Questions FP3 (473 questions)

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AQA FP3 2012 June Q2
2
  1. Write down the expansion of \(\sin 2 x\) in ascending powers of \(x\) up to and including the term in \(x ^ { 5 }\).
  2. Show that, for some value of \(k\), $$\lim _ { x \rightarrow 0 } \left[ \frac { 2 x - \sin 2 x } { x ^ { 2 } \ln ( 1 + k x ) } \right] = 16$$ and state this value of \(k\).
AQA FP3 2012 June Q3
3 The diagram shows a sketch of a curve \(C\), the pole \(O\) and the initial line.
\includegraphics[max width=\textwidth, alt={}, center]{c4bce668-61f1-4be0-97ee-c635df7e1fc6-2_380_735_1827_648} The polar equation of \(C\) is $$r = 2 \sqrt { 1 + \tan \theta } , \quad - \frac { \pi } { 4 } \leqslant \theta \leqslant \frac { \pi } { 4 }$$ Show that the area of the shaded region, bounded by the curve \(C\) and the initial line, is \(\frac { \pi } { 2 } - \ln 2\).
(4 marks)
AQA FP3 2012 June Q4
4
  1. By using an integrating factor, find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 4 } { 2 x + 1 } y = 4 ( 2 x + 1 ) ^ { 5 }$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
  2. The gradient of a curve at any point \(( x , y )\) on the curve is given by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 4 ( 2 x + 1 ) ^ { 5 } - \frac { 4 } { 2 x + 1 } y$$ The point whose \(x\)-coordinate is zero is a stationary point of the curve. Using your answer to part (a), find the equation of the curve.
AQA FP3 2012 June Q5
5
  1. Find \(\int x ^ { 2 } \mathrm { e } ^ { - x } \mathrm {~d} x\).
  2. Hence evaluate \(\int _ { 0 } ^ { \infty } x ^ { 2 } \mathrm { e } ^ { - x } \mathrm {~d} x\), showing the limiting process used.
AQA FP3 2012 June Q6
6 It is given that \(y = \ln ( 1 + \sin x )\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \mathrm { e } ^ { - y }\).
  3. Express \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\) in terms of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\mathrm { e } ^ { - y }\).
  4. Hence, by using Maclaurin's theorem, find the first four non-zero terms in the expansion, in ascending powers of \(x\), of \(\ln ( 1 + \sin x )\).
AQA FP3 2012 June Q7
7
  1. Show that the substitution \(x = \mathrm { e } ^ { t }\) transforms the differential equation $$\text { into } \quad \begin{aligned} x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 6 y & = 3 + 20 \sin ( \ln x )
    \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 5 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 6 y & = 3 + 20 \sin t \end{aligned}$$ (7 marks)
  2. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 5 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 6 y = 3 + 20 \sin t$$ (11 marks)
  3. Write down the general solution of the differential equation $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 6 y = 3 + 20 \sin ( \ln x )$$
AQA FP3 2012 June Q8
8
  1. A curve has cartesian equation \(x y = 8\). Show that the polar equation of the curve is \(r ^ { 2 } = 16 \operatorname { cosec } 2 \theta\).
  2. The diagram shows a sketch of the curve, \(C\), whose polar equation is $$r ^ { 2 } = 16 \operatorname { cosec } 2 \theta , \quad 0 < \theta < \frac { \pi } { 2 }$$ \includegraphics[max width=\textwidth, alt={}, center]{c4bce668-61f1-4be0-97ee-c635df7e1fc6-4_364_567_1635_726}
    1. Find the polar coordinates of the point \(N\) which lies on the curve \(C\) and is closest to the pole \(O\).
    2. The circle whose polar equation is \(r = 4 \sqrt { 2 }\) intersects the curve \(C\) at the points \(P\) and \(Q\). Find, in an exact form, the polar coordinates of \(P\) and \(Q\).
