Moderate -0.5 This is a verification question requiring substitution of given values into a provided expression and checking against a diagram. It involves algebraic manipulation but no derivation, proof, or problem-solving—students simply need to substitute and compute. This is easier than average as it's purely computational with the answer structure already given.
14 Substitute appropriate values of \(t _ { 1 }\) and \(t _ { 2 }\) to verify that the expression \(t _ { 1 } ^ { 2 } + t _ { 2 } ^ { 2 } + t _ { 1 } t _ { 2 } + \frac { 1 } { 2 }\) gives the correct value for the \(y\)-coordinate of the point of intersection of the normals at the points A and B in Fig. C2.
Using \(t_1\) and \(t_2\) with values 1 and \(-3\) and showing substitution to get \(7.5\)
B1
Using \(-1\) and \(3\) is incorrect and scores B0
Mark Scheme Guidance Notes (H640/03, June 2024)
Question 15a:
Answer
Marks
Guidance
Working/Answer
Marks
Guidance
Differentiate to get gradient as \(2at + b\) (or \(2ax + b\))
M1
Writing \(2ax+b/2at+b\) without differentiating scores M0
Substitute gradient and \(x = t\) into straight line equation using \((y - y_1) = m(x - x_1)\) or \(y = mx + c\)
M1
Condone lack of brackets around \(m\) if recovered
Correct answer obtained with working shown
A1
Answer is given — working must be seen
Question 15b:
Answer
Marks
Guidance
Working/Answer
Marks
Guidance
Equate two tangents with distinct values of \(t\) (e.g. \(x_p\) and \(x_q\), or \(t_1\) and \(t_2\), or \(m\) and \(n\))
M1
Other pairs of symbols acceptable
Rearrange to get \(x\) terms on one side and non-\(x\) terms on the other
M1
Show difference of two squares and cancelling to reach required result
A1
Allow final result in terms of their variable; working must be shown
## Question 14:
| Using $t_1$ and $t_2$ with values 1 and $-3$ and showing substitution to get $7.5$ | B1 | Using $-1$ and $3$ is incorrect and scores B0 |
# Mark Scheme Guidance Notes (H640/03, June 2024)
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## Question 15a:
| Working/Answer | Marks | Guidance |
|---|---|---|
| Differentiate to get gradient as $2at + b$ (or $2ax + b$) | M1 | Writing $2ax+b/2at+b$ without differentiating scores M0 |
| Substitute gradient and $x = t$ into straight line equation using $(y - y_1) = m(x - x_1)$ or $y = mx + c$ | M1 | Condone lack of brackets around $m$ if recovered |
| Correct answer obtained with working shown | A1 | Answer is given — working must be seen |
---
## Question 15b:
| Working/Answer | Marks | Guidance |
|---|---|---|
| Equate two tangents with distinct values of $t$ (e.g. $x_p$ and $x_q$, or $t_1$ and $t_2$, or $m$ and $n$) | M1 | Other pairs of symbols acceptable |
| Rearrange to get $x$ terms on one side and non-$x$ terms on the other | M1 | |
| Show difference of two squares and cancelling to reach required result | A1 | Allow final result in terms of their variable; working must be shown |
---
14 Substitute appropriate values of $t _ { 1 }$ and $t _ { 2 }$ to verify that the expression $t _ { 1 } ^ { 2 } + t _ { 2 } ^ { 2 } + t _ { 1 } t _ { 2 } + \frac { 1 } { 2 }$ gives the correct value for the $y$-coordinate of the point of intersection of the normals at the points A and B in Fig. C2.
\hfill \mbox{\textit{OCR MEI Paper 3 2024 Q14 [1]}}