# Question 10(a)(i):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $I_{\text{disc}} = \frac{1}{2}(\frac{1}{4}ma^2) + \frac{1}{2}m\{(3a/2)^2 + (2a)^2\}$ | M1 A1 | Verify MI of either disc about axis $l$ at $O$ |
| $= ma^2/8 + 25ma^2/8 = 13ma^2/4$ | A1 | A.G. |

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# Question 10(a)(ii):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $I_{OC} = (4/3)\,ma^2$ | B1 | Find or state MI of rod $OC$ about $l$ |
| $I_{AB} = \frac{1}{3}\cdot 2m\,(3a/2)^2 + 2m\,(2a)^2 = (19/2)\,ma^2$ | M1 A1 | Find MI of rod $AB$ about $l$ |
| $I = (2\times 13/4 + 4/3 + 19/2)\,ma^2 = (52/3)\,ma^2$ | A1 | A.G. Verify MI of object about $l$ |

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# Question 10(a)(iii):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\omega_0 = \sqrt{(2ag)}/2a$ or $\sqrt{(g/2a)}$ | B1 | Find and use initial angular speed |
| $\frac{1}{2}I\,\omega_0^2 = \frac{1}{2}(52/3)\,ma^2 \times (g/2a) = (13/3)\,mga$ | M1 A1 | Find initial rotational KE |
| $(3mg\times 2a + mga)(1-\cos\theta)$ or $(4mg\times 7a/4)(1-\cos\theta) = 7mga(1-\cos\theta)$ | M1 A1 | Find gain in P.E. at instantaneous rest |
| $1 - \cos\theta = (13/3)\,mga/7mga = 13/21,\ \cos\theta = 8/21$ | A.G. M1 A1 | Verify $\cos\theta$ by equating KE and PE |

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# Question 10(b)(i):

| Answer/Working | Marks | Guidance |
|---|---|---|
| EITHER: $\bar{x}=5,\ \bar{y}=6$ by solving simultaneous equations | M1, A1; A1 | Find $\bar{x},\ \bar{y}$ |
| $22 + x_5 = 5\times 5,\ x_5=3$; $25+y_5=5\times 6,\ y_5=5$ | A1, A1 | Hence find missing values $x_5,\ y_5$ |
| OR: $25+y_5 = 5\times 4.5 + 0.3(22+x_5)$; $22+x_5=3(25+y_5)-5\times 13$ | (M1 A1)(A1) | Formulate simultaneous equations for $x_5,\ y_5$ |
| $y_5 = 0.3\,x_5 + 4.1,\ x_5 = 3\,y_5-12$; $x_5=3,\ y_5=5$ | (A1 A1) | Hence find missing values |

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# Question 10(b)(ii):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $r = \sqrt{(0.3\times 3)}$ or $12/\sqrt{(40\times 4)} = 0.949$ | M1 A1 | Find correlation coefficient $r$; A0 for $-0.949$ or $\pm 0.949$ |

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# Question 10(b)(iii):

| Answer/Working | Marks | Guidance |
|---|---|---|
| $\sum x = 25+20=45,\ \sum x^2=165+100=265$ | B1 | Find corresponding summations for combined data (B1 needs all 5 correct) |
| $\sum y=30+17=47,\ \sum y^2=184+69=253,\ \sum xy=162+75=237$ | | |
| $S_{xy}=237-45\times 47/10=25.5$ | | |
| $S_{xx}=265-45^2/10=62.5$ | | |
| $S_{yy}=253-47^2/10=32.1$ | | |
| $r' = S_{xy}/\sqrt{(S_{xx}S_{yy})} = 25.5/44.79 = 0.569$ | M1 *A1 | Find correlation coefficient $r'$ |
| $H_0: \rho=0,\ H_1: \rho\neq 0$ | B1 | State both hypotheses (B0 for $r\ldots$) |
| $r_{10,\ 5\%} = 0.632$ | *B1 | State or use correct tabular two-tail $r$-value |
| Accept $H_0$ if $|r'| <$ tabular value (AEF) | M1 | |
| $0.569 < 0.632$ so popln. pmcc not different from 0 | DA1 | Correct conclusion (AEF, dep *A1, *B1) |