Questions FP3 (539 questions)

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AQA FP3 2010 June Q2
7 marks Standard +0.8
2
  1. Find the value of the constant \(k\) for which \(k \sin 2 x\) is a particular integral of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y = \sin 2 x$$
  2. Hence find the general solution of this differential equation.
AQA FP3 2010 June Q3
7 marks Standard +0.3
3
  1. Explain why \(\int _ { 1 } ^ { \infty } 4 x \mathrm { e } ^ { - 4 x } \mathrm {~d} x\) is an improper integral.
  2. Find \(\quad \int 4 x \mathrm { e } ^ { - 4 x } \mathrm {~d} x\).
  3. Hence evaluate \(\int _ { 1 } ^ { \infty } 4 x \mathrm { e } ^ { - 4 x } \mathrm {~d} x\), showing the limiting process used.
AQA FP3 2010 June Q4
9 marks Standard +0.8
4 By using an integrating factor, find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 3 } { x } y = \left( x ^ { 4 } + 3 \right) ^ { \frac { 3 } { 2 } }$$ given that \(y = \frac { 1 } { 5 }\) when \(x = 1\).
(9 marks)
AQA FP3 2010 June Q5
13 marks Standard +0.8
5
  1. Write down the expansion of \(\cos 4 x\) in ascending powers of \(x\) up to and including the term in \(x ^ { 4 }\). Give your answer in its simplest form.
    1. Given that \(y = \ln \left( 2 - \mathrm { e } ^ { x } \right)\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\).
      (You may leave your expression for \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) unsimplified.)
    2. Hence, by using Maclaurin's theorem, show that the first three non-zero terms in the expansion, in ascending powers of \(x\), of \(\ln \left( 2 - \mathrm { e } ^ { x } \right)\) are $$- x - x ^ { 2 } - x ^ { 3 }$$
  3. Find $$\lim _ { x \rightarrow 0 } \left[ \frac { x \ln \left( 2 - \mathrm { e } ^ { x } \right) } { 1 - \cos 4 x } \right]$$
AQA FP3 2010 June Q6
19 marks Challenging +1.2
6 The polar equation of a curve \(C _ { 1 }\) is $$r = 2 ( \cos \theta - \sin \theta ) , \quad 0 \leqslant \theta \leqslant 2 \pi$$
    1. Find the cartesian equation of \(C _ { 1 }\).
    2. Deduce that \(C _ { 1 }\) is a circle and find its radius and the cartesian coordinates of its centre.
  2. The diagram shows the curve \(C _ { 2 }\) with polar equation $$r = 4 + \sin \theta , \quad 0 \leqslant \theta \leqslant 2 \pi$$ \includegraphics[max width=\textwidth, alt={}, center]{90a59b47-3799-46a2-b76b-ced5cc3e1aac-4_519_847_443_593}
    1. Find the area of the region that is bounded by \(C _ { 2 }\).
    2. Prove that the curves \(C _ { 1 }\) and \(C _ { 2 }\) do not intersect.
    3. Find the area of the region that is outside \(C _ { 1 }\) but inside \(C _ { 2 }\).
AQA FP3 2010 June Q7
14 marks Challenging +1.2
7
  1. Given that \(x = t ^ { \frac { 1 } { 2 } } , x > 0 , t > 0\) and \(y\) is a function of \(x\), show that:
    1. \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 t ^ { \frac { 1 } { 2 } } \frac { \mathrm {~d} y } { \mathrm {~d} t }\);
      (2 marks)
    2. \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 4 t \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} t }\).
      (3 marks)
  2. Hence show that the substitution \(x = t ^ { \frac { 1 } { 2 } }\) transforms the differential equation $$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \left( 8 x ^ { 2 } + 1 \right) \frac { \mathrm { d } y } { \mathrm {~d} x } + 12 x ^ { 3 } y = 12 x ^ { 5 }$$ into $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 4 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 3 y = 3 t$$ (2 marks)
  3. Hence find the general solution of the differential equation $$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \left( 8 x ^ { 2 } + 1 \right) \frac { \mathrm { d } y } { \mathrm {~d} x } + 12 x ^ { 3 } y = 12 x ^ { 5 }$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
AQA FP3 2011 June Q1
5 marks Standard +0.3
1 The function \(y ( x )\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = x + \ln ( 1 + y )$$ 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 four decimal places.
