Questions — SPS (1106 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 2024 February Q10
10. Prove by contradiction that there are infinitely many prime numbers.
SPS SPS SM Pure 2024 February Q11
11. The curves \(y = \mathrm { h } ( x )\) and \(y = \mathrm { h } ^ { - 1 } ( x )\), where \(\mathrm { h } ( x ) = x ^ { 3 } - 8\), are shown below.
The curve \(y = \mathrm { h } ( x )\) crosses the \(x\)-axis at B and the \(y\)-axis at A.
The curve \(y = \mathrm { h } ^ { - 1 } ( x )\) crosses the \(x\)-axis at D and the \(y\)-axis at C .
\includegraphics[max width=\textwidth, alt={}, center]{ede204ac-09c3-486b-8877-df935e6ed015-22_789_798_568_242}
  1. Find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).
  2. Determine the coordinates of A, B, C and D.
  3. Determine the equation of the perpendicular bisector of AB . Give your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be determined.
  4. Points \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D lie on a circle. Determine the equation of the circle. Give your answer in the form \(( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }\), where \(a , b\) and \(r ^ { 2 }\) are constants to be determined.
SPS SPS SM Pure 2024 February Q12
12.
  1. Sketch the graph with equation $$y = | 3 x - 2 a |$$ where \(a\) is a positive constant.
    State the coordinates of each point where the graph cuts or meets the coordinate axes.
  2. Solve, in terms of \(a\), the inequality $$| 3 x - 2 a | \leqslant x + a$$ Given that \(| 3 x - 2 a | \leqslant x + a\)
  3. find, in terms of \(a\), the range of possible values of \(\mathrm { g } ( x )\), where $$\mathrm { g } ( x ) = 5 a - \left| \frac { 1 } { 2 } a - x \right|$$
SPS SPS SM Pure 2024 February Q13
13. A particle moves in the \(x - y\) plane so that its position at time \(t\) s is given by \(x = t ^ { 3 } - 8 t , y = t ^ { 2 }\) for \(- 3.5 < t < 3.5\). The units of distance are metres. The graph shows the path of the particle and the direction of travel at the point \(\mathrm { P } ( 8,4 )\).
\includegraphics[max width=\textwidth, alt={}, center]{ede204ac-09c3-486b-8877-df935e6ed015-28_485_917_445_210}
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Hence show that the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at P is - 1 .
  3. Find the time at which the particle is travelling in the direction opposite to that at P .
  4. Find the cartesian equation of the path, giving \(x ^ { 2 }\) as a function of \(y\).
SPS SPS SM Pure 2024 February Q14
14.
  1. Use the substitution \(x = u ^ { 2 } + 1\) to show that $$\int _ { 5 } ^ { 10 } \frac { 3 \mathrm {~d} x } { ( x - 1 ) ( 3 + 2 \sqrt { x - 1 } ) } = \int _ { p } ^ { q } \frac { 6 \mathrm {~d} u } { u ( 3 + 2 u ) }$$ where \(p\) and \(q\) are positive constants to be found.
  2. Hence, using algebraic integration, show that $$\int _ { 5 } ^ { 10 } \frac { 3 \mathrm {~d} x } { ( x - 1 ) ( 3 + 2 \sqrt { x - 1 } ) } = \ln a$$ where \(a\) is a rational constant to be found.
    (6) Use this page for any additional working. Use this page for any additional working. Use this page for any additional working. Use this page for any additional working.
SPS SPS FM Pure 2024 February Q1
  1. The plane \(x + 2 y + c z = 4\) is perpendicular to the plane \(2 x - c y + 6 z = 9\), where \(c\) is a constant. Find the value of \(c\).
  2. Find the mean value of \(\mathrm { f } ( x ) = x ^ { 2 } + 6 x\) over the interval \([ 0,3 ]\).
  3. It is given that \(1 - 3 \mathrm { i }\) is one root of the quartic equation
$$z ^ { 4 } - 2 z ^ { 3 } + p z ^ { 2 } + r z + 80 = 0$$ where \(p\) and \(r\) are real numbers.
  1. Express \(z ^ { 4 } - 2 z ^ { 3 } + p z ^ { 2 } + r z + 80\) as the product of two quadratic factors with real coefficients.
  2. Find the value of \(p\) and the value of \(r\).
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SPS SPS FM Pure 2024 February Q4
4. Using standard summation of series formulae, determine the sum of the first \(n\) terms of the series $$( 1 \times 2 \times 4 ) + ( 2 \times 3 \times 5 ) + ( 3 \times 4 \times 6 ) + \ldots$$ where \(n\) is a positive integer. Give your answer in fully factorised form.
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SPS SPS FM Pure 2024 February Q5
6 marks
5. The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 0 \quad u _ { n + 1 } = \frac { 5 } { 6 - u _ { n } }$$ Prove by induction that, for all integers \(n \geq 1\), $$u _ { n } = \frac { 5 ^ { n } - 5 } { 5 ^ { n } - 1 }$$ [6 marks]
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SPS SPS FM Pure 2024 February Q6
6.
