Moderate -0.3 This is a straightforward double integration problem requiring application of the power rule twice and using two given conditions to find constants of integration. While it involves fractional powers and requires careful algebraic manipulation, it's a standard textbook exercise with no novel problem-solving required, making it slightly easier than average.
5 A curve has an equation which satisfies \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 3 x ^ { - \frac { 1 } { 2 } }\) for all positive values of \(x\). The point \(P ( 4,1 )\) lies on the curve, and the gradient of the curve at \(P\) is 5 . Find the equation of the curve.
Must be of form \(px^{0.5}\), any non-zero numerical \(p\), and no other algebraic terms
Obtain \(6x^{0.5}\) (allow no \(+c\))
A1
Allow unsimplified coeff ie \(\frac{3}{0.5}\), even if subsequently incorrect
\(5 = 12 + c\)
M1d*
Attempt to use \(x=4\), gradient \(= 5\); M0 if no \(+c\); attempt to use \(x=4\), \(\frac{dy}{dx}=5\) — allow slip as long as intention clear
\(c = -7\)
A1
No need to see explicit expression for \(\frac{dy}{dx}\)
\(y = 4x^{1.5} - 7x + k\)
M1dd*
Must be of form \(qx^{1.5} + rx\), any non-zero numerical \(q\), \(r\), and no other algebraic terms; dependent on at least M1 M1 awarded
\(1 = 32 - 28 + k\), hence \(k = -3\)
M1ddd*
Condone notation for constant of integration being same as previously used; dependent on all previous M marks; attempt to use \(x=4\), \(y=1\)
\(y = 4x^{1.5} - 7x - 3\)
A1
Coefficients must now be simplified; must be an equation ie \(y = ...\); so A0 for \(f(x) = ...\) or 'equation \(= ...\)'
## Question 5:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx} = 6x^{0.5} + c$ | M1* | Must be of form $px^{0.5}$, any non-zero numerical $p$, and no other algebraic terms |
| Obtain $6x^{0.5}$ (allow no $+c$) | A1 | Allow unsimplified coeff ie $\frac{3}{0.5}$, even if subsequently incorrect |
| $5 = 12 + c$ | M1d* | Attempt to use $x=4$, gradient $= 5$; M0 if no $+c$; attempt to use $x=4$, $\frac{dy}{dx}=5$ — allow slip as long as intention clear |
| $c = -7$ | A1 | No need to see explicit expression for $\frac{dy}{dx}$ |
| $y = 4x^{1.5} - 7x + k$ | M1dd* | Must be of form $qx^{1.5} + rx$, any non-zero numerical $q$, $r$, and no other algebraic terms; dependent on at least M1 M1 awarded |
| $1 = 32 - 28 + k$, hence $k = -3$ | M1ddd* | Condone notation for constant of integration being same as previously used; dependent on all previous M marks; attempt to use $x=4$, $y=1$ |
| $y = 4x^{1.5} - 7x - 3$ | A1 | Coefficients must now be simplified; must be an equation ie $y = ...$; so A0 for $f(x) = ...$ or 'equation $= ...$' |
5 A curve has an equation which satisfies $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 3 x ^ { - \frac { 1 } { 2 } }$ for all positive values of $x$. The point $P ( 4,1 )$ lies on the curve, and the gradient of the curve at $P$ is 5 . Find the equation of the curve.
\hfill \mbox{\textit{OCR C2 2015 Q5 [7]}}