SPS SPS SM Pure (SPS SM Pure) 2023 February

1. $$f ( x ) = x ^ { 3 } + x ^ { 2 } - 12 x - 18$$

1. Use the factor theorem to show that \(( x + 3 )\) is a factor of \(f ( x )\).
   (2)
2. Factorise \(f ( x )\) to a linear and quadratic factor.
   (2)
3. Hence find exact values for all the solutions of the equation \(\mathrm { f } ( x ) = 0\)
   (3)

2.
\(f ( x ) = 3 x ^ { 2 } + 2 x . \quad\) Find \(f ^ { \prime } ( x )\) from first principles.
(4)

3.
a) Show that when \(x\) is small, \(2 \cos x - 3 \sin x\) can be written as \(a + b x + c x ^ { 2 }\), where \(a , b\) and \(c\) are integers to be found.
b) Hence find a small positive value of \(x\) that is an approximate solution to \(2 \cos x - 3 \sin x = 7 x\)

4. The curve \(C\) has equation $$y = 12 x ^ { \frac { 5 } { 4 } } - \frac { 5 } { 18 } x ^ { 2 } - 1000 , \quad x > 0$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. Hence find the coordinates of the stationary point on \(C\).
3. Use \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) to determine the nature of this stationary point.

5. \begin{figure}[h] \end{figure} The straight line \(l\) with equation \(y = \frac { 1 } { 2 } x + 1\) cuts the curve \(C\), with equation \(y = x ^ { 2 } - 4 x + 3\), at the points \(P\) and \(Q\), as shown in Figure 2 The finite region \(R\) is shown shaded in Figure 2. This region \(R\) is bounded by the line segment \(P Q\), the line segment \(T S\), and the arcs \(P T\) and \(S Q\) of the curve. Use integration to find the exact area of the shaded region \(R\).

6. $$f ( x ) = ( 3 - 2 x ) ^ { - 4 }$$ a) Find the binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), giving each coefficient as a simplified fraction.
(3)
b) For what values of \(x\) is the expansion valid?

7. The function f is defined by $$\mathrm { f } ( x ) = 3 ^ { x } \sqrt { x } - 1 \quad \text { where } x \geq 0$$

1. \(\quad \mathrm { f } ( x ) = 0\) has a single solution at the point \(x = \alpha\) By considering a suitable change of sign, show that \(\alpha\) lies between 0 and 1
   [0pt] [2 marks]
2. 1. Show that $$\mathrm { f } ^ { \prime } ( x ) = \frac { 3 ^ { x } ( 1 + x \ln 9 ) } { 2 \sqrt { x } }$$
3. (ii) Use the Newton-Raphson method with \(x _ { 1 } = 1\) to find \(x _ { 3 }\), an approximation for \(\alpha\). Give your answer to five decimal places.
4. (iii) Explain why the Newton-Raphson method fails to find \(\alpha\) with \(x _ { 1 } = 0\)

8. \begin{figure}[h] \end{figure} Figure 3 shows part of the curve \(C\) with equation $$y = \frac { 3 \ln \left( x ^ { 2 } + 1 \right) } { \left( x ^ { 2 } + 1 \right) } , \quad x \in \mathbb { R }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
2. Using your answer to (a), find the exact coordinates of the stationary point on the curve \(C\) for which \(x > 0\). Write each coordinate in its simplest form.
   (4) The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the \(x\)-axis and the line \(x = 3\)
3. Complete the table below with the value of \(y\) corresponding to \(x = 1\)

   \(x\)	0	1	2	3
   \(y\)	0		\(\frac { 3 } { 5 } \ln 5\)	\(\frac { 3 } { 10 } \ln 10\)

   (1)
4. Use the trapezium rule with all the \(y\) values in the completed table to find an approximate value for the area of \(R\), giving your answer to 4 significant figures.
   (2)

9. The function g is defined by $$\mathrm { g } : x \mapsto | 8 - 2 x | , \quad x \in \mathbb { R } , \quad x \geqslant 0$$

1. Sketch the graph with equation \(y = \mathrm { g } ( x )\), showing the coordinates of the points where the graph cuts or meets the axes.
   (2)
2. Solve the equation $$| 8 - 2 x | = x + 5$$ The function \(f\) is defined by $$\mathrm { f } : x \mapsto x ^ { 2 } - 3 x + 1 , \quad x \in \mathbb { R } , \quad 0 \leqslant x \leqslant 4$$
3. Find fg(5).
4. Find the range of f . You must make your method clear.

10.
a) Show that $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \frac { 4 \sin x } { 1 + 2 \cos x } \mathrm {~d} x = \ln a$$ where \(a\) is a rational constant to be found.
b) By using a suitable substitution, find the exact value of $$\int _ { 0 } ^ { \sqrt { 3 } } \frac { 1 } { \left( 4 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } d x$$

11.

1. Given that $$2 \cos ( x + 30 ) ^ { \circ } = \sin ( x - 30 ) ^ { \circ }$$ without using a calculator, show that $$\tan x ^ { \circ } = 3 \sqrt { 3 } - 4$$ (4)
2. Hence or otherwise solve, for \(0 \leqslant \theta < 180\), $$2 \cos ( 2 \theta + 40 ) ^ { \circ } = \sin ( 2 \theta - 20 ) ^ { \circ }$$ Give your answers to one decimal place.
   (3)

12. Given that $$4 x ^ { 3 } + 2 x ^ { 2 } + 17 x + 8 \equiv ( A x + B ) \left( x ^ { 2 } + 4 \right) + C x + D$$

1. find the values of the constants \(A , B , C\) and \(D\).
   (3)
2. Hence find $$\int _ { 1 } ^ { 4 } \frac { 4 x ^ { 3 } + 2 x ^ { 2 } + 17 x + 8 } { x ^ { 2 } + 4 } d x$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are integers.

13. The curve \(C\) has parametric equations $$x = 10 \cos 2 t , \quad y = 6 \sin t , \quad - \frac { \pi } { 2 } \leqslant t \leqslant \frac { \pi } { 2 }$$ The point \(A\) with coordinates \(( 5,3 )\) lies on \(C\).

1. Find the value of \(t\) at the point \(A\).
2. Show that an equation of the normal to \(C\) at \(A\) is $$3 y = 10 x - 41$$ The normal to \(C\) at \(A\) cuts \(C\) again at the point \(B\).
3. Find the exact coordinates of \(B\).

14. The sum to infinity of a geometric series is 96
The first term of the series is less than 30
The second term of the series is 18

1. Find the first term and common ratio of the series.
   [0pt] [4 marks]
2. 1. Show that the \(n\)th term of the series, \(u _ { n }\), can be written as $$u _ { n } = \frac { 3 ^ { n } } { 2 ^ { 2 n - 5 } }$$ [3 marks]
3. (ii) Hence show that $$\log _ { 3 } u _ { n } = n \left( 1 - 2 \log _ { 3 } 2 \right) + 5 \log _ { 3 } 2$$