SAT Math mixed review practice: 4 mixed sets with answer keys

Mixed review 1: algebra, geometry, statistics and modelling

Mixed review · 15 questions · 15 mixed-topic questions spanning the whole SAT Math paper — a HL AA warm-up sheet before you sit a mock.

Practice Questions

Level 1: Easy (Foundational)

1. A car rental company charges a fixed base fee plus a daily rate. If renting a car for $3$ days costs $\$150$ and renting it for $5$ days costs $\$210$, determine the fixed base fee in dollars. (SPR)
2. A circular logo has an area of $36\pi$ square centimeters. Investigate the circumference of this logo. Which of the following represents the circumference in centimeters?
   A) $6\pi$
   B) $12\pi$
   C) $18\pi$
   D) $36\pi$
3. A population of bacteria initially consists of $800$ cells. If the population increases by $25\%$, what is the new population?
   A) $900$
   B) $1000$
   C) $1050$
   D) $1200$
4. The height of a projectile $h(t)$ at time $t$ is given by $h(t) = -5(t - 2)^2 + 20$. Determine the maximum height reached by the projectile.
   A) $2$
   B) $5$
   C) $15$
   D) $20$
5. A student has a weekly budget of strictly less than $\$40$. They spend $\$15$ on a textbook and want to buy notebooks that cost $\$4$ each. What is the maximum integer number of notebooks they can purchase? (SPR)

Level 2: Medium (Standard)

1. A wheelchair ramp rises $3$ meters vertically for every $4$ meters of horizontal distance. If the ramp must reach a vertical height of $6$ meters, show the calculation for the total length of the ramp's surface. What is this length in meters?
   A) $8$
   B) $10$
   C) $12$
   D) $14$
2. The mean of a dataset of $5$ integers is $12$. When a $6$th integer is added to the dataset, the new mean becomes $14$. Determine the value of the $6$th integer.
   A) $14$
   B) $20$
   C) $22$
   D) $24$
3. The polynomial $p(x) = x^3 - 2x^2 + kx + 6$ has a factor of $(x - 3)$. Explain how to use the Factor Theorem to find the value of $k$. What is $k$?
   A) $-5$
   B) $-3$
   C) $3$
   D) $5$
4. A bakery sells muffins for $\$2$ each and croissants for $\$3$ each. In one morning, they sold a total of $50$ items and made $\$120$. How many muffins did they sell?
   A) $20$
   B) $30$
   C) $40$
   D) $50$
5. Two gears are interconnected. Gear A has $12$ teeth and Gear B has $30$ teeth. If Gear A rotates at $60$ revolutions per minute (rpm), at what rate, in rpm, does Gear B rotate? (SPR)

Level 3: Hard (Advanced)

1. The mass of a radioactive substance is modelled by $M(t) = M_0 \left(\frac{1}{2}\right)^{\frac{t}{h}}$, where $t$ is time in days and $h$ is the half-life. If the mass decays to exactly $\frac{1}{16}$ of its initial amount in $20$ days, determine the half-life $h$ in days.
   A) $2$
   B) $4$
   C) $5$
   D) $10$
2. A solid right cylinder has a radius of $r$ and a height of $3r$. If the total surface area of the cylinder is exactly $32\pi$, investigate the volume of the cylinder. Which of the following is the volume?
   A) $8\pi$
   B) $16\pi$
   C) $24\pi$
   D) $48\pi$
3. A dataset of $n=5$ pairs $(x,y)$ has a mean $\bar{x} = 4$ and a mean $\bar{y} = 10$. The line of best fit passes through the point $(2, 14)$. Determine the exact value of the gradient $m$ of this regression line. (SPR)
4. The straight line $y = 2x + k$ intersects the parabola $y = x^2 - 4x + 5$ at exactly one point. What is the value of the constant $k$?
   A) $-4$
   B) $-1$
   C) $1$
   D) $4$
5. A regular square-based pyramid has a base side length of $6$ and a vertical height of $4$. Determine the exact length of the slant edge connecting a base corner to the top vertex.
   A) $5$
   B) $\sqrt{34}$
   C) $\sqrt{41}$
   D) $\sqrt{52}$

Mixed review 2: algebra, modelling, statistics and geometry

Mixed review · 12 questions · 12 questions spanning every SAT Math domain — the first HL AI mixed review sheet, matched to full-mock difficulty.

Worked Examples: SAT Strategy

Practice Questions

Level 1: Easy (Foundational)

1. Solve $2x + 7 = 21$.
   A) $5$
   B) $6$
   C) $7$
   D) $14$
2. A triangle has angles $50^\circ$ and $60^\circ$. Determine the exact third angle.
   A) $60^\circ$
   B) $70^\circ$
   C) $80^\circ$
   D) $90^\circ$
3. What is the mean of $\{4, 6, 8, 10\}$?
   A) $6$
   B) $7$
   C) $8$
   D) $28$
4. Determine $\sqrt{81}$. (SPR)

Level 2: Medium (Standard)

1. A cone has base radius $3$ and height $4$. Determine the exact volume.
   A) $4\pi$
   B) $9\pi$
   C) $12\pi$
   D) $36\pi$
2. A bag has $2$ red, $3$ blue, $5$ green marbles. Determine the exact probability of drawing a blue.
   A) $\tfrac{1}{10}$
   B) $\tfrac{3}{10}$
   C) $\tfrac{5}{10}$
   D) $\tfrac{3}{5}$
3. The exponential $y = 100 \cdot 0.8^{t}$ models decay. Determine the exact value of $y$ at $t = 2$.
   A) $64$
   B) $72$
   C) $80$
   D) $100$
4. A car depreciates $15\%$ each year from $\$20{,}000$. Determine the exact value, in dollars, after $2$ years. (SPR)

