# Question 6:

## Part (a)
$[v] = LT^{-1}$ | B1 (AO 1.2) | Used in solution; penalise wrong basic terms only once
$[u^\alpha a^\beta t^\gamma] = L^\alpha T^{-\alpha} L^\beta T^{-2\beta} T^\gamma$ | B1 (AO 3.3) | Correctly finding the dimensions of $u^\alpha a^\beta t^\gamma$ in terms of $\alpha$, $\beta$ and $\gamma$; allow unsimplified
$1 = \alpha + \beta$ or $-1 = -\alpha - 2\beta + \gamma$ | M1 (AO 3.4) | Equating their dimensions L and T
$\Rightarrow \alpha = 1 - \beta$ | A1 (AO 1.1) |
$\Rightarrow \gamma = \beta$ | A1 [5] (AO 1.1) | www; if extra term such as M is included, then B1B0M1A0A0

## Part (b)
For straight line graph $t^\gamma$ must be 1 (or constant or $t^0$) or $t$ (or $t^1$)... | M1 (AO 3.1b) | For clear understanding that the relationship must be of the form $v = mt + c$ where both $mt$ and $c$ must take the form $[k]u^\alpha a^\beta t^\gamma$; or could see e.g. $v = u^{1-\beta}a^\beta t^\beta$ with $\beta = 1 \Rightarrow v = at$ and $\beta = 0 \Rightarrow v = u$
...(so $\gamma = 0$ or 1) so $\beta = 0$ or 1 | A1 [2] (AO 1.1) | SC1 for $\beta = 1$ using direct proportion or unsupported but www e.g. $\beta = -1$

## Part (c)
$v$ must be the sum of terms like $ku^\alpha a^\beta t^\gamma$ | M1 (AO 2.1) | AG (or $k_1u + k_2at$ or $mt + c$); award if at least one term seen, must have $k$, $u$, $a$ and $t$
$v = k_1u + k_2at$ and $v = u$ when $t = 0 \Rightarrow k_1 = 1$ | A1 (AO 3.4) |
and $v = u + a$ when $t = 1 \Rightarrow k_1 = 1$ so $v = u + at$ | A1 [3] (AO 2.2a) |

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