2021 Paper 2, Q20
The random variable $B$ is normally distributed with a mean of 0 and a standard deviation of 1.
Determine the probability that the quadratic equation $x^2+3x+2B=0$ has real roots.
Three marks, two sentences, and no mention of a discriminant, a graph or an inequality. The difficulty is in the collision between two topics that never meet in class.
One random number decides whether the parabola has real roots
$B$ is drawn from the standard normal, and it only moves the parabola up and down. Roots exist while the curve still reaches the axis, and it stops reaching the axis at $B=1.125$.
$B$ runs from −0.5 to 2.6 and back, pausing where the roots disappear
An algebra condition becomes a probability event
$b^2-4ac\ge 0$ with $a=1$, $b=3$, $c=2B$: the discriminant is $9-8B$. Nothing here is Unit 4 mathematics.
$9-8B\ge 0\ \Rightarrow\ B\le\frac98=1.125$. The condition on the equation has become a condition on a random variable.
$P(B\le 1.125)$ on the standard normal is one calculator entry: 0.8697.
The same B, read two ways
Drag $B$ and watch both pictures at once. The parabola loses its roots at the same moment the value of $B$ slides out of the shaded 87%.
What is random here?
QCAA marking guide · 3 marks
The quadratic has real roots when
$$b^2-4ac\ge 0$$
Marker: correctly identifies the need to use the discriminant. One of three marks for knowing which tool the phrase “real roots” names.
$$9-8B\ge 0\ \Rightarrow\ 9\ge 8B\ \Rightarrow\ \frac98\ge B$$
Marker: correctly determines the range of values for $B$. Note the inequality does not flip, because you are dividing by $+8$, but it does end up reversed in appearance, so write it as $B\le\frac98$ before using it.
$$P\left(B\le\tfrac98\right)=0.8697$$
About an 87% chance of real roots
Marker: determines the probability, using the standard normal distribution given in the stem. Equivalent decimals or percentages accepted, e.g. 86.97% or 0.87. FT mark allowed for earlier errors.
The answer is 0.8697, and the common errors either side of it
The complement, the chance of no real roots. Easy to produce by taking the wrong tail after a correct inequality.
This is what you get from $B\le 1.25$, by dividing 9 by 8 carelessly, or by reading the coefficient of $x$ as 2 instead of 3.
1.125 is just past one standard deviation, so the answer had to be a little under 0.84 + 0.05. If your figure came out near 0.5 or 0.99, the inequality is wrong.
This is the purest fusion question in the set, and the shortest. In two sentences, the QCAA put a Year 10 algebra fact and a Unit 4 distribution side by side, and the only hard part is realising they belong together. “Has real roots” is a condition on the discriminant. The discriminant contains the random variable, so “has real roots” is an event with a probability. Students who notice the quadratic tend to try to solve it. Students who notice the normal distribution tend to look for a mean and standard deviation to use. The mark is for seeing both at once, and I think this is one of the best three-mark questions the QCAA have written.
Question wording and marking-guide steps are from the 2021 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2021, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.