Questions FP2 (1157 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
AQA FP2 2013 June Q8
8
    1. Use de Moivre's theorem to show that $$\cos 4 \theta = \cos ^ { 4 } \theta - 6 \cos ^ { 2 } \theta \sin ^ { 2 } \theta + \sin ^ { 4 } \theta$$ and find a similar expression for \(\sin 4 \theta\).
    2. Deduce that $$\tan 4 \theta = \frac { 4 \tan \theta - 4 \tan ^ { 3 } \theta } { 1 - 6 \tan ^ { 2 } \theta + \tan ^ { 4 } \theta }$$
  2. Explain why \(t = \tan \frac { \pi } { 16 }\) is a root of the equation $$t ^ { 4 } + 4 t ^ { 3 } - 6 t ^ { 2 } - 4 t + 1 = 0$$ and write down the three other roots in trigonometric form.
  3. Hence show that $$\tan ^ { 2 } \frac { \pi } { 16 } + \tan ^ { 2 } \frac { 3 \pi } { 16 } + \tan ^ { 2 } \frac { 5 \pi } { 16 } + \tan ^ { 2 } \frac { 7 \pi } { 16 } = 28$$
AQA FP2 2014 June Q1
7 marks
1
  1. Express - 9 i in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
    [0pt] [2 marks]
  2. Solve the equation \(z ^ { 4 } + 9 \mathrm { i } = 0\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
    [0pt] [5 marks]
AQA FP2 2014 June Q2
3 marks
2
  1. Sketch, on the Argand diagram below, the locus \(L\) of points satisfying $$\arg ( z - 2 \mathrm { i } ) = \frac { 2 \pi } { 3 }$$
    1. A circle \(C\), of radius 3, has its centre lying on \(L\) and touches the line \(\operatorname { Im } ( z ) = 2\). Sketch \(C\) on the Argand diagram used in part (a).
    2. Find the centre of \(C\), giving your answer in the form \(a + b \mathrm { i }\).
      [0pt] [3 marks]
AQA FP2 2014 June Q3
3
  1. Express \(( k + 1 ) ^ { 2 } + 5 ( k + 1 ) + 8\) in the form \(k ^ { 2 } + a k + b\), where \(a\) and \(b\) are constants.
  2. Prove by induction that, for all integers \(n \geqslant 1\), $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) \left( \frac { 1 } { 2 } \right) ^ { r - 1 } = 16 - \left( n ^ { 2 } + 5 n + 8 \right) \left( \frac { 1 } { 2 } \right) ^ { n - 1 }$$
AQA FP2 2014 June Q4
4 The roots of the equation $$z ^ { 3 } + 2 z ^ { 2 } + 3 z - 4 = 0$$ are \(\alpha , \beta\) and \(\gamma\).
    1. Write down the value of \(\alpha + \beta + \gamma\) and the value of \(\alpha \beta + \beta \gamma + \gamma \alpha\).
    2. Hence show that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 2\).
  2. Find the value of:
    1. \(( \alpha + \beta ) ( \beta + \gamma ) + ( \beta + \gamma ) ( \gamma + \alpha ) + ( \gamma + \alpha ) ( \alpha + \beta )\);
    2. \(( \alpha + \beta ) ( \beta + \gamma ) ( \gamma + \alpha )\).
  3. Find a cubic equation whose roots are \(\alpha + \beta , \beta + \gamma\) and \(\gamma + \alpha\).
AQA FP2 2014 June Q5
2 marks
5
  1. Using the definition \(\sinh \theta = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \theta } - \mathrm { e } ^ { - \theta } \right)\), prove the identity $$4 \sinh ^ { 3 } \theta + 3 \sinh \theta = \sinh 3 \theta$$
  2. Given that \(x = \sinh \theta\) and \(16 x ^ { 3 } + 12 x - 3 = 0\), find the value of \(\theta\) in terms of a natural logarithm.
  3. Hence find the real root of the equation \(16 x ^ { 3 } + 12 x - 3 = 0\), giving your answer in the form \(2 ^ { p } - 2 ^ { q }\), where \(p\) and \(q\) are rational numbers.
    [0pt] [2 marks]
AQA FP2 2014 June Q6
6
    1. Use De Moivre's Theorem to show that if \(z = \cos \theta + \mathrm { i } \sin \theta\), then $$z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta$$
    2. Write down a similar expression for \(z ^ { n } + \frac { 1 } { z ^ { n } }\).
    1. Expand \(\left( z - \frac { 1 } { z } \right) ^ { 2 } \left( z + \frac { 1 } { z } \right) ^ { 2 }\) in terms of \(z\).
