| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Session | Specimen |
| Marks | 9 |
| Topic | Polynomial Division & Manipulation |
| Type | Division then Solve Polynomial Equation |
| Difficulty | Standard +0.3 Part (a) is a straightforward polynomial long division with clear structure. Part (b)(i) is trivial substitution to verify a factor. Part (b)(ii) requires recognizing the connection to part (i) and factoring, but once the (x-1) factor is removed, solving the resulting cubic is routine. This is a standard multi-part question testing polynomial techniques without requiring novel insight or extended problem-solving. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Attempt division obtaining at least \(2x^2 - kx\) | M1 | Complete the division obtaining a quadratic quotient and linear remainder |
| (b) (i) Substitute \(x = 1\) and make a correct statement invoking the Factor Theorem | B1 | 1 mark |
| (ii) Obtain \(3x^4 - 4x^3 + 1 = (x - 1)(3x^3 - x^2 - x - 1)\) | B1 | \(= (x - 1)^2(3x^2 + 2x + 1)\) |
(a) Attempt division obtaining at least $2x^2 - kx$ | M1 | Complete the division obtaining a quadratic quotient and linear remainder | M1(dep) | $Q = 2x^2 - 7x + 15$ | A1 | $R = -27x + 27$ | A1 | 4 marks
(b) (i) Substitute $x = 1$ and make a correct statement invoking the Factor Theorem | B1 | 1 mark
(ii) Obtain $3x^4 - 4x^3 + 1 = (x - 1)(3x^3 - x^2 - x - 1)$ | B1 | $= (x - 1)^2(3x^2 + 2x + 1)$ | B1 | Roots are $1, 1, \frac{-1 \pm \sqrt{2}i}{3}$ | M1A1 | 4 marks\begin{enumerate}[label=(\alph*)]
\item Divide the quartic $2x^4 - 5x^3 + 4x^2 + 2x - 3$ by the quadratic $x^2 + x - 2$, identifying the quotient and the remainder. [4]
\item \begin{enumerate}[label=(\roman*)]
\item Show that $(x - 1)$ is a factor of $nx^{n+1} - (n + 1)x^n + 1$, where $n$ is a positive integer. [1]
\item Hence, or otherwise, find all the roots of $3x^4 - 4x^3 + 1 = 0$. [4]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 Q5 [9]}}