exactly one output f(x). Vertical Line Test confirms this graphically.
β–ΈDomain: all valid inputs. Avoid division by zero and square roots of negatives.
β–ΈTransformations: f(x βˆ’ h) + k shifts right h, up k; βˆ’f(x) reflects over x-axis; f(βˆ’x) reflects over y-axis.
β–ΈComposition: (f ∘ g)(x) = f(g(x)). Evaluate inner function first.
Even: f(βˆ’x) = f(x)   |   Odd: f(βˆ’x) = βˆ’f(x)
Inverse: f(f⁻¹(x)) = x   (swap x and y, solve)

✦ Worked Example

Given f(x) = 2x + 1 and g(x) = x², find (f ∘ g)(3).

Step 1: g(3) = 9

Step 2: f(9) = 2(9) + 1 = 19

Answer: 19

Functions Β· Domain
Q1 What is the domain of f(x) = √(x βˆ’ 3) / (x βˆ’ 7)?
Functions Β· Composition
Q2 If f(x) = xΒ² βˆ’ 1 and g(x) = 2x + 3, what is (g ∘ f)(2)?
UNIT 02 Polynomial Functions

πŸ“Œ Key Concepts to Memorize

β–ΈRemainder Theorem: Dividing p(x) by (x βˆ’ c) gives remainder p(c).
β–ΈFactor Theorem: (x βˆ’ c) is a factor of p(x) if and only if p(c) = 0.
β–ΈEnd behavior: determined by leading term. Even degree β†’ same ends; Odd degree β†’ opposite ends.
β–ΈMultiplicity: Even multiplicity β†’ graph touches x-axis and bounces; Odd multiplicity β†’ graph crosses.
Fundamental Theorem: Degree n polynomial has exactly n zeros (counting multiplicity, in β„‚)
Rational Root Theorem: possible rational roots = Β±(factors of constant) / (factors of leading coeff)

✦ Worked Example

Find all real zeros of p(x) = xΒ³ βˆ’ 6xΒ² + 11x βˆ’ 6.

Try x = 1: 1 βˆ’ 6 + 11 βˆ’ 6 = 0 βœ“

Factor: (x βˆ’ 1)(xΒ² βˆ’ 5x + 6) = (x βˆ’ 1)(x βˆ’ 2)(x βˆ’ 3)

Zeros: x = 1, 2, 3

Polynomials Β· Remainder Theorem
Q3 When p(x) = 2xΒ³ βˆ’ 3xΒ² + x βˆ’ 5 is divided by (x βˆ’ 2), what is the remainder?
Polynomials Β· End Behavior
Q4 Which describes the end behavior of f(x) = βˆ’3x⁴ + 5xΒ² βˆ’ 2?
UNIT 03 Rational Functions

πŸ“Œ Key Concepts to Memorize

β–ΈVertical asymptote: occurs where denominator = 0 (and numerator β‰  0).
β–ΈHorizontal asymptote rules: if deg(num) < deg(den) β†’ y = 0; if degrees equal β†’ y = ratio of leading coefficients; if deg(num) > deg(den) β†’ no horizontal asymptote (oblique instead).
β–ΈHoles: occur where a common factor cancels from numerator and denominator.
Rational Functions Β· Asymptotes
Q5 What is the horizontal asymptote of f(x) = (3xΒ² βˆ’ 2) / (xΒ² + 5)?
UNIT 04 Exponential & Logarithmic Functions

πŸ“Œ Key Concepts to Memorize

β–ΈDefinition: log_b(x) = y means b^y = x. Always: base b > 0, b β‰  1, and x > 0.
β–ΈLog properties: Product β†’ log(MN) = log M + log N; Quotient β†’ log(M/N) = log M βˆ’ log N; Power β†’ log(M^p) = pΒ·log M.
β–ΈChange of base: log_b(x) = ln(x)/ln(b).
β–ΈNatural exponential: e β‰ˆ 2.718. Inverse of ln.
Compound Interest: A = P(1 + r/n)^(nt)
Continuous: A = Pe^(rt)

✦ Worked Example

Solve: logβ‚‚(x + 3) = 4

Convert: x + 3 = 2⁴ = 16

Answer: x = 13

Logarithms Β· Properties
Q6 Which expression is equal to logβ‚‚(8) + logβ‚‚(4)?
Exponential Β· Equations
Q7 Solve for x:   5^(2x) = 125
Logarithms Β· Solving
Q8 Solve: ln(x) + ln(x βˆ’ 2) = ln(3)
UNIT 05 Trigonometry