    2. Hence show that $$8 \sin ^ { 2 } \theta \cos ^ { 2 } \theta = A + B \cos 4 \theta$$ where \(A\) and \(B\) are integers.
  3. Hence, by means of the substitution \(x = 2 \sin \theta\), find the exact value of $$\int _ { 1 } ^ { 2 } x ^ { 2 } \sqrt { 4 - x ^ { 2 } } \mathrm {~d} x$$ \includegraphics[max width=\textwidth, alt={}, center]{5287255f-5ac4-401a-b850-758257412ff7-14_1180_1707_1525_153}
AQA FP2 2014 June Q7
7 marks
7
  1. Given that \(y = \tan ^ { - 1 } \left( \frac { 1 + x } { 1 - x } \right)\) and \(x \neq 1\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 1 + x ^ { 2 } }\).
    [0pt] [4 marks]
  2. Hence, given that \(x < 1\), show that \(\tan ^ { - 1 } \left( \frac { 1 + x } { 1 - x } \right) - \tan ^ { - 1 } x = \frac { \pi } { 4 }\).
    [0pt] [3 marks]
AQA FP2 2014 June Q8
6 marks
8 A curve has equation \(y = 2 \sqrt { x - 1 }\), where \(x > 1\). The length of the arc of the curve between the points on the curve where \(x = 2\) and \(x = 9\) is denoted by \(s\).
  1. Show that \(s = \int _ { 2 } ^ { 9 } \sqrt { \frac { x } { x - 1 } } \mathrm {~d} x\).
    1. Show that \(\cosh ^ { - 1 } 3 = 2 \ln ( 1 + \sqrt { 2 } )\).
    2. Use the substitution \(x = \cosh ^ { 2 } \theta\) to show that $$s = m \sqrt { 2 } + \ln ( 1 + \sqrt { 2 } )$$ where \(m\) is an integer.
      [0pt] [6 marks]
      \includegraphics[max width=\textwidth, alt={}]{5287255f-5ac4-401a-b850-758257412ff7-20_1638_1709_1069_153}
      \includegraphics[max width=\textwidth, alt={}, center]{5287255f-5ac4-401a-b850-758257412ff7-24_2489_1728_221_141}
AQA FP2 2016 June Q1
4 marks
1
  1. Given that \(\mathrm { f } ( r ) = \frac { 1 } { 4 r - 1 }\), show that $$\mathrm { f } ( r ) - \mathrm { f } ( r + 1 ) = \frac { A } { ( 4 r - 1 ) ( 4 r + 3 ) }$$ where \(A\) is an integer.
  2. Use the method of differences to find the value of \(\sum _ { r = 1 } ^ { 50 } \frac { 1 } { ( 4 r - 1 ) ( 4 r + 3 ) }\), giving your answer as a fraction in its simplest form.
    [0pt] [4 marks]
AQA FP2 2016 June Q2
2 The cubic equation \(3 z ^ { 3 } + p z ^ { 2 } + 17 z + q = 0\), where \(p\) and \(q\) are real, has a root \(\alpha = 1 + 2 \mathrm { i }\).
    1. Write down the value of another non-real root, \(\beta\), of this equation.
    2. Hence find the value of \(\alpha \beta\).
  2. Find the value of the third root, \(\gamma\), of this equation.
  3. Find the values of \(p\) and \(q\).
AQA FP2 2016 June Q3
6 marks
3 The arc of the curve with equation \(y = 4 - \ln \left( 1 - x ^ { 2 } \right)\) from \(x = 0\) to \(x = \frac { 3 } { 4 }\) has length \(s\).
  1. Show that \(s = \int _ { 0 } ^ { \frac { 3 } { 4 } } \left( \frac { 1 + x ^ { 2 } } { 1 - x ^ { 2 } } \right) \mathrm { d } x\).
  2. Find the value of \(s\), giving your answer in the form \(p + \ln N\), where \(p\) is a rational number and \(N\) is an integer.
    [0pt] [6 marks]
    \includegraphics[max width=\textwidth, alt={}]{a629b09d-3633-4dbd-83db-7eb89577438c-06_1870_1717_840_150}
AQA FP2 2016 June Q4
4 marks
4
  1. Given that \(y = \tan ^ { - 1 } \sqrt { ( 3 x ) }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving your answer in terms of \(x\).
