Questions — SPS SPS SM Pure (200 questions)

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SPS SPS SM Pure 2023 October Q3
3.
  1. Given that \(\cos A = \frac { 3 } { 4 }\), where \(270 ^ { \circ } < A < 360 ^ { \circ }\), find the exact value of \(\sin 2 A\).
    1. Show that \(\cos \left( 2 x + \frac { \pi } { 3 } \right) + \cos \left( 2 x - \frac { \pi } { 3 } \right) \equiv \cos 2 x\). Given that $$y = 3 \sin ^ { 2 } x + \cos \left( 2 x + \frac { \pi } { 3 } \right) + \cos \left( 2 x - \frac { \pi } { 3 } \right)$$
    2. show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sin 2 x\).
      [0pt] [BLANK PAGE]
SPS SPS SM Pure 2023 October Q4
4. $$\mathrm { f } ( x ) = 12 \cos x - 4 \sin x$$ Given that \(\mathrm { f } ( x ) = R \cos ( x + \alpha )\), where \(R \geqslant 0\) and \(0 \leqslant \alpha \leqslant 90 ^ { \circ }\),
  1. find the value of \(R\) and the value of \(\alpha\).
    (4)
  2. Hence solve the equation $$12 \cos x - 4 \sin x = 7$$ for \(0 \leqslant x < 360 ^ { \circ }\), giving your answers to one decimal place.
    1. Write down the minimum value of \(12 \cos x - 4 \sin x\).
    2. Find, to 2 decimal places, the smallest positive value of \(x\) for which this minimum value occurs.
      [0pt] [BLANK PAGE]
SPS SPS SM Pure 2023 October Q5
5. The curve \(C\) has equation $$y = \frac { 3 + \sin 2 x } { 2 + \cos 2 x }$$
  1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 \sin 2 x + 4 \cos 2 x + 2 } { ( 2 + \cos 2 x ) ^ { 2 } }$$
  2. Find an equation of the tangent to \(C\) at the point on \(C\) where \(x = \frac { \pi } { 2 }\). Write your answer in the form \(y = a x + b\), where \(a\) and \(b\) are exact constants.
    [0pt] [BLANK PAGE]
SPS SPS SM Pure 2023 September Q1
1. a) Find the first four terms, in ascending powers of \(x\), of the binomial expansion of \(\left( 1 + \frac { x } { 2 } \right) ^ { 7 }\), giving each coefficient in exact simplified form.
b) Hence determine the coefficient of \(x\) in the expansion of $$\left( 1 + \frac { 2 } { x } \right) ^ { 2 } \left( 1 + \frac { x } { 2 } \right) ^ { 7 }$$ [BLANK PAGE]
SPS SPS SM Pure 2023 September Q2
2.
\includegraphics[max width=\textwidth, alt={}, center]{e0c7b6f3-9c28-48bd-a401-635efb2521e3-06_562_1118_185_614} The figure above shows a triangle with vertices at \(A ( 2,6 ) , B ( 11,6 )\) and \(C ( p , q )\).
a) Given that the point \(D ( 6,2 )\) is the midpoint of \(A C\), determine the value of \(p\) and the value of \(q\). The straight line \(l\), passes through \(D\) and is perpendicular to \(A C\).
The point \(E\) is the intersection of \(l\) and \(A B\).
b) Find the coordinates of \(E\).
(4)
[0pt] [BLANK PAGE]
SPS SPS SM Pure 2023 September Q3
3. $$x ^ { 2 } + y ^ { 2 } - 2 x - 2 y = 8$$ The circle with the above equation has radius \(r\) and has its centre at the point \(C\).
a) Determine the value of \(r\) and the coordinates of \(C\). The point \(P ( 4,2 )\) lies on the circle.
b) Show that an equation of the tangent to the circle at \(P\) is $$y = 14 - 3 x$$ [BLANK PAGE]
SPS SPS SM Pure 2023 September Q4
4. $$\begin{aligned} & f ( x ) = \mathrm { e } ^ { x } , x \in \mathbb { R } , x > 0
& g ( x ) = 2 x ^ { 3 } + 11 , x \in \mathbb { R } \end{aligned}$$ a) Find and simplify an expression for the composite function \(g f ( x )\).
b) State the domain and range of \(g f ( x )\).
c) Solve the equation $$g f ( x ) = 27$$ The equation \(g f ( x ) = k\), where \(k\) is a constant, has solutions.
d) State the range of the possible values of \(k\).
