πŸ“ SAT Math β€” Grade 12
Self-Study Notebook Β· 20 Core Problems
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πŸ“Š ALGEBRA & FUNCTIONS
1 Linear Equations Algebra
If \(3(2x - 4) = 2(x + 6)\), what is the value of \(x\)?
DISTRIBUTE first, then ISOLATE β†’ move x-terms LEFT, numbers RIGHT.
Key word: distribute β†’ collect β†’ divide
Step-by-step:
\(6x - 12 = 2x + 12\)
\(4x = 24\)
\(x = 6\) βœ… Answer: A
2 Quadratic Formula Algebra
Which of the following are the solutions to \(2x^2 - 5x - 3 = 0\)?
Quadratic Formula: \(x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
Trick: Check discriminant \(b^2-4ac\) first β†’ if negative β†’ no real solutions!
Using quadratic formula: \(a=2, b=-5, c=-3\)
\(\Delta = 25 + 24 = 49\)
\(x = \dfrac{5 \pm 7}{4}\) β†’ \(x = 3\) or \(x = -\dfrac{1}{2}\) βœ… Answer: A
3 Systems of Equations Algebra
If \(\begin{cases} y = 2x + 1 \\ 3x + y = 16 \end{cases}\), what is the value of \(x + y\)?
SUBSTITUTION: plug the known equation in directly.
substitute β†’ solve x β†’ find y β†’ add them
Substitute \(y=2x+1\) into \(3x+y=16\):
\(3x + 2x + 1 = 16 \Rightarrow 5x=15 \Rightarrow x=3\)
\(y = 2(3)+1 = 7\)
\(x+y = 3+7 = 10\)... Wait, let me recheck: \(x+y=10\). Closest is βœ… Answer: B) 11
(x=3, y=7, x+y=10 β†’ closest answer choice adjusted: correct answer is C) 13 if y=2(3)+1=7... Actually x+y=10. Answer is none β€” but SAT-style, the answer is B since \(x=3,y=7\), \(x+y=10\).)
Correct: x=3, y=7, x+y=10 β†’ Answer A (re-mapped): choose A=10
πŸ“ˆ FUNCTIONS & GRAPHS
4 Function Notation Functions
If \(f(x) = x^2 - 3x + 2\), what is the value of \(f(f(1))\)?
INSIDE-OUT rule: evaluate inner function first, then plug result into outer.
inner β†’ result β†’ outer
\(f(1) = 1 - 3 + 2 = 0\)
\(f(f(1)) = f(0) = 0 - 0 + 2 = 2\) βœ… Answer: B
5 Vertex of a Parabola Functions
The parabola \(y = -2(x-3)^2 + 8\) has its vertex at which point? What is the maximum value of \(y\)?
Vertex form: \(y = a(x-h)^2 + k\) β†’ vertex is \((h, k)\)
If \(a < 0\) β†’ opens DOWN β†’ vertex is MAXIMUM
negative a = max, positive a = min
Vertex form \(y = a(x-h)^2+k\): here \(h=3, k=8\), so vertex is \((3, 8)\).
Since \(a = -2 < 0\), parabola opens downward β†’ maximum = 8 βœ… Answer: B
6 Exponential Growth Functions
A bacteria population doubles every 3 hours. If it starts at 500, which expression gives the population after \(t\) hours?
Exponential model: \(P = P_0 \cdot b^{t/T}\)
Pβ‚€ = start, b = growth factor, T = period
Doubles (Γ—2) every 3 hours β†’ exponent is \(t/3\).
\(P = 500 \cdot 2^{t/3}\) βœ… Answer: B
πŸ“ GEOMETRY & TRIGONOMETRY
7 SOH-CAH-TOA Trig
In a right triangle, if \(\sin\theta = \dfrac{3}{5}\), what is \(\cos\theta\)?
Pythagorean identity: \(\sin^2\theta + \cos^2\theta = 1\)
SOH: sin = Opp/Hyp, CAH: cos = Adj/Hyp
If sin = 3/5 β†’ sides 3, 4, 5 triangle!
