| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2012 |
| Session | November |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Hemisphere or sphere resting on plane or wall |
| Difficulty | Challenging +1.8 This is a challenging 3-sphere equilibrium problem requiring systematic force analysis at multiple contact points, friction inequalities, and geometric reasoning with angles. While the setup is complex with 6 unknowns (normal and friction forces at 4 contacts), the symmetry simplifies the algebra significantly. The problem demands careful free-body diagrams, resolution in two directions, and understanding limiting friction conditions, but follows a standard mechanics approach without requiring novel insight. It's harder than typical A-level mechanics due to the multi-body system and Further Maths context, but more straightforward than problems requiring geometric breakthroughs or extended proof chains. |
| Spec | 3.03r Friction: concept and vector form3.03t Coefficient of friction: F <= mu*R model3.03u Static equilibrium: on rough surfaces3.04a Calculate moments: about a point3.04b Equilibrium: zero resultant moment and force |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Stating or implying reactions \(R_P\), \(R_S\) same as for \(B\) | B1 | |
| Stating or implying \(F_P = F_S\) by moments about \(O_A\) | B1 | |
| Stating or implying 3 independent equations for \(F\), \(R_P\), \(R_S\) | \(3 \times\) M1 A1 | |
| \(2R_P = 3W\) | Up to 2 resolutions of forces | |
| \(R_P = W + R_S \cos\theta + F_S \sin\theta\) | \(\uparrow\) for \(A\) | |
| \(2R_S \cos\theta + 2F_S \sin\theta = W\) | \(\uparrow\) for \(C\) | |
| \(R_P \cos\theta + F_P \sin\theta = R_S + W\cos\theta\) | \(O_A \rightarrow O_C\) for \(A\) | |
| Moments about \(S\) for \(A\): \(F_P(r + r\cos\theta) + Wr\sin\theta = R_P r\sin\theta\) | ||
| \(R_P = 3W/2\) | A1 | |
| \(R_S = W/2\) | A1 | |
| \(F = (W\sin\theta)/2(1+\cos\theta)\) | A1 | Find \(F\) at \(P\) and/or \(S\) |
| \(\mu \geq \sin\theta \ / \ 3(1+\cos\theta)\) | A.G. M1 A1 | Use \(F \leq \mu R_P\) to find bound for \(\mu\) |
| \(\mu' \geq \sin\theta \ / \ (1+\cos\theta)\) | A.G. M1 | Use \(F \leq \mu' R_S\) to find bound for \(\mu'\) |
| Total | 14 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(E(X) = \int_2^4 (5x^2 - x^3 - 4x)/10 \ dx\) | M1 A1 | Find \(E(X)\) using \(\int xf(x)dx\) |
| \(= \frac{1}{2}(4^3 - 2^3) - 3(4^4 - 2^4)/40 - 3(4^2 - 2^2)/5\) | ||
| \(= 28 - 18 - 7.2 = 2.8\) | *A1 | |
| \((E(X) - 2.69)/2.69 = 0.041 < 0.1\) | M1 A1 | Verify \(E(X)\) within 10% of \(2.69\), A1 dep *A1 |
| *or* \(1.1 \times 2.69 = 2.96 > E(X)\) | ||
| Subtotal | 5 | |
| \(60\int_{2.5}^{3.6}(5x - x^2 - 4)/10 \ dx\) | M1 | Show derivation of tabular entry |
| \(= 60[3(5x^2/2 - x^3/3 - 4x)/10]_{3.2}^{3.6}\) | ||
| *or* \([45x^2 - 6x^3 - 72x]_{3.2}^{3.6}\) | ||
| \(= 122.4 - 83.328 - 28.8\) | ||
| *or* \(60 \times 0.1712 = 10.272\) | A.G. A1 | |
| Subtotal | 2 | |
| \(H_0\): \(f(x)\) fits data (A.E.F.) | B1 | State null hypothesis |
| \(O: \ldots 8\) ; \(E: \ldots 14.208\) | B1 | Combine last 2 cells since exp. value \(< 5\) |
| \(\chi^2 = 0.8126 + 0.0584 + 0.2011 + 2.7135 = 3.78[47]\) | M1 *A1 | Calculate \(\chi^2\) to 2 d.p. |
| \(\chi^2_{3,0.9} = 6.25[1]\) | ||
| \(\chi^2_{4,0.9} = 7.78\) | *B1 | |
| Accept \(H_0\) if \(\chi^2 <\) tabular value | M1 | Valid method for conclusion |
