| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2007 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Generalised Binomial Theorem |
| Type | Expand and state validity |
| Difficulty | Moderate -0.3 This is a straightforward application of the binomial expansion for negative powers with standard follow-through work. Part (i) requires routine substitution into the binomial formula and stating |x/2| < 1, while part (ii) involves simple multiplication of polynomials to find one coefficient. The question is slightly easier than average as it's a textbook-style exercise with clear steps and no conceptual challenges beyond remembering the formula. |
| Spec | 1.04c Extend binomial expansion: rational n, |x|<11.04d Binomial expansion validity: convergence conditions |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \((1 + \frac{x}{2})^{-2}\) | M1 | Clear indication of method of \(\geq 3\) terms |
| \(= 1 + (-2)(\frac{x}{2}) + \frac{-2}{2}3(\frac{x}{2})^2 + \frac{-2-3-4}{3!}(\frac{x}{2})^3\) | B1 | First two terms, not dependent on M1 |
| \(= 1 - x + \frac{3}{4}x^2 - \frac{1}{8}x^3\) | A1 | For both third and fourth terms |
| \((2 + x)^{-2} = \frac{1}{4}\)(their exp of \((1 + ax)^{-2}\)) mult out | √B1 | Correct: \(\frac{1}{4} - \frac{1}{4}x + \frac{3}{16}x^2 - \frac{1}{8}x^3\) |
| \( | x | < 2\) or \(-2 < x < 2\) (but not \( |
| (ii) If (i) is \(a + bx + cx^2 + dx^3\) evaluate \(b + d\) | M1 | Follow-through from \(D + d\) |
| \(-\frac{3}{8}\) \((\frac{1}{8})\) | √A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{dy}{dx} = \frac{dy/dt}{dx/dt}\) | M1 | |
| \(= \frac{-4\sin 2t}{-\sin t}\) | A1 | Accept \(\frac{4\sin 2t}{\sin t}\) WWW |
| \(= 8\cos t\) | A1 | |
| \(\leq 8\) | A1 | with brief explanation eg \(\cos t \leq 1\) |
| Answer | Marks | Guidance |
|---|---|---|
| Use \(\cos 2t = 2\cos^2 t +/- 1\) or \(1 - 2\cos^2 t\) | M1 | If starting with \(y = 4x^2 + 1\), then |
| Use correct version \(\cos 2t = 2\cos^2 t - 1\) | A1 | Subst \(x = \cos t, y = 3 + 2\cos 2t\) |
| Produce WWW \(y = 4x^2 + 1\) AG | A1 | Either substitute a formula for \(\cos 2t\) |
| Obtain \(0 = 0\) or \(4\cos^2 t + 1 = 4\cos^2 t + 1\) | A1 | |
| Or Manip to give formula for \(\cos 2t\) | M1 | |
| Obtain correct formula & say it's correct | A1 | |
| Any labelling must be correct | ||
| either \(x = \pm 1\) or \(y = 5\) must be marked | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| U-shaped parabola abve x-axis, sym abt y-axis | B1 | |
| Portion between \((-1, 5)\) and \((1, 5)\) | B1 | |
| N.B. If (ii) answered or quoted before (i) attempted, allow in part (i) B2 for \(\frac{dy}{dx} = 8x\) +B1, B1 if earned. |
(i) $(1 + \frac{x}{2})^{-2}$ | M1 | Clear indication of method of $\geq 3$ terms
$= 1 + (-2)(\frac{x}{2}) + \frac{-2}{2}3(\frac{x}{2})^2 + \frac{-2-3-4}{3!}(\frac{x}{2})^3$ | B1 | First two terms, not dependent on M1
$= 1 - x + \frac{3}{4}x^2 - \frac{1}{8}x^3$ | A1 | For both third and fourth terms
$(2 + x)^{-2} = \frac{1}{4}$(their exp of $(1 + ax)^{-2}$) mult out | √B1 | Correct: $\frac{1}{4} - \frac{1}{4}x + \frac{3}{16}x^2 - \frac{1}{8}x^3$
$|x| < 2$ or $-2 < x < 2$ (but not $|\frac{1}{2}x| < 1$) | B1 | | 5
(ii) If (i) is $a + bx + cx^2 + dx^3$ evaluate $b + d$ | M1 | Follow-through from $D + d$
$-\frac{3}{8}$ $(\frac{1}{8})$ | √A1 | | 2
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# Question 5(i):
$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$ | M1 |
$= \frac{-4\sin 2t}{-\sin t}$ | A1 | Accept $\frac{4\sin 2t}{\sin t}$ WWW
$= 8\cos t$ | A1 |
$\leq 8$ | A1 | with brief explanation eg $\cos t \leq 1$ | 4
# Question 5(ii):
Use $\cos 2t = 2\cos^2 t +/- 1$ or $1 - 2\cos^2 t$ | M1 | If starting with $y = 4x^2 + 1$, then
Use correct version $\cos 2t = 2\cos^2 t - 1$ | A1 | Subst $x = \cos t, y = 3 + 2\cos 2t$ | M1
Produce WWW $y = 4x^2 + 1$ AG | A1 | Either substitute a formula for $\cos 2t$ | M1
| | | Obtain $0 = 0$ or $4\cos^2 t + 1 = 4\cos^2 t + 1$ | A1
| | | Or Manip to give formula for $\cos 2t$ | M1
| | | Obtain correct formula & say it's correct | A1
| | | Any labelling must be correct
| | | either $x = \pm 1$ or $y = 5$ must be marked | 3
# Question 5(iii):
U-shaped parabola abve x-axis, sym abt y-axis | B1 |
Portion between $(-1, 5)$ and $(1, 5)$ | B1 | | 2
N.B. If (ii) answered or quoted before (i) attempted, allow in part (i) B2 for $\frac{dy}{dx} = 8x$ +B1, B1 if earned. | | | 9
---4 (i) Expand $( 2 + x ) ^ { - 2 }$ in ascending powers of $x$ up to and including the term in $x ^ { 3 }$, and state the set of values of $x$ for which the expansion is valid.\\
(ii) Hence find the coefficient of $x ^ { 3 }$ in the expansion of $\frac { 1 + x ^ { 2 } } { ( 2 + x ) ^ { 2 } }$.
\hfill \mbox{\textit{OCR C4 2007 Q4 [7]}}