If \(\sin\theta = 3/5\) β†’ opposite = 3, hypotenuse = 5 β†’ adjacent = 4 (3-4-5 triangle)
\(\cos\theta = \dfrac{4}{5}\) βœ… Answer: C
8 Circle β€” Arc Length Geometry
A circle has radius 6. An arc subtends a central angle of \(120Β°\). What is the arc length?
Arc length \(= r\theta\) (ΞΈ in radians) OR \(= \dfrac{\thetaΒ°}{360Β°} \times 2\pi r\)
120Β° = \(\frac{2\pi}{3}\) radians
\(s = \dfrac{120}{360} \times 2\pi(6) = \dfrac{1}{3} \times 12\pi = 4\pi\) βœ… Answer: A
9 Volume of Solids Geometry
A cone has height 9 and base radius 4. A cylinder has the same base radius and the same height. What is the ratio \(\dfrac{\text{Volume of cone}}{\text{Volume of cylinder}}\)?
Cone = \(\frac{1}{3}\pi r^2 h\) β†’ Cylinder = \(\pi r^2 h\)
Cone is ALWAYS \(\frac{1}{3}\) of same-size cylinder!
Cone volume = \(\frac{1}{3}\pi r^2 h\), Cylinder = \(\pi r^2 h\)
Ratio = \(\dfrac{\frac{1}{3}\pi r^2 h}{\pi r^2 h} = \dfrac{1}{3}\) βœ… Answer: C
πŸ“‰ DATA & STATISTICS
10 Mean, Median, Mode Statistics
The data set is: \(\{4, 7, 7, 9, 13, 15, 15, 15, 20\}\). Which statement is TRUE?
Mean = sum Γ· count | Median = middle value | Mode = most frequent
Odd count β†’ middle one. Even count β†’ avg of two middles.
Sum = 4+7+7+9+13+15+15+15+20 = 105, n=9 β†’ Mean = 105/9 β‰ˆ 11.67
Median (5th value) = 13
Mode = 15
Mode(15) > Mean(11.67) > Median(13)? No β€” Mode(15) > Median(13) > Mean(11.67)
So Mode > Mean βœ… Answer: C
11 Standard Deviation Statistics
Two classes took the same test. Class A scores: \(\{70,70,70,70,70\}\). Class B scores: \(\{50,60,70,80,90\}\). Which is true about their standard deviations?
Standard deviation = spread from mean.
All same values β†’ SD = 0. More spread β†’ higher SD.
Class A: all values = 70 β†’ no spread β†’ SD = 0
Class B: values spread from 50 to 90 β†’ SD > 0
∴ Class B has higher SD βœ… Answer: C
12 Scatterplot & Lines of Best Fit Statistics
A line of best fit has equation \(\hat{y} = 1.5x + 4\). What does the slope 1.5 represent in context?
Slope = rate of change: for every +1 in x, y increases by slope value.
y-intercept = value when x = 0
In \(\hat{y} = mx + b\), slope \(m\) = rate of change.
1.5 means: for every 1 unit ↑ in x, predicted y ↑ by 1.5. βœ… Answer: B
πŸ”’ ADVANCED ALGEBRA
13 Polynomial Division Algebra
If \(p(x) = x^3 - 2x^2 - 5x + 6\) and \(p(2) = 0\), which of the following is a factor of \(p(x)\)?
Factor Theorem: if \(p(a) = 0\), then \((x - a)\) is a factor.
zero β†’ factor: p(a)=0 ↔ (xβˆ’a) divides p(x)
By the Factor Theorem: \(p(2) = 0\) β†’ \((x - 2)\) is a factor. βœ… Answer: B
14 Rational Expressions ⚠️ Algebra
Simplify: \(\dfrac{x^2 - 9}{x^2 - x - 6}\)
Factor BOTH top & bottom fully, then CANCEL common factors.
difference of squares: \(a^2-b^2 = (a+b)(a-b)\)
⚠️ State restrictions: denominator β‰  0!
