Questions — AQA FP3 (144 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 FP3 2006 January Q2
2
  1. Find \(\int _ { 0 } ^ { a } x \mathrm { e } ^ { - 2 x } \mathrm {~d} x\), where \(a > 0\).
  2. Write down the value of \(\lim _ { a \rightarrow \infty } a ^ { k } \mathrm { e } ^ { - 2 a }\), where \(k\) is a positive constant.
  3. Hence find \(\int _ { 0 } ^ { \infty } x \mathrm { e } ^ { - 2 x } \mathrm {~d} x\).
AQA FP3 2006 January Q3
3
  1. Show that \(y = x ^ { 3 } - x\) is a particular integral of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 x y } { x ^ { 2 } - 1 } = 5 x ^ { 2 } - 1$$
  2. By differentiating \(\left( x ^ { 2 } - 1 \right) y = c\) implicitly, where \(y\) is a function of \(x\) and \(c\) is a constant, show that \(y = \frac { c } { x ^ { 2 } - 1 }\) is a solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 x y } { x ^ { 2 } - 1 } = 0$$
  3. Hence find the general solution of $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 x y } { x ^ { 2 } - 1 } = 5 x ^ { 2 } - 1$$
AQA FP3 2006 January Q4
4
  1. Use the series expansion $$\ln ( 1 + x ) = x - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 } - \frac { 1 } { 4 } x ^ { 4 } + \ldots$$ to write down the first four terms in the expansion, in ascending powers of \(x\), of \(\ln ( 1 - x )\).
  2. The function f is defined by $$\mathrm { f } ( x ) = \mathrm { e } ^ { \sin x }$$ Use Maclaurin's theorem to show that when \(\mathrm { f } ( x )\) is expanded in ascending powers of \(x\) :
    1. the first three terms are $$1 + x + \frac { 1 } { 2 } x ^ { 2 }$$
    2. the coefficient of \(x ^ { 3 }\) is zero.
  3. Find $$\lim _ { x \rightarrow 0 } \frac { \mathrm { e } ^ { \sin x } - 1 + \ln ( 1 - x ) } { x ^ { 2 } \sin x }$$ (4 marks)
AQA FP3 2006 January Q5
5
  1. The function \(y ( x )\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = x \ln x + \frac { y } { x }$$ and $$y ( 1 ) = 1$$
    1. Use the Euler formula $$y _ { r + 1 } = y _ { r } + h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$ with \(h = 0.1\), to obtain an approximation to \(y ( 1.1 )\).
    2. Use the formula $$y _ { r + 1 } = y _ { r - 1 } + 2 h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$ with your answer to part (a)(i) to obtain an approximation to \(y ( 1.2 )\), giving your answer to three decimal places.
    1. Show that \(\frac { 1 } { x }\) is an integrating factor for the first-order differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } - \frac { 1 } { x } y = x \ln x$$
    2. Solve this differential equation, given that \(y = 1\) when \(x = 1\).
    3. Calculate the value of \(y\) when \(x = 1.2\), giving your answer to three decimal places.
AQA FP3 2006 January Q6
6
  1. A circle \(C _ { 1 }\) has cartesian equation \(x ^ { 2 } + ( y - 6 ) ^ { 2 } = 36\). Show that the polar equation of \(C _ { 1 }\) is \(r = 12 \sin \theta\).
  2. A curve \(C _ { 2 }\) with polar equation \(r = 2 \sin \theta + 5,0 \leqslant \theta \leqslant 2 \pi\) is shown in the diagram.
    \includegraphics[max width=\textwidth, alt={}, center]{b572aeb5-bcbb-4d50-964c-7f37e223f51d-5_545_837_559_651} Calculate the area bounded by \(C _ { 2 }\).
  3. The circle \(C _ { 1 }\) intersects the curve \(C _ { 2 }\) at the points \(P\) and \(Q\). Find, in surd form, the area of the quadrilateral \(O P M Q\), where \(M\) is the centre of the circle and \(O\) is the pole.
    (6 marks)
AQA FP3 2007 January Q1
1 The function \(y ( x )\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = \ln \left( 1 + x ^ { 2 } + y \right)$$ and $$y ( 1 ) = 0.6$$
  1. Use the Euler formula $$y _ { r + 1 } = y _ { r } + h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$ with \(h = 0.05\), to obtain an approximation to \(y ( 1.05 )\), giving your answer to four decimal places.