AQA FP3 2011 June Q2
12 marks Standard +0.8
2
  1. Find the values of the constants \(p\) and \(q\) for which \(p + q x \mathrm { e } ^ { - 2 x }\) is a particular integral of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + \frac { \mathrm { d } y } { \mathrm {~d} x } - 2 y = 4 - 9 \mathrm { e } ^ { - 2 x }$$
  2. Hence find the general solution of this differential equation.
  3. Hence express \(y\) in terms of \(x\), given that \(y = 4\) when \(x = 0\) and that \(\frac { \mathrm { d } y } { \mathrm {~d} x } \rightarrow 0\) as \(x \rightarrow \infty\).
AQA FP3 2011 June Q3
7 marks Standard +0.8
3
  1. Find \(\int x ^ { 2 } \ln x \mathrm {~d} x\).
  2. Explain why \(\int _ { 0 } ^ { \mathrm { e } } x ^ { 2 } \ln x \mathrm {~d} x\) is an improper integral.
  3. Evaluate \(\int _ { 0 } ^ { \mathrm { e } } x ^ { 2 } \ln x \mathrm {~d} x\), showing the limiting process used.
AQA FP3 2011 June Q4
10 marks Standard +0.8
4 By using an integrating factor, find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + ( \cot x ) y = \sin 2 x , \quad 0 < x < \frac { \pi } { 2 }$$ given that \(y = \frac { 1 } { 2 }\) when \(x = \frac { \pi } { 6 }\).
(10 marks)
AQA FP3 2011 June Q5
10 marks Challenging +1.2
5
  1. Given that \(y = \ln ( 1 + 2 \tan x )\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
    (You may leave your expression for \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) unsimplified.)
  2. Hence, using Maclaurin's theorem, find the first two non-zero terms in the expansion, in ascending powers of \(x\), of \(\ln ( 1 + 2 \tan x )\).
    (2 marks)
  3. Find $$\lim _ { x \rightarrow 0 } \left[ \frac { \ln ( 1 + 2 \tan x ) } { \ln ( 1 - x ) } \right]$$ (4 marks)
AQA FP3 2011 June Q6
12 marks Challenging +1.2
6 A differential equation is given by $$\left( x ^ { 3 } + 1 \right) \frac { d ^ { 2 } y } { d x ^ { 2 } } - 3 x ^ { 2 } \frac { d y } { d x } = 2 - 4 x ^ { 3 }$$
  1. Show that the substitution $$u = \frac { \mathrm { d } y } { \mathrm {~d} x } - 2 x$$ transforms this differential equation into $$\left( x ^ { 3 } + 1 \right) \frac { \mathrm { d } u } { \mathrm {~d} x } = 3 x ^ { 2 } u$$ (4 marks)
  2. Hence find the general solution of the differential equation $$\left( x ^ { 3 } + 1 \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 3 x ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = 2 - 4 x ^ { 3 }$$ giving your answer in the form \(y = \mathrm { f } ( x )\). \(7 \quad\) The curve \(C _ { 1 }\) is defined by \(r = 2 \sin \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }\). The curve \(C _ { 2 }\) is defined by \(r = \tan \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }\).
    1. Find a cartesian equation of \(C _ { 1 }\).
      1. Prove that the curves \(C _ { 1 }\) and \(C _ { 2 }\) meet at the pole \(O\) and at one other point, \(P\), in the given domain. State the polar coordinates of \(P\).
      2. The point \(A\) is the point on the curve \(C _ { 1 }\) at which \(\theta = \frac { \pi } { 4 }\). The point \(B\) is the point on the curve \(C _ { 2 }\) at which \(\theta = \frac { \pi } { 4 }\). Determine which of the points \(A\) or \(B\) is further away from the pole \(O\), justifying your answer.
      3. Show that the area of the region bounded by the arc \(O P\) of \(C _ { 1 }\) and the arc \(O P\) of \(C _ { 2 }\) is \(a \pi + b \sqrt { 3 }\), where \(a\) and \(b\) are rational numbers.
AQA FP3 2012 June Q1
5 marks Standard +0.3
1 The function \(y ( x )\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = \sqrt { ( 2 x ) } + \sqrt { y }$$ and $$y ( 2 ) = 9$$ 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 \(h = 0.25\), to obtain an approximation to \(y ( 2.25 )\), giving your answer to two decimal places.
AQA FP3 2012 June Q2
5 marks Standard +0.8
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
4 marks Challenging +1.2
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
10 marks Standard +0.3
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
7 marks Standard +0.8
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
11 marks Challenging +1.3
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
19 marks Challenging +1.3
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
14 marks Challenging +1.2
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
5 marks Standard +0.3
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
4 marks Standard +0.3
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
12 marks Standard +0.3
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
6 marks Challenging +1.2
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
9 marks Standard +0.3
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)