Questions — SPS SPS SM Pure (200 questions)

1. In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable. Solve $$2 \times 4 ^ { x } - 2 ^ { x + 3 } = 17 \times 2 ^ { x - 1 } - 4$$ (Total for Question 4 is 4 marks)

5. Find the coefficient of the term in \(x ^ { 7 }\) of the binomial expansion of $$\left( \frac { 3 } { 8 } + 4 x \right) ^ { 12 }$$ giving your answer in simplest form.
(3) \section*{(Total for Question 5 is 3 marks)}

1. The curve \(C _ { 1 }\) has equation \(y = \mathrm { f } ( x )\).

Given that

• \(\mathrm { f } ( x )\) is a quadratic expression
• \(C _ { 1 }\) has a maximum turning point at \(( 2,20 )\)
• \(C _ { 1 }\) passes through the origin
  1. sketch a graph of \(C _ { 1 }\) showing the coordinates of any points where \(C _ { 1 }\) cuts the coordinate axes,
  2. find an expression for \(\mathrm { f } ( x )\).

The curve \(C _ { 2 }\) has equation \(y = x \left( x ^ { 2 } - 4 \right)\)
Curve \(C _ { 1 }\) and \(C _ { 2 }\) meet at the origin, and at the points \(P\) and \(Q\)
Given that the \(x\) coordinate of the point \(P\) is negative,

using algebra and showing all stages of your working, find the coordinates of \(P\)
(3)

7. The circle \(C\)

• has centre \(A ( 3,5 )\)
• passes through the point \(B ( 8 , - 7 )\)

The points \(M\) and \(N\) lie on \(C\) such that \(M N\) is a chord of \(C\).
Given that \(M N\)

• lies above the \(x\)-axis
• is parallel to the \(x\)-axis
• has length \(4 \sqrt { 22 }\)

Find an equation for the line passing through points \(M\) and \(N\).
(5)
(Total for Question 7 is 5 marks)

8. (a) Sketch the curve with equation $$y = a ^ { - x } + 4$$ where \(a\) is a constant and \(a > 1\)
On your sketch show

• the coordinates of the point of intersection of the curve with the \(y\)-axis
• the equation of any asymptotes to the curve.
  (3)
  (b) Use the trapezium rule with 5 trapeziums to find an approximate value for

$$\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } + 4 \right) d x$$ giving your answer to two significant figures.
(3)
(c) Using the answer to part (b), find an approximate value for

1. \(\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } \right) \mathrm { d } x\)
2. \(\int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } + 4 \right) \mathrm { d } x + \int _ { - 4 } ^ { 8.5 } \left( 3 ^ { - \frac { 1 } { 2 } x } + 4 \right) \mathrm { d } x\)

9. The sum to infinity of the geometric series $$a + a r + a r ^ { 2 } + \ldots$$ is 10 .
The sum to infinity of the series formed by the squares of the terms is 100/9.
a) Show that \(r = 4 / 5\) and find \(a\).
b) Find the sum to infinity of the series formed by the cubes of the terms. \section*{(Total for Question 9 is 5 marks)}

1. In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.

1. Given that $$2 \log _ { 4 } ( x + 3 ) + \log _ { 4 } x = \log _ { 4 } ( 4 x + 2 ) + \frac { 1 } { 2 }$$ show that $$x ^ { 3 } + 6 x ^ { 2 } + x - 4 = 0$$
2. Given also that - 1 is a root of the equation $$x ^ { 3 } + 6 x ^ { 2 } + x - 4 = 0$$ solve $$2 \log _ { 4 } ( x + 3 ) + \log _ { 4 } x = \log _ { 4 } ( 4 x + 2 ) + \frac { 1 } { 2 }$$ \section*{(Total for Question 10 is 6 marks)}

1. In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.

1. Solve, for \(0 \leq x < 360 ^ { \circ }\), the equation $$\sin x \tan x = 5$$ giving your answers to one decimal place.
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8b616e3c-db87-430e-91c1-63a24e2f9593-24_641_732_778_680} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows a sketch of part of the curve with equation $$y = A \sin \left( 2 \theta - \frac { 3 \pi } { 8 } \right) + 2$$ where \(A\) is a constant and \(\theta\) is measured in radians.
   The points \(P , Q\) and \(R\) lie on the curve and are shown in Figure 1.
   Given that the \(y\) coordinate of \(P\) is 7
   (a) state the value of \(A\),
   (b) find the exact coordinates of \(Q\),
   (c) find the value of \(\theta\) at \(R\), giving your answer to 3 significant figures.

