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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 SM 2020 June Q1
1. A curve has equation $$y = 2 x ^ { 3 } - 2 x ^ { 2 } - 2 x + 8$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
  2. Hence find the range of values of \(x\) for which \(y\) is increasing. Write your answer in set notation.
SPS SPS SM 2020 June Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f2033889-3cc5-48de-9bdb-cb1861921a2a-04_556_1052_1119_552} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Diagram not drawn to scale Figure 1 shows the plan view of a design for a stage at a concert.
The stage is modelled as a sector \(B C D F\), of a circle centre \(F\), joined to two congruent triangles \(A B F\) and \(E D F\). Given that
\(A F E\) is a straight line $$\begin{aligned} & A F = F E = 10.7 \mathrm {~m}
& B F = F D = 9.2 \mathrm {~m} \end{aligned}$$ angle \(B F D = 1.82\) radians
find
  1. the perimeter of the stage, in metres, to one decimal place,
  2. the area of the stage, in \(\mathrm { m } ^ { 2 }\), to one decimal place.
SPS SPS SM 2020 June Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f2033889-3cc5-48de-9bdb-cb1861921a2a-05_702_700_278_712} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 is a sketch showing the line \(l _ { 1 }\) with equation \(y = 2 x - 1\) and the point \(A\) with coordinates \(( - 2,3 )\). The line \(l _ { 2 }\) passes through \(A\) and is perpendicular to \(l _ { 1 }\)
  1. Find the equation of \(l _ { 2 }\) writing your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found. The point \(B\) and the point \(C\) lie on \(l _ { 1 }\) such that \(A B C\) is an isosceles triangle with \(A B = A C = 2 \sqrt { 13 }\)
  2. Show that the \(x\) coordinates of points \(B\) and \(C\) satisfy the equation $$5 x ^ { 2 } - 12 x - 32 = 0$$ Given that \(B\) lies in the 3rd quadrant
  3. find, using algebra and showing your working, the coordinates of \(B\).
SPS SPS SM 2020 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f2033889-3cc5-48de-9bdb-cb1861921a2a-06_803_816_269_676} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { g } ( x )\).
The curve has a single turning point, a minimum, at the point \(M ( 4 , - 1.5 )\).
The curve crosses the \(x\)-axis at two points, \(P ( 2,0 )\) and \(Q ( 7,0 )\).
The curve crosses the \(y\)-axis at a single point \(R ( 0,5 )\).
  1. State the coordinates of the turning point on the curve with equation \(y = 2 \mathrm {~g} ( x )\).
  2. State the largest root of the equation $$\mathrm { g } ( x + 1 ) = 0$$
  3. State the range of values of \(x\) for which \(\mathrm { g } ^ { \prime } ( x ) \leqslant 0\) Given that the equation \(\mathrm { g } ( x ) + k = 0\), where \(k\) is a constant, has no real roots,
  4. state the range of possible values for \(k\). Use the binomial expansion to find, in ascending powers of \(x\), the first four terms in the expansion of $$\left( 1 + \frac { 3 } { 4 } x \right) ^ { 6 }$$ simplifying each term.
SPS SPS SM 2020 June Q6
6. A company which makes batteries for electric cars has a 10 -year plan for growth.
  • In year 1 the company will make 2600 batteries
  • In year 10 the company aims to make 12000 batteries
In order to calculate the number of batteries it will need to make each year, from year 2 to year 9 , the company considers the following model: \section*{the number of batteries made will increase by the same percentage each year} Showing detailed reasoning, calculate the total number of batteries made from year 1 to year 10.
SPS SPS SM 2020 June Q7
7.
  1. Solve, for \(- 90 ^ { \circ } \leqslant \theta < 270 ^ { \circ }\), the equation, $$\sin \left( 2 \theta + 10 ^ { \circ } \right) = - 0.6$$ giving your answers to one decimal place.
  2. (a) A student's attempt at the question
    "Solve, for \(- 90 ^ { \circ } < x < 90 ^ { \circ }\), the equation \(7 \tan x = 8 \sin x\) " is set out below. $$\begin{gathered} 7 \tan x = 8 \sin x
    7 \times \frac { \sin x } { \cos x } = 8 \sin x
    7 \sin x = 8 \sin x \cos x
    7 = 8 \cos x
    \cos x = \frac { 7 } { 8 }
    x = 29.0 ^ { \circ } \text { (to } 3 \mathrm { sf } \text { ) } \end{gathered}$$ Identify two mistakes made by this student, giving a brief explanation of each mistake.