  1. Explain why \(\int _ { 1 } ^ { \infty } \frac { 1 } { x ( 2 x + 5 ) } \mathrm { d } x\) is an improper integral.
  2. Prove that $$\int _ { 1 } ^ { \infty } \frac { 1 } { x ( 2 x + 5 ) } \mathrm { d } x = a \ln b$$ where \(a\) and \(b\) are rational numbers to be determined.
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SPS SPS FM Pure 2024 February Q7
7. In an Argand diagram the points representing the numbers \(2 + 3 \mathrm { i }\) and \(1 - \mathrm { i }\) are two adjacent vertices of a square, \(S\).
  1. Find the area of \(S\).
  2. Find all the possible pairs of numbers represented by the other two vertices of \(S\).
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SPS SPS FM Pure 2024 February Q8
8. A linear transformation of the plane is represented by the matrix \(\mathbf { M } = \left( \begin{array} { r r } 1 & - 2
\lambda & 3 \end{array} \right)\), where \(\lambda\) is a
constant.
  1. Find the set of values of \(\lambda\) for which the linear transformation has no invariant lines through the origin.
  2. Given that the transformation multiplies areas by 5 and reverses orientation, find the invariant lines.
    [0pt] [BLANK PAGE] \section*{9. In this question you must show detailed reasoning.} The complex number \(- 4 + \mathrm { i } \sqrt { 48 }\) is denoted by \(z\).
  3. Determine the cube roots of \(z\), giving the roots in exponential form. The points which represent the cube roots of \(z\) are denoted by \(A , B\) and \(C\) and these form a triangle in an Argand diagram.
  4. Write down the angles that any lines of symmetry of triangle \(A B C\) make with the positive real axis, justifying your answer.
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SPS SPS FM Pure 2024 February Q10
11 marks
10. The diagram shows the polar curve \(C _ { 1 }\) with equation \(r = 2 \sin \theta\) The diagram also shows part of the polar curve \(C _ { 2 }\) with equation \(r = 1 + \cos 2 \theta\)
\includegraphics[max width=\textwidth, alt={}, center]{44ff7962-1982-46f0-aa02-be7127485bde-18_405_959_331_995}
  1. On the diagram above, complete the sketch of \(C _ { 2 }\)
    [0pt] [2 marks]
  2. Show that the area of the region shaded in the diagram is equal to $$k \pi + m \alpha - \sin 2 \alpha + q \sin 4 \alpha$$ where \(\alpha = \sin ^ { - 1 } \left( \frac { \sqrt { 5 } - 1 } { 2 } \right)\), and \(k , m\) and \(q\) are rational numbers.
    [0pt] [9 marks]
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SPS SPS FM Pure 2024 February Q11
11. Three planes have equations $$\begin{aligned} ( 4 k + 1 ) x - 3 y + ( k - 5 ) z & = 3
( k - 1 ) x + ( 3 - k ) y + 2 z & = 1
7 x - 3 y + 4 z & = 2 \end{aligned}$$
  1. The planes do not meet at a unique point. Show that \(k = 4.5\) is one possible value of \(k\), and find the other possible value of \(k\).
  2. For each value of \(k\) found in part (a), identify the configuration of the given planes. In each case fully justify your answer, stating whether or not the equations of the planes form a consistent system.
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SPS SPS FM Pure 2024 February Q12
7 marks
12. Find the general solution of the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } - 2 y = \frac { x ^ { 3 } } { \sqrt { 4 - 2 x - x ^ { 2 } } }$$ where \(0 < x < \sqrt { 5 } - 1\)
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SPS SPS FM Pure 2024 February Q13
13. In this question you must show detailed reasoning. The region in the first quadrant bounded by curve \(y = \cosh \frac { 1 } { 2 } x ^ { 2 }\), the \(y\)-axis, and the line \(y = 2\) is rotated through \(360 ^ { \circ }\) about the \(y\)-axis. Find the exact volume of revolution generated, expressing your answer in a form involving a logarithm.
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SPS SPS FM Pure 2024 February Q14
14. Show that \(\int _ { 0 } ^ { \frac { 1 } { \sqrt { 3 } } } \frac { 4 } { 1 - x ^ { 4 } } \mathrm {~d} x = \ln ( a + \sqrt { b } ) + \frac { \pi } { c }\) where \(a , b\) and \(c\) are integers to be determined.
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SPS SPS FM Pure 2024 February Q15
15. $$y = \cosh ^ { n } x \quad n \geqslant 5$$
    1. Show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = n ^ { 2 } \cosh ^ { n } x - n ( n - 1 ) \cosh ^ { n - 2 } x$$
    2. Determine an expression for \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\)
  2. Hence, or otherwise, determine the first three non-zero terms of the Maclaurin series for \(y\), simplifying each coefficient and justifying your answer.
