| **Answer/Working** | **Mark** | **Guidance** |
|---|---|---|
| Use of $s = ut + \frac{1}{2}at^2$ | M1 | Form equation in $u$ and $a$. N.B. Marks are available if they use two other unknowns, rather than $u$ and $a$ |
| $20 = 3u + \frac{9a}{2}$ | A1 | Correct unsimplified equation |
| Use of $suvat$ | M1 | Form second equation in $u$ and $a$. N.B. Marks are available if they use the same two other unknowns, rather than $u$ and $a$ |
| $10 = (u+3a) + \frac{a}{2}$ or $30 = 4u + 8a$ | A1 | Correct unsimplified equation |
| $30 = 3u + \frac{21a}{2} \Rightarrow 10 = 6a, a = \frac{5}{3}$ | M1 | Solve for $u$ or $a$ Or for one of their unknowns. |
| $u = \frac{25}{6}$ | A1 | $u$ and $a$ both correct or both their unknowns correct. Accept equivalent forms. 1.7, 4.2 or better |
| Use of $v = u + at$, $20 = \frac{25}{6} + \frac{5}{3}t$ | M1 | Complete method using $suvat$ to find $t$. Correct unsimplified for their $u, a$. cao |
| $t = 9.5$ (s) | A1 | |
| | | (8) |

# Question 3a:

| **Answer/Working** | **Mark** | **Guidance** |
|---|---|---|
| Allow use of column vectors $(5\mathbf{i}+2\mathbf{j})+(-3\mathbf{i}+\mathbf{j})+\mathbf{F}_3 = \mathbf{0}$ oe | M1 | Use equilibrium to find $\mathbf{F}_3$ |
| $\mathbf{F}_3 = -2\mathbf{i}-3\mathbf{j}$ ($\Rightarrow a = -2, b = -3$) | A1 | Correct $\mathbf{F}_3$ |
| $\tan\theta = \frac{2}{3}$ | M1 | For an equation in a relevant angle using their $a$ and $b$ |
| $\theta = 33.7°$ | A1 | 34° or better. 0.588 (0.59) rads |
| | | (4) |

# Question 3b:

| **Answer/Working** | **Mark** | **Guidance** |
|---|---|---|
| Resultant force $= (2+\lambda)\mathbf{i}+(3+3\lambda)\mathbf{j}$ | B1 | Seen or implied. They must collect the $\mathbf{i}$'s and $\mathbf{j}$'s. |
| $\mathbf{F} = 4a$ oe, where $\mathbf{F}$ is their resultant, seen or implied ($\text{could be implied by } \|\mathbf{F}\| = 13$) | M1 | Must have attempted to add all 3 forces. N.B. $3.25 = \frac{1}{4}[(2+\lambda)\mathbf{i}+(3+3\lambda)\mathbf{j}]$ oe Scores B1M1M0A0 but allow recovery. |
| Finding magnitude of their $a$ or $\mathbf{F}$ $\sqrt{\left(\frac{2+\lambda}{4}\right)^2+\left(\frac{3+3\lambda}{4}\right)^2}$ or $\sqrt{(2+\lambda)^2+(3+3\lambda)^2}$ | M1 | |
| Use of $\|\mathbf{a}\| = 3.25$ or $\|\mathbf{F}\| = 13$ to form (3 term quadratic in $\lambda$) = 0 $(10\lambda^2+22\lambda-156 = 0)$ | M1 | |
| $\lambda = 3$ | A1 | A0 if they give 2 values. |
| | | (5) [9] |

# Question 4a:

| **Answer/Working** | **Mark** | **Guidance** |
|---|---|---|
| $\uparrow T-(15g+25g) = (15+25)\times0.2$ | M1 | All terms required. Must be in $T$ only. Condone sign errors |
| $T = 400$ (N) | A1 | Correct unsimplified equation in $T$ Must be positive |
| | | (3) |

# Question 4b:

| **Answer/Working** | **Mark** | **Guidance** |
|---|---|---|
| $\uparrow 12g-R = -0.1\times12$ | M1 | All terms required. Condone sign errors |
| $R = 119$ (N) (120) | A1 | Correct unsimplified equation in $R$ only. Allow $+ R$ at this stage Must be positive |
| | | (3) [6] |

