AQA Paper 1 (Paper 1) 2018 June

2 The graph of \(y = 5 ^ { x }\) is transformed by a stretch in the \(y\)-direction, scale factor 5 State the equation of the transformed graph. Circle your answer.
[0pt] [1 mark]
\(y = 5 \times 5 ^ { x }\)
\(y = 5 ^ { \frac { x } { 5 } }\)
\(y = \frac { 1 } { 5 } \times 5 ^ { x }\)
\(y = 5 ^ { 5 x }\)

3 A periodic sequence is defined by \(U _ { n } = \sin \left( \frac { n \pi } { 2 } \right)\)
State the period of this sequence. Circle your answer.
\(82 \pi \quad 4 \quad \pi\)

4 The function f is defined by \(\mathrm { f } ( x ) = \mathrm { e } ^ { x - 4 } , x \in \mathbb { R }\)
Find \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.

5 A curve is defined by the parametric equations $$\begin{aligned} & x = 4 \times 2 ^ { - t } + 3
& y = 3 \times 2 ^ { t } - 5 \end{aligned}$$ 5

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 3 } { 4 } \times 2 ^ { 2 t }\)
   5
2. Find the Cartesian equation of the curve in the form \(x y + a x + b y = c\), where \(a , b\) and \(c\) are integers.

6

1. Find the first three terms, in ascending powers of \(x\), of the binomial expansion of \(\frac { 1 } { \sqrt { 4 + x } }\)
   6
2. Hence, find the first three terms of the binomial expansion of \(\frac { 1 } { \sqrt { 4 - x ^ { 3 } } }\)
   6
3. 1. Edward, a student, decides to use this method to find a more accurate value for the integral by increasing the number of terms of the binomial expansion used. Explain clearly whether Edward's approximation will be an overestimate, an underestimate, or if it is impossible to tell.
      [0pt] [2 marks]
      6
4. (ii) Edward goes on to use the expansion from part (b) to find an approximation for \(\int _ { - 2 } ^ { 0 } \frac { 1 } { \sqrt { 4 - x ^ { 3 } } } \mathrm {~d} x\) Explain why Edward's approximation is invalid.

7 Three points \(A , B\) and \(C\) have coordinates \(A ( 8,17 ) , B ( 15,10 )\) and \(C ( - 2 , - 7 )\) 7

1. Show that angle \(A B C\) is a right angle.
   7
2. \(\quad A , B\) and \(C\) lie on a circle.
   7
3. 1. Explain why \(A C\) is a diameter of the circle.
      7
4. (ii) Determine whether the point \(D ( - 8 , - 2 )\) lies inside the circle, on the circle or outside the circle. Fully justify your answer.

8 The diagram shows a sector of a circle \(O A B\).
\(C\) is the midpoint of \(O B\).
Angle \(A O B\) is \(\theta\) radians.
\includegraphics[max width=\textwidth, alt={}, center]{85b10472-8149-4387-999f-2ef153f1a105-10_700_963_536_534} 8

1. Given that the area of the triangle \(O A C\) is equal to one quarter of the area of the sector \(O A B\), show that \(\theta = 2 \sin \theta\)
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2. Use the Newton-Raphson method with \(\theta _ { 1 } = \pi\), to find \(\theta _ { 3 }\) as an approximation for \(\theta\). Give your answer correct to five decimal places.
   8
3. Given that \(\theta = 1.89549\) to five decimal places, find an estimate for the percentage error in the approximation found in part (b).
   Turn over for the next question

9 An arithmetic sequence has first term \(a\) and common difference \(d\). The sum of the first 36 terms of the sequence is equal to the square of the sum of the first 6 terms. 9

1. Show that \(4 a + 70 d = 4 a ^ { 2 } + 20 a d + 25 d ^ { 2 }\) 9
2. Given that the sixth term of the sequence is 25 , find the smallest possible value of \(a\).

10 A scientist is researching the effects of caffeine. She models the mass of caffeine in the body using $$m = m _ { 0 } \mathrm { e } ^ { - k t }$$ where \(m _ { 0 }\) milligrams is the initial mass of caffeine in the body and \(m\) milligrams is the mass of caffeine in the body after \(t\) hours. On average, it takes 5.7 hours for the mass of caffeine in the body to halve.
One cup of strong coffee contains 200 mg of caffeine.
10

1. The scientist drinks two strong cups of coffee at 8 am. Use the model to estimate the mass of caffeine in the scientist's body at midday.
   10
2. The scientist wants the mass of caffeine in her body to stay below 480 mg

   10

   Use the model to find the earliest time

   coffee.