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SPS SPS FM 2024 October Q1
  1. Given the function \(f ( x ) = x - x ^ { 2 }\), defined for all real values of \(x\),
    1. Show that \(f ^ { \prime } ( x ) = 1 - 2 x\) by differentiating \(f ( x )\) from first principles.
    2. Find the maximum value of \(f ( x )\).
    3. Explain why \(f ^ { - 1 } ( x )\) does not exist.
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    4. The quadratic equation \(k x ^ { 2 } + 2 k x + 2 k = 3 x - 1\), where \(k\) is a constant, has no real roots.
    5. Show that \(k\) satisfies the inequality
    $$4 k ^ { 2 } + 16 k - 9 > 0$$
  2. Hence find the set of possible values of \(k\). Give your answer in set notation.
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SPS SPS FM 2024 October Q3
3. (a) Find and simplify the first three terms in the expansion, in ascending powers of \(x\), of \(\left( 2 + \frac { 1 } { 3 } k x \right) ^ { 6 }\), where \(k\) is a constant.
(b) In the expansion of \(( 3 - 4 x ) \left( 2 + \frac { 1 } { 3 } k x \right) ^ { 6 }\), the constant term is equal to the coefficient of \(x ^ { 2 }\). Determine the exact value of \(k\), given that \(k\) is positive.
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SPS SPS FM 2024 October Q4
4. The curve \(y = \sqrt { 2 x - 1 }\) is stretched by scale factor \(\frac { 1 } { 4 }\) parallel to the \(x\)-axis and by scale factor \(\frac { 1 } { 2 }\) parallel to the \(y\)-axis. Find the resulting equation of the curve, giving your answer in the form \(\sqrt { a x - b }\) where \(a\) and \(b\) are rational numbers.
[0pt] [BLANK PAGE] \section*{5. In this question you must show detailed reasoning.} The polynomial \(\mathrm { f } ( x )\) is given by $$f ( x ) = x ^ { 3 } + 6 x ^ { 2 } + x - 4$$
  1. (a) Show that \(( x + 1 )\) is a factor of \(\mathrm { f } ( x )\).
    (b) Hence find the exact roots of the equation \(\mathrm { f } ( x ) = 0\).
  2. (a) Show that the equation $$2 \log _ { 2 } ( x + 3 ) + \log _ { 2 } x - \log _ { 2 } ( 4 x + 2 ) = 1$$ can be written in the form \(\mathrm { f } ( x ) = 0\).
    (b) Explain why the equation $$2 \log _ { 2 } ( x + 3 ) + \log _ { 2 } x - \log _ { 2 } ( 4 x + 2 ) = 1$$ has only one real root and state the exact value of this root.
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SPS SPS FM 2024 October Q6
6. The first three terms of a geometric sequence are $$u _ { 1 } = 3 k + 4 \quad u _ { 2 } = 12 - 3 k \quad u _ { 3 } = k + 16$$ where \(k\) is a constant. Given that the sequence converges,
  1. Find the value of \(k\), giving a reason for your answer.
  2. Find the value of \(\sum _ { r = 2 } ^ { \infty } u _ { r }\)
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SPS SPS FM 2024 October Q7
7. The diagram shows part of the graph of \(y = x ^ { 2 }\). The normal to the curve at the point \(A ( 1,1 )\) meets the curve again at \(B\). Angle \(A O B\) is denoted by \(\alpha\).
\includegraphics[max width=\textwidth, alt={}, center]{1e5d102a-955d-4968-8328-339f12665e01-16_506_741_283_217}
  1. Determine the coordinates of \(B\).
  2. Hence determine the exact value of \(\tan \alpha\).
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SPS SPS FM 2024 October Q8
8. Prove by induction that \(11 \times 7 ^ { n } - 13 ^ { n } - 1\) is divisible by 3 , for all integers \(n \geqslant 0\).
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SPS SPS FM 2024 October Q9
9. A circle has centre \(C\) which lies on the \(x\)-axis, as shown in the diagram. The line \(y = x\) meets the circle at \(A\) and \(B\). The midpoint of \(A B\) is \(M\).
\includegraphics[max width=\textwidth, alt={}, center]{1e5d102a-955d-4968-8328-339f12665e01-20_776_730_280_214} The equation of the circle is \(x ^ { 2 } - 6 x + y ^ { 2 } + a = 0\), where \(a\) is a constant.
  1. In this question you must show detailed reasoning. Find the \(x\)-coordinate of M and hence show that the area of triangle ABC is \(\frac { 3 } { 2 } \sqrt { 9 - 2 a }\).
    1. Find the value of \(a\) when the area of triangle \(A B C\) is zero.
    2. Give a geometrical interpretation of the case in part (b)(i).
  3. Give a geometrical interpretation of the case where \(a = 5\).
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SPS SPS FM 2024 October Q1
  1. Solve the following simultaneous equations:
$$\begin{aligned} & y = 4 x ^ { 2 } + 2 x - 5
& y = | 4 x + 1 | \end{aligned}$$