  2. Hence, or otherwise, show that \(\int _ { \frac { 1 } { 3 } } ^ { 1 } \frac { 1 } { ( 1 + 3 x ) \sqrt { x } } \mathrm {~d} x = \frac { \sqrt { 3 } \pi } { n }\), where \(n\) is an integer.
    [0pt] [4 marks]
AQA FP2 2016 June Q5
5 marks
5
  1. Find the modulus of the complex number \(- 4 \sqrt { 3 } + 4 \mathrm { i }\), giving your answer as an integer.
  2. The locus of points, \(L\), satisfies the equation \(| z + 4 \sqrt { 3 } - 4 \mathrm { i } | = 4\).
    1. Sketch the locus \(L\) on the Argand diagram below.
    2. The complex number \(w\) lies on \(L\) so that \(- \pi < \arg w \leqslant \pi\). Find the least possible value of arg \(w\), giving your answer in terms of \(\pi\).
  3. Solve the equation \(z ^ { 3 } = - 4 \sqrt { 3 } + 4 \mathrm { i }\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
    [0pt] [5 marks]
AQA FP2 2016 June Q6
11 marks
6
  1. Given that \(y = \sinh x\), use the definition of \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\) to show that \(x = \ln \left( y + \sqrt { y ^ { 2 } + 1 } \right)\).
  2. A curve has equation \(y = 6 \cosh ^ { 2 } x + 5 \sinh x\).
    1. Show that the curve has a single stationary point and find its \(x\)-coordinate, giving your answer in the form \(\ln p\), where \(p\) is a rational number.
    2. The curve lies entirely above the \(x\)-axis. The region bounded by the curve, the coordinate axes and the line \(x = \cosh ^ { - 1 } 2\) has area \(A\). Show that $$A = a \cosh ^ { - 1 } 2 + b \sqrt { 3 } + c$$ where \(a\), \(b\) and \(c\) are integers.
      [0pt] [5 marks] \(7 \quad\) Given that \(p \geqslant - 1\), prove by induction that, for all integers \(n \geqslant 1\), $$( 1 + p ) ^ { n } \geqslant 1 + n p$$ [6 marks]
AQA FP2 2016 June Q8
8
  1. By applying de Moivre's theorem to \(( \cos \theta + i \sin \theta ) ^ { 4 }\), where \(\cos \theta \neq 0\), show that $$( 1 + i \tan \theta ) ^ { 4 } + ( 1 - i \tan \theta ) ^ { 4 } = \frac { 2 \cos 4 \theta } { \cos ^ { 4 } \theta }$$
  2. Hence show that \(z = \mathrm { i } \tan \frac { \pi } { 8 }\) satisfies the equation \(( 1 + z ) ^ { 4 } + ( 1 - z ) ^ { 4 } = 0\), and express the three other roots of this equation in the form \(\mathrm { i } \tan \phi\), where \(0 < \phi < \pi\).
  3. Use the results from part (b) to find the values of:
    1. \(\tan ^ { 2 } \frac { \pi } { 8 } \tan ^ { 2 } \frac { 3 \pi } { 8 }\);
    2. \(\tan ^ { 2 } \frac { \pi } { 8 } + \tan ^ { 2 } \frac { 3 \pi } { 8 }\).
      \includegraphics[max width=\textwidth, alt={}]{a629b09d-3633-4dbd-83db-7eb89577438c-23_2488_1709_219_153}
      \section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}
Edexcel FP2 2019 June Q1
  1. A complex number \(z = x + \mathrm { i } y\) is represented by the point \(P\) in an Argand diagram.
Given that $$| z - 3 | = 4 | z + 1 |$$
  1. show that the locus of \(P\) has equation $$15 x ^ { 2 } + 15 y ^ { 2 } + 38 x + 7 = 0$$
  2. Hence find the maximum value of \(| z |\)
Edexcel FP2 2019 June Q2
  1. The matrix \(\mathbf { A }\) is given by
$$\mathbf { A } = \left( \begin{array} { r r r } 6 & - 2 & 2
- 2 & 3 & - 1
2 & - 1 & 3 \end{array} \right)$$
  1. Show that 2 is a repeated eigenvalue of \(\mathbf { A }\) and find the other eigenvalue.
  2. Hence find three non-parallel eigenvectors of \(\mathbf { A }\).
  3. Find a matrix \(\mathbf { P }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P }\) is a diagonal matrix.
Edexcel FP2 2019 June Q3
  1. The number of visits to a website, in any particular month, is modelled as the number of visits received in the previous month plus \(k\) times the number of visits received in the month before that, where \(k\) is a positive constant.
