Theory question 113
Coordinate Geometry(a) Find the equation of a straight line which passes through the point (2, −3) and is parallel to the line 2x + y = 6.
(b) The operation Δ is defined on the set T = {2, 3, 5, 7} by x Δ y = (x + y + xy) mod 8. (i) Construct modulo 8 table for the operation Δ on the set T. (ii) Use the table to find: I. 2 Δ (5 Δ 7); II. 2 Δ n = 5 Δ 7.
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Model answer
(a) Gradient of 2x + y = 6 is −2; y + 3 = −2(x − 2) → y = −2x + 1 (or 2x + y = 1)
(b)(i) Rows x = 2, 3, 5, 7; columns y = 2, 3, 5, 7:
2: 0, 3, 1, 7
3: 3, 7, 7, 7
5: 1, 7, 3, 7
7: 7, 7, 7, 7
(ii) I. 5 Δ 7 = 7, so 2 Δ 7 = 7
II. 5 Δ 7 = 7; from row 2, 2 Δ n = 7 when n = 7
Explanation
Parallel lines have equal gradients.