| \(3.78 < 6.25\) so \(f(x)\) does fit | A1 | Conclusion (A.E.F., dep *A1, *B1) |
| Subtotal | 7 |
# Question 10:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Stating or implying reactions $R_P$, $R_S$ same as for $B$ | B1 | |
| Stating or implying $F_P = F_S$ by moments about $O_A$ | B1 | |
| Stating or implying 3 independent equations for $F$, $R_P$, $R_S$ | $3 \times$ M1 A1 | |
| $2R_P = 3W$ | | Up to 2 resolutions of forces |
| $R_P = W + R_S \cos\theta + F_S \sin\theta$ | | $\uparrow$ for $A$ |
| $2R_S \cos\theta + 2F_S \sin\theta = W$ | | $\uparrow$ for $C$ |
| $R_P \cos\theta + F_P \sin\theta = R_S + W\cos\theta$ | | $O_A \rightarrow O_C$ for $A$ |
| Moments about $S$ for $A$: $F_P(r + r\cos\theta) + Wr\sin\theta = R_P r\sin\theta$ | | |
| $R_P = 3W/2$ | A1 | |
| $R_S = W/2$ | A1 | |
| $F = (W\sin\theta)/2(1+\cos\theta)$ | A1 | Find $F$ at $P$ and/or $S$ |
| $\mu \geq \sin\theta \ / \ 3(1+\cos\theta)$ | A.G. M1 A1 | Use $F \leq \mu R_P$ to find bound for $\mu$ |
| $\mu' \geq \sin\theta \ / \ (1+\cos\theta)$ | A.G. M1 | Use $F \leq \mu' R_S$ to find bound for $\mu'$ |
| **Total** | **14** | |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $E(X) = \int_2^4 (5x^2 - x^3 - 4x)/10 \ dx$ | M1 A1 | Find $E(X)$ using $\int xf(x)dx$ |
| $= \frac{1}{2}(4^3 - 2^3) - 3(4^4 - 2^4)/40 - 3(4^2 - 2^2)/5$ | | |
| $= 28 - 18 - 7.2 = 2.8$ | *A1 | |
| $(E(X) - 2.69)/2.69 = 0.041 < 0.1$ | M1 A1 | Verify $E(X)$ within 10% of $2.69$, A1 dep *A1 |
| *or* $1.1 \times 2.69 = 2.96 > E(X)$ | | |
| **Subtotal** | **5** | |
| $60\int_{2.5}^{3.6}(5x - x^2 - 4)/10 \ dx$ | M1 | Show derivation of tabular entry |
| $= 60[3(5x^2/2 - x^3/3 - 4x)/10]_{3.2}^{3.6}$ | | |
| *or* $[45x^2 - 6x^3 - 72x]_{3.2}^{3.6}$ | | |
| $= 122.4 - 83.328 - 28.8$ | | |
| *or* $60 \times 0.1712 = 10.272$ | A.G. A1 | |
| **Subtotal** | **2** | |
| $H_0$: $f(x)$ fits data (A.E.F.) | B1 | State null hypothesis |
| $O: \ldots 8$ ; $E: \ldots 14.208$ | B1 | Combine last 2 cells since exp. value $< 5$ |
| $\chi^2 = 0.8126 + 0.0584 + 0.2011 + 2.7135 = 3.78[47]$ | M1 *A1 | Calculate $\chi^2$ to 2 d.p. |
| $\chi^2_{3,0.9} = 6.25[1]$ | | |
| $\chi^2_{4,0.9} = 7.78$ | *B1 | |
| Accept $H_0$ if $\chi^2 <$ tabular value | M1 | Valid method for conclusion |
| $3.78 < 6.25$ so $f(x)$ does fit | A1 | Conclusion (A.E.F., dep *A1, *B1) |
| **Subtotal** | **7** | |\begin{center}
\includegraphics[max width=\textwidth, alt={}]{d3e9a568-a9ea-483e-8e65-90fdc4a69781-5_389_702_484_719}
\end{center}
Two identical uniform rough spheres $A$ and $B$, each of weight $W$ and radius $a$, are at rest on a rough horizontal plane, and are not in contact with each other. A third identical sphere $C$ rests on $A$ and $B$ with its centre in the same vertical plane as the centres of $A$ and $B$. The line joining the centres of $A$ and $C$ and the line joining the centres of $B$ and $C$ are each inclined at an angle $\theta$ to the vertical (see diagram). The coefficient of friction between each sphere and the plane is $\mu$. The coefficient of friction between $C$ and $A$, and between $C$ and $B$, is $\mu ^ { \prime }$. The system remains in equilibrium. Show that
$$\mu \geqslant \frac { \sin \theta } { 3 ( 1 + \cos \theta ) } \quad \text { and } \quad \mu ^ { \prime } \geqslant \frac { \sin \theta } { 1 + \cos \theta } .$$
\hfill \mbox{\textit{CAIE FP2 2012 Q10 EITHER}}