  2. Use the improved Euler formula $$y _ { r + 1 } = y _ { r } + \frac { 1 } { 2 } \left( k _ { 1 } + k _ { 2 } \right)$$ where \(k _ { 1 } = h \mathrm { f } \left( x _ { r } , y _ { r } \right)\) and \(k _ { 2 } = h \mathrm { f } \left( x _ { r } + h , y _ { r } + k _ { 1 } \right)\) and \(h = 0.05\), to obtain an approximation to \(y ( 1.05 )\), giving your answer to four decimal places.
AQA FP3 2007 January Q2
2 A curve has polar equation \(r ( 1 - \sin \theta ) = 4\). Find its cartesian equation in the form \(y = \mathrm { f } ( x )\).
AQA FP3 2007 January Q3
3
  1. Show that \(x ^ { 2 }\) is an integrating factor for the first-order differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 } { x } y = 3 \left( x ^ { 3 } + 1 \right) ^ { \frac { 1 } { 2 } }$$
  2. Solve this differential equation, given that \(y = 1\) when \(x = 2\).
AQA FP3 2007 January Q4
4
  1. Explain why \(\int _ { 0 } ^ { \mathrm { e } } \frac { \ln x } { \sqrt { x } } \mathrm {~d} x\) is an improper integral.
    (1 mark)
  2. Use integration by parts to find \(\int x ^ { - \frac { 1 } { 2 } } \ln x \mathrm {~d} x\).
    (3 marks)
  3. Show that \(\int _ { 0 } ^ { \mathrm { e } } \frac { \ln x } { \sqrt { x } } \mathrm {~d} x\) exists and find its value.
    (4 marks)
AQA FP3 2007 January Q5
5 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 3 y = 6 + 5 \sin x$$ (12 marks)
AQA FP3 2007 January Q6
6 The function f is defined by \(\mathrm { f } ( x ) = ( 1 + 2 x ) ^ { \frac { 1 } { 2 } }\).
    1. Find f'''(x).
    2. Using Maclaurin's theorem, show that, for small values of \(x\), $$\mathrm { f } ( x ) \approx 1 + x - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 2 } x ^ { 3 }$$
  2. Use the expansion of \(\mathrm { e } ^ { x }\) together with the result in part (a)(ii) to show that, for small values of \(x\), $$\mathrm { e } ^ { x } ( 1 + 2 x ) ^ { \frac { 1 } { 2 } } \approx 1 + 2 x + x ^ { 2 } + k x ^ { 3 }$$ where \(k\) is a rational number to be found.
  3. Write down the first four terms in the expansion, in ascending powers of \(x\), of \(\mathrm { e } ^ { 2 x }\).
  4. Find $$\lim _ { x \rightarrow 0 } \frac { \mathrm { e } ^ { x } ( 1 + 2 x ) ^ { \frac { 1 } { 2 } } - \mathrm { e } ^ { 2 x } } { 1 - \cos x }$$ (4 marks)
AQA FP3 2007 January Q7
7 A curve \(C\) has polar equation $$r = 6 + 4 \cos \theta , \quad - \pi \leqslant \theta \leqslant \pi$$ The diagram shows a sketch of the curve \(C\), the pole \(O\) and the initial line.
\includegraphics[max width=\textwidth, alt={}, center]{0d894ac0-8d96-4182-8454-c306e1fdad8f-4_599_866_612_587}
  1. Calculate the area of the region bounded by the curve \(C\).
  2. The point \(P\) is the point on the curve \(C\) for which \(\theta = \frac { 2 \pi } { 3 }\). The point \(Q\) is the point on \(C\) for which \(\theta = \pi\).
    Show that \(Q P\) is parallel to the line \(\theta = \frac { \pi } { 2 }\).
  3. The line \(P Q\) intersects the curve \(C\) again at a point \(R\). The line \(R O\) intersects \(C\) again at a point \(S\).
    1. Find, in surd form, the length of \(P S\).
    2. Show that the angle \(O P S\) is a right angle.
AQA FP3 2007 June Q1
1
  1. Find the value of the constant \(k\) for which \(k x ^ { 2 } \mathrm { e } ^ { 5 x }\) is a particular integral of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 10 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 25 y = 6 \mathrm { e } ^ { 5 x }$$
  2. Hence find the general solution of this differential equation.
AQA FP3 2007 June Q2
2 The function \(y ( x )\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = \sqrt { x ^ { 2 } + y ^ { 2 } + 3 }$$ and $$y ( 1 ) = 2$$
  1. Use the Euler formula $$y _ { r + 1 } = y _ { r } + h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$ with \(h = 0.1\), to obtain an approximation to \(y ( 1.1 )\), giving your answer to four decimal places.