    (b) Find the smallest positive solution to the equation $$7 \tan \left( 4 \alpha + 199 ^ { \circ } \right) = 8 \sin \left( 4 \alpha + 199 ^ { \circ } \right)$$
SPS SPS SM 2020 June Q8
8. Prove by contradiction that there are no positive integers \(a\) and \(b\) with \(a\) odd such that $$a + 2 b = \sqrt { 8 a b }$$
SPS SPS SM 2020 June Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f2033889-3cc5-48de-9bdb-cb1861921a2a-09_639_1007_808_561} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Red squirrels were introduced into a large wood in Northumberland on 1st June 1996.
Scientists counted the number of red squirrels in the wood, \(P\), on 1st June each year for \(t\) years after 1996. Scientists counted the number of red squirrels in the wood, \(P\), on 1st June each year for \(t\) years after 1996. The scientists found that over time the number of red squirrels can be modelled by the formula $$P = a b ^ { t }$$ where \(a\) and \(b\) are constants.
The line \(l\), shown in Figure 1, illustrates the linear relationship between \(\log _ { 10 } P\) and \(t\) over a period of 20 years. Using the information given on the graph and using the model,
find a complete equation for the model giving the value of b to 4 significant figures.
SPS SPS SM 2020 June Q10
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f2033889-3cc5-48de-9bdb-cb1861921a2a-10_883_885_283_644} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of the curve \(C\) with equation \(y = 3 x - 2 \sqrt { x } , x \geqslant 0\) and the line \(l\) with equation \(y = 8 x - 16\) The line cuts the curve at point \(A\) as shown in Figure 3.
  1. Using algebra, find the \(x\) coordinate of point \(A\).
    (5)
  2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f2033889-3cc5-48de-9bdb-cb1861921a2a-10_656_814_1786_662} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} The region \(R\) is shown unshaded in Figure 4. Identify the inequalities that define \(R\).
SPS SPS SM 2020 June Q11
11.
  1. Sketch the curve with equation $$y = k - \frac { 1 } { 2 x } \quad \text { where } k \text { is a positive constant }$$ State, in terms of \(k\), the coordinates of any points of intersection with the coordinate axes and the equation of the horizontal asymptote. The straight line \(l\) has equation \(y = 2 x + 3\)
    Given that \(l\) cuts the curve in two distinct places,
  2. find the range of values of \(k\), writing your answer in set notation.
SPS SPS SM 2020 June Q12
12. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f2033889-3cc5-48de-9bdb-cb1861921a2a-11_572_675_1098_772} \captionsetup{labelformat=empty} \caption{Figure 6}
\end{figure} \section*{In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.} Figure 6 shows a sketch of part of the curve with equation $$y = 3 \times 2 ^ { 2 x }$$ The point \(P ( a , 96 \sqrt { 2 } )\) lies on the curve.
  1. Find the exact value of \(a\). The curve with equation \(y = 3 \times 2 ^ { 2 x }\) meets the curve with equation \(y = 6 ^ { 3 - x }\) at the point \(Q\).
  2. Show that the \(x\) coordinate of \(Q\) is $$\frac { 3 + 2 \log _ { 2 } 3 } { 3 + \log _ { 2 } 3 }$$
SPS SPS SM 2020 June Q13
13. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f2033889-3cc5-48de-9bdb-cb1861921a2a-12_803_981_301_651} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows a sketch of the curve \(C\) with equation \(y = ( x - 2 ) ^ { 2 } ( x + 3 )\)
The region \(R\), shown shaded in Figure 5, is bounded by \(C\), the vertical line passing through the maximum turning point of \(C\) and the \(x\)-axis. Find the exact area of \(R\).
(Solutions based entirely on graphical or numerical methods are not acceptable.)
SPS SPS SM Pure 2020 February Q1
4 marks
1
  1. Given that $$2 \ln ( 3 - x ) - \ln ( 21 - 2 x ) = 0$$ show that $$x ^ { 2 } - 4 x - 12 = 0$$ [4]
    1. Write down the roots of the equation \(x ^ { 2 } - 4 x - 12 = 0\).
    2. State which of these two roots is not a solution of $$2 \ln ( 3 - x ) - \ln ( 21 - 2 x ) = 0$$ giving a reason for your answer.