### 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 $$\left ( a+bx \right )\sqrt{1 + 4x}$$ , where a and b are constants, is expanded in ascending powers of x, the coefficients of x and x2 are 3 and −6 respectively.

Find the values of a and b.

Ans:

State or imply 1 + 2x as first terms of the expansion of  $$\sqrt{1 +4x}$$

State or imply  − 2x2 as third term of the expansion of $$\sqrt{1 +4x}$$

Form an expression for the coefficient of x or coefficient of  x2 in the expansion of (a + bx) $$\sqrt{1 +4x}$$ and equate to given coefficient

Obtain 2a + b = 3, or equivalent

Obtain– 2a + 2b = – 6 or equivalent

Obtain answer a = 2 and b = – 1