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 \(\left ( a+bx \right )\sqrt{1 + 4x}\) , 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.
Answer/Explanation
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