Moderate -0.8 This is a straightforward application of the remainder theorem requiring direct substitution f(2)=42 to find k, followed by finding one root. It's more routine than average A-level questions since it explicitly tells students to use the remainder theorem and guides them through both steps with minimal problem-solving required.
8 You are given that \(\mathrm { f } ( x ) = 4 x ^ { 3 } + k x + 6\), where \(k\) is a constant. When \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\), the remainder is 42 . Use the remainder theorem to find the value of \(k\). Hence find a root of \(\mathrm { f } ( x ) = 0\).
2 substituted in \(f(x)\) or \(f(2) = 42\) seen; or correct division of \(4x^3 + kx + 6\) by \(x-2\) as far as obtaining \(4x^2 + 8x + (k+16)\) oe [may have \(4x^2 + 8x + 18\)]
\(4 \times 2^3 + 2k + 6 = 42\)
M1
Or \(6 + 2(k+16) = 42\) oe; or finding (usually after division) that the constant term is 36 and then working with the \(x\) term to find \(k\) eg \(kx + 16x = 18x\)
\(k = 2\)
A1
\([x =]\ -1\)
A1
As their answer, not just a trial; accept with no working since it can be found by inspection; A0 for just \(f(-1) = 0\) with no further statement; A0 if confusion between roots and factors in final statement eg '\(x+1\) is a root', even if they also state \(x = -1\)
[4]
## Question 8:
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use of $f(2)$ | M1 | 2 substituted in $f(x)$ or $f(2) = 42$ seen; or correct division of $4x^3 + kx + 6$ by $x-2$ as far as obtaining $4x^2 + 8x + (k+16)$ oe [may have $4x^2 + 8x + 18$] |
| $4 \times 2^3 + 2k + 6 = 42$ | M1 | Or $6 + 2(k+16) = 42$ oe; or finding (usually after division) that the constant term is 36 and then working with the $x$ term to find $k$ eg $kx + 16x = 18x$ |
| $k = 2$ | A1 | |
| $[x =]\ -1$ | A1 | As their answer, not just a trial; accept with no working since it can be found by inspection; A0 for just $f(-1) = 0$ with no further statement; A0 if confusion between roots and factors in final statement eg '$x+1$ is a root', even if they also state $x = -1$ |
| | **[4]** | |
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8 You are given that $\mathrm { f } ( x ) = 4 x ^ { 3 } + k x + 6$, where $k$ is a constant. When $\mathrm { f } ( x )$ is divided by $( x - 2 )$, the remainder is 42 . Use the remainder theorem to find the value of $k$. Hence find a root of $\mathrm { f } ( x ) = 0$.
\hfill \mbox{\textit{OCR MEI C1 2014 Q8 [4]}}