# Question 5a:

| **Answer/Working** | **Mark** | **Guidance** |
|---|---|---|
| Allow use of column vectors $\mathbf{a} = \frac{(\mathbf{i}+7\mathbf{j})-(-3\mathbf{i}+5\mathbf{j})}{0.5}$ oe | M1 | Use of $\mathbf{a} = \frac{\mathbf{v}-\mathbf{u}}{t}$ Allow $\mathbf{u}$ and $\mathbf{v}$ reversed Or equivalent |
| $\mathbf{a} = -4\mathbf{i}+4\mathbf{j}$ | A1 | |
| $\Rightarrow \mathbf{v}_p = (3\mathbf{i}+5\mathbf{j})+(-4\mathbf{i}+4\mathbf{j})t$ | M1 | For their $\mathbf{a}$ |
| $= (3-4t)\mathbf{i}+(5+4t)\mathbf{j}$ | A1ft | Follow their $\mathbf{a}$. Must collect $\mathbf{i}$'s and $\mathbf{j}$'s This could be implied in subsequent working |
| $\Rightarrow 5+4T = -2(3-4T)$ | M1 | Use of correct ratio to form equation in $T$ (allow $t$) |
| $T = \frac{11}{4}$ oe | A1 | cao |
| | | (6) |

# Question 5b:

| **Answer/Working** | **Mark** | **Guidance** |
|---|---|---|
| $\mathbf{v}_p = \mathbf{v}_Q \Rightarrow \begin{pmatrix} 3-4t \\ 5+4t \end{pmatrix} = \begin{pmatrix} -4-2t \\ \mu+3t \end{pmatrix}$ | M1 | Equate velocities and form two equations in $t$ and $\mu$ i.e. equate coefficients of $\mathbf{i}$ and $\mathbf{j}$ oe Follow their $\mathbf{v}_p$ |
| $\Rightarrow 3-4t = -4-2t$ and $5+4t = \mu+3t$ | M1 | |
| $\mu = 8.5$ oe | M1 | Solve for $\mu$. Follow their $\mathbf{v}_p$ |
| | A1 | cao |
| | | (3) [9] |

# Question 6a:

| **Answer/Working** | **Mark** | **Guidance** |
|---|---|---|
| Resolve perpendicular to the plane $R = 6g\cos\theta$ | M1 | Condone sin/cos confusion |
| $F = \frac{1}{4}R = \frac{18g}{13} = 13.6(\text{N})$ or 14(N) | A1 | Correct resolution 2 sf or 3 sf for decimal answer |
| | | (3) |

# Question 6b:

| **Answer/Working** | **Mark** | **Guidance** |
|---|---|---|
| Equation of motion parallel to the plane | M1 | Need all terms and dimensionally correct. Condone sign errors and sin/cos confusion. |
| $-F-6g\sin\theta = 6a$ | A1 | Correct unsimplified equation in $F$ Allow $-6a$ on RHS |
| $0 = 5^2+2\times as$ | M1 | Complete method using $suvat$ and calculated $a$ ($a \neq g$) to find $s$ This is independent of previous M mark but they must have found a value for $a$. |
| $0 = 5^2-2\times\frac{8g}{13}s$ | A1 | Correct unsimplified equation. Allow $(-s)$ |
| $s = 2.07(\text{m})$ or $2.1(\text{m})$ | A1 | Must be positive. |
| | | (5) |

# Question 6c:

| **Answer/Working** | **Mark** | **Guidance** |
|---|---|---|
| Equation of motion parallel to the plane | M1 | Need all terms and dimensionally correct. Condone sign errors and sin/cos confusion. |
| $6g\sin\theta - F = 6a'$ | A1 | Correct unsimplified equation in $F$ |
| $5^2 = 0 + 2a's$ | M1 | Complete method using $suvat$, with $a' \neq a$ and $a' \neq g$ to find $s$ |
| $5^2 = 0+2\times\frac{2g}{13}\times s$ | A1 | Correct unsimplified equation |
| $8.29(\text{m})$ or $8.3(\text{m})$ | A1 | |
| | | (5) [13] |

