Year: 2025
Paper: 3
Question Number: 6
Course: UFM Pure
Section: Roots of polynomials
The majority of candidates focused solely on the pure questions, with questions 1, 2 and 8 the most popular. The statistics questions were more popular than the mechanics questions in this exam series. Candidates who did well on this paper generally: were careful to explain and justify the steps in their arguments, explaining what they had done rather than expecting the examiner to infer what had been done from disjointed groups of calculations; paid close attention to what was required by the questions; made fewer unnecessary mistakes with calculations; thought carefully about how to present rigorous arguments involving trig functions and their inverse functions, especially in relation to domain considerations; understood that questions set on the STEP papers require sufficient justification to earn full credit; knew the difference between 'positive' and 'non-negative'; attempted all parts of a question, picking up marks for later parts even when they had not necessarily attempted or completed previous parts. Candidates who did less well on this paper generally: did not pay attention to 'Hence' instructions: this means that you must use the previous part; presented explanations that were not precise enough (e.g. in Question 3 describing the transformations but not in the context of the graphs or in Question 8 not explaining use of trigonometric relationships sufficiently well); made additional assumptions, e.g. that a function was differentiable when this had not been given; tried to present if and only if arguments in a single argument when dealing with each direction separately would have been more appropriate and safer (note that this is not always the case; in general candidates need to consider what is the most appropriate presentation of an if and only if argument); tried to carry out too many steps in one go, resulting in them not justifying the key steps sufficiently; did not take sufficient care with graphs/curve sketching.
Difficulty Rating: 1500.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
\begin{questionparts}
\item Let $a$, $b$ and $c$ be three non-zero complex numbers with the properties $a + b + c = 0$ and $a^2 + b^2 + c^2 = 0$.
Show that $a$, $b$ and $c$ cannot all be real.
Show further that $a$, $b$ and $c$ all have the same modulus.
\item Show that it is not possible to find three non-zero complex numbers $a$, $b$ and $c$ with the properties $a + b + c = 0$ and $a^3 + b^3 + c^3 = 0$.
\item Show that if any four non-zero complex numbers $a$, $b$, $c$ and $d$ have the properties $a + b + c + d = 0$ and $a^3 + b^3 + c^3 + d^3 = 0$, then at least two of them must have the same modulus.
\item Show, by taking $c = 1$, $d = -2$ and $e = 3$ that it is possible to find five real numbers $a$, $b$, $c$, $d$ and $e$ with distinct magnitudes and with the properties $a + b + c + d + e = 0$ and $a^3 + b^3 + c^3 + d^3 + e^3 = 0$.
\end{questionparts}\begin{questionparts}
\item If $a,b,c$ were all real then $a^2+b^2+c^2 = 0 \Rightarrow a,b,c = 0$ but they are non-zero. Therefore they cannot all be real.
Since $(a+b+c)^2 = 0$ we must have $ab+bc+ca = 0$. Therefore $a,b,c$ must satisfy $x^3 -abc = 0 \Rightarrow$ they all have the same modulus, since they are all cube roots of the same number.
\item Notice that $a^3+b^3+c^3 - 3abc = (a+b+c)(a^2+b^2+c^2 - ab-bc-ca) \Rightarrow abc = 0$ but therefore they cannot all be non-zero.
\item Suppose $a+b+c+d = 0$ then note that $\displaystyle a^2+b^2+c^2+d^2 = (a+b+c+d)^2 - 2\sum_{sym} ab$ and
$\displaystyle a^3+b^3+c^3+d^3 = (a+b+c+d)^3 - 3(a+b+c+d)(ab+ac+ad+bc+bd+cd) + 3(abc+abd+acd+bcd) \Rightarrow abc+abd+acd+bcd = 0$. Therefore $a,b,c,d$ are roots of a polynomial of the form $x^4 -kx^2 + l = 0$, but this means they must come in pairs with the same modulus.
\item Suppose $c = 1, d = -2, e = 3$ so $c+d+e = 2$ and $c^3 + d^3 + e^3 = 1 - 8 + 27 = 20$, so we need to find $a,b$ satisfying $a+b = -2, a^2+b^2 = -20$, ie $4 = (a+b)^2 = -20 + 2ab \Rightarrow ab = 12$, so we need the roots of $x^2 +2x + 12= 0$ which clearly have different modulus.
\end{questionparts}
Question 6 proved to be quite challenging for many candidates, with a significant number scoring fewer than 5 marks and only attempting part (i) or part (i) and part (ii). In part (i), the first mark was easily earned for strict inequalities or stating the only solution is the zero solution but was not earned if it was stated that 'the square of any real is positive' rather than any non-zero real. For the rest of the question, it was very common for candidates to attempt to consider each of a, b, c in the form x + yi, and then substitute in to obtain four equations in six variables. Those that tried this invariably made no progress. While it is possible to answer the question using real and imaginary parts, it requires far more work and so no credit was awarded for just writing down these four equations. Those who left the algebra in terms of a, b, c or used the roots of a quadratic tended to answer this part well. Part (ii) also saw attempts to split a, b, c into real and imaginary parts. This saw no further progress, or credit. The most common way that this question was answered was by writing down an identity relating the sum of cubes to the cube of the sum. This identity could be written in several equivalent forms, and saw many errors in the coefficients and signs, for which candidates were penalised accuracy marks. Careful thought about presentation was required before commencing the algebraic manipulation to part (iii) to avoid introducing sign and arithmetic errors. Complicated identities were common and often contained errors. Establishing that abc + bcd + acd + abd = 0 was a common approach and led to considering the roots of a quartic. Part (iv) was generally well answered. Most candidates that attempted it were able to identify a quadratic and solve it. Several candidates that could not solve some of parts (i), (ii), (iii) skipped straight to this part and picked up some marks. This is good general exam practice and STEP candidates should remember that subsequent parts of a question can often still be answered even if an early part seems challenging.