PRE-CALCULUS Master Quiz · 20 Problems
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📚 Concepts
✏️ Quiz
📋 Answer Key
Core Topics & Key Formulas
1
Functions & Their Properties
Must Memorize
Vertical Line Test: A graph represents a function if every vertical line intersects the graph at most once.
Domain = all valid x-values; Range = all resulting y-values
Even function: f(−x) = f(x) → symmetric about y-axis
Odd function: f(−x) = −f(x) → symmetric about origin
Composite: (f∘g)(x) = f(g(x)); domain: x in domain of g where g(x) in domain of f
Inverse: (f⁻¹∘f)(x) = x; switch x and y, then solve
Example
Is f(x) = x⁴ − 3x² even, odd, or neither?
f(−x) = (−x)⁴ − 3(−x)² = x⁴ − 3x² = f(x) → EVEN ✓
2
Polynomial & Rational Functions
Remainder & Factor Theorem
If p(x) ÷ (x−c): remainder = p(c). If p(c) = 0, then (x−c) is a factor.
Rational Zero Theorem: possible zeros = ±(factors of constant / factors of leading coeff)
Vertical asymptote of f(x)/g(x): where g(x) = 0 (and f(x) ≠ 0)
Horizontal asymptote: compare degrees of numerator (n) and denominator (m)
n < m → y = 0; n = m → y = ratio of leading coefficients; n > m → no H.A.
Example
H.A. of f(x) = (3x² + 1)/(x² − 4)?
n = m = 2; H.A.: y = 3/1 = 3 ✓
3
Exponential & Logarithmic Functions
Log Laws (Memorize These!)
logb(MN) = logbM + logbN
logb(M/N) = logbM − logbN
logb(Mⁿ) = n·logbM
logbx = ln x / ln b (change of base)
logₐx = y ↔ aʸ = x (definition)
Compound interest: A = P(1 + r/n)^(nt)
Continuous: A = Pe^(rt)
Natural log: ln(eˣ) = x and e^(ln x) = x
Example
Solve: log₂(x+3) + log₂(x−1) = 5
log₂[(x+3)(x−1)] = 5 → (x+3)(x−1) = 32 → x²+2x−3=32 → x²+2x−35=0 → (x+7)(x−5)=0; x=5 (x=−7 rejected) ✓
4
Trigonometry
Unit Circle Key Angles
sin 30°=½, cos 30°=√3/2, tan 30°=1/√3
sin 45°=√2/2, cos 45°=√2/2, tan 45°=1
sin 60°=√3/2, cos 60°=½, tan 60°=√3
sin 90°=1, cos 90°=0
Pythagorean: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = csc²θ
Double angle: sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ
Sum: sin(A±B) = sinA cosB ± cosA sinB
Law of Cosines: c² = a² + b² − 2ab cos C
Example
Exact value of sin(7π/6)?
7π/6 = π + π/6 → in Q III, sin negative → sin(7π/6) = −sin(π/6) = −1/2 ✓
5
Conic Sections
Standard Forms
Circle: (x−h)²+(y−k)²=r²
Ellipse: (x−h)²/a² + (y−k)²/b² = 1 (a²>b²: major axis horizontal)
Hyperbola: (x−h)²/a² − (y−k)²/b² = 1
Parabola: (x−h)² = 4p(y−k)
Ellipse: c² = a² − b²; foci at (h±c, k) if a²>b²
Hyperbola: c² = a² + b²; asymptotes: y−k = ±(b/a)(x−h)
Parabola: focus at (h, k+p), directrix y = k−p
Example
Center and radius of x² + y² − 6x + 4y − 3 = 0?
(x−3)²+(y+2)²=16 → center (3,−2), radius 4 ✓
6
Sequences & Series
Key Formulas
Arithmetic: aₙ = a₁ + (n−1)d; Sₙ = n(a₁+aₙ)/2
Geometric: aₙ = a₁·rⁿ⁻¹; Sₙ = a₁(1−rⁿ)/(1−r)
Infinite Geometric (|r|<1): S∞ = a₁/(1−r)
Binomial: (a+b)ⁿ = Σ C(n,k)·aⁿ⁻ᵏ·bᵏ
Arithmetic: constant difference d between terms
Geometric: constant ratio r between consecutive terms
Binomial Theorem: coefficient of (n,k) = n!/(k!(n−k)!)
Example
Sum of infinite geometric: first term 8, r = 1/4?