  2. Use the improved Euler formula $$y _ { r + 1 } = y _ { r } + \frac { 1 } { 2 } \left( k _ { 1 } + k _ { 2 } \right)$$ where \(k _ { 1 } = h \mathrm { f } \left( x _ { r } , y _ { r } \right)\) and \(k _ { 2 } = h \mathrm { f } \left( x _ { r } + h , y _ { r } + k _ { 1 } \right)\) and \(h = 0.1\), to obtain an approximation to \(y ( 1.1 )\), giving your answer to four decimal places.
AQA FP3 2007 June Q3
3 By using an integrating factor, find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + ( \tan x ) y = \sec x$$ given that \(y = 3\) when \(x = 0\).
AQA FP3 2007 June Q4
4
  1. Show that \(( \cos \theta + \sin \theta ) ^ { 2 } = 1 + \sin 2 \theta\).
  2. A curve has cartesian equation $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 3 } = ( x + y ) ^ { 4 }$$ Given that \(r \geqslant 0\), show that the polar equation of the curve is $$r = 1 + \sin 2 \theta$$
  3. The curve with polar equation $$r = 1 + \sin 2 \theta , \quad - \pi \leqslant \theta \leqslant \pi$$ is shown in the diagram.
    \includegraphics[max width=\textwidth, alt={}, center]{f90167c3-2ffd-464a-b2d2-9f86a8d64887-3_389_611_1062_708}
    1. Find the two values of \(\theta\) for which \(r = 0\).
    2. Find the area of one of the loops.
AQA FP3 2007 June Q5
5
  1. A differential equation is given by $$\left( x ^ { 2 } - 1 \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = x ^ { 2 } + 1$$ Show that the substitution $$u = \frac { \mathrm { d } y } { \mathrm {~d} x } + x$$ transforms this differential equation into $$\frac { \mathrm { d } u } { \mathrm {~d} x } = \frac { 2 x u } { x ^ { 2 } - 1 }$$ (4 marks)
  2. Find the general solution of $$\frac { \mathrm { d } u } { \mathrm {~d} x } = \frac { 2 x u } { x ^ { 2 } - 1 }$$ giving your answer in the form \(u = \mathrm { f } ( x )\).
  3. Hence find the general solution of the differential equation $$\left( x ^ { 2 } - 1 \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = x ^ { 2 } + 1$$ giving your answer in the form \(y = \mathrm { g } ( x )\).
AQA FP3 2007 June Q6
6
  1. The function f is defined by $$\mathrm { f } ( x ) = \ln \left( 1 + \mathrm { e } ^ { x } \right)$$ Use Maclaurin's theorem to show that when \(\mathrm { f } ( x )\) is expanded in ascending powers of \(x\) :
    1. the first three terms are $$\ln 2 + \frac { 1 } { 2 } x + \frac { 1 } { 8 } x ^ { 2 }$$
    2. the coefficient of \(x ^ { 3 }\) is zero.
  2. Hence write down the first two non-zero terms in the expansion, in ascending powers of \(x\), of \(\ln \left( \frac { 1 + \mathrm { e } ^ { x } } { 2 } \right)\).
  3. Use the series expansion $$\ln ( 1 + x ) = x - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 3 } x ^ { 3 } - \ldots$$ to write down the first three terms in the expansion, in ascending powers of \(x\), of \(\ln \left( 1 - \frac { x } { 2 } \right)\).
  4. Use your answers to parts (b) and (c) to find $$\lim _ { x \rightarrow 0 } \left[ \frac { \ln \left( \frac { 1 + \mathrm { e } ^ { x } } { 2 } \right) + \ln \left( 1 - \frac { x } { 2 } \right) } { x - \sin x } \right]$$
AQA FP3 2007 June Q7
7
  1. Write down the value of $$\lim _ { x \rightarrow \infty } x \mathrm { e } ^ { - x }$$
  2. Use the substitution \(u = x \mathrm { e } ^ { - x } + 1\) to find \(\int \frac { \mathrm { e } ^ { - x } ( 1 - x ) } { x \mathrm { e } ^ { - x } + 1 } \mathrm {~d} x\).
  3. Hence evaluate \(\int _ { 1 } ^ { \infty } \frac { 1 - x } { x + \mathrm { e } ^ { x } } \mathrm {~d} x\), showing the limiting process used.