Moderate -0.3 This is a straightforward binomial expansion problem requiring students to expand two binomial expressions, multiply them, and equate coefficients. While it involves multiple steps and some algebraic manipulation, it's a standard textbook exercise testing routine application of the binomial theorem with no novel insight required. The 6-mark allocation reflects the working needed rather than conceptual difficulty.
The first three terms in the expansion of \((1 - 2x)^2(1 + ax)^6\), in ascending powers of \(x\), are \(1 - x + bx^2\). Find the values of the constants \(a\) and \(b\). [6]
Coeff of \(x\) in \((1+ax)^6 = 6ax\); Coeff of \(x^2\) in \((1+ax)^6 = 15a^2x^2\)
Answer
Marks
Guidance
Multiplies by \((1 - 4x + 4x^2)\); 2 terms in \(x\); \(6a - 4 = -1 \to a = \frac{1}{2}\)
M1, A1
Needs to consider 2 terms in equation; Co
3 terms in \(x^2\); \(15a^2 - 24a + 4 = b \to b = -4\frac{1}{4}\)
M1, A1
Needs to consider 3 terms in equation
$(1-2x)^6(1+ax)^6$
Coeff of $x$ in $(1+ax)^6 = 6ax$; Coeff of $x^2$ in $(1+ax)^6 = 15a^2x^2$
Multiplies by $(1 - 4x + 4x^2)$; 2 terms in $x$; $6a - 4 = -1 \to a = \frac{1}{2}$ | M1, A1 | Needs to consider 2 terms in equation; Co
3 terms in $x^2$; $15a^2 - 24a + 4 = b \to b = -4\frac{1}{4}$ | M1, A1 | Needs to consider 3 terms in equation
The first three terms in the expansion of $(1 - 2x)^2(1 + ax)^6$, in ascending powers of $x$, are $1 - x + bx^2$. Find the values of the constants $a$ and $b$. [6]
\hfill \mbox{\textit{CAIE P1 2012 Q3 [6]}}