S∞ = 8/(1−1/4) = 8/(3/4) = 32/3 ✓
7
Introduction to Limits
Key Limit Facts
lim(x→a) f(x) = L means f(x) approaches L as x→a (from both sides)
lim(x→0) sin(x)/x = 1
lim(x→∞) (1 + 1/n)ⁿ = e
Direct substitution: try f(a) first — if defined, that's the limit
0/0 form: factor and cancel, or use conjugate (for radicals)
Squeeze theorem: if g(x) ≤ f(x) ≤ h(x) near a and lim g=lim h=L, then lim f=L
Example
lim(x→2) (x²−4)/(x−2)?
= lim(x→2) (x+2)(x−2)/(x−2) = lim(x→2) (x+2) = 4 ✓
Practice Quiz
20 exam-style questions
01
Functions
★☆☆ Easy
Which of the following relations is NOT a function?
A
{(1,2), (2,3), (3,4), (4,5)}
B
{(1,2), (1,3), (2,4), (3,5)}
C
{(−1,1), (0,0), (1,1), (2,4)}
D
{(2,5), (3,5), (4,5), (5,5)}
📝 Solution
Answer: B
A function assigns exactly one output for each input.
In option B, the input x = 1 maps to both y = 2 and y = 3. This violates the definition of a function.
A, C, D: each x-value appears only once → these are functions ✓
02
Functions
★★☆ Medium
If f(x) = 2x + 1 and g(x) = x² − 3, what is (f∘g)(2)?
A
1
B
5
C
3
D
−1
📝 Solution
Answer: C (3)
(f∘g)(2) means f applied to g(2):
Step 1: g(2) = (2)² − 3 = 4 − 3 = 1 Step 2: f(1) = 2(1) + 1 = 3 Therefore (f∘g)(2) = 3 ✓
03
Functions
★★☆ Medium
What is the inverse of f(x) = 3x − 5?
A
f⁻¹(x) = (x + 5)/3
B
f⁻¹(x) = (x − 5)/3
C
f⁻¹(x) = 3x + 5
D
f⁻¹(x) = 1/(3x − 5)
📝 Solution
Answer: A
To find the inverse, swap x and y, then solve for y:
y = 3x − 5 → x = 3y − 5 x + 5 = 3y → y = (x + 5)/3 So f⁻¹(x) = (x + 5)/3 ✓
Check: f(f⁻¹(x)) = 3·(x+5)/3 − 5 = x+5−5 = x ✓
04
Polynomials
★★☆ Medium
When p(x) = x³ − 2x² + 3x − 4 is divided by (x − 2), what is the remainder?
A
0
B
2
C
6
D
−2
📝 Solution
Answer: B
By the Remainder Theorem: remainder = p(2)
p(2) = (2)³ − 2(2)² + 3(2) − 4 = 8 − 8 + 6 − 4 = 2 Remainder = 2 ✓
05
Rational Functions
★★☆ Medium
What is the horizontal asymptote of f(x) = (4x² − 1)/(2x² + 3x)?
A
y = 0
B
y = 4
C
y = 2
D
No horizontal asymptote
📝 Solution
Answer: C
Both numerator and denominator have degree 2 (n = m).
When degrees are equal, horizontal asymptote = ratio of leading coefficients:
Leading coeff of numerator: 4 Leading coeff of denominator: 2 H.A.: y = 4/2 = 2 ✓
06
Logarithms
★★☆ Medium
Which expression is equal to log₃ 8 + log₃ 9 − log₃ 24?
A
log₃ 3 = 1
B
log₃ 6
C
2
D
log₃ 2
📝 Solution
Answer: A
Use log rules: log M + log N − log K = log(MN/K)
log₃(8 × 9 / 24) = log₃(72/24) = log₃(3) log₃(3) = 1 ✓ 8 × 9 = 72; 72/24 = 3; log₃(3) = 1
07
Exponentials
★★★ Hard
Solve for x: 2^(x+1) = 5^(x−1). Which expression equals x?
A
x = (log 5 − log 2)/(log 2 − log 5)
B
x = (log 10)/(log 2.5)
C
x = (log 5 + log 2)/(log 5 − log 2)
D
x = log(5/2) + 1
📝 Solution
Answer: C
Take log of both sides:
(x+1)log 2 = (x−1)log 5 x·log 2 + log 2 = x·log 5 − log 5 log 2 + log 5 = x·log 5 − x·log 2 log 10 = x(log 5 − log 2) 1 = x(log 5 − log 2) x = 1/(log 5 − log 2) = log 10/(log 5 − log 2) Note: log 2 + log 5 = log 10 = 1
Rewriting: x = (log 2 + log 5)/(log 5 − log 2) = (log 5 + log 2)/(log 5 − log 2) ✓
08
Trigonometry
★☆☆ Easy
What is the exact value of cos(5π/6)?
