| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Binomial Theorem (positive integer n) |
| Type | Product of expansions |
| Difficulty | Moderate -0.3 Part (i) is a standard binomial expansion exercise requiring identification of the r=4 term to get x^0. Part (ii) adds one layer of complexity by multiplying two expressions and combining like terms, but remains a routine textbook question with clear methodology and minimal problem-solving demand. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\left(x - \frac{2}{x}\right)^6\); Term is \({}_6C_3 \times (-2)^3 = (-)160\) | B1 | \(\pm 160\) seen anywhere |
| \(-160\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Term in \(x^2 = {}_6C_2(-2)^2 x^2 = 60\ (x^2)\) | B1 | \(\pm 60\) seen anywhere |
| B1 | ||
| Term independent of \(x\): \(= 2 \times (\text{their } {-160}) + 3 \times (\text{their } 60)\) | M1 | Using 2 products correctly |
| \(-140\) | A1 |
## Question 4:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\left(x - \frac{2}{x}\right)^6$; Term is ${}_6C_3 \times (-2)^3 = (-)160$ | B1 | $\pm 160$ seen anywhere |
| $-160$ | B1 | |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Term in $x^2 = {}_6C_2(-2)^2 x^2 = 60\ (x^2)$ | B1 | $\pm 60$ seen anywhere |
| | B1 | |
| Term independent of $x$: $= 2 \times (\text{their } {-160}) + 3 \times (\text{their } 60)$ | M1 | Using 2 products correctly |
| $-140$ | A1 | |4 Find the term that is independent of $x$ in the expansion of\\
(i) $\left( x - \frac { 2 } { x } \right) ^ { 6 }$,\\
(ii) $\left( 2 + \frac { 3 } { x ^ { 2 } } \right) \left( x - \frac { 2 } { x } \right) ^ { 6 }$.
\hfill \mbox{\textit{CAIE P1 2016 Q4 [6]}}