Questions — SPS SPS FM (245 questions)

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SPS SPS FM 2020 June Q1
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 \(\sqrt { 26 }\)
SPS SPS FM 2020 June Q2
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 $$\int \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } d x$$
SPS SPS FM 2020 June Q3
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 \(y = \mathrm { g } ( x )\).
SPS SPS FM 2020 June Q4
4. Use the identity for \(\tan ( A + B )\) to show that $$\tan 3 \theta \equiv \frac { 3 \tan \theta - \tan ^ { 3 } \theta } { 1 - 3 \tan ^ { 2 } \theta }$$
SPS SPS FM 2020 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0e7cab3d-c1e6-4420-93b4-eca5af704432-05_700_1281_884_488} \captionsetup{labelformat=empty} \caption{Figure 2}
\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 $$5 \cos ( 2 x - 30 ) ^ { \circ } = 4 \sin 2 x ^ { \circ } \text { for } 0 \leqslant x < 3600$$
SPS SPS FM 2020 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0e7cab3d-c1e6-4420-93b4-eca5af704432-06_758_1227_280_443} \captionsetup{labelformat=empty} \caption{Figure 3}
\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 \(R\).
SPS SPS FM 2020 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0e7cab3d-c1e6-4420-93b4-eca5af704432-07_591_730_294_735} \captionsetup{labelformat=empty} \caption{Diagram not drawn to scale}
\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.
SPS SPS FM 2020 June Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0e7cab3d-c1e6-4420-93b4-eca5af704432-08_890_919_248_630} \captionsetup{labelformat=empty} \caption{Figure 5}
\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.
SPS SPS FM 2020 June Q9
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 \(k\).
SPS SPS FM 2020 June Q10
10. Prove by induction that for \(n \in \mathbb { Z } ^ { + }\) $$2 \times 4 + 4 \times 5 + 6 \times 6 + \ldots + 2 n ( n + 3 ) = \frac { 2 } { 3 } n ( n + 1 ) ( n + 5 )$$
SPS SPS FM 2020 June Q11
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0e7cab3d-c1e6-4420-93b4-eca5af704432-10_766_791_283_701} \captionsetup{labelformat=empty} \caption{Figure 1}
\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 \(a , b \in \mathbb { R }\).
SPS SPS FM 2020 June Q12
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 \(A B C\).
SPS SPS FM 2020 May Q1
4 marks
1. Solve the equation \(2 z - 5 \mathrm { i } z ^ { * } = 12\)
[0pt] [4 marks]
SPS SPS FM 2020 May Q2
3 marks
2. A plane has equation \(\mathbf { r } \cdot \left[ \begin{array} { l } 1
1
1 \end{array} \right] = 7\)
A line has equation \(\mathbf { r } = \left[ \begin{array} { l } 2
0
1 \end{array} \right] + \mu \left[ \begin{array} { l } 1
0
1 \end{array} \right]\)
Calculate the acute angle between the line and the plane.
Give your answer to the nearest \(0.1 ^ { \circ }\)
[0pt] [3 marks]
SPS SPS FM 2020 May Q3
3 marks
3. Show that $$\cosh ^ { 3 } x + \sinh ^ { 3 } x = \frac { 1 } { 4 } \mathrm { e } ^ { m x } + \frac { 3 } { 4 } \mathrm { e } ^ { n x }$$ where \(m\) and \(n\) are integers.
[0pt] [3 marks]
SPS SPS FM 2020 May Q4
7 marks
4.
  1. If \(z = \cos \theta + \mathrm { i } \sin \theta\), use de Moivre's theorem to prove that $$z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta$$ [3 marks]
  2. Express \(\sin ^ { 5 } \theta\) in terms of \(\sin 5 \theta , \sin 3 \theta\) and \(\sin \theta\)
    [0pt] [4 marks]
  3. Hence show that $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \sin ^ { 5 } \theta \mathrm {~d} \theta = \frac { 53 } { 480 }$$
SPS SPS FM 2020 May Q5
5. The equation \(z ^ { 3 } + k z ^ { 2 } + 9 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
    1. Show that $$\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = k ^ { 2 }$$
  2. (ii) Show that $$\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 } = - 18 k$$
SPS SPS FM 2020 May Q6
4 marks
6. The points \(A , B\) and \(C\) have coordinates \(A ( 4,5,2 ) , B ( - 3,2 , - 4 )\) and \(C ( 2,6,1 )\)
Use a vector product to show that the area of triangle \(A B C\) is \(\frac { 5 \sqrt { 11 } } { 2 }\)
[0pt] [4 marks]
SPS SPS FM 2020 May Q7
7. Prove by induction that \(\mathrm { f } ( n ) = n ^ { 3 } + 3 n ^ { 2 } + 8 n\) is divisible by 6 for all integers \(n \geq 1\)
SPS SPS FM 2020 May Q8
6 marks
8. Let $$S _ { n } = \sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 3 ) }$$ where \(n \geq 1\) Use the method of differences to show that $$S _ { n } = \frac { 5 n ^ { 2 } + a n } { 12 ( n + b ) ( n + c ) }$$ where \(a\), \(b\) and \(c\) are integers.
