Questions — SPS SPS SM Pure (200 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
SPS SPS SM Pure 2022 June Q4
4. The function f is defined by $$f ( x ) = \frac { 5 x } { 7 x - 5 }$$
  1. The domain of f is the set \(\{ x \in \mathbb { R } : x \neq a \}\) State the value of \(a\)
  2. Prove that f is a self-inverse function
  3. Find the range of f
    [0pt] [BLANK PAGE]
SPS SPS SM Pure 2022 June Q5
5. Relative to a fixed origin \(O\),
  • the point \(A\) has position vector \(- 2 \mathbf { i } + 3 \mathbf { j }\),
  • the point \(B\) has position vector \(3 \mathbf { i } + p \mathbf { j }\), where \(p\) is constant,
Given that \(| \overrightarrow { A B } | = 5 \sqrt { 2 }\), find the possible values for \(p\).
[0pt] [BLANK PAGE]
SPS SPS SM Pure 2022 June Q6
6. A small 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 two models. Model \(A\) assumes that the number of batteries it will make each year will increase by the same number each year.
  1. According to model \(A\), determine the number of batteries the company will make in year 2 . Give your answer to the nearest whole number of batteries. Model \(B\) assumes that the numbers of batteries it will make each year will increase by the same percentage each year.
  2. According to model \(B\), determine the number of batteries the company will make in year 2 . Give your answer to the nearest 10 batteries. Sam calculates the total number of batteries made from year 1 to year 10 inclusive, using each of the two models.
  3. Calculate the difference between the two totals, giving your answer to the nearest 100 batteries.
    [0pt] [BLANK PAGE]
SPS SPS SM Pure 2022 June Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8053bd07-c2b2-4ada-ae0e-8ab6b8466c78-16_504_951_199_578} \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, \(A F = F E = 10.7 \mathrm {~m} , B F = F D = 9.2 \mathrm {~m}\) and angle \(B F D = 1.82\) radians, find the perimeter of the stage, in metres, to one decimal place,
[0pt] [BLANK PAGE]
SPS SPS SM Pure 2022 June Q8
4 marks
8. The function \(\mathrm { f } ( x )\) is such that \(\mathrm { f } ( x ) = - x ^ { 3 } + 2 x ^ { 2 } + k x - 10\) The graph of \(y = \mathrm { f } ( x )\) crosses the \(x\)-axis at the points with coordinates \(( a , 0 ) , ( 2,0 )\) and \(( b , 0 )\) where \(a < b\)
  1. Show that \(k = 5\)
    [0pt] [1 mark]
  2. Find the exact value of \(a\) and the exact value of \(b\)
    [0pt] [3 marks]
  3. The functions \(\mathrm { g } ( x )\) and \(\mathrm { h } ( x )\) are such that $$\begin{aligned} & g ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 5 x - 10
    & h ( x ) = - 8 x ^ { 3 } + 8 x ^ { 2 } + 10 x - 10 \end{aligned}$$
    1. Explain how the graph of \(y = \mathrm { f } ( x )\) can be transformed into the graph of \(y = \mathrm { g } ( x )\) Fully justify your answer.
  5. (ii) Explain how the graph of \(y = \mathrm { f } ( x )\) can be transformed into the graph of \(y = \mathrm { h } ( x )\) Fully justify your answer.
    [0pt] [BLANK PAGE] A geometric series has second term 16 and fourth term 8 All the terms of the series are positive. The \(n\)th term of the series is \(u _ { n }\)
    Find the exact value of \(\sum _ { n = 5 } ^ { \infty } u _ { n }\)
    [0pt] [BLANK PAGE]
SPS SPS SM Pure 2022 June Q10
10.
  1. Sketch the graph of \(y = \cos x\) in the interval \(0 \leqslant x \leqslant 2 \pi\). State the values of the intercepts with the coordinate axes.
    1. Given that $$\sin ^ { 2 } \theta = \cos \theta ( 2 - \cos \theta )$$ prove that \(\cos \theta = \frac { 1 } { 2 }\).
    2. Hence solve the equation $$\sin ^ { 2 } 2 x = \cos 2 x ( 2 - \cos 2 x )$$ in the interval \(0 \leqslant x \leqslant \pi\)
      [0pt] [BLANK PAGE]
SPS SPS SM Pure 2022 June Q11
11. A sequence is defined by $$\begin{aligned} u _ { 1 } & = 600
u _ { n + 1 } & = p u _ { n } + q \end{aligned}$$ where \(p\) and \(q\) are constants.
It is given that \(u _ { 2 } = 500\) and \(u _ { 4 } = 356\)
  1. Find the two possible values of \(u _ { 3 }\)
  2. When \(u _ { n }\) is a decreasing sequence, the limit of \(u _ { n }\) as \(n\) tends to infinity is \(L\). Write down an equation for \(L\) and hence find the value of \(L\).
    [0pt] [BLANK PAGE]
SPS SPS SM Pure 2022 June Q12
5 marks
12. A curve is defined for \(x \geq 0\) by the equation $$y = 6 x - 2 x ^ { \frac { 3 } { 2 } }$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    [0pt] [2 marks]
  2. The curve has one stationary point. Find the coordinates of the stationary point and determine whether it is a maximum or minimum point. Fully justify your answer.