    3. The obtuse angle \(P N Q\) is \(\alpha\) radians. Find the value of \(\alpha\), giving your answer to three significant figures.
      (5 marks)
AQA FP3 2013 June Q1
1 It is given that \(y ( x )\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = ( x - y ) \sqrt { x + y }$$ and $$y ( 2 ) = 1$$ 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 \(h = 0.2\), to obtain an approximation to \(y ( 2.2 )\), giving your answer to three decimal places.
AQA FP3 2013 June Q2
2 The Cartesian equation of a circle is \(( x + 8 ) ^ { 2 } + ( y - 6 ) ^ { 2 } = 100\).
Using the origin \(O\) as the pole and the positive \(x\)-axis as the initial line, find the polar equation of this circle, giving your answer in the form \(r = p \sin \theta + q \cos \theta\).
(4 marks)
AQA FP3 2013 June Q3
3
  1. Find the values of the constants \(a , b\) and \(c\) for which \(a + b x + c x \mathrm { e } ^ { - 3 x }\) is a particular integral of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 3 y = 3 x - 8 \mathrm { e } ^ { - 3 x }$$
  2. Hence find the general solution of this differential equation.
  3. Hence express \(y\) in terms of \(x\), given that \(y = 1\) when \(x = 0\) and that \(\frac { \mathrm { d } y } { \mathrm {~d} x } \rightarrow - 1\) as \(x \rightarrow \infty\).
AQA FP3 2013 June Q4
4 Evaluate the improper integral $$\int _ { 0 } ^ { \infty } \left( \frac { 2 x } { x ^ { 2 } + 4 } - \frac { 4 } { 2 x + 3 } \right) \mathrm { d } x$$ showing the limiting process used and giving your answer in the form \(\ln k\), where \(k\) is a constant.
AQA FP3 2013 June Q5
5
  1. Differentiate \(\ln ( \ln x )\) with respect to \(x\).
    1. Show that \(\ln x\) is an integrating factor for the first-order differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 1 } { x \ln x } y = 9 x ^ { 2 } , \quad x > 1$$
    2. Hence find the solution of this differential equation, given that \(y = 4 \mathrm { e } ^ { 3 }\) when \(x = \mathrm { e }\).
      (6 marks)
AQA FP3 2013 June Q6
6 It is given that \(y = ( 4 + \sin x ) ^ { \frac { 1 } { 2 } }\).
  1. Express \(y \frac { \mathrm {~d} y } { \mathrm {~d} x }\) in terms of \(\cos x\).
  2. Find the value of \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) when \(x = 0\).
  3. Hence, by using Maclaurin's theorem, find the first four terms in the expansion, in ascending powers of \(x\), of \(( 4 + \sin x ) ^ { \frac { 1 } { 2 } }\).
    (2 marks)
AQA FP3 2013 June Q7
7 A differential equation is given by $$\sin ^ { 2 } x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \sin x \cos x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 2 \sin ^ { 4 } x \cos x , \quad 0 < x < \pi$$
  1. Show that the substitution $$y = u \sin x$$ where \(u\) is a function of \(x\), transforms this differential equation into $$\frac { \mathrm { d } ^ { 2 } u } { \mathrm {~d} x ^ { 2 } } + u = \sin 2 x$$
  2. Hence find the general solution of the differential equation $$\sin ^ { 2 } x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \sin x \cos x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 2 \sin ^ { 4 } x \cos x$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
    (6 marks)
AQA FP3 2013 June Q8
8 The diagram shows a sketch of a curve and a circle.
\includegraphics[max width=\textwidth, alt={}, center]{a2bc95fe-5588-4ff7-a8a3-0cd07df412c9-4_460_693_370_680} The polar equation of the curve is $$r = 3 + 2 \sin \theta , \quad 0 \leqslant \theta \leqslant 2 \pi$$ The circle, whose polar equation is \(r = 2\), intersects the curve at the points \(P\) and \(Q\), as shown in the diagram.