A
√3/2
B
−1/2
C
−√3/2
D
1/2
📝 Solution
Answer: C
5π/6 is in Quadrant II (between π/2 and π).
Reference angle: π − 5π/6 = π/6
cos(π/6) = √3/2 In Q II, cosine is negative → cos(5π/6) = −√3/2 ✓
09
Trigonometry
★★☆ Medium
If sin θ = 3/5 and θ is in Quadrant II, what is cos(2θ)?
A
7/25
B
−7/25
C
24/25
D
−24/25
📝 Solution
Answer: A (7/25)
Given sin θ = 3/5, Q II → cos θ = −4/5 (since cos²θ = 1 − 9/25 = 16/25, negative in Q II)
Double angle formula: cos(2θ) = cos²θ − sin²θ
= (−4/5)² − (3/5)² = 16/25 − 9/25 = 7/25 Verify: 1 − 2sin²θ = 1 − 2(9/25) = 1 − 18/25 = 7/25 ✓ Note: cos(2θ) = 7/25 is positive because we squared the negative cosine.
10
Trig Identities
★★★ Hard
Simplify: (sin²x − 1)/cos x
A
cos x
B
−cos x
C
sin x
D
−sin x
📝 Solution
Answer: B
Use Pythagorean identity: sin²x + cos²x = 1 → sin²x − 1 = −cos²x
(sin²x − 1)/cos x = −cos²x / cos x = −cos x ✓
11
Conic Sections
★★☆ Medium
The equation x²/9 + y²/4 = 1 represents an ellipse. What are the coordinates of the foci?
A
(±√5, 0)
B
(±√13, 0)
C
(0, ±√5)
D
(±3, 0)
📝 Solution
Answer: A
a² = 9, b² = 4 (a² > b², so major axis is horizontal)
For ellipse: c² = a² − b² = 9 − 4 = 5
c = √5 Foci at (±√5, 0) ✓
12
Conic Sections
★★★ Hard
What are the equations of the asymptotes of the hyperbola x²/16 − y²/9 = 1?
A
y = ±(4/3)x
B
y = ±(3/4)x
C
y = ±(9/4)x
D
y = ±4x
📝 Solution
Answer: B
Hyperbola: x²/a² − y²/b² = 1 where a² = 16 → a = 4, b² = 9 → b = 3
Asymptotes formula: y = ±(b/a)x
y = ±(3/4)x ✓ Note: b/a, not a/b — it's the ratio b over a for horizontal hyperbola.
13
Sequences
★☆☆ Easy
The first term of an arithmetic sequence is 7 and the common difference is −3. What is the 10th term?
A
−20
B
−17
C
−24
D
−14
📝 Solution
Answer: A
Arithmetic: aₙ = a₁ + (n − 1)d
a₁₀ = 7 + (10 − 1)(−3) = 7 + 9(−3) = 7 − 27 = −20 ✓
14
Series
★★☆ Medium
Find the sum of the infinite geometric series: 12 + 4 + 4/3 + 4/9 + ···
A
16
B
18
C
20
D
24
📝 Solution
Answer: B
a₁ = 12, r = 4/12 = 1/3
Since |r| = 1/3 < 1, sum converges:
S∞ = a₁/(1 − r) = 12/(1 − 1/3) = 12/(2/3) = 12 × 3/2 = 18 ✓
15
Binomial Theorem
★★★ Hard
What is the coefficient of x³ in the expansion of (2x + 3)⁵?
A
720
B
1080
C
360
D
2160
📝 Solution
Answer: A
(2x + 3)⁵: General term = C(5,k)·(2x)^(5−k)·3^k
For x³: need 5−k = 3 → k = 2
Term = C(5,2)·(2x)³·3² = 10·8x³·9 = 720x³ Coefficient of x³ = 720 ✓ C(5,2) = 10; 2³ = 8; 3² = 9; 10×8×9 = 720
16
Limits
★★☆ Medium
Evaluate: lim(x→3) (x² − 9)/(x − 3)
A
0
B
3
C
6
D
undefined
📝 Solution
Answer: C
Direct substitution gives 0/0 (indeterminate). Factor:
x² − 9 = (x+3)(x−3) lim(x→3) (x+3)(x−3)/(x−3) = lim(x→3) (x+3) = 3 + 3 = 6 ✓
17
Limits
★★★ Hard
What is lim(x→∞) (3x² + 2x)/(5x² − 7)?