12. \begin{figure}[h] \end{figure} The diagram shows a quadrilateral \(O A C B\) in which $$\overrightarrow { O A } = 4 \mathbf { a } \quad \overrightarrow { O B } = 3 \mathbf { b } \quad \overrightarrow { B C } = 2 \mathbf { a } + \mathbf { b }$$ The point \(P\) lies on \(A C\) such that \(A P : P C = 3 : 2\)
The point \(Q\) is such that \(O P Q\) and \(B C Q\) are straight lines.
Using a vector method, find \(\overrightarrow { O Q }\) in terms of \(\mathbf { a }\) and \(\mathbf { b }\)
Give your answer in its simplest form.
Show your working clearly.

13. In this question you must show detailed reasoning.
Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with equation $$y = \frac { 1 } { 2 } x ^ { 2 } + \frac { 1458 } { \sqrt { x ^ { 3 } } } - 74 \quad x > 0$$ The point \(P\) is the only stationary point on the curve.
The line \(l\) passes through the point \(P\) and is parallel to the \(x\)-axis.
The region \(R\), shown shaded in Figure 2, is bounded by the curve, the line \(l\) and the line with equation \(x = 4\) Use algebraic integration to find the exact area of \(R\).
(8) \section*{ADDITIONAL SHEET } \section*{ADDITIONAL SHEET } \section*{ADDITIONAL SHEET }

1. The circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } + 4 x - 30 y + 209 = 0$$ Find

1. the coordinates of the centre of \(C\),
2. the exact value of the radius of \(C\).
   coordinates of the centre of C
   radius of \(C\)

2. (a) Find, in terms of \(a\), the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 2 + a x ) ^ { 6 }$$ where \(a\) is a non-zero constant. Give each term in simplest form. $$f ( x ) = \left( 3 + \frac { 1 } { x } \right) ^ { 2 } ( 2 + a x ) ^ { 6 }$$ Given that the constant term in the expansion of \(\mathrm { f } ( x )\) is 576
(b) find the value of \(a\). \section*{3. In this question you must show all stages of your working.} \section*{Solutions relying entirely on calculator technology are not acceptable.} The curve \(C\) has equation $$y = 4 x ^ { \frac { 1 } { 2 } } + 9 x ^ { - \frac { 1 } { 2 } } + 3 \quad x > 0$$ (a) Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) giving each term in simplest form.
(b) Hence find the \(x\) coordinate of the stationary point of \(C\).
(c) Determine the nature of the stationary point of \(C\), giving a reason for your answer.
(d) State the range of values of \(x\) for which \(y\) is decreasing.
(Total for Question 3 is 7 marks)

4. In this question you must show all stages of your working. \section*{Solutions relying entirely on calculator technology are not acceptable.} Solve, for \(0 < \theta \leq 360 ^ { \circ }\), the equation $$3 \tan ^ { 2 } \theta + 7 \sec \theta - 3 = 0$$ giving your answers to one decimal place.
(Total for Question 4 is 4 marks)

5. The number of bacteria on a surface is being monitored. The number of bacteria, \(N\), on the surface, \(t\) hours after monitoring began is modelled by the equation $$\log _ { 10 } N = 0.35 t + 2$$ Use the equation of the model to answer parts (a) to (c).

1. Find the initial number of bacteria on the surface.
2. Show that the equation of the model can be written in the form $$N = a b ^ { t }$$ where \(a\) and \(b\) are constants to be found. Give the value of \(b\) to 2 decimal places.
3. Hence find the rate of growth of bacteria on the surface exactly 5 hours after monitoring began.

6. The region bounded by the curve $$y = ( 2 x - 8 ) \ln x$$ and the \(x\)-axis is shaded in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{bc7fb499-9462-40ae-88f4-87fc60f6a005-12_871_913_422_575} Show that the exact area is given by $$32 \ln 2 - \frac { 33 } { 2 }$$ Fully justify your answer.