  1. Find the polar coordinates of \(P\) and the polar coordinates of \(Q\).
  2. A straight line, drawn from the point \(P\) through the pole \(O\), intersects the curve again at the point \(A\).
    1. Find the polar coordinates of \(A\).
    2. Find, in surd form, the length of \(A Q\).
    3. Hence, or otherwise, explain why the line \(A Q\) is a tangent to the circle \(r = 2\).
  3. Find the area of the shaded region which lies inside the circle \(r = 2\) but outside the curve \(r = 3 + 2 \sin \theta\). Give your answer in the form \(\frac { 1 } { 6 } ( m \sqrt { 3 } + n \pi )\), where \(m\) and \(n\) are integers.
AQA FP3 2014 June Q1
5 marks
1 It is given that \(y ( x )\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = \frac { \ln ( x + y ) } { \ln y }$$ and $$y ( 6 ) = 3$$ 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 \(h = 0.4\), to obtain an approximation to \(y ( 6.4 )\), giving your answer to three decimal places.
[0pt] [5 marks]
AQA FP3 2014 June Q2
8 marks
2
  1. Find the values of the constants \(a\), \(b\) and \(c\) for which \(a + b \sin 2 x + c \cos 2 x\) is a particular integral of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 4 y = 20 - 20 \cos 2 x$$ [4 marks]
  2. Hence find the solution of this differential equation, given that \(y = 4\) when \(x = 0\).
    [0pt] [4 marks]
    \includegraphics[max width=\textwidth, alt={}]{0eb3e96e-528c-4a99-b164-31cc865f0d68-04_1974_1709_733_153}
AQA FP3 2014 June Q3
4 marks
3 A curve has polar equation \(r ( 4 - 3 \cos \theta ) = 4\). Find its Cartesian equation in the form \(y ^ { 2 } = \mathrm { f } ( x )\).
[0pt] [4 marks]
AQA FP3 2014 June Q4
10 marks
4 Solve the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 3 y = 2 \mathrm { e } ^ { - x }$$ given that \(y \rightarrow 0\) as \(x \rightarrow \infty\) and that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 3\) when \(x = 0\).
[0pt] [10 marks]
\includegraphics[max width=\textwidth, alt={}, center]{0eb3e96e-528c-4a99-b164-31cc865f0d68-08_46_145_829_159}
AQA FP3 2014 June Q5
4 marks
5
  1. Find \(\int x \cos 8 x \mathrm {~d} x\).
  2. Find \(\lim _ { x \rightarrow 0 } \left[ \frac { 1 } { x } \sin 2 x \right]\).
  3. Explain why \(\int _ { 0 } ^ { \frac { \pi } { 4 } } \left( 2 \cot 2 x - \frac { 1 } { x } + x \cos 8 x \right) \mathrm { d } x\) is an improper integral.
  4. Evaluate \(\int _ { 0 } ^ { \frac { \pi } { 4 } } \left( 2 \cot 2 x - \frac { 1 } { x } + x \cos 8 x \right) \mathrm { d } x\), showing the limiting process used. Give your answer as a single term.
    [0pt] [4 marks]
AQA FP3 2014 June Q6
8 marks
6
  1. By using an integrating factor, find the general solution of the differential equation $$\frac { \mathrm { d } u } { \mathrm {~d} x } - \frac { 2 x } { x ^ { 2 } + 4 } u = 3 \left( x ^ { 2 } + 4 \right)$$ giving your answer in the form \(u = \mathrm { f } ( x )\).
