It is given instead that in the expansion of $$\((4+2 x)(2-a x)^{5}\)$$, the coefficient of $$\(x^{2}\)$$ is $$\(k\)$$. It is also given that there is only one value of $$\(a\)$$ which leads to this value of $$\(k\)$$. Find the values of $$\(k\)$$ and $$\(a\)$$. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................
Exam No：9709_w21_qp_12 Year：2021 Question No：8(b)
Answer:
\(320 a^{2}-160 a=k \Rightarrow 320 a^{2}-160 a-k[=0]\)
Their \(b^{2}-4 a c[=0],\left[160^{2}-4 \times 320 \times(-k)=0\right]\)
\(k=-20\)
\(a=\frac{1}{4}\)
Alternative method for question
\(320 a^{2}-160 a=k\) and divide by \(320\left[a^{2}-\frac{a}{2}=\frac{k}{320}\right]\)
Attempt to complete the square \(\left[\left(a-\frac{1}{4}\right)^{2}-\frac{1}{16}=\frac{k}{320}\right]\)
\(a=\frac{1}{4}\)
\(k=-20\)
Alternative method for question
\(320 a^{2}-160 a=k\) and attempt to differentiate LHS \([640 a-160]\)
Setting their \((640 a-160)=0\) and attempt to solve.
\(a=\frac{1}{4}\)
\(k=-20\)
Their \(b^{2}-4 a c[=0],\left[160^{2}-4 \times 320 \times(-k)=0\right]\)
\(k=-20\)
\(a=\frac{1}{4}\)
Alternative method for question
\(320 a^{2}-160 a=k\) and divide by \(320\left[a^{2}-\frac{a}{2}=\frac{k}{320}\right]\)
Attempt to complete the square \(\left[\left(a-\frac{1}{4}\right)^{2}-\frac{1}{16}=\frac{k}{320}\right]\)
\(a=\frac{1}{4}\)
\(k=-20\)
Alternative method for question
\(320 a^{2}-160 a=k\) and attempt to differentiate LHS \([640 a-160]\)
Setting their \((640 a-160)=0\) and attempt to solve.
\(a=\frac{1}{4}\)
\(k=-20\)
Knowledge points:
1.1.2 find the discriminant of a quadratic polynomial and use the discriminant
1.1.3 solve quadratic equations, and quadratic inequalities, in one unknown (By factorising, completing the square and using the formula.)
Solution:
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