[0pt] [6 marks]
SPS SPS FM 2020 May Q9
9.
\includegraphics[max width=\textwidth, alt={}, center]{ab2949b2-11f2-4682-ab0c-25ecee2d665a-4_268_648_1169_623} Two tanks, \(A\) and \(B\), each have a capacity of 800 litres. At time \(t = 0\) both tanks are full of pure water. When \(t > 0\), water flows in the following ways:
  • Water with a salt concentration of \(\mu\) grams per litre flows into tank \(A\) at a constant rate
  • Water flows from tank \(A\) to tank \(B\) at a rate of 16 litres per minute
  • Water flows from tank \(B\) to tank \(A\) at a rate of \(r\) litres per minute
  • Water flows out of tank \(B\) through a waste pipe
  • The amount of water in each tank remains at 800 litres.
This system is represented by the coupled differential equations $$\begin{aligned} & \frac { \mathrm { d } x } { \mathrm {~d} t } = 36 - 0.02 x + 0.005 y
& \frac { \mathrm {~d} y } { \mathrm {~d} t } = 0.02 x - 0.02 y \end{aligned}$$ Solve the coupled differential equations to find both \(x\) and \(y\) in terms of \(t\).
SPS SPS FM 2020 May Q10
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ab2949b2-11f2-4682-ab0c-25ecee2d665a-5_643_325_388_822} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A child's toy is a uniform solid consisting of a hemisphere of radius \(r \mathrm {~cm}\) joined to a cone of base radius \(r \mathrm {~cm}\). The curved surface of the cone makes an angle \(\alpha\) with its base. The two shapes are joined at the plane faces with their circumferences coinciding (see Fig. 1). The distance of the centre of mass of the toy above the common circular plane face is \(x \mathrm {~cm}\).
[0pt] [The volume of a sphere is \(\frac { 4 } { 3 } \pi r ^ { 3 }\) and the volume of a cone is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]
  1. Show that \(x = \frac { r \left( \tan ^ { 2 } \alpha - 3 \right) } { 8 + 4 \tan \alpha }\).
SPS SPS FM 2020 May Q11
11. A particle, \(P\), of mass 0.4 kg is moving along the positive \(x\)-axis, in the positive \(x\) direction under the action of a single force. At time \(t\) seconds, \(t > 0 , P\) is \(x\) metres from the origin \(O\) and the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The force is acting in the direction of \(x\) increasing and has magnitude \(\frac { k } { v }\) newtons, where \(k\) is a constant. At \(x = 3 , v = 2\) and at \(x = 6 , v = 2.5\)
  1. Show that \(v ^ { 3 } = \frac { 61 x + 9 } { 24 }\) The time taken for the speed of \(P\) to increase from \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is \(T\) seconds.
  2. Use algebraic integration to show that \(T = \frac { 81 } { 61 }\)
SPS SPS FM 2020 May Q12
12.
[0pt] [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]
A smooth uniform sphere \(A\) has mass 0.2 kg and another smooth uniform sphere \(B\), with the same radius as \(A\), has mass 0.4 kg . The spheres are moving on a smooth horizontal surface when they collide obliquely. Immediately before the collision, the velocity of \(A\) is \(( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(B\) is \(( - 4 \mathbf { i } - \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) At the instant of collision, the line joining the centres of the spheres is parallel to \(\mathbf { i }\) The coefficient of restitution between the spheres is \(\frac { 3 } { 7 }\)
  1. Find the velocity of \(A\) immediately after the collision.
  2. Find the magnitude of the impulse received by \(A\) in the collision.
  3. Find, to the nearest degree, the size of the angle through which the direction of motion of \(A\) is deflected as a result of the collision.
SPS SPS FM 2020 May Q13
13. Six women and five men stand in a line for a photo.
  1. In how many arrangements will all the men stand next to each other and all the women stand next to each other?
  2. In how many arrangements will all the men be apart?