Standard +0.3 This is a straightforward application of the binomial expansion for non-integer powers, requiring students to expand two expressions to the x^4 term, subtract them, and observe that lower-order terms cancel. While it involves two expansions and careful algebraic manipulation, it's a standard textbook exercise with no novel insight required—slightly easier than average due to its mechanical nature.
3 Show that, for small values of \(x ^ { 2 }\),
$$\left( 1 - 2 x ^ { 2 } \right) ^ { - 2 } - \left( 1 + 6 x ^ { 2 } \right) ^ { \frac { 2 } { 3 } } \approx k x ^ { 4 }$$
where the value of the constant \(k\) is to be determined.
Obtain correct (unsimplified) version of \(x^2\) or \(x^4\) term in \((1-2x^2)^{-2}\)
M1
Obtain \(1 + 4x^2\)
A1
Obtain \(\ldots + 12x^4\)
A1
Obtain correct (unsimplified) version of \(x^2\) or \(x^4\) term in \((1 + 6x^2)^{\frac{2}{3}}\)
M1
Obtain \(1 + 4x^2 - 4x^4\)
A1
Combine expansions to obtain \(k = 16\) with no error seen
A1
Or
Obtain correct (unsimplified) version of \(x^2\) or \(x^4\) term in \((1 + 6x^2)^{\frac{2}{3}}\)
M1
Obtain \(1 + 4x^2\)
A1
Obtain \(\ldots - 4x^4\)
A1
Obtain correct (unsimplified) version of \(x^2\) or \(x^4\) term in \((1-2x^2)^{-2}\)
M1
Obtain \(1 + 4x^2 + 12x^4\)
A1
Combine expansions to obtain \(k = 16\) with no error seen
A1
[6]
**Either** |
Obtain correct (unsimplified) version of $x^2$ or $x^4$ term in $(1-2x^2)^{-2}$ | M1 |
Obtain $1 + 4x^2$ | A1 |
Obtain $\ldots + 12x^4$ | A1 |
Obtain correct (unsimplified) version of $x^2$ or $x^4$ term in $(1 + 6x^2)^{\frac{2}{3}}$ | M1 |
Obtain $1 + 4x^2 - 4x^4$ | A1 |
Combine expansions to obtain $k = 16$ with no error seen | A1 |
**Or** |
Obtain correct (unsimplified) version of $x^2$ or $x^4$ term in $(1 + 6x^2)^{\frac{2}{3}}$ | M1 |
Obtain $1 + 4x^2$ | A1 |
Obtain $\ldots - 4x^4$ | A1 |
Obtain correct (unsimplified) version of $x^2$ or $x^4$ term in $(1-2x^2)^{-2}$ | M1 |
Obtain $1 + 4x^2 + 12x^4$ | A1 |
Combine expansions to obtain $k = 16$ with no error seen | A1 | [6]
3 Show that, for small values of $x ^ { 2 }$,
$$\left( 1 - 2 x ^ { 2 } \right) ^ { - 2 } - \left( 1 + 6 x ^ { 2 } \right) ^ { \frac { 2 } { 3 } } \approx k x ^ { 4 }$$
where the value of the constant $k$ is to be determined.
\hfill \mbox{\textit{CAIE P3 2015 Q3 [6]}}