Level 3: Hard (Advanced)

1. For what $k$ does $x^2 - 6x + k = 0$ have exactly one real root?
   A) $-9$
   B) $3$
   C) $9$
   D) $36$
2. A trapezium has parallel sides $6$ and $10$ and height $4$. Determine the exact area.
   A) $24$
   B) $32$
   C) $40$
   D) $60$
3. The vectors $\mathbf{u} = (2, k)$ and $\mathbf{v} = (3, -4)$ are perpendicular. Determine the exact value of $k$.
   A) $\tfrac{3}{2}$
   B) $\tfrac{4}{3}$
   C) $\tfrac{3}{4}$
   D) $2$
4. A savings account of $\$1000$ earns $5\%$ compound interest annually. Determine, to the nearest whole dollar, the balance after $10$ years. (SPR)

Mixed review 3: full-domain sprint

Mixed review · 12 questions · A second 12-question mixed sheet spanning every SAT Math domain — a warm-up for full mock papers.

Practice Questions

Level 1: Easy (Foundational)

1. If $3x + 5 = 20$, determine the exact value of $x$.
   A) $3$
   B) $5$
   C) $\tfrac{15}{3}$
   D) $\tfrac{25}{3}$
2. A circle has radius $5$. Determine the exact area.
   A) $10\pi$
   B) $25\pi$
   C) $50\pi$
   D) $100\pi$
3. What is the median of $\{2, 5, 7, 10, 12\}$?
   A) $5$
   B) $7$
   C) $10$
   D) $36/5$
4. Determine $\sqrt{144}$. (SPR)

Level 2: Medium (Standard)

1. Solve $x^2 - 5x + 6 = 0$. Which is the sum of the roots?
   A) $5$
   B) $6$
   C) $-5$
   D) $-6$
2. A right triangle has legs $5$ and $12$. Determine the exact length of the hypotenuse.
   A) $13$
   B) $17$
   C) $\sqrt{60}$
   D) $\sqrt{119}$
3. A jar holds $6$ red, $3$ blue, $1$ green ball. Determine the exact probability of drawing a red ball.
   A) $\tfrac{1}{10}$
   B) $\tfrac{3}{10}$
   C) $\tfrac{6}{10}$
   D) $\tfrac{9}{10}$
4. The line through $(1,2)$ and $(4,11)$ has slope $m$. Determine the exact value of $m$. (SPR)

Level 3: Hard (Advanced)

1. For what value of $k$ does $y = x^2 - 4x + k$ touch the $x$-axis at exactly one point?
   A) $1$
   B) $2$
   C) $3$
   D) $4$
2. Vectors $\mathbf{u}=(3,4)$ and $\mathbf{v}=(a,-3)$ are perpendicular. Determine the exact value of $a$.
   A) $3$
   B) $4$
   C) $\tfrac{9}{4}$
   D) $\tfrac{4}{3}$
3. An investment of $\$1000$ grows by a compounded $6\%$ annually. Determine the exact balance after $2$ years.
   A) $\$1060$
   B) $\$1120$
   C) $\$1123.60$
   D) $\$1136$
4. Determine the exact value of $\sin(30^\circ) + \cos(60^\circ)$. (SPR)

Mixed review 4: applied modelling sprint

Mixed review · 12 questions · 12 more full-domain SAT questions with an HL AI applied-modelling slant — statistics, exponential decay, and geometric reasoning.

Practice Questions

Level 1: Easy (Foundational)

1. Simplify: $\tfrac{8x^2}{4x}$ for $x \ne 0$.
   A) $2x$
   B) $\tfrac{1}{2}x$
   C) $4x$
   D) $2x^2$
2. A rectangle has length $10$ and width $6$. Determine the exact perimeter.
   A) $16$
   B) $26$
   C) $32$
   D) $60$
3. The median of $\{1, 3, 3, 5, 7, 9, 11\}$ is:
   A) $3$
   B) $5$
   C) $7$
   D) $9$
4. Determine $2^{10}$. (SPR)

Level 2: Medium (Standard)

1. A car accelerates from $0$ to $30$ m/s in $12$ s. Determine, to $2$ s.f., the acceleration in m/s².
   A) $1.5$
   B) $2.0$
   C) $2.5$
   D) $3.0$
2. Two events $A, B$ have $P(A)=0.5$, $P(B)=0.4$, $P(A \cap B) = 0.2$. Determine $P(A \cup B)$.
   A) $0.6$
   B) $0.7$
   C) $0.8$
   D) $0.9$
3. If $f(x) = 3x^2 - 2$, determine $f(3)$.
   A) $16$
   B) $19$
   C) $25$
   D) $27$
4. A pyramid has square base side $6$ and vertical height $8$. Determine the exact volume. (SPR)

Level 3: Hard (Advanced)

1. A best-fit line has equation $y = 0.6x + 5$. Determine the predicted $y$ at $x = 25$.
   A) $17$
   B) $18$
   C) $20$
   D) $22$
2. A binomial random variable $X \sim B(20, 0.4)$. Determine the exact expected value.
   A) $4$
   B) $6$
   C) $8$
   D) $10$
3. A cylinder has volume $V = \pi r^2 h$. If $V = 100\pi$ and $r = 5$, determine the exact height.
   A) $2$
   B) $4$
   C) $5$
   D) $10$
4. A dataset has values $\{10, 12, 14, 16, 18\}$. Determine the exact standard deviation (population, $\sigma$). (SPR)