Standard +0.3 This is a straightforward coordinate geometry problem requiring finding the midpoint, calculating the gradient of AB, using the perpendicular gradient property, and solving simultaneous equations. While it involves multiple steps (7 marks), each step uses standard A-level techniques with no novel insight required, making it slightly easier than average.
Points \(A\) and \(B\) have coordinates \((h, h)\) and \((4h + 6, 5h)\) respectively. The equation of the perpendicular bisector of \(AB\) is \(3x + 2y = k\). Find the values of the constants \(h\) and \(k\). [7]
Integrate by parts and reach a ( x+1 ) e − 3 x +b∫e − 3 x dx
M1*
1 1
Answer
Marks
Guidance
Obtain −3 ( x+1 ) e − 3 x +3∫e − 3 x dx, or equivalent
A1
−3xe −1 3 x + ∫3e −1 3 x dx−3e −1 3 x
1 1
Complete integration and obtain −3 ( x+1 ) e − 3 x −9e − 3 x , or
Answer
Marks
equivalent
A1
Use correct limits x = – 1 and x = 0 in the correct order,
Answer
Marks
having integrated twice
M1(dep*)
1
Answer
Marks
Obtain answer 9e3 −12, or equivalent
A1
5
Answer
Marks
Guidance
Question
Answer
Marks
Question 8:
--- 8(i) ---
8(i) | Use correct product or quotient rule | M1 | dy = 1 ( x+1 ) e −1 3 x +e −1 3 x
−
dx 3
e 1 3 x − ( x+1 )1 e 1 3 x
or dy = 3
dx 2x
e3
Obtain complete correct derivative in any form | A1
Equate derivative to zero and solve for x | M1
Obtain answer x = 2 with no errors seen | A1
4
--- 8(ii) ---
8(ii) | 1 1
Integrate by parts and reach a ( x+1 ) e − 3 x +b∫e − 3 x dx | M1*
1 1
Obtain −3 ( x+1 ) e − 3 x +3∫e − 3 x dx, or equivalent | A1 | −3xe −1 3 x + ∫3e −1 3 x dx−3e −1 3 x
1 1
Complete integration and obtain −3 ( x+1 ) e − 3 x −9e − 3 x , or
equivalent | A1
Use correct limits x = – 1 and x = 0 in the correct order,
having integrated twice | M1(dep*)
1
Obtain answer 9e3 −12, or equivalent | A1
5
Question | Answer | Marks | Guidance
Points $A$ and $B$ have coordinates $(h, h)$ and $(4h + 6, 5h)$ respectively. The equation of the perpendicular bisector of $AB$ is $3x + 2y = k$. Find the values of the constants $h$ and $k$. [7]
\hfill \mbox{\textit{CAIE P3 2018 Q8 [7]}}