SAT Math — Tricky 20

Question 01 Linear Equations ⚠ TRAP

💡 Memory NO SOLUTION: same slope, different y-intercept → parallel lines (never meet). INFINITE: identical lines.

If the system below has no solution, what is the value of $k$?

\[ \begin{cases} 2x + 4y = 8 \\ kx + 6y = 3 \end{cases} \]

📖 Explanation

No solution = parallel lines. Parallel means the ratio of x-coefficients equals the ratio of y-coefficients, but NOT the ratio of constants.

Equation 1: $2x+4y=8$ → coefficients $(2,4,8)$
Equation 2: $kx+6y=3$ → coefficients $(k,6,3)$

Set $\dfrac{k}{2} = \dfrac{6}{4}$ → $k = 3$.
Check constant ratio: $\dfrac{3}{8} \ne \dfrac{6}{4}$. ✓ Not equal, so no solution confirmed.

Question 02 Absolute Value ⚠ TRAP

💡 Memory ABS VALUE: |x| = a gives TWO answers: x = +a or x = −a. Always check BOTH in original equation!

How many solutions does $|2x - 3| = x + 1$ have?

📖 Explanation

Case 1: $2x-3 = x+1 \Rightarrow x=4$. Check: $|5|=5=4+1$ ✓

Case 2: $2x-3 = -(x+1) \Rightarrow 2x-3=-x-1 \Rightarrow 3x=2 \Rightarrow x=\tfrac{2}{3}$. Check: $|{-5/3}|=\tfrac{5}{3}=\tfrac{2}{3}+1$ ✓

Both valid → 2 solutions. Many students forget Case 2!

Question 03 Inequalities ⚠ TRAP

💡 Memory FLIP THE SIGN when multiplying or dividing by a NEGATIVE number in an inequality!

Which value of $x$ satisfies $-3x + 5 > 14$?

📖 Explanation

$-3x+5 > 14 \Rightarrow -3x > 9 \Rightarrow x < -3$ (sign flips when dividing by −3).

Only $x=-4$ satisfies $x < -3$.
Trap: students often forget to flip the inequality sign → pick $x=0$ or $x=2$ incorrectly.

Question 04 Quadratics ⚠ TRAP

💡 Memory DISCRIMINANT: b²−4ac. Positive=2 real roots, Zero=1 root, Negative=no real roots.

For what value of $c$ does $x^2 - 6x + c = 0$ have exactly one real solution?

📖 Explanation

One solution → discriminant = 0: $b^2 - 4ac = 0$.
Here $a=1, b=-6$: $(-6)^2 - 4(1)(c) = 0 \Rightarrow 36 = 4c \Rightarrow c=9$.

Check: $x^2-6x+9=(x-3)^2=0 \Rightarrow x=3$ (one solution ✓).

Question 05 Functions ⚠ TRAP

💡 Memory f(a+b) ≠ f(a)+f(b) in general! Always substitute the WHOLE expression inside.

If $f(x) = x^2 + 1$, what is $f(x+2)$?

📖 Explanation

Replace $x$ with $(x+2)$:
$f(x+2) = (x+2)^2 + 1 = x^2+4x+4+1 = x^2+4x+5$.

Common mistake: Writing $f(x)+f(2) = x^2+1+5 = x^2+6$. Wrong! You must substitute the entire $(x+2)$.

Question 06 Exponents & Radicals ⚠ TRAP

💡 Memory x^(a/b) = b-th root of x^a. Negative exponent = reciprocal: x^(−n) = 1/xⁿ.

Which expression is equal to $x^{-\frac{2}{3}}$ for all positive $x$?

📖 Explanation

$x^{-2/3} = \dfrac{1}{x^{2/3}} = \dfrac{1}{\sqrt[3]{x^2}}$.

The negative exponent makes it a reciprocal; the fractional exponent $\tfrac{2}{3}$ means "cube root of x squared".

Question 07 Polynomial Division

💡 Memory REMAINDER THEOREM: remainder of p(x)÷(x−a) = p(a). Just plug in!

