(i) Robot 1 starts from the origin and traces a path by repeatedly moving 3 units to the right and then 4 units upward. Robot 2 starts from the point (10, 0) and repeatedly moves 1 unit to the right and then 2 units upward. Will the paths traced by these two robots intersect and if so, where?
(ii) Robot 1 starts from (3, 0) and traces a path by repeatedly moving 5 units to the right and then 5 units upward. Robot 2 starts from the point (7, 0) and repeatedly moves 5 units to the right and then 3 units downwards. Will the paths traced by these two robots intersect and if so, where?
- (i) Robot 1 starts from the origin and traces a path by repeatedly moving 3 units to the right and then 4 units upward. Robot 2 starts from the point (10, 0) and repeatedly moves 1 unit to the right and then 2 units upward. Will the paths traced by these two robots intersect and if so, where?
- (ii) Robot 1 starts from (3, 0) and traces a path by repeatedly moving 5 units to the right and then 5 units upward. Robot 2 starts from the point (7, 0) and repeatedly moves 5 units to the right and then 3 units downwards. Will the paths traced by these two robots intersect and if so, where?
Answer: (i) Yes, at (30, 40): Robot 1 follows y = 43x and Robot 2 follows y = 2(x − 10). (ii) No, as straight tracks: the lines y = x − 3 and y = −35(x − 7) would meet only at (4.5, 1.5), behind Robot 2’s start, where Robot 2 never goes. (Traced step by step, Robot 1’s first move along the x-axis, from (3, 0) to (8, 0), passes over Robot 2’s starting point (7, 0), so the two step paths share only that short stretch from (7, 0) to (8, 0).)
Step-by-step solution
Idea: Each robot’s steady pattern of “right, then up/down” gives a straight track with slope = rise per roundrun per round through its starting point. Find where the two lines meet, then check that the point is on the part of each line the robot actually travels (both robots only move to the right).
(i) Robot 1 starts from the origin and traces a path by repeatedly moving 3 units to the right and then 4 units upward. Robot 2 starts from the point (10, 0) and repeatedly moves 1 unit to the right and then 2 units upward. Will the paths traced by these two robots intersect and if so, where?
- Robot 1: slope 43 through (0, 0): y = 43x. Robot 2: slope 21 = 2 through (10, 0): y = 2(x − 10) = 2x − 20.1 mark
- Equate: 43x = 2x − 20 ⇒ 4x = 6x − 60 ⇒ x = 30, y = 40.1 mark
- Both robots reach this point: Robot 1 after 10 rounds (10 × 3 = 30, 10 × 4 = 40) and Robot 2 after 20 rounds (10 + 20 = 30, 20 × 2 = 40). Yes, the paths meet at (30, 40).½ mark
- Note: if the zig-zag steps themselves are drawn exactly, the corners stick out on either side of the straight tracks, and the step paths already touch a little earlier, first at (24, 28). The straight-line answer (30, 40) is the meeting point of the two tracks.
Yes, at (30, 40)
(ii) Robot 1 starts from (3, 0) and traces a path by repeatedly moving 5 units to the right and then 5 units upward. Robot 2 starts from the point (7, 0) and repeatedly moves 5 units to the right and then 3 units downwards. Will the paths traced by these two robots intersect and if so, where?
- Robot 1: slope 55 = 1 through (3, 0): y = x − 3, for x ≥ 3. Robot 2: slope −35 through (7, 0): y = −35(x − 7), for x ≥ 7.1 mark
- As full lines: x − 3 = −35(x − 7) ⇒ 5x − 15 = −3x + 21 ⇒ x = 4.5, y = 1.5.1 mark
- But Robot 2 starts at x = 7 and only moves right, so it never reaches x = 4.5. For x ≥ 7, Robot 1 is above the x-axis and climbing while Robot 2 is on or below it and falling. So, as straight tracks, the paths do not intersect (they only would if Robot 2’s line were extended backwards to (4.5, 1.5)).½ mark
- Note: traced as exact steps, Robot 1’s first move runs along the x-axis from (3, 0) to (8, 0), so it passes over Robot 2’s start (7, 0) and the stretch to (8, 0). Apart from that shared bit of the starting line, the tracks separate.
No, as straight tracks: the lines meet only at (4.5, 1.5), which is behind Robot 2’s start. (Traced step by step, Robot 1’s first move passes over Robot 2’s start (7, 0), so the step paths share only the stretch (7, 0) to (8, 0).)
(i) Yes, at (30, 40). (ii) No, as straight tracks: they would meet only at (4.5, 1.5), a point Robot 2 never reaches. Traced step by step, the paths share only the start stretch from (7, 0) to (8, 0), where Robot 1’s first move passes Robot 2’s starting point.
Check: (i) (30, 40): 43 × 30 = 40 ✓ and 2(30) − 20 = 40 ✓. (ii) (4.5, 1.5): 4.5 − 3 = 1.5 ✓ and −35(−2.5) = 1.5 ✓, but 4.5 < 7.
Answer to write in the exam
(i)
Robot 1: slope 43, through (0, 0) ⇒ y = 43x
Robot 2: slope 2, through (10, 0) ⇒ y = 2x − 20
43x = 2x − 20 ⇒ x = 30, y = 40
∴ The paths intersect at (30, 40)
(ii)
Robot 1: y = x − 3 (x ≥ 3); Robot 2: y = −35(x − 7) (x ≥ 7)
x − 3 = −35(x − 7) ⇒ 8x = 36 ⇒ x = 4.5, y = 1.5
Robot 2 never goes left of x = 7, so (4.5, 1.5) is not on its path
Step paths: Robot 1’s first move (3, 0) → (8, 0) passes Robot 2’s start (7, 0)
∴ No, as straight tracks; the step paths share only the stretch (7, 0) to (8, 0)
Common mistakes that cost marks
- Using slope 34 (run over rise) for Robot 1.
- In (ii), stopping at (4.5, 1.5) without checking that Robot 2 can actually be there.
- Writing Robot 2’s line as y = 2x and forgetting it starts at (10, 0).
How this can come in the exam
MCQ (1 mark)
A robot starts at (2, 1) and repeatedly moves 2 right and 6 up. Its track is
- y = 3x − 5
- y = 3x + 1
- y = 13x + 1
- y = 3x − 2
Show answer
(A) y = 3x − 5
Slope 3 through (2, 1): y − 1 = 3(x − 2) ⇒ y = 3x − 5.
Try one yourself
Ant A starts at (0, 0) moving 1 right, 1 up repeatedly; ant B starts at (6, 0) moving 1 right, 2 up repeatedly. Where do their straight tracks meet?
Show answer
y = x and y = 2x − 12 ⇒ (12, 12).
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