7. (i) Prove by counter example that the statement
"If \(n\) is a prime number then \(3 ^ { n } + 2\) is also a prime number."
is false.
(ii) Use proof by exhaustion to prove that if \(m\) is an integer that is not divisible by 3 , then $$m ^ { 2 } - 1$$ is divisible by 3
(3)
(Total for Question 7 is 5 marks)

8. In this question you must show all stages of your working. \section*{Solutions relying entirely on calculator technology are not acceptable.}

1. Show that \(\sin 3 x\) can be written in the form $$P \sin x + Q \sin ^ { 3 } x$$ where \(P\) and \(Q\) are constants to be found.
2. Hence or otherwise, solve, for \(0 < \theta \leq 360 ^ { \circ }\), the equation $$2 \sin 3 \theta = 5 \sin 2 \theta$$ giving your answers, in degrees, to one decimal place as appropriate.

9. \begin{figure}[h] \end{figure} The curve \(C\) has equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = x ^ { 3 } \sqrt { 4 x + 7 } \quad x \geq - \frac { 7 } { 4 }$$

1. Show that $$\mathrm { f } ^ { \prime } ( x ) = \frac { k x ^ { 2 } ( 2 x + 3 ) } { \sqrt { 4 x + 7 } }$$ where \(k\) is a constant to be found. The point \(P\), shown in Figure 3, is the minimum turning point on \(C\).
2. Find the coordinates of \(P\).
3. Hence find the range of the function g defined by $$\operatorname { g } ( x ) = - 4 \mathrm { f } ( x ) \quad x \geq - \frac { 7 } { 4 }$$ The point \(Q\) with coordinates \(\left( \frac { 1 } { 2 } , \frac { 3 } { 8 } \right)\) lies on \(C\).
4. Find the coordinates of the point to which \(Q\) is mapped when \(C\) is transformed to the curve with equation $$y = 40 f \left( x - \frac { 3 } { 2 } \right) - 8$$

10. The function f is defined by \(\mathrm { f } ( x ) = \arccos x\) for \(0 \leq x \leq a\)
The curve with equation \(y = \mathrm { f } ( x )\) is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{bc7fb499-9462-40ae-88f4-87fc60f6a005-22_769_771_317_648}

1. State the value of \(a\)
2. 1. On the diagram above, sketch the curve with equation $$y = \cos x \text { for } 0 \leq x \leq \frac { \pi } { 2 }$$ and
      sketch the line with equation $$y = x \text { for } 0 \leq x \leq \frac { \pi } { 2 }$$
   2. Explain why the solution to the equation $$x - \cos x = 0$$ must also be a solution to the equation $$\cos x = \arccos x$$
3. Use the Newton-Raphson method with \(x _ { 0 } = 0\) to find an approximate solution, \(x _ { 3 }\), to the equation $$x - \cos x = 0$$ Give your answer to four decimal places. CONTINUE YOUR ANSWER HERE CONTINUE YOUR ANSWER HERE

11. \begin{figure}[h] \end{figure} The heart rate of a horse is being monitored.
The heart rate \(H\), measured in beats per minute (bpm), is modelled by the equation $$H = 32 + 40 \mathrm { e } ^ { - 0.2 t } - 20 \mathrm { e } ^ { - 0.9 t }$$ where \(t\) minutes is the time after monitoring began.
Figure 4 is a sketch of \(H\) against \(t\). \section*{Use the equation of the model to answer parts (a) to (e).}

1. State the initial heart rate of the horse. In the long term, the heart rate of the horse approaches \(L \mathrm { bpm }\).
2. State the value of \(L\). The heart rate of the horse reaches its maximum value after \(T\) minutes.
3. Find the value of \(T\), giving your answer to 3 decimal places.
   (Solutions based entirely on calculator technology are not acceptable.) The heart rate of the horse is 37 bpm after \(M\) minutes.
4. Show that \(M\) is a solution of the equation $$t = 5 \ln \left( \frac { 8 } { 1 + 4 \mathrm { e } ^ { - 0.9 t } } \right)$$ Using the iteration formula $$t _ { n + 1 } = 5 \ln \left( \frac { 8 } { 1 + 4 \mathrm { e } ^ { - 0.9 t _ { n } } } \right) \quad \text { with } \quad t _ { 1 } = 10$$
5. 1. find, to 4 decimal places, the value of \(t _ { 2 }\)
   2. find, to 4 decimal places, the value of \(M\)

12.