πŸ“Œ Key Concepts to Memorize

β–ΈUnit circle: sin(ΞΈ) = y-coord, cos(ΞΈ) = x-coord at angle ΞΈ. sinΒ²ΞΈ + cosΒ²ΞΈ = 1.
β–ΈSpecial angles (radians): 0, Ο€/6, Ο€/4, Ο€/3, Ο€/2 β†’ cos values: 1, √3/2, √2/2, 1/2, 0.
β–ΈPeriod & amplitude: For y = A sin(Bx + C) + D, amplitude = |A|, period = 2Ο€/|B|.
β–ΈASTC rule (All Students Take Calculus): Quadrant I all positive; II sin+; III tan+; IV cos+.
tan ΞΈ = sin ΞΈ / cos ΞΈ   |   sec ΞΈ = 1/cos ΞΈ   |   csc ΞΈ = 1/sin ΞΈ   |   cot ΞΈ = 1/tan ΞΈ

✦ Worked Example

Find the exact value of sin(150Β°).

150Β° is in Quadrant II (sin positive). Reference angle = 30Β°.

sin(150Β°) = sin(30Β°) = 1/2

Trigonometry Β· Unit Circle
Q9 What is the exact value of tan(225Β°)?
Trigonometry Β· Graphs
Q10 What is the period of f(x) = 3sin(2x βˆ’ Ο€/4)?
Trigonometry Β· Law of Cosines
Q11 In triangle ABC, sides a = 7, b = 5, and angle C = 60Β°. What is the length of side c?
Trigonometry Β· Inverse
Q12 Evaluate: arcsin(βˆ’βˆš2/2), giving your answer in radians in the range [βˆ’Ο€/2, Ο€/2].
UNIT 06 Analytic Trigonometry

πŸ“Œ Key Identities to Memorize

β–ΈPythagorean: sinΒ²ΞΈ + cosΒ²ΞΈ = 1,  1 + tanΒ²ΞΈ = secΒ²ΞΈ,  1 + cotΒ²ΞΈ = cscΒ²ΞΈ.
β–ΈDouble angle: sin(2ΞΈ) = 2sinΞΈcosΞΈ,  cos(2ΞΈ) = cosΒ²ΞΈ βˆ’ sinΒ²ΞΈ = 1 βˆ’ 2sinΒ²ΞΈ = 2cosΒ²ΞΈ βˆ’ 1.
β–ΈSum formulas: sin(AΒ±B) = sinAcosB Β± cosAsinB,  cos(AΒ±B) = cosAcosB βˆ“ sinAsinB.
Half-angle: sin(ΞΈ/2) = ±√((1 βˆ’ cosΞΈ)/2)   |   cos(ΞΈ/2) = ±√((1 + cosΞΈ)/2)
Analytic Trig Β· Identities
Q13 If sin ΞΈ = 3/5 and ΞΈ is in Quadrant II, what is cos(2ΞΈ)?
Analytic Trig Β· Equations
Q14 Solve on [0, 2Ο€): 2cosΒ²x βˆ’ cosx βˆ’ 1 = 0. How many solutions are there?
UNIT 07 Sequences & Series

πŸ“Œ Key Concepts to Memorize

β–ΈArithmetic: common difference d. General term: aβ‚™ = a₁ + (nβˆ’1)d. Sum: Sβ‚™ = n(a₁ + aβ‚™)/2.
β–ΈGeometric: common ratio r. General term: aβ‚™ = a₁ Β· r^(nβˆ’1). Sum: Sβ‚™ = a₁(1 βˆ’ rⁿ)/(1 βˆ’ r).
β–ΈInfinite geometric series: converges if |r| < 1, then S = a₁/(1 βˆ’ r).
Binomial Theorem: (a + b)ⁿ = Ξ£ C(n,k) Β· aⁿ⁻ᡏ Β· bᡏ   where   C(n,k) = n! / (k!(nβˆ’k)!)
Sequences Β· Geometric
Q15 What is the sum of the infinite geometric series: 12 + 4 + 4/3 + 4/9 + Β·Β·Β·?
Binomial Theorem
Q16 What is the coefficient of x² in the expansion of (x + 3)⁡?
UNIT 08 Conic Sections

