Question
Expand \(\frac{1+3x}{\sqrt{\left ( 1+2x \right )}}\) in ascending powers of x up to and including the term in x2, simplifying the coefficients. [4]
Answer/Explanation
Ans:
Obtain 1− x as first two terms of \(\left ( 1+2x \right )^{-\frac{1}{2}}\)
Obtain \(+\frac{3}{2}x^{2}\) or unsimplified equivalent as third term of \(\left ( 1+2x \right )^{-\frac{1}{2}}\)
Multiply 1 + 3x by attempt at \(\left ( 1+2x \right )^{-\frac{1}{2}}\) , obtaining sufficient terms
Obtain final answer \(1+2x-\frac{3}{2}x^{2}\)
Question
Expand\(\frac{1}{3\sqrt{(1+6x)}}\) in ascending powers of x, up to and including the term in \(x^{3}\) , simplifying the
coefficients.
Answer/Explanation
EITHER:
State a correct unsimplified version of the x or\( x^{2}\) or \(x^{3}\) term in the expansion of \( (1+6x)^{-\frac{1}{3}}\)
State correct first two terms 1-2x
Obtain term \(8x^{2}\)
Obtain term \( -\frac{112}{3}x^{3}\left ( 37\frac{1}{3}x^{2} \right ) \)in final answer
OR:
Differentiate expression and evaluate f (0) and f (0) ′where, \(f'(x)=k(1+6x)^{-\frac{4}{3}}\)
Obtain correct first two terms 1-2x
Obtain term \(8x^{2}\)
Obtain term\(-\frac{112}{3}x^{3}\) in final answer
Question
(a) Expand (2 − 3x)−2 in ascending powers of x, up to and including the term in x2, simplifying the
coefficients. [4]
(b) State the set of values of x for which the expansion is valid. [1]
Answer/Explanation
Ans
2 (a) State a correct unsimplified version of the x or x2 term of the expansion of \((2-3x)^{-2}\ or \ \left ( 1-\frac{3}{2}x \right )^{-2}\)
State correct first term \(\frac{1}{4}\)
Obtain the next two terms \(\frac{3}{4}x+\frac{27}{16}x^{2}\)
2 (b) State answer \(|x|< \frac{2}{3}, \ or \ equivalent\)
Question
Expand\( (2-x)(1+2x)^{\frac{3}{2}}\) in ascending powers of x, up to and including the term in\( x^{2}\) simplifying the coefficients.
Answer/Explanation
State correct unsimplified first two terms of the expansion of\( (1+2x)^{-\frac{3}{2}},eg.1+(-\frac{3}{2})(2x)\)
State correct unsimplified term in\( x^{2}\),e.g.\( (-\frac{3}{2})(-\frac{3}{2}-1)(2x)^{2}/2!\)
Obtain sufficient terms of the product of (2 – x) and the expansion up to the term in\( x^{2}\)
Obtain final answer 2-\(7x+18x^{2}\)Do not ISW
Question
(i) Expand\(\frac{1}{\sqrt{(1-4x)}}\) in ascending powers of x, up to and including the term in \(x^{2}\), simplifying the
coefficients.
(ii) Hence find the coefficient of\( x^{2}\) in the expansion of\( \frac{1+2x}{\sqrt{(4-16x)}}\)
Answer/Explanation
(i) Either Obtain correct (unsimplified) version of x or\( x^{2}\) term from\( (1-4x)^{\frac{1}{2}}\)
Obtain 1 + 2x
Obtain +\( 6x^{2}\)
Or Differentiate and evaluate f(0) and f′(0) where f′(x) =\( k(1-4x)^{-\frac{3}{2}}\)
Obtain 1 + 2x
Obtain + 6x^{2}
(ii) Combine both x^{2}
terms from product of 1 + 2x and answer from part (i)
Obtain 5
Question
When (1 + ax)^{−2} , where a is a positive constant, is expanded in ascending powers of x, the coefficients of x and\( x^{3}\) are equal.
(i) Find the exact value of a.
(ii) When a has this value, obtain the expansion up to and including the term in \(x^{2}\), simplifying the
coefficients.
Answer/Explanation
(i) Obtain correct unsimplified terms in x and \(x^{3}\)
Equate coefficients and solve for a
Obtain final answer a=\(\frac{1}{\sqrt{2}}\) , or exact equivalent
(ii) Use correct method and value of a to find the first two terms of the expansion \((1 + ax)^{–2} \)
Obtain 1 – √2x, or equivalent
Obtain term \(\frac{3}{2}x^{2}\)
[Symbolic coefficients, e.g.\(\begin{pmatrix}
-2 & \\
1&
\end{pmatrix}\) a, are not sufficient for the first B marks]
[The f.t. is solely on the value of a.]