### 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. 

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.

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.                                                                                                                                                          

(b) State the set of values of x for which the expansion is valid.                                                                     

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.

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$$\frac{1+2x}{\sqrt{(4-16x)}}$$

(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.

(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.]

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