Sketch the graph of y = 2^x for \( - 2 \leqslant x \leqslant 3\). Indicate clearly where the curve intersects the y-axis.

Write down the equation of the asymptote of the graph of y = 2^x.

On the same axes sketch the graph of y = 3 + 2x − x². Indicate clearly where this curve intersects the x and y axes.

Using your graphic display calculator, solve the equation 3 + 2x − x² = 2^x.

Write down the maximum value of the function f (x) = 3 + 2x − x².

Use Differential Calculus to verify that your answer to (e) is correct.

The curve y = px² + qx − 4 passes through the point (2, –10).

Use the above information to write down an equation in p and q.

The gradient of the curve \(y = p{x^2} + qx - 4\) at the point (2, –10) is 1.

Find \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\).

The gradient of the curve \(y = p{x^2} + qx - 4\) at the point (2, –10) is 1.

Hence, find a second equation in p and q.

The gradient of the curve \(y = p{x^2} + qx - 4\) at the point (2, –10) is 1.

Solve the equations to find the value of p and of q.

     (A1)(A1)(A1)

Note: Award (A1) for correct domain, (A1) for smooth curve, (A1) for y-intercept clearly indicated.

[3 marks]

y = 0     (A1)(A1)

Note: Award (A1) for y = constant, (A1) for 0.

[2 marks]

Note: Award (A1) for smooth parabola,

(A1) for vertex (maximum) in correct quadrant.

(A1) for all clearly indicated intercepts x = −1, x = 3 and y = 3.

The final mark is to be applied very strictly.     (A1)(A1)(A1)

[3 marks]

x = −0.857   x = 1.77     (G1)(G1)

Note: Award a maximum of (G1) if x and y coordinates are both given.

[2 marks]

4     (G1)

Note: Award (G0) for (1, 4).

[1 mark]

\(f'(x) = 2 - 2x\)     (A1)(A1)

Note: Award (A1) for each correct term.

Award at most (A1)(A0) if any extra terms seen.

\(2 - 2x = 0\)     (M1)

Note: Award (M1) for equating their gradient function to zero.

\(x = 1\)     (A1)(ft)

\(f (1) = 3 + 2(1) - (1)^2 = 4\)     (A1)

Note: The final (A1) is for substitution of x = 1 into \(f (x)\) and subsequent correct answer. Working must be seen for final (A1) to be awarded.

[5 marks]

2² × p + 2q − 4 = −10     (M1)

Note: Award (M1) for correct substitution in the equation.

4p + 2q = −6     or     2p + q = −3     (A1)

Note: Accept equivalent simplified forms.

[2 marks]

\(\frac{{{\text{d}}y}}{{{\text{d}}x}} = 2px + q\)     (A1)(A1)

Note: Award (A1) for each correct term.

Award at most (A1)(A0) if any extra terms seen.

[2 marks]

4p + q = 1     (A1)(ft)

[1 mark]

4p + 2q = −6

4p + q = 1     (M1)

Note: Award (M1) for sensible attempt to solve the equations.

p = 2, q = −7     (A1)(A1)(ft)(G3)

[3 marks]

Undoubtedly, this question caused the most difficulty in terms of its content. Where there was no alternative to using the calculus, the majority of candidates struggled. However, for those with a sound grasp of the topic, there were many very successful attempts.

The most common error was using the incorrect domain.

Undoubtedly, this question caused the most difficulty in terms of its content. Where there was no alternative to using the calculus, the majority of candidates struggled. However, for those with a sound grasp of the topic, there were many very successful attempts.

Many had little idea of asymptotes. Others did not write their answer as an equation.

Undoubtedly, this question caused the most difficulty in terms of its content. Where there was no alternative to using the calculus, the majority of candidates struggled. However, for those with a sound grasp of the topic, there were many very successful attempts.

The intercepts being inexact or unlabelled was the most frequent cause of loss of marks.

Undoubtedly, this question caused the most difficulty in terms of its content. Where there was no alternative to using the calculus, the majority of candidates struggled. However, for those with a sound grasp of the topic, there were many very successful attempts.

Often, only one solution to the equation was given. Elsewhere, a lack of appreciation that the solutions were the x coordinates was a common mistake.

Undoubtedly, this question caused the most difficulty in terms of its content. Where there was no alternative to using the calculus, the majority of candidates struggled. However, for those with a sound grasp of the topic, there were many very successful attempts.

The maximum is the y coordinate only; again a common misapprehension was the answer “(1, 4)”.

Undoubtedly, this question caused the most difficulty in terms of its content. Where there was no alternative to using the calculus, the majority of candidates struggled. However, for those with a sound grasp of the topic, there were many very successful attempts.

This was a major discriminator in the paper. Many candidates were unable to follow the analytic approach to finding a maximum point.

Undoubtedly, this question caused the most difficulty in terms of its content. Where there was no alternative to using the calculus, the majority of candidates struggled. However, for those with a sound grasp of the topic, there were many very successful attempts.

This part was challenging to the majority, with a large number not attempting the question at all. However, there were a pleasing number of correct attempts that showed a fine understanding of the calculus.

Undoubtedly, this question caused the most difficulty in terms of its content. Where there was no alternative to using the calculus, the majority of candidates struggled. However, for those with a sound grasp of the topic, there were many very successful attempts.

This part was challenging to the majority, with a large number not attempting the question at all. However, there were a pleasing number of correct attempts that showed a fine understanding of the calculus.

Undoubtedly, this question caused the most difficulty in terms of its content. Where there was no alternative to using the calculus, the majority of candidates struggled. However, for those with a sound grasp of the topic, there were many very successful attempts.

This part was challenging to the majority, with a large number not attempting the question at all. However, there were a pleasing number of correct attempts that showed a fine understanding of the calculus.

Undoubtedly, this question caused the most difficulty in terms of its content. Where there was no alternative to using the calculus, the majority of candidates struggled. However, for those with a sound grasp of the topic, there were many very successful attempts.

This part was challenging to the majority, with a large number not attempting the question at all. However, there were a pleasing number of correct attempts that showed a fine understanding of the calculus.