Moderate -0.3 This is a straightforward integration question requiring students to integrate powers of x (with fractional indices) between given limits. The integrand is already provided, the limits are clearly stated, and the technique is standard application of the power rule for integration. It's slightly easier than average because it's a direct single-method problem with no setup required, though the fractional powers require careful handling.
\includegraphics{figure_5}
The diagram shows the curve with equation \(y = 10x^{\frac{1}{2}} - \frac{5}{2}x^{\frac{3}{2}}\) for \(x > 0\). The curve meets the \(x\)-axis at the points \((0, 0)\) and \((4, 0)\).
Find the area of the shaded region. [4]
Where a candidate has misread a number in the question and used that value consistently throughout, provided that number does not alter the difficulty or
the method required, award all marks earned and deduct just 1 mark for the misread.
Answer
Marks
5
1 5 3 10 3 5 5 20 3 5
[10x2 x2] x2 x2 x2 x2
2 3 2 5 3
Answer
Marks
Guidance
2 2
B1 B1
B1 for contents of each { } then ISW.
20
their 832 0
Answer
Marks
Guidance
3
M1
Using limit(s) correctly in an integrated expression
(defined by one correct power). Minimum acceptable
160
working is their ( 32).
3
64 1
[Area of shaded region =] , 21 or 21.3[333…]
Answer
Marks
Guidance
3 3
A1
Condone the presence of π for the first 3 marks.
Condone using the limits the wrong way around for the M
mark and if 21.3 is corrected to 21.3 allow the A mark.
SC: if M0 scored SCB1 is available for correct final
answer
1 5 3
If 10x2 x2 21.3 and no integration seen B1 only.
2
4
Answer
Marks
Guidance
Question
Answer
Marks
Question 5:
5 | Where a candidate has misread a number in the question and used that value consistently throughout, provided that number does not alter the difficulty or
the method required, award all marks earned and deduct just 1 mark for the misread.
5 |
1 5 3 10 3 5 5 20 3 5
[10x2 x2] x2 x2 x2 x2
2 3 2 5 3
2 2 | B1 B1 | B1 for contents of each { } then ISW.
20
their 832 0
3 | M1 | Using limit(s) correctly in an integrated expression
(defined by one correct power). Minimum acceptable
160
working is their ( 32).
3
64 1
[Area of shaded region =] , 21 or 21.3[333…]
3 3 | A1 | Condone the presence of π for the first 3 marks.
Condone using the limits the wrong way around for the M
mark and if 21.3 is corrected to 21.3 allow the A mark.
SC: if M0 scored SCB1 is available for correct final
answer
1 5 3
If 10x2 x2 21.3 and no integration seen B1 only.
2
4
Question | Answer | Marks | Guidance
\includegraphics{figure_5}
The diagram shows the curve with equation $y = 10x^{\frac{1}{2}} - \frac{5}{2}x^{\frac{3}{2}}$ for $x > 0$. The curve meets the $x$-axis at the points $(0, 0)$ and $(4, 0)$.
Find the area of the shaded region. [4]
\hfill \mbox{\textit{CAIE P1 2023 Q5 [4]}}