Write down

(i)     the equation of the vertical asymptote to the graph of \(y = f (x)\) ;

(ii)    the solution to the equation \(f (x) = 0\) ;

(iii)   the coordinates of the local minimum point.

Find  \(f'(x)\) .

Show that \(f'(x)\) can be written as \(f'(x) = \frac{{2{x^3} - 2{x^2} - 9}}{{{x^2}}}\) .

Find the gradient of the tangent to \(y = f (x)\) at the point \({\text{A}}(1{\text{, }}8)\) .

The line, \(L\), passes through the point A and is perpendicular to the tangent at A.

Write down the gradient of \(L\) .

The line, \(L\) , passes through the point A and is perpendicular to the tangent at A.

Find the equation of \(L\) . Give your answer in the form \(y = mx + c\) .

The line, \(L\) , passes through the point A and is perpendicular to the tangent at A.

\(L\) also intersects the graph of \(y = f (x)\) at points B and C . Write down the x-coordinate of B and of C .

(i)     \(x = 0\)     (A1)(A1)

Note: Award (A1) for \(x = \) a constant, (A1) for the constant in their equation being \(0\).

(ii)    \( - 1.58\) (\( - 1.58454 \ldots \))     (G1)

Note: Accept \( - 1.6\), do not accept \( - 2\) or \( - 1.59\).

(iii)   \((2.06{\text{, }}4.49)\) \((2.06020 \ldots {\text{, }}4.49253 \ldots )\)     (G1)(G1)

Note: Award at most (G1)(G0) if brackets not used. Award (G0)(G1)(ft) if coordinates are reversed.

Note: Accept \(x = 2.06\), \(y = 4.49\) .

Note: Accept \(2.1\), do not accept \(2.0\) or \(2\). Accept \(4.5\), do not accept \(5\) or \(4.50\).

[5 marks]

\(f'(x) = 2x - 2 - \frac{9}{{{x^2}}}\)     (A1)(A1)(A1)(A1)

Notes: Award (A1) for \(2x\), (A1) for \( - 2\), (A1) for \( - 9\), (A1) for \({x^{ - 2}}\) . Award a maximum of (A1)(A1)(A1)(A0) if there are extra terms present.

[4 marks]

\(f'(x) = \frac{{{x^2}(2x - 2)}}{{{x^2}}} - \frac{9}{{{x^2}}}\)     (M1)

Note: Award (M1) for taking the correct common denominator.

\( = \frac{{(2{x^3} - 2{x^2})}}{{{x^2}}} - \frac{9}{{{x^2}}}\)     (M1)

Note: Award (M1) for multiplying brackets or equivalent.

\( = \frac{{2{x^3} - 2{x^2} - 9}}{{{x^2}}}\)     (AG)

Note: The final (M1) is not awarded if the given answer is not seen.

[2 marks]

\(f'(1) = \frac{{2{{(1)}^3} - 2(1) - 9}}{{{{(1)}^2}}}\)     (M1)

\( = - 9\)     (A1)(G2)

Note: Award (M1) for substitution into given (or their correct) \(f'(x)\) . There is no follow through for use of their incorrect derivative.

[2 marks]

\(\frac{1}{9}\)     (A1)(ft)

Note: Follow through from part (d).

[1 mark]

\(y - 8 = \frac{1}{9}(x - 1)\)     (M1)(M1)

Notes: Award (M1) for substitution of their gradient from (e), (M1) for substitution of given point. Accept all forms of straight line.

\(y = \frac{1}{9}x + \frac{{71}}{9}\) (\(y = 0.111111 \ldots x + 7.88888 \ldots \))     (A1)(ft)(G3)

Note: Award the final (A1)(ft) for a correctly rearranged formula of their straight line in (f). Accept \(0.11x\), do not accept \(0.1x\). Accept \(7.9\), do not accept \(7.88\), do not accept \(7.8\).

[3 marks]

\( - 2.50\), \(3.61\) (\( - 2.49545 \ldots \), \(3.60656 \ldots \))     (A1)(ft)(A1)(ft)

Notes: Follow through from their line \(L\) from part (f) even if no working shown. Award at most (A0)(A1)(ft) if their correct coordinate pairs given.

Note: Accept \( - 2.5\), do not accept \( - 2.49\). Accept \(3.6\), do not accept \(3.60\).

[2 marks]

As usual, the content in this question caused difficulty for many candidates. However, for those with a sound grasp of the topic, there were many very successful attempts. The curve was given so that a comparison could be made to a GDC version and the correct form of the derivative was also given to permit weaker candidates to progress to the latter stages. Unfortunately, some decided to proceed with their own incorrect versions, in which case very limited follow through accrued. It should be emphasized to candidates that when an answer is given in this way it should be used in subsequent parts of the question.

