| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Sum/difference of two binomials simplification |
| Difficulty | Moderate -0.8 This is a straightforward binomial expansion question requiring routine application of the binomial theorem, basic algebraic manipulation to identify coefficients, and solving a simple equation. All parts follow standard C2 procedures with no problem-solving insight needed, making it easier than average but not trivial due to the multi-step nature and potential for arithmetic errors. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks |
|---|---|
| \(= 2^4 + 4(2^3)(x) + 6(2^2)(x^2) + 4(2)(x^3) + x^4 = 16 + 32x + 24x^2 + 8x^3 + x^4\) | M1 A1, B1 A1 |
| Answer | Marks |
|---|---|
| \((2-x)^4 = 16 - 32x + 24x^2 - 8x^3 + x^4\) | M1 |
| \((2+x)^4 + (2-x)^4 = 32 + 48x^2 + 2x^4\), \(A = 32\), \(B = 48\), \(C = 2\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(32 + 48x^2 + 2x^4 = 136\) | M1 | |
| \(x^4 + 24x^2 - 52 = 0\) | A1 | |
| \((x^2 + 26)(x^2 - 2) = 0\) | M1 | |
| \(x^2 = -26\) (no real solutions) or \(2\) | A1 | |
| \(x = \pm\sqrt{2}\) | A1 | (9) |
## Part (i)
$= 2^4 + 4(2^3)(x) + 6(2^2)(x^2) + 4(2)(x^3) + x^4 = 16 + 32x + 24x^2 + 8x^3 + x^4$ | M1 A1, B1 A1 |
## Part (ii)
$(2-x)^4 = 16 - 32x + 24x^2 - 8x^3 + x^4$ | M1 |
$(2+x)^4 + (2-x)^4 = 32 + 48x^2 + 2x^4$, $A = 32$, $B = 48$, $C = 2$ | A1 |
## Part (iii)
$32 + 48x^2 + 2x^4 = 136$ | M1 |
$x^4 + 24x^2 - 52 = 0$ | A1 |
$(x^2 + 26)(x^2 - 2) = 0$ | M1 |
$x^2 = -26$ (no real solutions) or $2$ | A1 |
$x = \pm\sqrt{2}$ | A1 | **(9)**
---\begin{enumerate}[label=(\roman*)]
\item Expand $(2 + x)^4$ in ascending powers of $x$, simplifying each coefficient. [4]
\item Find the integers $A$, $B$ and $C$ such that
$$(2 + x)^4 + (2 - x)^4 = A + Bx^2 + Cx^4.$$ [2]
\item Find the real values of $x$ for which
$$(2 + x)^4 + (2 - x)^4 = 136.$$ [3]
\end{enumerate}
\hfill \mbox{\textit{OCR C2 Q7 [9]}}