    [0pt] [6 marks]
  2. Show that the substitution \(u = x ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x }\) transforms the differential equation $$x ^ { 2 } \left( x ^ { 2 } + 4 \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 8 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 3 \left( x ^ { 2 } + 4 \right) ^ { 2 }$$ into $$\frac { \mathrm { d } u } { \mathrm {~d} x } - \frac { 2 x } { x ^ { 2 } + 4 } u = 3 \left( x ^ { 2 } + 4 \right)$$
  3. Hence, given that \(x > 0\), find the general solution of the differential equation $$x ^ { 2 } \left( x ^ { 2 } + 4 \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 8 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 3 \left( x ^ { 2 } + 4 \right) ^ { 2 }$$ [2 marks]
AQA FP3 2014 June Q7
4 marks
7
  1. It is given that \(y = \ln ( \cos x + \sin x )\).
    1. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \frac { 2 } { 1 + \sin 2 x }\).
    2. Find \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\).
    1. Hence use Maclaurin's theorem to show that the first three non-zero terms in the expansion, in ascending powers of \(x\), of \(\ln ( \cos x + \sin x )\) are \(x - x ^ { 2 } + \frac { 2 } { 3 } x ^ { 3 }\).
    2. Write down the first three non-zero terms in the expansion, in ascending powers of \(x\), of \(\ln ( \cos x - \sin x )\).
  3. Hence find the first three non-zero terms in the expansion, in ascending powers of \(x\), of \(\ln \left( \frac { \cos 2 x } { \mathrm { e } ^ { 3 x - 1 } } \right)\).
    [0pt] [4 marks]
    \includegraphics[max width=\textwidth, alt={}]{0eb3e96e-528c-4a99-b164-31cc865f0d68-17_2484_1707_221_153}
    \includegraphics[max width=\textwidth, alt={}]{0eb3e96e-528c-4a99-b164-31cc865f0d68-19_2484_1707_221_153}
AQA FP3 2014 June Q8
1 marks
8 The diagram shows a sketch of a curve \(C\), the pole \(O\) and the initial line. The curve \(C\) intersects the initial line at the point \(P\).
\includegraphics[max width=\textwidth, alt={}, center]{0eb3e96e-528c-4a99-b164-31cc865f0d68-20_432_949_402_525} The polar equation of \(C\) is \(r = \left( 1 - \tan ^ { 2 } \theta \right) \sec \theta , - \frac { \pi } { 4 } \leqslant \theta \leqslant \frac { \pi } { 4 }\).
  1. Show that the area of the region bounded by the curve \(C\) is \(\frac { 8 } { 15 }\).
  2. The curve whose polar equation is $$r = \frac { 1 } { 2 } \sec ^ { 3 } \theta , \quad - \frac { \pi } { 4 } \leqslant \theta \leqslant \frac { \pi } { 4 }$$ intersects \(C\) at the points \(A\) and \(B\).
    1. Find the polar coordinates of \(A\) and \(B\).
    2. Given that angle \(O A P =\) angle \(O B P = \alpha\), show that \(\tan \alpha = k \sqrt { 3 }\), where \(k\) is an integer.
    3. Using your value of \(k\) from part (b)(ii), state whether the point \(A\) lies inside or lies outside the circle whose diameter is \(O P\). Give a reason for your answer.
      [0pt] [1 mark]
      \includegraphics[max width=\textwidth, alt={}]{0eb3e96e-528c-4a99-b164-31cc865f0d68-21_2484_1707_221_153}
      \includegraphics[max width=\textwidth, alt={}]{0eb3e96e-528c-4a99-b164-31cc865f0d68-23_2484_1707_221_153}
AQA FP3 2015 June Q1
3 marks
1 It is given that \(y ( x )\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = \frac { x + y ^ { 2 } } { x }$$ and $$y ( 2 ) = 5$$
  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 \(y ( 2.05 )\).
  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), to obtain an approximation to \(y ( 2.1 )\), giving your answer to three significant figures.
    [0pt] [3 marks]
AQA FP3 2015 June Q2
9 marks
2 By using an integrating factor, find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + ( \tan x ) y = \tan ^ { 3 } x \sec x$$ given that \(y = 2\) when \(x = \frac { \pi } { 3 }\).
[0pt] [9 marks]