| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 12 |
| 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 for n=6, followed by algebraic simplification where odd powers cancel. Part (iii) involves basic substitution and error estimation. All steps are standard textbook exercises with no problem-solving insight required, making it easier than average. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| \((1+2x)^6=1+6\cdot2x+15(2x)^2+20(2x)^3+15(2x)^4+6(2x)^5+(2x)^6\) | B1 | Coefficients |
| B1 | Powers | |
| \(=1+12x+60x^2+160x^3+240x^4+192x^5+64x^6\) | B1 | Answer |
| Answer | Marks | Guidance |
|---|---|---|
| \((1+2x)^6=1+12x+60x^2+160x^3+240x^4+192x^5+64x^6\) | B1 | Signs |
| \((1-2x)^6=1-12x+60x^2-160x^3+240x^4-192x^5+64x^6\) | B1 | Terms |
| \(\Rightarrow (1+2x)^6+(1-2x)^6=2+120x^2+480x^4+128x^6\) | B1 | Answer |
| Answer | Marks |
|---|---|
| \(x=0.01 \Rightarrow (1+2x)^6+(1-2x)^6=1.02^6+0.98^6\) | M1 |
| \(=2+120(0.01)^2\) | A1 |
| \(=2+0.012=2.012\) | A1 |
| 3rd term \(=0.0000048\) | B1 |
| B1 | \(3^{\text{rd}}\) term; Accept 5 decimal places |
| \(\Rightarrow\) 4 decimal places (i.e. \(2.01200\)) | B1 |
## Question 12(i):
$(1+2x)^6=1+6\cdot2x+15(2x)^2+20(2x)^3+15(2x)^4+6(2x)^5+(2x)^6$ | B1 | Coefficients
| B1 | Powers
$=1+12x+60x^2+160x^3+240x^4+192x^5+64x^6$ | B1 | Answer
**Total: 3**
## Question 12(ii):
$(1+2x)^6=1+12x+60x^2+160x^3+240x^4+192x^5+64x^6$ | B1 | Signs
$(1-2x)^6=1-12x+60x^2-160x^3+240x^4-192x^5+64x^6$ | B1 | Terms
$\Rightarrow (1+2x)^6+(1-2x)^6=2+120x^2+480x^4+128x^6$ | B1 | Answer
**Total: 3**
## Question 12(iii):
$x=0.01 \Rightarrow (1+2x)^6+(1-2x)^6=1.02^6+0.98^6$ | M1 |
$=2+120(0.01)^2$ | A1 |
$=2+0.012=2.012$ | A1 |
3rd term $=0.0000048$ | B1 |
| B1 | $3^{\text{rd}}$ term; Accept 5 decimal places
$\Rightarrow$ 4 decimal places (i.e. $2.01200$) | B1 |
**Total: 6**12 (i) Expand $( 1 + 2 x ) ^ { 6 }$, simplifying all the terms.\\
(ii) Hence find an expression for $\mathrm { f } ( x ) = ( 1 + 2 x ) ^ { 6 } + ( 1 - 2 x ) ^ { 6 }$ in its simplest form.\\
(iii) Substituting $x = 0.01$ into the first two terms of $\mathrm { f } ( x )$ gives an approximate value, z for $1.02 ^ { 6 } + 0.98 ^ { 6 }$. Find $z$.
By considering the value of the third term, comment on the accuracy of $z$ as an approximation for $1.02 ^ { 6 } + 0.98 ^ { 6 }$.
\hfill \mbox{\textit{OCR MEI C1 Q12 [12]}}