Moderate -0.3 This is a straightforward application of Vieta's formulas requiring students to express a symmetric function of roots in terms of coefficients. While it involves algebraic manipulation (finding a common denominator and using α+β and αβ), it's a standard textbook exercise with a clear method. Slightly easier than average due to being a direct application of well-practiced techniques, though the algebraic simplification prevents it from being routine recall.
2 In this question you must show detailed reasoning.
The equation \(\mathrm { x } ^ { 2 } - \mathrm { kx } + 2 \mathrm { k } = 0\), where \(k\) is a non-zero constant, has roots \(\alpha\) and \(\beta\).
Find \(\frac { \alpha } { \beta } + \frac { \beta } { \alpha }\) in terms of \(k\), simplifying your answer.
Question 2:
2 | DR
+ = k, = 2k
2+2
+ =
( ) 2 2 + −
=
k2−4k 1
= = k−2
2k 2 | B1
M1
M1
A1 | 1.1a
2.3
3.1a
1.1 | for combining fractions correctly
2 + 2 = ( + )2 −2 (soi)
𝑘−4
must be simplified (accept )
2
Alternative solution | B1 | by formula or completing the square
k k 2 − 8 k
x =
2
2 8 2 8 k + k − k k − k − k
+ = +
2 8 2 8 k − k − k k + k − k
( ) 2 ( ) 2
k + k 2 − 8 k + k − k 2 − 8 k
=
( k − k 2 − 8 k ) ( k + k 2 − 8 k ) | M1 | combining fractions
2 ( k 2 + k 2 − 8 k )
=
k 2 − k 2 + 8 k | A1 | expanding correctly
2 ( 2 k 2 − 8 k ) 1
= = k − 2
8 k 2 | A1 | 𝑘−4
must be simplified (accept )
2
[4]
B1
by formula or completing the square
M1
combining fractions
2 In this question you must show detailed reasoning.\\
The equation $\mathrm { x } ^ { 2 } - \mathrm { kx } + 2 \mathrm { k } = 0$, where $k$ is a non-zero constant, has roots $\alpha$ and $\beta$.\\
Find $\frac { \alpha } { \beta } + \frac { \beta } { \alpha }$ in terms of $k$, simplifying your answer.
\hfill \mbox{\textit{OCR MEI Further Pure Core AS 2023 Q2 [4]}}