| Exam Board | Edexcel |
|---|---|
| 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 Pascal's triangle or the binomial theorem, followed by simple algebraic manipulation. Part (a) is standard recall, part (b) involves recognizing odd powers cancel, and part (c) is a basic quadratic equation. All steps are mechanical with no problem-solving insight required, making it easier than average but not trivial due to the multi-part structure. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(= 2^4 + 4(2^3)(x) + 6(2^2)(x^2) + 4(2)(x^3) + x^4\) | M1 A1 | |
| \(= 16 + 32x + 24x^2 + 8x^3 + x^4\) | B1 A1 | |
| (b) \((2 - x)^4 = 16 - 32x + 24x^2 - 8x^3 + x^4\) | M1 | |
| \((2 + x)^4 + (2 - x)^4 = 32 + 48x^2 + 2x^4\), \(A = 32, B = 48, C = 2\) | A1 | |
| (c) \(32 + 48x^2 + 2x^4 = 136\) | ||
| \(x^4 + 24x^2 - 52 = 0\) | M1 | |
| \((x^2 + 26)(x^2 - 2) = 0\) | A1 | |
| \(x^2 = -26\) (no real solutions) or \(2\) | ||
| \(x = \pm\sqrt{2}\) | A1 | (9) |
**(a)** $= 2^4 + 4(2^3)(x) + 6(2^2)(x^2) + 4(2)(x^3) + x^4$ | M1 A1 |
$= 16 + 32x + 24x^2 + 8x^3 + x^4$ | B1 A1 |
**(b)** $(2 - x)^4 = 16 - 32x + 24x^2 - 8x^3 + x^4$ | M1 |
$(2 + x)^4 + (2 - x)^4 = 32 + 48x^2 + 2x^4$, $A = 32, B = 48, C = 2$ | A1 |
**(c)** $32 + 48x^2 + 2x^4 = 136$ | |
$x^4 + 24x^2 - 52 = 0$ | M1 |
$(x^2 + 26)(x^2 - 2) = 0$ | A1 |
$x^2 = -26$ (no real solutions) or $2$ | |
$x = \pm\sqrt{2}$ | A1 | (9)\begin{enumerate}[label=(\alph*)]
\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{Edexcel C2 Q6 [9]}}