Theory question 115
Linear and Simultaneous Equations(a) Under the mapping h(x) = px2 − qx + 2, the image of 3 is 14 and the image of −2 is 24. Find: (i) the values of p and q; (ii) the elements whose image is 4.
(b) In the pentagon ABCDE, AE = (2, 0), DE = (−1, 2), DC = (−2, −5) and CB = (−3, 2) (column vectors). Show that ABCD is a parallelogram.
Show the model answer
(a)(i) h(3) = 9p − 3q + 2 = 14 gives 3p − q = 4. h(−2) = 4p + 2q + 2 = 24 gives 2p + q = 11. Adding: 5p = 15, p = 3, q = 5.
(ii) 3x2 − 5x + 2 = 4 → 3x2 − 5x − 2 = 0 → (3x + 1)(x − 2) = 0, so x = 2 or x = −⅓.
(b) AB = AE + ED + DC + CB = (2, 0) + (1, −2) + (−2, −5) + (−3, 2) = (−2, −5).
AB = DC = (−2, −5), so AB and DC are equal and parallel; therefore ABCD is a parallelogram.
A quadrilateral with one pair of opposite sides equal and parallel is a parallelogram.