Given that \(V _ { n }\) is the number of visits to the website in month \(n\),
  1. write down a general recurrence relation for \(V _ { n + 2 }\) in terms of \(V _ { n + 1 } , V _ { n }\) and \(k\). For a particular website you are given that
    • \(k = 0.24\)
    • In month 1 , there were 65 visits to the website.
    • In month 2 , there were 71 visits to the website.
    • Show that
    $$V _ { n } = 50 ( 1.2 ) ^ { n } - 25 ( - 0.2 ) ^ { n }$$ This model predicts that the number of visits to this website will exceed one million for the first time in month \(N\).
  2. Find the value of \(N\).
Edexcel FP2 2019 June Q4
    1. Use Fermat's Little Theorem to find the least positive residue of \(6 ^ { 542 }\) modulo 13
    2. Seven students, Alan, Brenda, Charles, Devindra, Enid, Felix and Graham, are attending a concert and will sit in a particular row of 7 seats. Find the number of ways they can be seated if
      1. there are no restrictions where they sit in the row,
    3. Alan, Enid, Felix and Graham sit together,
    4. Brenda sits at one end of the row and Graham sits at the other end of the row,
    5. Charles and Devindra do not sit together.
Edexcel FP2 2019 June Q5
5. $$I _ { n } = \int \operatorname { cosec } ^ { n } x \mathrm {~d} x \quad n \in \mathbb { Z }$$
  1. Prove that, for \(n \geqslant 2\) $$I _ { n } = \frac { n - 2 } { n - 1 } I _ { n - 2 } - \frac { \operatorname { cosec } ^ { n - 2 } x \cot x } { n - 1 }$$
  2. Hence show that $$\int _ { \frac { \pi } { 3 } } ^ { \frac { \pi } { 2 } } \operatorname { cosec } ^ { 6 } x \mathrm {~d} x = \frac { 56 } { 135 } \sqrt { 3 }$$
Edexcel FP2 2019 June Q6
    1. A binary operation * is defined on positive real numbers by
$$a * b = a + b + a b$$ Prove that the operation * is associative.
(ii) The set \(G = \{ 1,2,3,4,5,6 \}\) forms a group under the operation of multiplication modulo 7
  1. Show that \(G\) is cyclic. The set \(H = \{ 1,5,7,11,13,17 \}\) forms a group under the operation of multiplication modulo 18
  2. List all the subgroups of \(H\).
  3. Describe an isomorphism between \(G\) and \(H\).
Edexcel FP2 2019 June Q7
  1. A transformation from the \(z\)-plane to the \(w\)-plane is given by
$$w = \frac { 3 \mathrm { i } z - 2 } { z + \mathrm { i } } \quad z \neq - \mathrm { i }$$
  1. Show that the circle \(C\) with equation \(| z + \mathrm { i } | = 1\) in the \(z\)-plane is mapped to a circle \(D\) in the \(w\)-plane, giving a Cartesian equation for \(D\).
  2. Sketch \(C\) and \(D\) on Argand diagrams.
Edexcel FP2 2019 June Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4ba4a815-f53d-4de2-810b-b06e145f457b-24_547_629_242_717} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the vertical cross section of a child's spinning top. The point \(A\) is vertically above the point \(B\) and the height of the spinning top is 5 cm . The line \(C D\) is perpendicular to \(A B\) such that \(C D\) is the maximum width of the spinning top.
The spinning top is modelled as the solid of revolution created when part of the curve with polar equation $$r ^ { 2 } = 25 \cos 2 \theta$$ is rotated through \(2 \pi\) radians about the initial line.
  1. Show that, according to the model, the surface area of the spinning top is $$k \pi ( 2 - \sqrt { 2 } ) \mathrm { cm } ^ { 2 }$$ where \(k\) is a constant to be determined.
  2. Show that, according to the model, the length \(C D\) is \(\frac { 5 \sqrt { 2 } } { 2 } \mathrm {~cm}\).
Edexcel FP2 2020 June Q1
  1. A small sports club has 12 adult members and 14 junior members.
The club needs to enter a team of 8 players for a particular competition.
Determine the number of ways in which the team can be selected if
  1. there are no restrictions on the team,
  2. the team must contain 4 adults and 4 juniors,
  3. more than half the team must be adults.