When $p(x) = x^3 - 2x^2 + 4x - 1$ is divided by $(x - 2)$, the remainder is:

📖 Explanation

By the Remainder Theorem, plug in $x=2$:
$p(2) = 8 - 8 + 8 - 1 = 7$.

No long division needed! The answer is 7.

Question 08 Circles ⚠ TRAP

💡 Memory CIRCLE EQUATION: (x−h)²+(y−k)²=r². Center=(h,k), radius=r. Don't forget the NEGATIVE signs for h and k!

What is the center of the circle $(x+3)^2 + (y-5)^2 = 16$?

📖 Explanation

Standard form: $(x-h)^2+(y-k)^2=r^2$, center $=(h,k)$.

$(x+3)^2 = (x-(-3))^2$ → $h=-3$
$(y-5)^2$ → $k=5$

Center $=(-3,5)$. Trap: students read $(x+3)$ and write $h=3$. The sign flips!

Question 09 Trigonometry ⚠ TRAP

💡 Memory CO-FUNCTION: sin(θ) = cos(90°−θ). Always: sin²θ + cos²θ = 1.

In a right triangle, $\sin(\theta) = \dfrac{3}{5}$. What is $\cos(\theta)$?

📖 Explanation

Use $\sin^2\theta + \cos^2\theta = 1$:
$\left(\tfrac{3}{5}\right)^2 + \cos^2\theta = 1 \Rightarrow \tfrac{9}{25} + \cos^2\theta = 1 \Rightarrow \cos^2\theta = \tfrac{16}{25} \Rightarrow \cos\theta = \tfrac{4}{5}$.

Or use 3-4-5 right triangle: if opp=3, hyp=5, then adj=4.

Question 10 Similar Triangles ⚠ TRAP

💡 Memory SIMILAR TRIANGLES: AREA scales by k², PERIMETER by k, VOLUME by k³ (where k = ratio of sides).

Two similar triangles have corresponding sides in ratio $2:3$. If the area of the smaller triangle is $12$, what is the area of the larger?

📖 Explanation

Areas scale by the square of the side ratio: $\left(\tfrac{3}{2}\right)^2 = \tfrac{9}{4}$.
Larger area $= 12 \times \tfrac{9}{4} = 27$.

Trap: multiplying by $\tfrac{3}{2}$ gives 18 — wrong! Area needs the ratio squared.

Question 11 Mean / Median ⚠ TRAP

💡 Memory OUTLIER: pulls MEAN strongly, barely moves MEDIAN. Mean = sensitive, Median = resistant.

Data set: $\{2, 4, 4, 5, 6, 100\}$. Which is true?

📖 Explanation

Mean $= (2+4+4+5+6+100)/6 = 121/6 \approx 20.2$
Median: middle two values are 4 and 5 → median $= 4.5$

Mean (20.2) $>$ Median (4.5). The outlier 100 drags the mean up while barely moving the median.

Question 12 Percent Change ⚠ TRAP

💡 Memory % CHANGE = (New − Old) ÷ Old × 100. Always divide by the ORIGINAL (old) value, NOT the new one!

A price increases from $\$80$ to $\$100$. What is the percent increase?

📖 Explanation

$\dfrac{100-80}{80} \times 100 = \dfrac{20}{80} \times 100 = 25\%$

Trap: dividing by 100 (the new value) gives 20%. Always divide by the original.

Question 13 Scatterplot / Correlation

💡 Memory CORRELATION ≠ CAUSATION. r close to ±1 = strong. r close to 0 = weak. r is always between −1 and +1.

A study shows a strong positive correlation between shoe size and reading level in children ages 5–12. The best explanation is:

📖 Explanation

Both shoe size and reading level increase with age. Age is a lurking (confounding) variable causing both to rise simultaneously. Neither causes the other.

This is a classic SAT reasoning question: correlation doesn't imply causation.

Question 14 Rate / Work ⚠ TRAP

💡 Memory COMBINED RATE: 1/A + 1/B = 1/T. Add the individual rates (jobs per hour), not the times!

