June 2024 Paper 3 Q6
6
(a) Find \(\displaystyle\int \left(6x^2 - \frac{5}{\sqrt{x}}\right) \mathrm{d}x\) [3 marks]
(b) The gradient of a curve is given by\[\frac{\mathrm{d}y}{\mathrm{d}x} = 6x^2 - \frac{5}{\sqrt{x}}\]
The curve passes through the point (4, 90).
Find the equation of the curve. [2 marks]
| Scheme | Marks | AO |
|---|---|---|
| Writes \(\dfrac{1}{\sqrt{x}}\) term as \(x^{-\frac{1}{2}}\) PI by \(\sqrt{x}\) or \(x^{\frac{1}{2}}\) in answer | B1 | 1.1b |
| Obtains one correctly integrated term May be unsimplified | M1 | 1.1a |
| Obtains \(2x^3 - 10x^{\frac{1}{2}} + c\) ISW Condone omission of \(+c\) Must be simplified | A1 | 1.1b |
| (3) |
Typical solution
\[\int \left(6x^2 - \frac{5}{\sqrt{x}}\right) \mathrm{d}x = \int \left(6x^2 - 5x^{-\frac{1}{2}}\right) \mathrm{d}x\]\[= 2x^3 - 10x^{\frac{1}{2}} + c\]| Scheme | Marks | AO |
|---|---|---|
| Substitutes \(x = 4\) into their integrated expression from part 6(a), with an arbitrary constant and sets equal to 90 PI by \(-18\) | M1 | 3.1a |
| Obtains \(y = 2x^3 - 10x^{\frac{1}{2}} - 18\) CAO Condone f(\(x\)) for \(y\) | A1 | 1.1b |
| (2) | ||
| (5 marks) |