# Question 7a:

| **Answer/Working** | **Mark** | **Guidance** |
|---|---|---|
| Possible equations: $\uparrow R+3R(=4R) = 60g$ M(A), $60gx = R\times a+3R\times 6a$ M(B), $60g(8a-x) = R\times 7a+3R\times 2a$ M(C), $60g(x-a) = 3R\times 5a$ M(D), $60g(6a-x) = R\times 5a$ | M1A1 M1A1 | Two equations required. For each equation, M1 for correct no. of terms, dim correct but condone sign errors. A1 for a correct unsimplified equation. Consistent omission of $g$ could score full marks. Inconsistent omission of $g$ is an A error. All four of these marks could be scored for consistent use of another unknown length which is clearly defined e.g. on a diagram N.B. M marks only available if using $R$ and $3R$ oe but allow if wrong way round. For vertical resolution, can score M1A1, even if wrong way round. |
| S.C. M(G), $R(x-a) = 3R(6a-x)$ | M2A2 | -1 each error |
| $x = \frac{19a}{4}$ oe | A1 | Or equivalent |
| | | (5) |

# Question 7b:

| **Answer/Working** | **Mark** | **Guidance** |
|---|---|---|
| Possible equations: ($\uparrow$), $60g+Mg = S+S$ M(A), $60g(8a-x)+Mg\times 2a = S\times 7a+S\times 2a$ M(B), $60g(x-a)+Mga = S\times 5a$ M(C), $60g(6a-x)+Mg\times 4a = S\times 5a$ M(D), M(G), $S(x-a) = S(6a-x)+Mg(x-2a)$ | M1A1ft M1A1ft | Two equations in two unknowns ($M$ and $S$) required. For each equation, M1 for correct no. of terms, dim correct but condone sign errors. A1ft for a correct unsimplified equation, follow their $x$. $x$ must be substituted to earn the A marks. Consistent omission of $g$ could score full marks. Inconsistent omission of $g$ is an A error. |
| $M = 50$ | A1 | Exact answer only. |
| | | (5) [10] |

# Question 8a:

| **Answer/Working** | **Mark** | **Guidance** |
|---|---|---|
| Correct shape for sketch for A, starting at the origin. Correct shape for sketch for B, must be correct relative to A, crossing it and ending at same time. Must be done on the same axes. | B1 B1 | B0 if solid vertical line at the end of either. Tram B starts later and acceleration greater. |
| 5, 20, 24 shown | DB1 | Dependent on previous two marks |
| | | (3) |

# Question 8b:

| **Answer/Working** | **Mark** | **Guidance** |
|---|---|---|
| $t = 20+\frac{10}{3}\left(=\frac{70}{3}\right)$ | B1 | |
| Distance travelled for either vehicle | M1 | |
| $\frac{1}{2}\times\frac{10}{3}\times10$ OR $\frac{1}{2}\times 5\times 10 + \frac{55}{3}\times 10$; $\frac{1}{70}\left(\frac{70}{3}+\frac{70}{3}-5\right)\times 10$ | A1 | |
| Find second distance and subtract | M1 | |
| $d = \frac{625}{3}-\frac{50}{3} = \frac{575}{3} = 191\frac{2}{3}$ | A1 | Accept 192 or better. |
| | | (5) |

# Question 8c:

| **Answer/Working** | **Mark** | **Guidance** |
|---|---|---|
| Equate distances from O | M1 | Find both distances at time $t$ seconds and equate, using correct structure – see examples. Correct unsimplified equation, –1each error (up to a maximum of 2) |
| $\left(\frac{t+t-5}{2}\right)\times 10 = \left(\frac{t-20+t-24}{2}\right)\times 12$ OR $\left(\frac{1}{2}\times 5\times 10\right)+10(t-5) = \left(\frac{1}{2}\times 4\times 12\right)+12(t-24)$ | A2 | |
| $t = 119.5$ | M1 | Solve for $t$ |
| Distance $= 5\times(6\times 44-30) = 1170$ (m) | A1 | Accept 1200 or better |
| | | (5) [13] |