SAT Math Premium Practice | 20 Hard Questions

01Linear Equations & Systems

📌 Must Memorize

Slope: $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ | Slope-intercept: $y = mx + b$
Parallel: same slope | Perpendicular: $m_1 \cdot m_2 = -1$

System of equations: solve by substitution or addition (elimination)

✏️ Example

$3x - 2y = 4$ and $x + 2y = 12$. Find $x+y$.
Add both: $4x = 16 \Rightarrow x = 4$. Sub: $4 + 2y = 12 \Rightarrow y = 4$. So $x + y = 8$.

Answer: $8$

02Quadratic Equations & Parabolas

📌 Must Memorize

Quadratic formula: $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
One real solution when discriminant $\Delta = b^2 - 4ac = 0$
Vertex: $x_v = -\dfrac{b}{2a}$, then find $y_v = f(x_v)$

Sum of roots $= -b/a$ | Product of roots $= c/a$

✏️ Example

$f(x) = 2x^2 - 8x + 6$. Minimum value?
$x_v = \dfrac{8}{4} = 2$, $\quad f(2) = 8 - 16 + 6 = -2$

Answer: $-2$

03Functions, Inverses & Exponentials

📌 Must Memorize

Inverse: swap $x$ and $y$, then solve for $y$.
Exponential growth: $A = A_0 \cdot r^{t/T}$ where $T$ = period
Composition: $(f \circ g)(x) = f(g(x))$

✏️ Example

Bacteria doubles every 3 hr, starts at 500. After $t$ hr: $500 \cdot 2^{t/3}$

At $t=3$: $500 \cdot 2^1 = 1000$ ✓

04Polynomials & Factoring

📌 Must Memorize

Remainder Theorem: $p(a)$ = remainder when dividing by $(x-a)$
$(a+b)^2 = a^2+2ab+b^2$ | $(a-b)(a+b) = a^2-b^2$
Synthetic division for quick polynomial division

✏️ Example

$p(x) = x^3 - 3x^2 + kx - 8$, remainder when divided by $(x-2)$ is $-6$.
$p(2) = 8-12+2k-8 = 2k-12 = -6 \Rightarrow k=3$

Answer: $k = 3$

05Ratios, Percents & Statistics

📌 Must Memorize

$\dfrac{x}{z} = \dfrac{x}{y} \cdot \dfrac{y}{z}$ (chain ratios)
Percent change $= \dfrac{\text{New} - \text{Old}}{\text{Old}} \times 100\%$
Mean $= \dfrac{\text{Sum}}{n}$ → Sum $= \text{Mean} \times n$

✏️ Example

30 students, mean = 72. Top 5 score 96 each. Remaining 25 average?
Total = $72 \times 30 = 2160$. Top 5 total = 480. Remaining = $1680 / 25 = 67.2$

Answer: $67.2$

06Geometry, Trig & Coordinate Geometry

📌 Must Memorize

Circle: $A = \pi r^2$, $(x-h)^2+(y-k)^2=r^2$
SOH-CAH-TOA | $\sin^2\theta + \cos^2\theta = 1$
Amplitude of $A\sin(Bx)$: $|A|$ | Period: $\dfrac{2\pi}{|B|}$
Midpoint: $\left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2}\right)$

✏️ Example

$4^x = 8^{x-1}$ → $2^{2x} = 2^{3x-3}$ → $2x = 3x-3$ → $x=3$. Check: $64=64$ ✓

Answer: $x = 3$

SAT Math
Master Class

Core Concepts & Key Formulas

Full Answer Key & Solutions

SAT MathMaster Class

Core Concepts & Key Formulas

Full Answer Key & Solutions

SAT Math
Master Class