Grade 12 Math practice questions — The Exam Room

Rates of Change

L1
Lesson 1
Reading the rate
which number is a rate, which is an amount
Grade 12 mathematics is, more than anything, the study of CHANGE — and every measure of change in this course is a quotient.
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L2
Lesson 2
Slope is a rate of change
the secant between two readings, computed
Mark two readings of a changing quantity, join them with a straight line, and that line is the secant. Its slope is the average rate of change over the interval between them — same number, three names, depending on who is in the room.
The formula in this lesson
$m = \dfrac{y_2 - y_1}{x_2 - x_1}$
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L3
Lesson 3
Average velocity
The trapezoid under the velocity line
Whichever of the four the question leaves blank is the one you solve for. Where it comes from is worth more than the formula.
The formula in this lesson
$d = \dfrac{v_0 + v}{2}\,t$
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L4
Lesson 4
Velocity as the derivative
The rate of the rate
Velocity is the rate of change of position.
The formula in this lesson
$v = v_0 + at$
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L5
Lesson 5
The position function
Where the half comes from
Notice which letter is absent: the final velocity. This is the relation for when you know how the motion BEGAN and how hard it was pushed, and want to know how far it got.
The formula in this lesson
$d = v_0 t + \tfrac{1}{2}at^2$
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L6
Lesson 6
The no-time shortcut
When the clock is missing
Sometimes nobody timed anything. You have a starting speed, a finishing speed, an acceleration and a distance — and not one second between them.
The formula in this lesson
$v^2 = v_0^2 + 2ad$
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Boss lesson
The Final Descent
the boss · no calculator · one runaway cart, line by line
No calculator, and every value is chosen to fit in your head. One cart, one ramp, and every answer feeds the next: the acceleration from the sensor, then the speed at the bottom, then the length of the ramp, then the total time of the run.
The formula in this lesson
$v = v_0 + at$
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Optimization & the Vertex

L8
Lesson 8
Reading the parabola
which coefficient is which, and where the optimum lives
In Grade 11 a parabola was a shape to describe. This year it is an objective function: something a business, an engineer or a physicist wants pushed as high or as low as it will go.
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L9
Lesson 9
Finding the turning point
The input that optimizes
That is worth knowing now, because it means this formula is not a trick from algebra class; it is calculus already solved for you.
The formula in this lesson
$h = -\dfrac{b}{2a}$
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L10
Lesson 10
The peak value
What the optimum is actually worth
Knowing WHEN a business peaks is half a mark.
The formula in this lesson
$k = c - \dfrac{b^2}{4a}$
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L11
Lesson 11
Roots and the discriminant
b² − 4ac decides how many answers exist
The vertex says where a model is best. The roots say where it is worth zero — for a profit model, the two break-even outputs, with profit living strictly between them.
The formulas in this lesson
$\Delta = b^2 - 4ac$ , $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
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L12
Lesson 12
What the roots whisper
sum and product of roots, without solving a thing
You can interrogate a quadratic's roots without ever finding them.
The formulas in this lesson
$S = -\dfrac{b}{a}$ , $P = \dfrac{c}{a}$ , $h = \dfrac{r_1 + r_2}{2}$ , $k = \left(\tfrac{b}{2}\right)^{2}$
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L13
Lesson 13
The break-even point
Optimization in business clothes
Not every optimization problem is a parabola. A firm's costs split in two, and the split is the whole idea. Fixed cost F — rent, insurance, salaries — is owed in dollars each period whether or not a single unit is made.
The formula in this lesson
$Q = \dfrac{F}{p - v}$
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Boss lesson
The Golden Vertex
the boss · no calculator · launch it, peak it, land it
Write each line down: one early slip rides the parabola all the way into the water.
The formula in this lesson
$h = -\dfrac{b}{2a}$
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Vectors