Pipe A fills a tank in 4 hours; Pipe B fills it in 6 hours. Together, how many hours to fill the tank?

📖 Explanation

Rate A $= \tfrac{1}{4}$, Rate B $= \tfrac{1}{6}$.
Combined: $\tfrac{1}{4}+\tfrac{1}{6} = \tfrac{3}{12}+\tfrac{2}{12}=\tfrac{5}{12}$ tanks/hour.
Time $= \tfrac{12}{5} = 2.4$ hours $= 2\tfrac{2}{5}$ hours.

Trap: averaging the times $(4+6)/2=5$ is always wrong for rate problems.

Question 15 Direct / Inverse Variation ⚠ TRAP

💡 Memory INVERSE VARIATION: xy = k (constant). As x↑, y↓. Don't confuse with direct variation y = kx.

$y$ varies inversely with $x$. When $x=4$, $y=9$. What is $y$ when $x=12$?

📖 Explanation

Inverse variation: $xy = k$. Find k: $4 \times 9 = 36$.
When $x=12$: $12 \cdot y = 36 \Rightarrow y=3$.

Trap: direct variation $y=kx$ would give $y=27$. Read "inversely" carefully!

Question 16 Exponential Growth ⚠ TRAP

💡 Memory EXPONENTIAL: P(t) = P₀ · bᵗ. Doubling? b=2. Half-life? b=½. Growth rate r%? b = (1+r/100).

A bacteria population doubles every 3 hours. Starting at 500, what is the population after 9 hours?

📖 Explanation

In 9 hours, there are $9 \div 3 = 3$ doubling periods.
Population $= 500 \times 2^3 = 500 \times 8 = 4000$.

Trap: multiplying by 3 instead of using $2^3$.

Question 17 Vertex Form ⚠ TRAP

💡 Memory VERTEX FORM: y = a(x−h)²+k. Vertex = (h, k). Watch sign of h! (x−h) means vertex is at +h.

A parabola has equation $y = 2(x-1)^2 - 3$. What is the minimum value of $y$?

📖 Explanation

Vertex form: $y=a(x-h)^2+k$. Vertex is at $(h,k)=(1,-3)$.
Since $a=2>0$, parabola opens upward → minimum at vertex.
Minimum value of $y = -3$.

Trap: students confuse vertex as $(-1,-3)$ or pick $k=1$.

Question 18 Rational Equations ⚠ TRAP

💡 Memory EXTRANEOUS SOLUTION: after solving, always check that the denominator ≠ 0. If it does, discard that answer!

How many valid solutions does $\dfrac{x}{x-2} = \dfrac{2}{x-2}$ have?

📖 Explanation

Multiply both sides by $(x-2)$: $x=2$.
But check: when $x=2$, denominator $= 2-2=0$. Undefined!

$x=2$ is an extraneous solution. There are no valid solutions.
Always check solutions in rational equations!

Question 19 Probability ⚠ TRAP

💡 Memory P(A or B) = P(A) + P(B) − P(A and B). Forgetting the overlap is the #1 mistake!

In a class of 30 students, 18 play soccer, 14 play basketball, and 6 play both. How many play neither?

📖 Explanation

Using inclusion-exclusion: $|S \cup B| = 18+14-6 = 26$ play at least one sport.
Neither: $30-26=4$.

Trap: forgetting to subtract the 6 who play both, giving $18+14=32 > 30$. That's impossible!

Question 20 Systems — Nonlinear ⚠ TRAP

💡 Memory LINE + PARABOLA: substitute the linear equation INTO the quadratic. The discriminant tells you how many intersections!

How many points of intersection does the system $y = x^2$ and $y = 2x - 1$ have?

📖 Explanation

Set equal: $x^2 = 2x-1 \Rightarrow x^2-2x+1=0 \Rightarrow (x-1)^2=0$.
Discriminant $= 4-4=0$ → exactly one solution: $x=1, y=1$.

The line is tangent to the parabola at $(1,1)$. One intersection point.

Tricky 20

Well done!