    [0pt] [3 marks]
    [0pt] [BLANK PAGE]
SPS SPS SM Pure 2022 June Q13
13. $$\frac { 1 + 11 x - 6 x ^ { 2 } } { ( x - 3 ) ( 1 - 2 x ) } \equiv A + \frac { B } { ( x - 3 ) } + \frac { C } { ( 1 - 2 x ) }$$ Find the values of the constants \(A , B\) and \(C\).
[0pt] [BLANK PAGE]
SPS SPS SM Pure 2022 June Q14
14. A region, R , is defined by \(x ^ { 2 } - 8 x + 12 \leq y \leq 12 - 2 x\)
a) Sketch a graph to show the region \(R\). Shade the region \(R\).
b) Find the area of R
[0pt] [BLANK PAGE]
SPS SPS SM Pure 2022 June Q15
15. a) Prove that $$n - 1 \text { is divisible by } 3 \Rightarrow n ^ { 3 } - 1 \text { is divisible by } 9$$ b) Show that if \(\log _ { 2 } 3 = \frac { p } { q }\), then $$2 ^ { p } = 3 ^ { q }$$ Use proof by contradiction to prove that \(\log _ { 2 } 3\) is irrational.
[0pt] [BLANK PAGE]
SPS SPS SM Pure 2022 June Q16
16. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8053bd07-c2b2-4ada-ae0e-8ab6b8466c78-34_606_737_146_760} \captionsetup{labelformat=empty} \caption{Figure 6
In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.}
\end{figure} 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 \(\frac { 3 + 2 \log _ { 2 } 3 } { 3 + \log _ { 2 } 3 }\).
    [0pt] [BLANK PAGE]
SPS SPS SM Pure 2022 June Q17
17. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8053bd07-c2b2-4ada-ae0e-8ab6b8466c78-36_613_860_189_653} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of the curve with equation \(y = 2 ^ { x ^ { 2 } } - x\).
The finite region \(R\), shown shaded in Figure 3, is bounded by the curve, the line with equation \(x = - 0.5\), the \(x\)-axis and the line with equation \(x = 1.5\).
  1. The trapezium rule with four strips is used to find an estimate for the area of \(R\). Explain whether the estimate for R is an underestimate or overestimate to the true value for the area of \(R\). The estimate for R is found to be 2.58 .
    Using this value, and showing your working,
  2. estimate the value of \(\int _ { - 0.5 } ^ { 1.5 } \left( 2 ^ { x ^ { 2 } + 1 } + 2 x \right) d x\).
    [0pt] [BLANK PAGE]
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SPS SPS SM Pure 2021 September Q1
5 marks
1.
  1. Find \(\int \left( \frac { 36 } { x ^ { 2 } } + a x \right) \mathrm { d } x\), where \(a\) is a constant.
    [0pt] [3 marks]
  2. Hence, given that \(\int _ { 1 } ^ { 3 } \left( \frac { 36 } { x ^ { 2 } } + a x \right) \mathrm { d } x = 16\), find the value of the constant \(a\).
    [0pt] [2 marks]
SPS SPS SM Pure 2021 September Q2
2. (a) (i) Using the binomial expansion, or otherwise, express \(( 2 + y ) ^ { 3 }\) in the form \(a + b y + c y ^ { 2 } + y ^ { 3 }\), where \(a , b\) and \(c\) are integers.
(ii) Hence show that \(\left( 2 + x ^ { - 2 } \right) ^ { 3 } + \left( 2 - x ^ { - 2 } \right) ^ { 3 }\) can be expressed in the form \(p + q x ^ { - 4 }\), where \(p\) and \(q\) are integers.
(b) (i) Hence find \(\int \left[ \left( 2 + x ^ { - 2 } \right) ^ { 3 } + \left( 2 - x ^ { - 2 } \right) ^ { 3 } \right] \mathrm { d } x\).
SPS SPS SM Pure 2021 September Q3
3. A circle with centre \(C ( 5 , - 3 )\) passes through the point \(A ( - 2,1 )\).
  1. Find the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
  2. Given that \(A B\) is a diameter of the circle, find the coordinates of the point \(B\).
  3. Find an equation of the tangent to the circle at the point \(A\), giving your answer in the form \(p x + q y + r = 0\), where \(p , q\) and \(r\) are integers.
SPS SPS SM Pure 2021 September Q4
4.
  1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{cee51b6b-40d2-4abb-acf7-47c73a919bf9-10_656_776_210_721} \captionsetup{labelformat=empty} \caption{Fig. 12}
    \end{figure} Fig. 12 shows part of the curve \(y = x ^ { 4 }\) and the line \(y = 8 x\), which intersect at the origin and the point P .
    (A) Find the coordinates of P , and show that the area of triangle OPQ is 16 square units.
    (B) Find the area of the region bounded by the line and the curve.
