Geometry and Trigonometry: Lines, Angles, and Triangles

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) $\sqrt{52}$