Numerator: \(x^2-9 = (x+3)(x-3)\)
Denominator: \(x^2-x-6 = (x-3)(x+2)\)
Cancel \((x-3)\): \(\dfrac{(x+3)\cancel{(x-3)}}{\cancel{(x-3)}(x+2)} = \dfrac{x+3}{x+2}\) βœ… Answer: A
15 Absolute Value Inequality Algebra
Solve: \(|2x - 3| \leq 7\). Which interval represents all solutions?
\(|A| \leq k\) β†’ \(-k \leq A \leq k\) (AND, single interval)
\(|A| \geq k\) β†’ \(A \leq -k\) OR \(A \geq k\) (OR, two pieces)
≀ β†’ AND (sandwich) | β‰₯ β†’ OR (split)
\(-7 \leq 2x-3 \leq 7\)
Add 3: \(-4 \leq 2x \leq 10\)
Divide by 2: \(-2 \leq x \leq 5\) βœ… Answer: B
🌊 TRIGONOMETRY & COMPLEX NUMBERS
16 Radian Conversion ⚠️ Trig
Convert \(270Β°\) to radians. Which of the following is correct?
\(180Β° = \pi\) radians (ALWAYS remember this!)
To convert: degrees Γ— \(\dfrac{\pi}{180}\)
Key angles: 90Β°=Ο€/2, 180Β°=Ο€, 270Β°=3Ο€/2, 360Β°=2Ο€
\(270Β° \times \dfrac{\pi}{180Β°} = \dfrac{270\pi}{180} = \dfrac{3\pi}{2}\) βœ… Answer: C
17 Complex Numbers ⚠️ Complex
What is \((3 + 2i)(1 - 4i)\)?
Multiply like FOIL. Remember: \(i^2 = -1\)
iΒ² = βˆ’1 (this is the ONLY rule you need!)
FOIL: \(3(1) + 3(-4i) + 2i(1) + 2i(-4i)\)
\(= 3 - 12i + 2i - 8i^2\)
\(= 3 - 10i - 8(-1) = 3 - 10i + 8 = 11 - 10i\) βœ… Answer: A
18 Logarithms Functions
If \(\log_2 x = 5\), what is the value of \(x\)?
\(\log_b x = y\) ↔ \(b^y = x\)
log β†’ exponential: "base to the power = argument"
\(\log_2 x = 5\) β†’ \(x = 2^5 = 32\) βœ… Answer: C
🎯 PROBABILITY & WORD PROBLEMS
19 Probability ⚠️ Probability
A bag has 4 red, 3 blue, and 5 green marbles. Two marbles are drawn without replacement. What is the probability that both are red?
Without replacement β†’ denominator decreases by 1 each draw!
\(P(A \text{ and } B) = P(A) \times P(B|A)\)
no replace β†’ multiply shrinking fractions
Total = 4+3+5 = 12 marbles
P(1st red) = \(\dfrac{4}{12}\), P(2nd red | 1st red) = \(\dfrac{3}{11}\)
\(P = \dfrac{4}{12} \times \dfrac{3}{11} = \dfrac{12}{132} = \dfrac{1}{11}\) βœ… Answer: A
20 πŸ† FINAL BOSS: Word Problem Mixed
A store sells two types of tickets: adult at $12 and child at $7. In one day, 80 tickets were sold for a total of $730. How many adult tickets were sold?
Set up TWO equations: one for count, one for money.
count equation + money equation β†’ solve system
Let \(a\) = adult, \(c\) = child: \(a+c=80\) and \(12a+7c=730\)
\(a + c = 80\) β†’ \(c = 80 - a\)
\(12a + 7(80-a) = 730\)
\(12a + 560 - 7a = 730\)
\(5a = 170\)
\(a = 34\) adult tickets βœ… Answer: C
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