[0pt] [BLANK PAGE]
SPS SPS SM Pure 2023 September Q5
5. Relative to the origin \(O\), the points \(A , B\) and \(C\) have position vectors \(4 \mathbf { i } + 2 \mathbf { j } , 3 \mathbf { i } + 4 \mathbf { j }\) and \(- \mathbf { i } + 12 \mathbf { j }\), respectively.
  1. Find the magnitude of the vector \(\overrightarrow { O C }\)
  2. Find the angle that the vector \(\overrightarrow { O B }\) makes with the vector \(\mathbf { j }\) to the nearest degree
  3. Show that the points \(A\), \(B\) and \(C\) are collinear
    [0pt] [BLANK PAGE]
SPS SPS SM Pure 2023 September Q6
6. Liquid is kept in containers, which due to evaporation and ongoing chemical reactions, at the end of each month the volume of the liquid in these containers reduces by \(10 \%\) compared with the volume at the start of the same month. One such container is filled up with 250 litres of liquid.
a) Show that the volume of the liquid in the container at the end of the second month is 202.5 litres.
b) Find the volume of the liquid in the container at the end of the twelfth month. (2) At the start of each month a new container is filled up with 250 litres of liquid, so that at the end of twelve months there are 12 containers with liquid.
c) Use an algebraic method to calculate the total amount of liquid in the 12 containers at the end of 12 months.
[0pt] [BLANK PAGE]
SPS SPS SM Pure 2023 September Q7
7.
\includegraphics[max width=\textwidth, alt={}, center]{e0c7b6f3-9c28-48bd-a401-635efb2521e3-16_581_978_210_699} The figure above shows a circular sector \(O A B\) whose centre is at \(O\). The radius of the sector is 60 cm . The points \(C\) and \(D\) lie on \(O A\) and \(O B\) respectively, so that \(| O C | = | O D | = 24 \mathrm {~cm}\). Given that the length of the arc \(A B\) is 48 cm , find the area of the shaded region \(A B D C\), correct to the nearest \(\mathrm { cm } ^ { 2 }\).
[0pt] [BLANK PAGE] $$y = ( 3 - x ) ( 4 + x ) ^ { 2 }$$ a) Sketch the graph of \(C\). The sketch must include any points where the graph meets the coordinate axes.
b) Sketch in separate diagrams the graph of ...
i. \(\quad \ldots \quad y = ( 3 - 2 x ) ( 4 + 2 x ) ^ { 2 }\).
ii. \(\ldots y = ( 3 + x ) ( 4 - x ) ^ { 2 }\).
iii. ... \(y = ( 2 - x ) ( 5 + x ) ^ { 2 }\). Each of the sketches must include any points where the graph meets the coordinate axes.
[0pt] [BLANK PAGE]
SPS SPS SM Pure 2023 September Q9
6 marks
9. Solve the following trigonometric equation in the range given. $$4 \tan ^ { 2 } \theta \cos \theta = 15,0 \leq \theta < 360 ^ { \circ } .$$ [6 marks]
[0pt] [BLANK PAGE]
SPS SPS SM Pure 2023 September Q10
10.
