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TMUA 2016 · Paper 1 · Question 3 of 20

TMUA 2016 Paper 1 Question 3

Differentiation and integration — Normal to a curve · intercepts and distance. Try it first; the answer and a full worked solution are below.

TMUA 2016 · Paper 1Differentiation and integrationNormal to a curve · intercepts and distance6 options
A line is drawn normal to the curve y = 2x2 at the point on the curve where x = 1.

This line cuts the x-axis at P and the y-axis at Q.

The length of PQ is

  1. A3√52
  2. B3√174
  3. C7√174
  4. D354
  5. E35√52
  6. F3√172
Show the answer and worked solution
answer · C
  1. A3√52
  2. B3√174
  3. C7√174
  4. D354
  5. E35√52
  6. F3√172
Write y = 2x−2, so dydx = −4x−3, which is −4 at x = 1. The normal therefore has gradient 14 and passes through (1,  2), giving y = x4 + 74. Setting y = 0 gives P = (−7,  0) and setting x = 0 gives Q = (0,  74). Then PQ = √49 + 4916 = 7√1716 = 7√174.