  2. If \(f ( x ) = x ^ { 3 }\), find \(f ^ { \prime } ( x )\) from first principles.
SPS SPS SM Pure 2021 September Q5
5. A curve has the equation $$y = \frac { 12 + x ^ { 2 } \sqrt { x } } { x } , \quad x > 0$$
  1. Express \(\frac { 12 + x ^ { 2 } \sqrt { x } } { x }\) in the form \(12 x ^ { p } + x ^ { q }\).
    1. Hence find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    2. Find an equation of the normal to the curve at the point on the curve where \(x = 4\).
    3. The curve has a stationary point \(P\). Show that the \(x\)-coordinate of \(P\) can be written in the form \(2 ^ { k }\), where \(k\) is a rational number.
SPS SPS SM Pure 2021 September Q6
8 marks
6. The diagram shows a triangle \(A B C\).
\includegraphics[max width=\textwidth, alt={}, center]{cee51b6b-40d2-4abb-acf7-47c73a919bf9-14_499_718_219_703} The lengths of \(A B , B C\) and \(A C\) are \(8 \mathrm {~cm} , 5 \mathrm {~cm}\) and 9 cm respectively.
Angle \(B A C\) is \(\theta\) radians.
  1. Show that \(\theta = 0.586\), correct to three significant figures.
    [0pt] [2 marks]
  2. Find the area of triangle \(A B C\), giving your answer, in \(\mathrm { cm } ^ { 2 }\), to three significant figures.
    [0pt] [2 marks]
  3. A circular sector, centre \(A\) and radius \(r \mathrm {~cm}\), is removed from triangle \(A B C\). The remaining shape is shown shaded in the diagram below.
    \includegraphics[max width=\textwidth, alt={}, center]{cee51b6b-40d2-4abb-acf7-47c73a919bf9-14_488_700_1409_685} Given that the area of the sector removed is equal to the area of the shaded shape, find the perimeter of the shaded shape. Give your answer in cm to three significant figures.
    [0pt] [4 marks]
SPS SPS SM Pure 2021 September Q7
7. The \(n\)th term of a sequence is \(u _ { n }\). The sequence is defined by $$u _ { n + 1 } = p u _ { n } + q$$ where \(p\) and \(q\) are constants.
The first two terms of the sequence are given by \(u _ { 1 } = 96\) and \(u _ { 2 } = 72\).
The limit of \(u _ { n }\) as \(n\) tends to infinity is 24 .
  1. Show that \(p = \frac { 2 } { 3 }\).
  2. Find the value of \(u _ { 3 }\).
SPS SPS SM Pure 2021 September Q8
8. (i) Given that \(6 \tan \theta \sin \theta = 5\), show that \(6 \cos ^ { 2 } \theta + 5 \cos \theta - 6 = 0\).
(3 marks)
(ii) Hence solve the equation \(6 \tan 3 x \sin 3 x = 5\), giving all values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
(2 marks)
SPS SPS SM Pure 2021 September Q9
4 marks
9. The gradient, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), at the point \(( x , y )\) on a curve is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 54 + 27 x - 6 x ^ { 2 }$$
    1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
      [0pt] [2 marks]
    2. The curve passes through the point \(P \left( - 1 \frac { 1 } { 2 } , 4 \right)\). Verify that the curve has a minimum point at \(P\).
      [0pt] [2 marks]
    1. Show that at the points on the curve where \(y\) is decreasing $$2 x ^ { 2 } - 9 x - 18 > 0$$
    2. Solve the inequality \(2 x ^ { 2 } - 9 x - 18 > 0\).
SPS SPS SM Pure 2021 September Q10
10. By first showing that \(\frac { 16 + 9 \sin ^ { 2 } \theta } { 5 - 3 \cos \theta }\) can be expressed in the form \(p + q \cos \theta\), where \(p\) and \(q\) are integers, find the least possible value of \(\frac { 16 + 9 \sin ^ { 2 } \theta } { 5 - 3 \cos \theta }\). State the exact value of \(\theta\), in radians in the interval \(0 \leqslant \theta < 2 \pi\), at which this least value occurs.
SPS SPS SM Pure 2021 September Q11
11.
  1. Given that \(\log _ { 3 } c = m\) and \(\log _ { 27 } d = n\), express \(\frac { \sqrt { c } } { d ^ { 2 } }\) in the form \(3 ^ { y }\), where \(y\) is an expression in terms of \(m\) and \(n\).
  2. Show that the equation $$\log _ { 4 } ( 2 x + 3 ) + \log _ { 4 } ( 2 x + 15 ) = 1 + \log _ { 4 } ( 14 x + 5 )$$ has only one solution and state its value.
SPS SPS SM Pure 2022 November Q1
  1. Do not use a calculator for this question
    a)
Find \(a\), given that \(a ^ { 3 } = 64 x ^ { 12 } y ^ { 3 }\).
b)
  1. Express \(\frac { 81 } { \sqrt { 3 } }\) in the form \(3 ^ { k }\).
  2. Express \(\frac { 5 + \sqrt { 3 } } { 5 - \sqrt { 3 } }\) in the form \(\frac { a + b \sqrt { 3 } } { c }\), where \(a , b\) and \(c\) are integers.