1. Show that the first two terms of the binomial expansion of \(\sqrt { 4 - 2 x ^ { 2 } }\) are $$2 - \frac { x ^ { 2 } } { 2 }$$
2. State the range of values of \(x\) for which the expansion found in part (a) is valid.
3. Hence, find an approximation for $$\int _ { 0 } ^ { 0.4 } \sqrt { \cos x } d x$$ giving your answer to five decimal places.
   Fully justify your answer.
4. A student decides to use this method to find an approximation for $$\int _ { 0 } ^ { 1.4 } \sqrt { \cos x } d x$$ Explain why this may not be a suitable method.

13. Use the substitution \(u = \sqrt { x ^ { 3 } + 1 }\) to show that $$\int \frac { 9 x ^ { 5 } } { \sqrt { x ^ { 3 } + 1 } } \mathrm {~d} x = 2 \left( x ^ { 3 } + 1 \right) ^ { k } \left( x ^ { 3 } - A \right) + C$$ where \(k\) and \(A\) are constants to be found and \(c\) is an arbitrary constant.
(Total for Question 13 is 4 marks)

14.

1. \(\quad y = \mathrm { e } ^ { - x } ( \sin x + \cos x )\) Find \(\frac { d y } { d x }\) and simplify your answer.
2. Hence, show that $$\int e ^ { - x } \sin x d x = a e ^ { - x } ( \sin x + \cos x ) + c$$ where \(a\) is a rational number.
3. A sketch of the graph of \(y = \mathrm { e } ^ { - x } \sin x\) for \(x \geq 0\) is shown below. The areas of the finite regions bounded by the curve and the \(x\)-axis are denoted by \(A _ { 1 }\), \(A _ { 2 } , \ldots , A _ { n } , \ldots\)
   \includegraphics[max width=\textwidth, alt={}, center]{bc7fb499-9462-40ae-88f4-87fc60f6a005-34_807_1246_959_406}
   1. Find the exact value of the area \(A _ { 1 }\)
   2. Show that $$\frac { A _ { 2 } } { A _ { 1 } } = e ^ { - \pi }$$
   3. Given that $$\frac { A _ { n + 1 } } { A _ { n } } = e ^ { - \pi }$$ show that the exact value of the total area enclosed between the curve and the \(x\)-axis is $$\frac { 1 + e ^ { \pi } } { 2 \left( e ^ { \pi } - 1 \right) }$$

15. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with parametric equations $$x = 3 \sin ^ { 3 } \theta \quad y = 1 + \cos 2 \theta \quad - \frac { \pi } { 2 } \leq \theta \leq \frac { \pi } { 2 }$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = k \operatorname { cosec } \theta \quad \theta \neq 0$$ where \(k\) is a constant to be found. The point \(P\) lies on \(C\) where \(\theta = \frac { \pi } { 6 }\)
2. Find the equation of the tangent to \(C\) at \(P\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
3. Show that \(C\) has Cartesian equation $$8 x ^ { 2 } = 9 ( 2 - y ) ^ { 3 } \quad - q \leq x \leq q$$ where \(q\) is a constant to be found.

