Standard +0.3 This is a straightforward binomial theorem application requiring students to find coefficients of x³ in two expansions, set up an equation, and solve for a constant. While it involves two binomial expansions and equation solving, the steps are mechanical and follow standard procedures with no conceptual challenges beyond basic binomial coefficient calculation.
Coeff of $x^3$ in $(a + x)^5 = 10 × a^2$ | B1 B1 | co co
Coeff of $x^3$ in $(2 - x)^6 = -160$ | B1 | co
$\to 10a^2 - 160 = 90$ | M1 | forms an equation from 2 terms co
$\to a = 5$ | A1 | [5]
The coefficient of $x^3$ in the expansion of $(a + x)^5 + (2 - x)^6$ is $90$. Find the value of the positive constant $a$. [5]
\hfill \mbox{\textit{CAIE P1 2012 Q3 [5]}}