   Give your answer to the nearest minute

   □

11 The daily world production of oil can be modelled using $$V = 10 + 100 \left( \frac { t } { 30 } \right) ^ { 3 } - 50 \left( \frac { t } { 30 } \right) ^ { 4 }$$ where \(V\) is volume of oil in millions of barrels, and \(t\) is time in years since 1 January 1980. 11

1. 1. The model is used to predict the time, \(T\), when oil production will fall to zero.
      Show that \(T\) satisfies the equation $$T = \sqrt [ 3 ] { 60 T ^ { 2 } + \frac { 162000 } { T } }$$ 11
2. (ii) Use the iterative formula \(T _ { n + 1 } = \sqrt [ 3 ] { 60 T _ { n } { } ^ { 2 } + \frac { 162000 } { T _ { n } } }\), with \(T _ { 0 } = 38\), to find the values of \(T _ { 1 } , T _ { 2 }\), and \(T _ { 3 }\), giving your answers to three decimal places.
   11
3. (iii) Explain the relevance of using \(T _ { 0 } = 38\) 11
4. From 1 January 1980 the daily use of oil by one technologically developing country can be modelled as $$V = 4.5 \times 1.063 ^ { t }$$ Use the models to show that the country's use of oil and the world production of oil will be equal during the year 2029.
   [0pt] [4 marks]
   \(12 \quad \mathrm { p } ( x ) = 30 x ^ { 3 } - 7 x ^ { 2 } - 7 x + 2\)

12

1. Prove that ( \(2 x + 1\) ) is a factor of \(\mathrm { p } ( x )\)
   12
2. Factorise \(\mathrm { p } ( x )\) completely.
   12
3. Prove that there are no real solutions to the equation $$\frac { 30 \sec ^ { 2 } x + 2 \cos x } { 7 } = \sec x + 1$$

13 A company is designing a logo. The logo is a circle of radius 4 inches with an inscribed rectangle. The rectangle must be as large as possible. The company models the logo on an \(x - y\) plane as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{85b10472-8149-4387-999f-2ef153f1a105-20_492_492_511_776} Use calculus to find the maximum area of the rectangle.
Fully justify your answer.

14 Some students are trying to prove an identity for \(\sin ( A + B )\). They start by drawing two right-angled triangles \(O D E\) and \(O E F\), as shown.
\includegraphics[max width=\textwidth, alt={}, center]{85b10472-8149-4387-999f-2ef153f1a105-22_695_662_477_689} The students' incomplete proof continues,
Let angle \(D O E = A\) and angle \(E O F = B\).
In triangle OFR,
Line \(1 \quad \sin ( A + B ) = \frac { R F } { O F }\) Line 2 $$= \frac { R P + P F } { O F }$$ Line 3 $$= \frac { D E } { O F } + \frac { P F } { O F } \text { since } D E = R P$$ Line 4 $$= \frac { D E } { \cdots \cdots } \times \frac { \cdots \cdots } { O F } + \frac { P F } { E F } \times \frac { E F } { O F }$$ Line 5
\(=\) \(\_\_\_\_\) \(+ \cos A \sin B\) 14

1. Explain why \(\frac { P F } { E F } \times \frac { E F } { O F }\) in Line 4 leads to \(\cos A \sin B\) in Line 5
   14
2. Complete Line 4 and Line 5 to prove the identity Line 4 $$= \frac { D E } { \ldots \ldots } \times \frac { \cdots \ldots } { O F } + \frac { P F } { E F } \times \frac { E F } { O F }$$ Line 5 = \(+ \cos A \sin B\) 14
3. Explain why the argument used in part (a) only proves the identity when \(A\) and \(B\) are acute angles. 14
4. Another student claims that by replacing \(B\) with \(- B\) in the identity for \(\sin ( A + B )\) it is possible to find an identity for \(\sin ( A - B )\). Assuming the identity for \(\sin ( A + B )\) is correct for all values of \(A\) and \(B\), prove a similar result for \(\sin ( A - B )\).

15 A curve has equation \(y = x ^ { 3 } - 48 x\) The point \(A\) on the curve has \(x\) coordinate - 4
The point \(B\) on the curve has \(x\) coordinate \(- 4 + h\)
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1. Show that the gradient of the line \(A B\) is \(h ^ { 2 } - 12 h\)
   15
2. Explain how the result of part (a) can be used to show that \(A\) is a stationary point on the curve.
   \includegraphics[max width=\textwidth, alt={}, center]{85b10472-8149-4387-999f-2ef153f1a105-25_2488_1719_219_150}
   \includegraphics[max width=\textwidth, alt={}, center]{85b10472-8149-4387-999f-2ef153f1a105-26_2488_1719_219_150}
   \includegraphics[max width=\textwidth, alt={}, center]{85b10472-8149-4387-999f-2ef153f1a105-27_2488_1719_219_150}
   \includegraphics[max width=\textwidth, alt={}, center]{85b10472-8149-4387-999f-2ef153f1a105-28_2498_1721_213_150}