1
\(\sqrt { 41 }\)
29
(1)
(b) The angle between the vector \(\mathbf { i }\) and the vector \(- 20 \mathbf { i } + 21 \mathbf { j }\) is \(\theta\). Which statement about \(\vartheta\) is true?
Circle your answer. $$0 ^ { \circ } < \theta < 45 ^ { \circ } \quad 45 ^ { \circ } < \theta < 90 ^ { \circ } \quad 90 ^ { \circ } < \theta < 135 ^ { \circ } \quad 135 ^ { \circ } < \theta < 180 ^ { \circ }$$ Q6.
The function \(f\) is defined for all real values of \(x\) by $$f ( x ) = x ^ { 3 } + x$$ (a) Express \(f ( 2 + h ) - f ( 2 )\) in the form $$p h + q h ^ { 2 } + r h ^ { 3 }$$ where \(p , q\) and \(r\) are integers.
(b) Using your answer to part (a) and showing detailed reasoning, find the value of \(f ^ { \prime } ( 2 )\). Q7. In this question you must show detailed reasoning. Do not use your calculator. Determine the set of values of \(x\) which satisfy the inequality $$3 x ^ { 2 } + 3 x > x + 6$$ Give your answer in exact form using set notation.
(Total 4 marks) \section*{ANSWER SHEET} Q8.
(a) Write each of the following in the form \(\log _ { a } k\), where \(k\) is an integer:
(i) \(\log _ { a } 4 + \log _ { a } 10\);
(ii) \(\log _ { a } 16 - \log _ { a } 2\);
(iii) \(3 \log _ { a } 5\).
(b) Show that, for \(x > 0\) $$\log _ { 10 } \frac { x ^ { 4 } } { 100 } + \log _ { 10 } 9 x - \log _ { 10 } x ^ { 3 } \equiv k \left( - 1 + \log _ { 10 } m x \right)$$ where \(k\) and \(m\) are constants to be found. \section*{ANSWER SHEET} Q9.
A curve has equation $$y = \frac { a } { \sqrt { x } } + b x ^ { 2 } + \frac { c } { x ^ { 3 } } \quad \text { for } x > 0$$ where \(a , b\) and \(c\) are positive constants.
The curve has a single turning point.
Use the second derivative of \(y\) to determine the nature of this turning point.
You do not need to find the coordinates of the turning point.
Fully justify your answer.
(Total 4 marks) \section*{ANSWER SHEET} Q10. In this question you must show detailed reasoning. A piece of wire of length 66 cm is bent to form the five sides of a pentagon.
The pentagon consists of three sides of a rectangle and two sides of an equilateral triangle. The sides of the rectangle measure \(x \mathrm {~cm}\) and \(y \mathrm {~cm}\) and the sides of the triangle measure \(x \mathrm {~cm}\), as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{f31b653d-7422-4961-a2fe-5de2a3d52492-18_410_494_657_861}
(a) Show that the area enclosed by the wire, \(A \mathrm {~cm} ^ { 2 }\), can be expressed by the formula $$A = 33 x - \frac { 1 } { 4 } ( 6 - \sqrt { 3 } ) x ^ { 2 }$$ (b) Use calculus to find the value of \(x\) for which the wire encloses the maximum area. Give your answer in the form \(p + q \sqrt { 3 }\), where \(p\) and \(q\) are integers. Fully justify your answer. \section*{ANSWER SHEET} Q11.
(a) Using \(y = 2 ^ { 2 x }\) as a substitution, show that $$16 ^ { x } - 2 ^ { ( 2 x + 3 ) } - 9 = 0$$ can be written as $$y ^ { 2 } - 8 y - 9 = 0$$ (b) Hence, show that the equation $$16 ^ { x } - 2 ^ { ( 2 x + 3 ) } - 9 = 0$$ has \(x = \log _ { 2 } 3\) as its only solution.
Fully justify your answer.
(3)
(Total 5 marks) \section*{ANSWER SHEET} Q12.
(a) (i) On the axes given below, sketch the graph of \(y = \tan x\) for \(0 ^ { \circ } \leq x \leq 360 ^ { \circ }\).
\includegraphics[max width=\textwidth, alt={}, center]{f31b653d-7422-4961-a2fe-5de2a3d52492-22_700_1319_299_541}
(b) Solve the equation
\(6 \tan 3 x \sin 3 x = 5\),
giving all values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } \leq x \leq 180 ^ { \circ }\).
(4)
(Total 6 marks) \section*{ANSWER SHEET} Q13.
A building has a leaking roof and, while it is raining, water drips into a 12 litre bucket. When the rain stops, the bucket is one third full.
Water continues to drip into the bucket from a puddle on the roof.
