# Question 2

## Part (a)
M1: First four arcs (AB, BC, CF, CE) correctly chosen, or first five nodes (ABCFE) correctly chosen in order. If any rejections seen at any point then M1 (max) only.

A1: First six arcs correctly chosen (AB, BC, CF, CE, FG, AD), or all nodes in order (ABCFEGDHI).

A1: CSO (must be arcs).

## Part (b)
B1: CAO

## Part (c)(i)
M1: Kruskal's - first three arcs (CF, AB, FG) correctly chosen and at least one rejection seen at some point.

A1: All arcs in tree selected correctly at correct time (CF, AB, FG, AD, EH, HI). Ignore any reference to BC and EF.

A1: CSO including all rejections correct and at the correct time. Ignore any reference to BC and EF.

## Part (c)(ii)
B1: Partially correct answer – e.g. an indication that the arcs (BC and EF) are not connected or any mention of the tree being (initially) disconnected - so in both of these examples a pertinent correct statement is made but no explicit mention is made to either of the two minimum connector algorithms (i.e. no mention is made of Prim requiring arcs to be connected or that Kruskal can grow in a disconnected fashion). Give bod but for this mark there must be some mention of the 'unconnected' nature of the two initial arcs or problem. Note: describing how Kruskal can be adapted to find the MST scores no marks.

B1: Fully correct answer (so either Kruskal allows a tree to be formed from initially unconnected arcs or Prim requires the arcs/tree to be connected at all times - so linking the correct algorithm with the issues of this particular problem) – do not condone incorrect technical language for this mark (e.g. vertex for arc, point for vertex etc.).

## Part (d)
B1: CAO

## Misread Notes
Starting at a node other than A scores M1 only – must have the first four arcs (or five nodes) correct.

| Starting at | Minimum arcs required for M1 only | Nodes |
|---|---|---|
| A | AB, BC, CF, CE | ABCFE |
| B | AB, BC, CF, CE | BACFE |
| C | CF, CE, FG, BC | CFEGB |
| D | AD, AB, BC, CE | DABCE |
| E | CE, CF, FG, BC | ECFGB |
| F | CF, CE, FG, BC | FCEGB |
| G | FG, CF, CE, BC | GFCEB |
| H | EH, CE, CF, FG | HECFG |
| I | HI, EH, CE, CF | IHECF |

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# Question 3

## Part (a)
B1: pairing or one to one

B1: element(s) from one set with element(s) of the other.

## Part (b)
M1: Alternating path from B to 6 - or vice versa

A1: CAO including change status (stated or shown), chosen path clear.

A1: CAO. Must follow from correct stated path, diagram okay (must be a clear diagram with only five arcs)

## Part (c)
M1: Alternating path from H to 3 (or vice versa)

A1: CAO including change status (stated or shown), chosen path clear.

A1: CAO. Must follow from two correct stated paths, diagram okay (must be a clear diagram with only six arcs). Must have scored both M marks in part (b) and (c).

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# Question 4

## Part (a)
M1: Three distinct pairings of their four odd nodes

A1: One row correct including pairing and total

A1: Two rows correct including pairing and total

A1: Three rows correct including pairing and total

A1: CAO correct arcs identified AB, BD, DH, HI, EG, GJ (accept ABDHI and EGJ).

## Part (b)
B1 ft: Must have a choice of at least two pairs seen in part (a). 379 + their least from (a).

## Part (c)
M1: Aim to include their AE (56) [ft from (a)] and add IJ (35) or 35 + '56' or 367 + 35 + '56'. Must see a numerical argument. Or if AE + IJ was the smallest pairing from (a) then comparing/mention of 35 with 38.

A1 ft: Correct calculation and conclusion from their working.

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# Question 5

## Part (a)
M1: Big replaced by smaller at least once in the working values at either C or D or T.

A1: S, A, B and E boxes all correct, including order of labelling.

A1: C and D boxes all correct (including working values in the correct order). Penalise order of labelling only once per question (so C and D labelled in that order with C labelled after S, A, B and E).

A1 ft: T correct ft (including working values in the correct order). Penalise order of labelling only once per question (so T labelled after all other nodes).

A1: Route (SAECDT) CAO

A1 ft: ft on their final value (if their answer is not 106 ft their final value at T) – ignore incorrect/lack of units.

## Part (b)
DM1: Must have scored the M mark in (a). Finding at least one correct path from S to T excluding arc CE.

A1: Both paths correct (SACDT and SBDT)

A1: Length (109) CAO (ignore incorrect/lack of units)

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# Question 6

## Part (a)
B3, 2, 1, 0: Lines must pass through one small square of points stated.

B1: for two lines drawn correctly

B1: for three lines drawn correctly

B1: for all four lines drawn correctly

B1: Region, R, labelled correctly - not just implied by shading - must have scored all three previous marks in this part.

**Lines information:**
- $x + y = 500$ passes through (0, 500), (250, 250), (500, 0)
- $5x + 4y = 4000$ passes through (0, 1000), (400, 500), (800, 0)
- $y = 2x$ passes through (0, 0), (200, 400), (400, 800)
- $y = x - 250$ passes through (250, 0), (500, 250), (700, 450)

## Part (b)
M1: Must see simultaneous equations being used to find 'exact' point (or correct to 2 dp).

M1: Must see simultaneous equations being used to find 'exact' point (or correct to 2 dp).

A1: accept awrt (555.56, 305.56)

A1: accept awrt (307.69, 615.38)

M1: Evaluating C at both of their points and clearly selecting their optimal point

A1: CAO, accept answer correct to 4 s.f. (either truncated or rounded)

**SC:** If no working shown and coordinates are given exactly or correct to 2dp then award M0M0A1A1 (if one coordinate correct then M0M0A1A0 or M0M0A0A1 – award in order as given in b1A1 and b2A1)

## Part (c)
M1: Seeking to find $x + y$ at their optimal point.

A1: CAO, accept awrt 861.11

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# Question 7

## Part (a)
M1: All top boxes complete, values generally increasing left to right, condone one rogue value.

A1: CAO

M1: All bottom boxes complete, values generally decreasing right to left, condone one rogue value. Condone missing 0 or 29 for the M only.

A1: CAO

## Part (b)
M1: Not a scheduling diagram. At least 9 activities including at least 4