Show that \(f’(1) = 1\).

Find the equation of \(L\) in the form \(y = ax + b\).

Find the \(x\)-coordinate of Q.

Find the area of the region enclosed by the graph of \(f\) and the line \(L\).

\(f’(x) = 2x - 1\)     A1A1

correct substitution     A1

eg\(\,\,\,\,\,\)\(2(1) - 1,{\text{ }}2 - 1\)

\(f’(1) = 1\)     AG     N0

[3 marks]

correct approach to find the gradient of the normal     (A1)

eg\(\,\,\,\,\,\)\(\frac{{ - 1}}{{f'(1)}},{\text{ }}{m_1}{m_2} =  - 1,{\text{ slope}} =  - 1\)

attempt to substitute correct normal gradient and coordinates into equation of a line     (M1)

eg\(\,\,\,\,\,\)\(y - 0 =  - 1(x - 1),{\text{ }}0 =  - 1 + b,{\text{ }}b = 1,{\text{ }}L =  - x + 1\)

\(y =  - x + 1\)     A1     N2

[3 marks]

equating expressions     (M1)

eg\(\,\,\,\,\,\)\(f(x) = L,{\text{ }} - x + 1 = {x^2} - x\)

correct working (must involve combining terms)     (A1)

eg\(\,\,\,\,\,\)\({x^2} - 1 = 0,{\text{ }}{x^2} = 1,{\text{ }}x = 1\)

\(x =  - 1\,\,\,\,\,\left( {{\text{accept }}Q( - 1,{\text{ }}2)} \right)\)     A2     N3

[4 marks]

valid approach     (M1)

eg\(\,\,\,\,\,\)\(\int {L - f,{\text{ }}\int_{ - 1}^1 {(1 - {x^2}){\text{d}}x} } \), splitting area into triangles and integrals

correct integration     (A1)(A1)

eg\(\,\,\,\,\,\)\(\left[ {x - \frac{{{x^3}}}{3}} \right]_{ - 1}^1,{\text{ }} - \frac{{{x^3}}}{3} - \frac{{{x^2}}}{2} + \frac{{{x^2}}}{2} + x\)

substituting their limits into their integrated function and subtracting (in any order)     (M1)

eg\(\,\,\,\,\,\)\(1 - \frac{1}{3} - \left( { - 1 - \frac{{ - 1}}{3}} \right)\)

Note:     Award M0 for substituting into original or differentiated function.

area \( = \frac{4}{3}\)     A2     N3

[6 marks]