Standard +0.3 This is a standard Further Maths question on transformed roots requiring the substitution y = 3 - 2x, rearranging to express x in terms of y, and substituting into the original equation. While it requires careful algebraic manipulation and understanding of root transformations, it follows a well-established procedure taught explicitly in Further Pure courses with no novel problem-solving required.
2 The equation \(3 x ^ { 2 } - 4 x + 2 = 0\) has roots \(\alpha\) and \(\beta\).
Find an equation with integer coefficients whose roots are \(3 - 2 \alpha\) and \(3 - 2 \beta\).
Question 2:
2 | 3−y
Let y=3−2x⇒x=
2
3(3−y)2 4(3−y)
⇒ − +2=0
4 2
⇒3y2 −10y+11(=0) | B1
M1
A1 | 1.1a
1.1
1.1 | soi
substituting their (3 − y)/2 for x
Alternative solution 1 | Attempt to find sum and product of
new roots
α+β= 4, αβ= 2
3 3 | B1
⇒3−2α+3−2β=6−2(α+β)=10
3
and (3−2α)(3−2β)=9−6(α+β)+4αβ=11
3 | M1
⇒3y2 −10y+11(=0)
A1
Any exact form
Alternative solution 2
4± 16−4×3×2 4± 8i 2± 2i
α,β are = =
2×3 6 3
5±2 2i
new roots are
3
B1
( 5+2 2i )( 5−2 2i )
sum = 10 , product = =11
3 9 3
M1
⇒3y2 −10y+11(=0)
A1
[3]
Attempt to find sum and product of
new roots
Any exact form
2 The equation $3 x ^ { 2 } - 4 x + 2 = 0$ has roots $\alpha$ and $\beta$.\\
Find an equation with integer coefficients whose roots are $3 - 2 \alpha$ and $3 - 2 \beta$.
\hfill \mbox{\textit{OCR MEI Further Pure Core AS 2021 Q2 [3]}}