\includegraphics[max width=\textwidth, alt={}, center]{e0c7b6f3-9c28-48bd-a401-635efb2521e3-22_451_506_178_767} The figure above shows solid right prism of height \(h \mathrm {~cm}\). The cross section of the prism is a circular sector of radius \(r \mathrm {~cm}\), subtending an angle of 2 radians at the centre.
a) Given that the volume of the prism is \(1000 \mathrm {~cm} ^ { 3 }\), show clearly that $$S = 2 r ^ { 2 } + \frac { 4000 } { r } ,$$ where \(S \mathrm {~cm} ^ { 2 }\) is the total surface area of the prism.
b) Hence determine the value of \(r\) and the value of \(h\) which make \(S\) least, fully justifying your answer.
[0pt] [BLANK PAGE]
SPS SPS SM Pure 2023 September Q11
11. It is given that $$f ( x ) = x ^ { 2 } - k x + ( k + 3 ) ,$$ where \(k\) is a constant. If the equation \(f ( x ) = 0\) has real roots find the range of the possible values of \(k\).
[0pt] [BLANK PAGE]
SPS SPS SM Pure 2023 September Q12
12.
\includegraphics[max width=\textwidth, alt={}, center]{e0c7b6f3-9c28-48bd-a401-635efb2521e3-26_504_856_239_657} The figure above shows the curve \(C\) with equation $$f ( x ) = \frac { x + 4 } { \sqrt { x } } , x > 0 .$$ a) Determine the coordinates of the minimum point of \(C\), labelled as \(M\). The point \(N\) lies on the \(x\) axis so that \(M N\) is parallel to the \(y\) axis. The finite region \(R\) is bounded by \(C\), the \(x\) axis, the straight line segment \(M N\) and the straight line with equation \(x = 1\).
b) Use the trapezium rule with 4 strips of equal width to estimate the area of \(R\).
[0pt] [BLANK PAGE]
SPS SPS SM Pure 2023 September Q13
13. Prove or disprove each of the following statements:
  1. If \(n\) is an integer, then \(3 n ^ { 2 } - 11 n + 13\) is a prime number.
  2. If \(x\) is a real number, then \(x ^ { 2 } - 8 x + 17\) is positive.
  3. If \(p\) and \(q\) are irrational numbers, then \(p q\) is irrational.
    [0pt] [BLANK PAGE]
SPS SPS SM Pure 2023 September Q14
14.
\includegraphics[max width=\textwidth, alt={}, center]{e0c7b6f3-9c28-48bd-a401-635efb2521e3-30_490_992_226_573} The diagram above shows the curve with equation $$y = ( x - 4 ) ^ { 2 } , x \in \mathbb { R }$$ intersected by the straight line with equation \(y = 4\), at the points \(A\) and \(B\). The curve meets the \(y\) axis at the point \(C\). Calculate the exact area of the shaded region, bounded by the curve and the straight line segments \(A B\) and \(B C\).
[0pt] [BLANK PAGE]
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SPS SPS SM Pure 2023 February Q1
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)
SPS SPS SM Pure 2023 February Q2
2.
\(f ( x ) = 3 x ^ { 2 } + 2 x . \quad\) Find \(f ^ { \prime } ( x )\) from first principles.
(4)
SPS SPS SM Pure 2023 February Q3
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 \(2 \cos x - 3 \sin x = 7 x\)
SPS SPS SM Pure 2023 February Q4
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.
SPS SPS SM Pure 2023 February Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8f97853c-812f-4b7b-9d40-2de7a85886c0-12_832_931_260_502} \captionsetup{labelformat=empty} \caption{Figure 2}
\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 \(R\).
SPS SPS SM Pure 2023 February Q6
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?
SPS SPS SM Pure 2023 February Q7
2 marks
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]
    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 \(x _ { 1 } = 0\)
SPS SPS SM Pure 2023 February Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8f97853c-812f-4b7b-9d40-2de7a85886c0-18_563_853_274_566} \captionsetup{labelformat=empty} \caption{Figure 3}
\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 \(x = 1\)
    \(x\)0123
    \(y\)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)
SPS SPS SM Pure 2023 February Q9
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.