As in previous years, much of the question could have been answered successfully by using the GDC. However, it was also clear that a large number of candidates did not attempt either to verify their work with their GDC or to use it in place of an algebraic approach.

Differentiation of terms with negative indices remains a testing process for the majority; it will continue to be tested. Some centres still do not teach the differential calculus.

As usual, the content in this question caused difficulty for many candidates. However, for those with a sound grasp of the topic, there were many very successful attempts. The curve was given so that a comparison could be made to a GDC version and the correct form of the derivative was also given to permit weaker candidates to progress to the latter stages. Unfortunately, some decided to proceed with their own incorrect versions, in which case very limited follow through accrued. It should be emphasized to candidates that when an answer is given in this way it should be used in subsequent parts of the question.

As in previous years, much of the question could have been answered successfully by using the GDC. However, it was also clear that a large number of candidates did not attempt either to verify their work with their GDC or to use it in place of an algebraic approach.

Differentiation of terms with negative indices remains a testing process for the majority; it will continue to be tested. Some centres still do not teach the differential calculus.

As usual, the content in this question caused difficulty for many candidates. However, for those with a sound grasp of the topic, there were many very successful attempts. The curve was given so that a comparison could be made to a GDC version and the correct form of the derivative was also given to permit weaker candidates to progress to the latter stages. Unfortunately, some decided to proceed with their own incorrect versions, in which case very limited follow through accrued. It should be emphasized to candidates that when an answer is given in this way it should be used in subsequent parts of the question.

As in previous years, much of the question could have been answered successfully by using the GDC. However, it was also clear that a large number of candidates did not attempt either to verify their work with their GDC or to use it in place of an algebraic approach.

Differentiation of terms with negative indices remains a testing process for the majority; it will continue to be tested. Some centres still do not teach the differential calculus.

As usual, the content in this question caused difficulty for many candidates. However, for those with a sound grasp of the topic, there were many very successful attempts. The curve was given so that a comparison could be made to a GDC version and the correct form of the derivative was also given to permit weaker candidates to progress to the latter stages. Unfortunately, some decided to proceed with their own incorrect versions, in which case very limited follow through accrued. It should be emphasized to candidates that when an answer is given in this way it should be used in subsequent parts of the question.

As in previous years, much of the question could have been answered successfully by using the GDC. However, it was also clear that a large number of candidates did not attempt either to verify their work with their GDC or to use it in place of an algebraic approach.

Differentiation of terms with negative indices remains a testing process for the majority; it will continue to be tested. Some centres still do not teach the differential calculus.

As usual, the content in this question caused difficulty for many candidates. However, for those with a sound grasp of the topic, there were many very successful attempts. The curve was given so that a comparison could be made to a GDC version and the correct form of the derivative was also given to permit weaker candidates to progress to the latter stages. Unfortunately, some decided to proceed with their own incorrect versions, in which case very limited follow through accrued. It should be emphasized to candidates that when an answer is given in this way it should be used in subsequent parts of the question.

As in previous years, much of the question could have been answered successfully by using the GDC. However, it was also clear that a large number of candidates did not attempt either to verify their work with their GDC or to use it in place of an algebraic approach.

Differentiation of terms with negative indices remains a testing process for the majority; it will continue to be tested. Some centres still do not teach the differential calculus.

As usual, the content in this question caused difficulty for many candidates. However, for those with a sound grasp of the topic, there were many very successful attempts. The curve was given so that a comparison could be made to a GDC version and the correct form of the derivative was also given to permit weaker candidates to progress to the latter stages. Unfortunately, some decided to proceed with their own incorrect versions, in which case very limited follow through accrued. It should be emphasized to candidates that when an answer is given in this way it should be used in subsequent parts of the question.

As in previous years, much of the question could have been answered successfully by using the GDC. However, it was also clear that a large number of candidates did not attempt either to verify their work with their GDC or to use it in place of an algebraic approach.

Differentiation of terms with negative indices remains a testing process for the majority; it will continue to be tested. Some centres still do not teach the differential calculus.

As usual, the content in this question caused difficulty for many candidates. However, for those with a sound grasp of the topic, there were many very successful attempts. The curve was given so that a comparison could be made to a GDC version and the correct form of the derivative was also given to permit weaker candidates to progress to the latter stages. Unfortunately, some decided to proceed with their own incorrect versions, in which case very limited follow through accrued. It should be emphasized to candidates that when an answer is given in this way it should be used in subsequent parts of the question.

As in previous years, much of the question could have been answered successfully by using the GDC. However, it was also clear that a large number of candidates did not attempt either to verify their work with their GDC or to use it in place of an algebraic approach.

Differentiation of terms with negative indices remains a testing process for the majority; it will continue to be tested. Some centres still do not teach the differential calculus.