πŸ“Œ Key Concepts to Memorize

β–ΈCircle: (xβˆ’h)Β² + (yβˆ’k)Β² = rΒ². Center (h,k), radius r.
β–ΈParabola: y = a(xβˆ’h)Β² + k (opens up/down) or x = a(yβˆ’k)Β² + h (opens left/right). Focus is 1/(4a) from vertex.
β–ΈEllipse: (xβˆ’h)Β²/aΒ² + (yβˆ’k)Β²/bΒ² = 1. Major axis along larger denominator. cΒ² = aΒ² βˆ’ bΒ².
β–ΈHyperbola: (xβˆ’h)Β²/aΒ² βˆ’ (yβˆ’k)Β²/bΒ² = 1. Asymptotes: y βˆ’ k = Β±(b/a)(x βˆ’ h). cΒ² = aΒ² + bΒ².
Conics Β· Ellipse
Q17 The ellipse xΒ²/25 + yΒ²/9 = 1 has foci at which coordinates?
Conics Β· Parabola
Q18 The parabola yΒ² = 12x has its focus at:
UNIT 09 Vectors

πŸ“Œ Key Concepts to Memorize

β–ΈMagnitude: |v| = √(aΒ² + bΒ²) for v = ⟨a, b⟩.
β–ΈDot product: u Β· v = a₁aβ‚‚ + b₁bβ‚‚ = |u||v|cosΞΈ. If u Β· v = 0, vectors are perpendicular.
β–ΈUnit vector: Γ» = v / |v|.
Vectors Β· Dot Product
Q19 Find the angle (in degrees) between vectors u = ⟨3, 4⟩ and v = ⟨4, βˆ’3⟩.
UNIT 10 Complex Numbers

πŸ“Œ Key Concepts to Memorize

β–ΈDefinition: i = √(βˆ’1), so iΒ² = βˆ’1, iΒ³ = βˆ’i, i⁴ = 1. Pattern repeats with period 4.
β–ΈModulus: |a + bi| = √(aΒ² + bΒ²).
β–ΈConjugate: (a + bi)(a βˆ’ bi) = aΒ² + bΒ². Use to divide complex numbers.
β–ΈPolar form: z = r(cosΞΈ + i sinΞΈ) where r = |z|.
De Moivre's Theorem: zⁿ = rⁿ(cos(nθ) + i sin(nθ))
Complex Numbers Β· Powers
Q20 What is the value of i⁴⁷?
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Final Score
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Answer Key & Full Explanations