In the first minute after the rain stops, 30 millilitres of water drip into the bucket. In each subsequent minute, the amount of water that drips into the bucket reduces by \(2 \%\). During the \(n\)th minute after the rain stops, the volume of water that drips into the bucket is \(W _ { n }\) millilitres.
(a) Find \(W _ { 2 }\)
(b) Explain why $$W _ { n } = A \times 0.98 ^ { n - 1 }$$ and state the value of \(A\).
(c) Find the increase in the water in the bucket 15 minutes after the rain stops. Give your answer to the nearest millilitre.
(d) Assuming it does not start to rain again, find the maximum amount of water in the bucket.
(e) After several hours the water has stopped dripping. Give one reason why the amount of water in the bucket is not as much as the answer found in part (d).
(1) \section*{ANSWER SHEET} Q14.
The line \(L\) has equation $$5 y + 12 x = 298$$ A circle, \(C\), has centre \(( 7,9 )\)
\(L\) is a tangent to \(C\).
(a) Find the coordinates of the point of intersection of \(L\) and \(C\).
(4)
(b) Find the equation of \(C\).
(3)
(Total 7 marks) \section*{ANSWER SHEET} Q15.
The curve with equation \(y = f ( x )\), where \(f ( x ) = \ln ( 2 x - 3 ) , x > \frac { 3 } { 2 }\), is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{f31b653d-7422-4961-a2fe-5de2a3d52492-28_606_1127_374_548}
(a) The inverse of \(f\) is \(f ^ { - 1 }\).
(i) Find \(f ^ { - 1 } ( x )\).
(ii) State the range of \(f ^ { - 1 }\).
(iii) Sketch, on the axes below, the curve with equation \(y = f ^ { - 1 } ( x )\), indicating the value of the \(y\)-coordinate of the point where the curve intersects the \(y\)-axis.
\includegraphics[max width=\textwidth, alt={}, center]{f31b653d-7422-4961-a2fe-5de2a3d52492-28_986_1294_1644_438}
(b) The function g is defined by $$g ( x ) = e ^ { 2 x } - 4 , \text { for all real values of } x$$ Write down an expression for \(\mathrm { g } ( x )\), and hence find the exact solution of the equation \(f g ( x ) = \ln 5\).
(3)
(Total 8 marks) Q16.
A new symmetric =design for a company logo is to be made from two sectors of a circle, \(O R P\) and \(O Q S\), and a rhombus \(O S T R\), as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{f31b653d-7422-4961-a2fe-5de2a3d52492-30_499_586_356_815} The points \(P , O\) and \(Q\) lie on a straight line and the angle \(R O S\) is \(\theta\) radians. A large copy of the logo, with \(P Q = 5\) metres, is to be put on a wall.
(a) Show that the area of the logo, \(A\) square metres, is given by $$A = k ( \pi - \theta + m \sin \theta )$$ where \(k\) and \(m\) are constants to be found.
(4)
(b) Find an expression for the perimeter of the logo. \section*{ANSWER SHEET} Q17.
In the expansion of \(( 3 + a x ) ^ { n }\), where \(a\) and \(n\) are integers, the coefficient of \(x ^ { 2 }\) is 4860
(a) Show that $$3 ^ { n } a ^ { 2 } n ( n - 1 ) = 87480$$ (b) The constant term in the expansion is 729 The coefficient of \(x\) in the expansion is negative.
(i) Verify that \(n = 6\)
(ii) Find the value of \(a\) \section*{ANSWER SHEET} Q18.
\includegraphics[max width=\textwidth, alt={}, center]{f31b653d-7422-4961-a2fe-5de2a3d52492-34_426_1170_349_468} The figure above shows the parabola with equation $$y = - ( x - a ) ( x - b ) , b > a > 0$$ The curve meets the \(x\) axis at the points \(A\) and \(B\).
a) Show that the area of the finite region \(R\), bounded by the parabola and the \(x\) axis is $$\frac { 1 } { 6 } ( b - a ) ^ { 3 } .$$ The midpoint of \(A B\) is \(N\). The point \(M\) is the maximum point of the parabola.
b) Show clearly that the area of \(R\) is given by $$k | A B \| M N | ,$$ Where k is a constant to be found and \(| A B |\) is the length of the line from A to B , and \(| M N |\) is the length of the line from M to N . \section*{ANSWER SHEET}