Question 1 - A-Level Maths

CAIE P1 2016 November — Question 1 4 marks

Exam Board	CAIE
Module	P1 (Pure Mathematics 1)
Year	2016
Session	November
Marks	4
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Integration by Substitution
Type	Finding curve equation from derivative
Difficulty	Moderate -0.8 This is a straightforward integration problem requiring a simple substitution (u = 4x + 1) to integrate (4x + 1)^(-1/2), followed by using the given point to find the constant of integration. It's a standard textbook exercise with clear steps and no conceptual challenges, making it easier than average for A-level.
Spec	1.08b Integrate x^n: where n != -1 and sums

1 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 8 } { \sqrt { } ( 4 x + 1 ) }\). The point \(( 2,5 )\) lies on the curve. Find the equation of the curve.

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Question 1:

Answer	Marks	Guidance
Answer/Working	Marks	Guidance
\((y) = 8(4x+1)^{\frac{1}{2}} \div \frac{1}{2} \div 4 \, (+c)\)	B1	Correct integrand (unsimplified) without \(\div 4\)
B1	\(\div 4\). Ignore \(c\)
Uses \(x = 2\) and \(y = 5\)	M1	Substitution of correct values into an integrand to find \(c\)
\(c = -7\)	A1	\(y = 4\sqrt{4x+1} - 7\)

## Question 1:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $(y) = 8(4x+1)^{\frac{1}{2}} \div \frac{1}{2} \div 4 \, (+c)$ | B1 | Correct integrand (unsimplified) without $\div 4$ |
| | B1 | $\div 4$. Ignore $c$ |
| Uses $x = 2$ and $y = 5$ | M1 | Substitution of correct values into an integrand to find $c$ |
| $c = -7$ | A1 | $y = 4\sqrt{4x+1} - 7$ |

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1 A curve is such that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 8 } { \sqrt { } ( 4 x + 1 ) }$. The point $( 2,5 )$ lies on the curve. Find the equation of the curve.

\hfill \mbox{\textit{CAIE P1 2016 Q1 [4]}}

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