L15
Lesson 15
Reading the vector
magnitude, component or angle — which is which
A number alone is a scalar: 40 km/h, 12 kg, three coffees. A vector carries a second fact — which way — and we draw it as an arrow.
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L16
Lesson 16
How long is the arrow
|v| = √(vₓ² + v_y²), asked every way
Walk 3 km east, then 4 km north.
The formula in this lesson
$|\vec{v}| = \sqrt{v_x^2 + v_y^2}$
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L17
Lesson 17
Casting the shadows
Resolve the vector
Last lesson built the arrow from its shadows.
The formulas in this lesson
$v_y = |\vec{v}|\sin\theta$ , $v_x = |\vec{v}|\cos\theta$
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L18
Lesson 18
The direction angle
Which way it points
Here is the trap that costs whole marks.
The formulas in this lesson
$\theta = \tan^{-1}\left(\dfrac{v_y}{v_x}\right)$ , $\tan\theta = \dfrac{o}{a}$
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L19
Lesson 19
Two forces, one resultant
Adding at an angle
Two vectors add head to tail: slide the second one's tail onto the first one's head, and the resultant is the arrow from the very start to the very end.
The formula in this lesson
$R = \sqrt{A^2 + B^2 + 2AB\cos\theta}$
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L20
Lesson 20
The dot product
a·b two ways — components, or lengths and the angle
Two vectors, and the answer is a plain number with no direction at all — which is why it is also called the scalar product.
The formulas in this lesson
$\vec{a}\cdot\vec{b} = a_x b_x + a_y b_y$ , $\vec{a}\cdot\vec{b} = |\vec{a}|\,|\vec{b}|\cos\theta$
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L21
Lesson 21
The angle between
Rearranged for the angle
Three ingredients, and this lesson makes you cook all three: the dot product on top, and BOTH magnitudes multiplied underneath.
The formula in this lesson
$\theta = \cos^{-1}\left(\dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}|\,|\vec{b}|}\right)$
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L22
Lesson 22
The cross product and area
The parallelogram it sweeps
The dot product asks how much two arrows AGREE.
The formulas in this lesson
$|\vec{a}\times\vec{b}| = |\vec{a}|\,|\vec{b}|\sin\theta$ , $A = \left|a_x b_y - a_y b_x\right|$ , $A = \tfrac{1}{2}\left|a_x b_y - a_y b_x\right|$
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L23
Lesson 23
Work is a dot product
Only the parallel part pushes
Push a sled with a rope angled up at the shoulder and part of your pull lifts rather than drags.
The formula in this lesson
$W = F_x d_x + F_y d_y$
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Boss lesson
The River Crossing
the boss · no calculator · current, heading and the true track
No calculator, and every value is pinned so it fits in your head.
The formula in this lesson
$|\vec{v}| = \sqrt{v_x^2 + v_y^2}$
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Counting & Probability

L25
Lesson 25
Reading the odds
probability, odds or a count — which is which
Nothing in this chapter carries a unit, so the usual unit check has nothing to grab. Its replacement is the range check, and it is just as absolute.
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L26
Lesson 26
Favourable over total
P = f / n, and the other side of it
Letter by letter: f is the number of favourable outcomes, the ones that count as a success; n is the number of all equally likely outcomes, winners included; and P is the resulting chance, a naked ratio with no unit attached.
The formulas in this lesson
$P = \dfrac{f}{n}$ , $P(A^{c}) = 1 - P(A)$
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L27
Lesson 27
Either, or, or both
the addition rules · subtract the overlap once
Two events, and the word or.
The formulas in this lesson
$P(A \cup B) = P(A) + P(B)$ , $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
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L28
Lesson 28
And then
multiplication rules · with replacement, without, and given
Where or adds, and multiplies. The question is only ever what the second factor should be.
The formulas in this lesson
$P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}$ , $P(A \cap B) = P(A) \, P(B)$ , $P(A \cap B) = P(A) \, P(B \mid A)$
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L29
Lesson 29
Counting the ways
factorials, permutations, combinations — order or not
Probability runs on counting, and counting runs on one question: does order matter? Answer that and the formula picks itself.
The formulas in this lesson
${}_{n}P_{r} = \dfrac{n!}{(n-r)!}$ , ${}_{n}C_{r} = \dfrac{n!}{r! \, (n-r)!}$ , $n! = n \times (n-1) \times \cdots \times 2 \times 1$
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L30
Lesson 30
The binomial distribution
exactly k successes in n tries
Repeat the same trial n times, independently, each succeeding with the same chance p, and count the successes. That is a binomial experiment, and it is the most common distribution on any exam paper.
The formulas in this lesson
$\mu = np$ , $P(X = k) = \binom{n}{k} p^{k} (1-p)^{\,n-k}$ , $\sigma^{2} = np(1-p)$
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L31
Lesson 31
At least one, and what it's worth
one minus none · the first success · the price of a bet
“At least one” is the phrase that separates students who compute from students who think.
The formulas in this lesson
$P = 1 - (1-p)^{n}$ , $P(X = k) = (1-p)^{\,k-1} p$ , $E = pW - (1-p)L$
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Boss lesson
The Full House
the boss · no calculator · count it, chance it, price it
One table, one long problem, and every answer feeds the next: how many plays exist, then the chance one play wins, then the chance the evening produces at least one win, then what a single ticket is actually worth.
The formulas in this lesson
${}_{n}C_{r} = \dfrac{n!}{r! \, (n-r)!}$ , ${}_{n}P_{r} = \dfrac{n!}{(n-r)!}$
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Statistics & Data