A
∞
B
0
C
2/5
D
3/5
📝 Solution
Answer: D
Divide numerator and denominator by x² (highest power):
(3 + 2/x)/(5 − 7/x²) As x→∞: 2/x→0 and 7/x²→0 Limit = 3/5 ✓ This is consistent with the H.A. rule: equal degrees → ratio of leading coefficients.
18
Trigonometry
★★★ Hard
Solve for θ in [0, 2π): 2cos²θ − cos θ − 1 = 0. How many solutions are there?
A
1
B
2
C
3
D
4
📝 Solution
Answer: C
Let u = cos θ. Factor: 2u² − u − 1 = 0 → (2u + 1)(u − 1) = 0
u = −1/2 or u = 1 cos θ = 1: θ = 0 → 1 solution
cos θ = −1/2: θ = 2π/3 and θ = 4π/3 → 2 solutions
Total: 1 + 2 = 3 solutions: {0, 2π/3, 4π/3} ✓
19
Logarithms
★★★ Hard
If log₂ x + log₂(x − 2) = 3, what is x?
A
x = 4
B
x = −2
C
x = 4 or x = −2
D
x = 8
📝 Solution
Answer: A
log₂[x(x−2)] = 3 → x(x−2) = 2³ = 8 x² − 2x − 8 = 0 → (x−4)(x+2) = 0 x = 4 or x = −2 Check domain: log requires positive arguments.
x = −2: log₂(−2) is undefined → rejected
x = 4: log₂(4) = 2 ✓ and log₂(2) = 1 ✓; sum = 3 ✓
Only x = 4 ✓
20
Parabola
★★★ Hard
The parabola (x − 2)² = 12(y + 1) has its focus at which point?
A
(2, 2)
B
(2, −4)
C
(2, 3)
D
(5, −1)
📝 Solution
Answer: A
Standard form: (x−h)² = 4p(y−k); vertex at (h, k)
Here: h = 2, k = −1, and 4p = 12 → p = 3
Vertex: (2, −1) Focus: (h, k+p) = (2, −1+3) = (2, 2) ✓ Directrix: y = k − p = −1 − 3 = −4
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Answer Key & Full Solutions
📋 Answer Key with Explanations
Q1 · Functions
B
x=1 maps to both y=2 and y=3 → not a function (each input must have exactly one output)
Q2 · Composition
C (3)
g(2)=1; f(1)=2(1)+1=3
Q3 · Inverse
A
Swap x↔y: x=3y−5 → y=(x+5)/3
Q4 · Remainder Thm
B (2)
p(2)=8−8+6−4=2
Q5 · Asymptotes
C (y=2)
Equal degrees → leading coeff ratio: 4/2=2
Q6 · Log Laws
A (1)
log₃(8×9/24)=log₃(3)=1
Q7 · Exp Equations
C
x=(log5+log2)/(log5−log2) since log2+log5=1
Q8 · Unit Circle
C (−√3/2)
5π/6 in Q II, ref angle π/6, cos negative
Q9 · Double Angle
A (7/25)
cos2θ=cos²θ−sin²θ=16/25−9/25=7/25
Q10 · Identities
B (−cos x)
sin²x−1=−cos²x; ÷cosx=−cosx
Q11 · Ellipse Foci
A (±√5, 0)
c²=9−4=5 → c=√5; foci on x-axis
Q12 · Hyperbola
B (y=±3/4·x)
a=4,b=3; asymptotes y=±(b/a)x=±(3/4)x
Q13 · Arithmetic
A (−20)
a₁₀=7+9(−3)=7−27=−20
Q14 · Infinite Series
B (18)
r=1/3; S∞=12/(2/3)=18
Q15 · Binomial
A (720)
C(5,2)·8·9=10×72=720
Q16 · Limits
C (6)
Factor (x+3)(x−3)/(x−3)→x+3→6
Q17 · Limits at ∞
D (3/5)
Equal degrees → ratio of leading coefficients
Q18 · Trig Equations
C (3)
θ=0, 2π/3, 4π/3 from cosθ=1 and cosθ=−½
Q19 · Log Equations
A (x=4)
x=−2 rejected (domain); x=4 verified
Q20 · Parabola Focus
A (2, 2)
4p=12→p=3; focus=(2,−1+3)=(2,2)