(i) Expand\(\frac{1}{\sqrt{(1-4x)}}\) in ascending powers of x, up to and including the term in  \(x^{2}\), simplifying the

(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


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.



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