L33
Lesson 33
Reading the data
centre, spread or position — which is which
A data set is a crowd, and there are only three useful things to ask about a crowd.
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L34
Lesson 34
Measuring the spread
variance, standard deviation, range and IQR
That squared unit is the tell, and it is the only tell you need. Which is why the headline slip of this lesson is handing in a variance where a standard deviation was asked for.
The formulas in this lesson
$\sigma = \sqrt{\sigma^2}$ , $\sigma^2 = \sigma \cdot \sigma$ , $R = x_{\max} - x_{\min}$ , $IQR = Q_3 - Q_1$
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L35
Lesson 35
How unusual is it
z = (x − μ)/σ, asked every way
Here is the move that makes statistics portable.
The formula in this lesson
$z = \dfrac{x - \mu}{\sigma}$
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L36
Lesson 36
Fences and outliers
Q₁ − 1.5·IQR and Q₃ + 1.5·IQR — who gets flagged
An outlier is a value far enough from the rest that it deserves a second look — not automatically an error, and never automatically deleted.
The formulas in this lesson
$LF = Q_1 - 1.5\,IQR$ , $UF = Q_3 + 1.5\,IQR$
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L37
Lesson 37
The line of best fit
slope from r, intercept through the means, then predict
A least-squares line is fitted, not guessed, and it arrives in three moves.
The formulas in this lesson
$b = r\,\dfrac{s_y}{s_x}$ , $a = \bar{y} - b\bar{x}$ , $\hat{y} = a + bx$
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L38
Lesson 38
How good is the fit
R², percent error, and the honest average
Fitting a line is easy; grading it is the skill.
The formulas in this lesson
$R^2 = r^2$ , $PE = \dfrac{|x_{\text{meas}} - x_{\text{acc}}|}{x_{\text{acc}}}$ , $\bar{x} = \dfrac{n_1\bar{x}_1 + n_2\bar{x}_2}{n_1 + n_2}$
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Boss lesson
The Data Defence
the boss · no calculator · one dataset, interrogated line by line
Every variance is a perfect square, every z lands on a half, and every correlation squares in your head — the values were chosen so that the arithmetic is never the difficulty.
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Financial Math

L40
Lesson 40
Percent change and simple interest
Money's first formula
Every formula in this chapter is built on one idea: a rate is a ratio, and a ratio has no units.
The formulas in this lesson
$c = \dfrac{x_{\text{new}} - x_{\text{old}}}{x_{\text{old}}}$ , $I = P\,r\,t$
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L41
Lesson 41
Compound growth
Interest on the interest
Chapter one of this course taught rates of change. Here is the most consequential one in your adult life: money whose growth rate is proportional to the balance itself.
The formulas in this lesson
$A = P\left(1 + \dfrac{r}{n}\right)^{nt}$ , $A = A_0 (1 + r)^{t}$
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L42
Lesson 42
Money travels in time
What a future amount is worth today
Compounding carries money forward.
The formula in this lesson
$\mathit{PV} = \dfrac{\mathit{FV}}{(1 + r)^{t}}$
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L43
Lesson 43
How long to double
the rule of 72, and solving for the exponent honestly
Doubling asks a different kind of question: not how much, but how long. The unknown moves into the exponent, and the only tool that reaches it is a logarithm.
The formulas in this lesson
$n \approx \dfrac{72}{R}$ , $t = \dfrac{\ln(A / P)}{\ln(1 + r)}$ , $N = N_0 \cdot 2^{t/T}$
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L44
Lesson 44
The annuity
regular deposits, compounding as they land
Nobody saves in one lump. They save $200 a month, and every instalment starts earning the day it lands — so the first deposit compounds for the whole term and the last one compounds for no time at all.
The formula in this lesson
$\mathit{FV} = D\,\dfrac{(1+i)^n - 1}{i}$
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L45
Lesson 45
Loans and honest rates
the payment formula, total interest, and the EAR
A loan is an annuity pointed the other way: you hold the balance, and equal payments grind it to exactly zero on the last one.
The formulas in this lesson
$M = \dfrac{P\,i}{1 - (1+i)^{-n}}$ , $\mathit{EAR} = \left(1 + \dfrac{r}{m}\right)^{m} - 1$
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Boss lesson
The Mortgage Final
the boss · no calculator · one household, one year, every formula
Every rate on this page is 6%, every number is chosen to fit in your head, and every answer feeds the next: the flat benchmark, then the compounding account, then the gap between them, then the savings plan, then the doubling check.
The formula in this lesson
$I = P\,r\,t$
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