# Question 10:

## Part (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.) | |

## Part (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 $l$ |
| $I = 2\times 13/4 + 4/3 + 19/2 = (52/3)\,ma^2$ | A1 (A.G.) | Verify MI of object about $l$ |

## Part (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 PE at instantaneous rest |
| $1 - \cos\theta = (13/3)\,mga/7mga = 13/21,\ \cos\theta = 8/21$ | M1 A1 (A.G.) | Verify $\cos\theta$ by equating KE and PE |

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# Question 10(b):

## Part (b)(i):

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

## Part (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 $\pm 0.949$ |

## Part (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) |
| $\sum y = 30+17=47,\ \sum y^2 = 184+69=253,\ \sum xy = 162+75=237$ | | |
| $S_{xy} = 237 - 45\times 47/10 = 25.5$ | M1 *A1 | Find correlation coefficient $r'$ |
| $S_{xx} = 265 - 45^2/10 = 62.5$ | | |
| $S_{yy} = 253 - 47^2/10 = 32.1$ | | |
| $r' = S_{xy}/\sqrt{(S_{xx}S_{yy})} = 25.5/44.79 = 0.569$ | | |
| $H_0: \rho=0,\ H_1: \rho\neq 0$ | 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 | State or imply valid method for conclusion |
| Population pmcc not different from 0 | DA1 | Correct conclusion (AEF, dep *A1, *B1) |