| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2005 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Area involving fractional powers |
| Difficulty | Moderate -0.3 This is a straightforward two-part question requiring standard techniques: finding a normal line equation (differentiation, negative reciprocal gradient, distance formula) and direct integration of a power function. Both parts are routine A-level exercises with no conceptual challenges, though the multi-step nature and need for careful arithmetic place it slightly below average difficulty. |
| Spec | 1.07l Derivative of ln(x): and related functions1.07m Tangents and normals: gradient and equations1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\frac{dy}{dx} = -2x^{-1.5}\) | M1 | Reasonable attempt at differentiation with his power of \(x\). |
| \(= -\frac{1}{4}\) | A1 | CAO |
| \(m\) of normal \(= 4\) | M1 | Use of \(m_1 m_2 = -1\) even if algebraic. |
| Eqn of normal: \(y - 2 = 4(x - 4)\) | M1 | Use of equation for a straight line + use of \(x = 0\) and \(y = 0\). |
| \(P(3.5, 0)\) and \(Q(0, -14)\) | M1 | Needs correct formula or method. |
| Length of \(PQ = \sqrt{(3.5^2 + 14^2)} = 14.4\) | A1 [6] | CAO |
| (ii) Area \(= \int_1^4 4x^{-0.5}dx = \left[\frac{4x^{0.5}}{0.5}\right]\) | M1 A1 | Attempt at integration. Correct unsimplified. |
| \(= [8\sqrt{x}] = 16 - 8 = 8\) | DM1A1 [4] | Correct use of limits. CAO |
$y = \frac{4}{\sqrt{x}}$
(i) $\frac{dy}{dx} = -2x^{-1.5}$ | M1 | Reasonable attempt at differentiation with his power of $x$.
$= -\frac{1}{4}$ | A1 | CAO
$m$ of normal $= 4$ | M1 | Use of $m_1 m_2 = -1$ even if algebraic.
Eqn of normal: $y - 2 = 4(x - 4)$ | M1 | Use of equation for a straight line + use of $x = 0$ and $y = 0$.
$P(3.5, 0)$ and $Q(0, -14)$ | M1 | Needs correct formula or method.
Length of $PQ = \sqrt{(3.5^2 + 14^2)} = 14.4$ | A1 [6] | CAO
(ii) Area $= \int_1^4 4x^{-0.5}dx = \left[\frac{4x^{0.5}}{0.5}\right]$ | M1 A1 | Attempt at integration. Correct unsimplified.
$= [8\sqrt{x}] = 16 - 8 = 8$ | DM1A1 [4] | Correct use of limits. CAO9 A curve has equation $y = \frac { 4 } { \sqrt { } x }$.\\
(i) The normal to the curve at the point $( 4,2 )$ meets the $x$-axis at $P$ and the $y$-axis at $Q$. Find the length of $P Q$, correct to 3 significant figures.\\
(ii) Find the area of the region enclosed by the curve, the $x$-axis and the lines $x = 1$ and $x = 4$.
\hfill \mbox{\textit{CAIE P1 2005 Q9 [10]}}