SPS SPS FM (SPS FM) 2020 June

1.

1. Find the first 4 terms, in ascending powers of \(x\), of the binomial expansion of $$\sqrt { 1 + 4 x }$$ giving each coefficient in its simplest form. The expansion can be used to find an approximation for \(\sqrt { 26 }\)
2. Explain why \(x = \frac { 25 } { 4 }\) should not be used in the expansion to find an approximation for \(\sqrt { 26 }\)

2. Show that the substitution \(x = \sin \theta\) transforms $$\int \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } d x$$ to $$\int \sec ^ { 2 } \theta d \theta$$ and hence find $$\int \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } d x$$

3. $$\mathrm { g } ( x ) = 4 x ^ { 3 } + a x ^ { 2 } + 4 x + b$$ where \(a\) and \(b\) are constants.
Given that

• ( \(2 x + 1\) ) is a factor of \(\mathrm { g } ( x )\)
• the curve with equation \(y = \mathrm { g } ( x )\) has a point of inflection at \(x = \frac { 1 } { 6 }\)
  1. find the value of \(a\) and the value of \(b\)
  2. Show that there are no stationary points on the curve with equation \(y = \mathrm { g } ( x )\).

4. Use the identity for \(\tan ( A + B )\) to show that $$\tan 3 \theta \equiv \frac { 3 \tan \theta - \tan ^ { 3 } \theta } { 1 - 3 \tan ^ { 2 } \theta }$$

5. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation $$y = 5 \cos ( x - 30 ) ^ { \circ } \quad x \geqslant 0$$ The point \(P\) on the curve is the minimum point with the smallest positive \(x\) coordinate.

1. State the coordinates of \(P\).
2. Solve, for \(0 \leqslant x < 360\), the equation $$5 \cos ( x - 30 ) ^ { \circ } = 4 \sin x ^ { \circ }$$ giving your answers to one decimal place.
   (4)
3. Deduce, giving reasons for your answer, the number of roots of the equation $$5 \cos ( 2 x - 30 ) ^ { \circ } = 4 \sin 2 x ^ { \circ } \text { for } 0 \leqslant x < 3600$$

6. \begin{figure}[h] \end{figure} In this question you must show all stages of your working. \section*{Solutions relying on calculator technology are not acceptable.} Figure 3 shows a sketch of part of the curve with equation $$y = x e ^ { - 2 x }$$ The point \(P ( a , b )\) is the turning point of the curve.

1. Find the value of \(a\) and the exact value of \(b\) The finite region \(R\), shown shaded in Figure 3, is bounded by the curve, the line with equation \(y = b\) and the \(y\)-axis.
2. Find the exact area of \(R\).

7. \begin{figure}[h] \end{figure} Figure 4
[0pt] [ The volume of a cone of base radius \(r\) and height \(h\) is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\) ]
Figure 4 shows a container in the shape of an inverted right circular cone which contains some water. The cone has an internal base radius of 2.5 m and a vertical height of 4 m .
At time \(t\) seconds

• the height of the water is \(h \mathrm {~m}\)
• the volume of the water is \(V \mathrm {~m} ^ { 3 }\)
• the water is modelled as leaking from a hole at the bottom of the container at a rate of

$$\left( \frac { \pi } { 512 } \sqrt { h } \right) m ^ { 3 } s ^ { - 1 }$$

1. Show that, while the water is leaking $$h ^ { \frac { 3 } { 2 } } \frac { \mathrm {~d} h } { \mathrm {~d} t } = - \frac { 1 } { 200 }$$ Given that the container was initially full of water
2. find an equation, in terms of \(h\) and \(t\), to model this situation.

8. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the curve \(C _ { 1 }\) with parametric equations $$x = 2 \sin t , \quad y = 3 \sin 2 t \quad 0 \leq t < 2 \pi$$

1. Show that the Cartesian equation of \(C _ { 1 }\) can be expressed in the form $$y ^ { 2 } = k x ^ { 2 } \left( 4 - x ^ { 2 } \right)$$ where \(k\) is a constant to be found. The circle \(C _ { 2 }\) with centre \(O\) touches \(C _ { 1 }\) at four points as shown in Figure 5.
2. Find the radius of this circle.

9. $$\mathbf { A } = \left( \begin{array} { c c } k & - 2
1 - k & k \end{array} \right) \quad \text { where } k \text { is a constant }$$

1. Show that the matrix \(\mathbf { A }\) is non-singular for all values of \(k\). A transformation \(T : \mathbb { R } ^ { 2 } \rightarrow \mathbb { R } ^ { 2 }\) is represented by the matrix \(\mathbf { A }\).
   The point \(P\) has position vector \(\binom { a } { 2 a }\) relative to an origin \(O\).
   The point \(Q\) has position vector \(\binom { 7 } { - 3 }\) relative to \(O\).
   Given that the point \(P\) is mapped onto the point \(Q\) under \(T\),
2. determine the value of \(a\) and the value of \(k\). Given that, for a different value of \(k , T\) maps the line \(y = 2 x\) onto itself,
3. determine this value of \(k\).

10. Prove by induction that for \(n \in \mathbb { Z } ^ { + }\) $$2 \times 4 + 4 \times 5 + 6 \times 6 + \ldots + 2 n ( n + 3 ) = \frac { 2 } { 3 } n ( n + 1 ) ( n + 5 )$$

11. \begin{figure}[h] \end{figure} The Argand diagram, shown in Figure 1, shows a circle \(C\) and a half-line \(l\).

1. Write down the equation of the locus of points represented in the complex plane by
   1. the circle \(C\),
   2. the half-line \(l\).
2. Use set notation to describe the set of points that lie on both \(C\) and \(l\).
3. Find the complex numbers that lie on both \(C\) and \(l\), giving your answers in the form \(a + \mathrm { i } b\), where \(a , b \in \mathbb { R }\).

12. The line \(l _ { 1 }\) has Cartesian equation $$x - 2 = \frac { y - 3 } { 2 } = z + 4$$ The line \(l _ { 2 }\) has Cartesian equation $$\frac { x } { 5 } = \frac { z + 3 } { 2 } , \quad y = 9$$ Given that \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point C , find

1. the coordinates of C . The point \(\mathrm { A } ( 2,3 , - 4 )\) is on the line \(l _ { 1 }\) and the point \(\mathrm { B } ( - 5,9 , - 5 )\) is on the line \(l _ { 2 }\).
2. find the area of the triangle \(A B C\).