SPS SPS SM Pure (SPS SM Pure) 2025 February

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.