Q1 βœ“ Answer: B Domain of √(xβˆ’3)/(xβˆ’7)
Require x βˆ’ 3 β‰₯ 0, so x β‰₯ 3. Also require denominator x βˆ’ 7 β‰  0, so x β‰  7. Combining these restrictions: domain = [3, 7) βˆͺ (7, ∞).
Q2 βœ“ Answer: C (g ∘ f)(2)
Evaluate the inner function first: f(2) = (2)Β² βˆ’ 1 = 3. Then apply g: g(3) = 2(3) + 3 = 9. Therefore (g ∘ f)(2) = 9.
Q3 βœ“ Answer: A Remainder when dividing by (xβˆ’2)
By the Remainder Theorem, evaluate p(2): p(2) = 2(8) βˆ’ 3(4) + 2 βˆ’ 5 = 16 βˆ’ 12 + 2 βˆ’ 5 = 1.
Q4 βœ“ Answer: B End behavior of βˆ’3x⁴ + 5xΒ² βˆ’ 2
Leading term is βˆ’3x⁴. Degree 4 is even β†’ both ends behave the same. Leading coefficient βˆ’3 < 0 β†’ both ends go to βˆ’βˆž. Therefore as x β†’ ±∞, f(x) β†’ βˆ’βˆž.
Q5 βœ“ Answer: B Horizontal asymptote of (3xΒ²βˆ’2)/(xΒ²+5)
The degrees of numerator and denominator are both 2 (equal). The horizontal asymptote is the ratio of the leading coefficients: 3/1 = 3. So y = 3.
Q6 βœ“ Answer: B logβ‚‚(8) + logβ‚‚(4)
Product rule: logβ‚‚(8) + logβ‚‚(4) = logβ‚‚(8 Γ— 4) = logβ‚‚(32). Since 2⁡ = 32, the answer is 5. Alternatively: logβ‚‚(8) = 3 and logβ‚‚(4) = 2, so 3 + 2 = 5.
Q7 βœ“ Answer: B Solve 5^(2x) = 125
Write 125 = 5Β³. Then 5^(2x) = 5Β³ implies 2x = 3, so x = 3/2.
Q8 βœ“ Answer: A Solve ln(x) + ln(xβˆ’2) = ln(3)
Product rule: ln(x(xβˆ’2)) = ln(3), so x(xβˆ’2) = 3, giving xΒ² βˆ’ 2x βˆ’ 3 = 0, which factors as (xβˆ’3)(x+1) = 0. Solutions: x = 3 or x = βˆ’1. But x = βˆ’1 makes ln(x) undefined. Therefore the only valid solution is x = 3.
Q9 βœ“ Answer: B tan(225Β°)
225Β° is in Quadrant III (reference angle = 45Β°). In Q III, tangent is positive. tan(45Β°) = 1, so tan(225Β°) = +1. (In Q III, both sin and cos are negative, so their ratio tan is positive.)
Q10 βœ“ Answer: B Period of 3sin(2x βˆ’ Ο€/4)
The general form is A sin(Bx + C). Here B = 2. Period = 2Ο€ / |B| = 2Ο€ / 2 = Ο€.
Q11 βœ“ Answer: A Law of Cosines: a=7, b=5, C=60Β°
Law of Cosines: cΒ² = aΒ² + bΒ² βˆ’ 2abΒ·cos(C).
cΒ² = 49 + 25 βˆ’ 2(7)(5)cos(60Β°) = 74 βˆ’ 70Β·(1/2) = 74 βˆ’ 35 = 39.
Therefore c = √39.
Q12 βœ“ Answer: A arcsin(βˆ’βˆš2/2)
We need the angle ΞΈ ∈ [βˆ’Ο€/2, Ο€/2] where sin(ΞΈ) = βˆ’βˆš2/2. Since sin(Ο€/4) = √2/2, and we need the negative value, the answer is ΞΈ = βˆ’Ο€/4.
Q13 βœ“ Answer: A cos(2ΞΈ) given sinΞΈ = 3/5, Quadrant II
In Q II: sin ΞΈ = 3/5, so cos ΞΈ = βˆ’4/5 (cos is negative in Q II; from cosΒ²ΞΈ = 1 βˆ’ 9/25 = 16/25).
cos(2ΞΈ) = cosΒ²ΞΈ βˆ’ sinΒ²ΞΈ = 16/25 βˆ’ 9/25 = 7/25.
Verification: 1 βˆ’ 2sinΒ²ΞΈ = 1 βˆ’ 2(9/25) = 1 βˆ’ 18/25 = 7/25 βœ“
Q14 βœ“ Answer: C 2cosΒ²x βˆ’ cosx βˆ’ 1 = 0 on [0, 2Ο€)
Factor as a quadratic in cos x: (2cosx + 1)(cosx βˆ’ 1) = 0.
Case 1: cosx = 1 β†’ x = 0. (1 solution)
Case 2: cosx = βˆ’1/2 β†’ x = 2Ο€/3 or x = 4Ο€/3. (2 solutions)
Total: 3 solutions: {0, 2Ο€/3, 4Ο€/3}.
Q15 βœ“ Answer: B Sum of 12 + 4 + 4/3 + Β·Β·Β·
First term a₁ = 12. Common ratio r = 4/12 = 1/3. Since |r| = 1/3 < 1, the series converges.
S = a₁/(1 βˆ’ r) = 12/(1 βˆ’ 1/3) = 12/(2/3) = 12 Γ— 3/2 = 18.
Q16 βœ“ Answer: B Coefficient of xΒ² in (x+3)⁡
The term with xΒ² in (x + 3)⁡ corresponds to k = 3 (since the power of x is 5 βˆ’ k = 2, so k = 3).
Term = C(5,3) Β· xΒ² Β· 3Β³ = 10 Β· xΒ² Β· 27 = 270xΒ².
Coefficient = 270.
Q17 βœ“ Answer: A Foci of xΒ²/25 + yΒ²/9 = 1
Here aΒ² = 25, bΒ² = 9. Since aΒ² > bΒ², major axis is along the x-axis.
cΒ² = aΒ² βˆ’ bΒ² = 25 βˆ’ 9 = 16, so c = 4.
Foci are at (Β±4, 0).
Q18 βœ“ Answer: A Focus of yΒ² = 12x
The form yΒ² = 4px is a rightward-opening parabola with vertex at origin. Here 4p = 12, so p = 3. The focus is at (3, 0).
Q19 βœ“ Answer: C Angle between u=⟨3,4⟩ and v=⟨4,βˆ’3⟩
Dot product: u Β· v = (3)(4) + (4)(βˆ’3) = 12 βˆ’ 12 = 0.
Since the dot product equals zero, the vectors are perpendicular. The angle between them is 90Β°.
Q20 βœ“ Answer: D i⁴⁷
Powers of i cycle with period 4: iΒΉ = i, iΒ² = βˆ’1, iΒ³ = βˆ’i, i⁴ = 1.
Divide: 47 Γ· 4 = 11 remainder 3. So i⁴⁷ = iΒ³ = βˆ’i.