Questions FP3 (539 questions)

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OCR MEI FP3 2016 June Q2
24 marks Challenging +1.2
2 A surface, S , has equation \(z = 3 x ^ { 2 } + 6 x y + y ^ { 3 }\).
  1. Find the equation of the section where \(y = 1\) in the form \(z = \mathrm { f } ( x )\). Sketch this section. Find in three-dimensional vector form the equation of the line of symmetry of this section.
  2. Show that there are two stationary points on S , at \(\mathrm { O } ( 0,0,0 )\) and at \(\mathrm { P } ( - 2,2 , - 4 )\).
  3. Given that the point ( \(- 2 + h , 2 + k , \lambda\) ) lies on the surface, show that $$\lambda = - 4 + 3 ( h + k ) ^ { 2 } + k ^ { 2 } ( k + 3 ) .$$ By considering small values of \(h\) and \(k\), deduce that there is a local minimum at P .
  4. By considering small values of \(x\) and \(y\), show that the stationary point at O is neither a maximum nor a minimum.
  5. Given that \(18 x + 18 y - z = d\) is a tangent plane to S , find the two possible values of \(d\).
OCR MEI FP3 2016 June Q3
24 marks Challenging +1.2
3 Fig. 3 shows the curve with parametric equations \(x = t - 3 t ^ { 3 } , y = 1 + 3 t ^ { 2 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{07eaad51-dc00-44d2-8bff-8652d62902ec-4_634_1294_388_386} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. Show that the values of \(t\) where the curve cuts the \(y\)-axis are \(t = 0 , \pm \frac { 1 } { \sqrt { 3 } }\). Write down the corresponding values of \(y\).
  2. Find the radius and centre of curvature when \(t = \frac { 1 } { \sqrt { 3 } }\). The arc of the curve given by \(0 \leqslant t \leqslant \frac { 1 } { \sqrt { 3 } }\) is denoted by \(C\).
  3. Find the length of \(C\).
  4. Show that the area of the curved surface generated when \(C\) is rotated about the \(y\)-axis through \(2 \pi\) radians is \(\frac { \pi } { 3 }\).
OCR MEI FP3 2016 June Q4
24 marks Challenging +1.2
4
  1. The elements of the set \(P = \{ 1,3,9,11 \}\) are combined under the binary operation, *, defined as multiplication modulo 16.
    1. Demonstrate associativity for the elements \(3,9,11\) in that order. Assuming associativity holds in general, show that \(P\) forms a group under the binary operation *.
    2. Write down the order of each element.
    3. Write down all subgroups of \(P\).
    4. Show that the group in part (i) is cyclic.
  2. Now consider a group of order 4 containing the identity element \(e\) and the two distinct elements, \(a\) and \(b\), where \(a ^ { 2 } = b ^ { 2 } = e\). Construct the composition table. Show that the group is non-cyclic.
  3. Now consider the four matrices \(\mathbf { I } , \mathbf { X } , \mathbf { Y }\) and \(\mathbf { Z }\) where $$\mathbf { I } = \left( \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right) , \mathbf { X } = \left( \begin{array} { r r } 1 & 0 \\ 0 & - 1 \end{array} \right) , \mathbf { Y } = \left( \begin{array} { r r } - 1 & 0 \\ 0 & 1 \end{array} \right) , \mathbf { Z } = \left( \begin{array} { r r } - 1 & 0 \\ 0 & - 1 \end{array} \right) .$$ The group G consists of the set \(\{ \mathbf { I } , \mathbf { X } , \mathbf { Y } , \mathbf { Z } \}\) with binary operation matrix multiplication. Determine which of the groups in parts (a) and (b) is isomorphic to G, and specify the isomorphism.
  4. The distinct elements \(\{ p , q , r , s \}\) are combined under the binary operation \({ } ^ { \circ }\). You are given that \(p ^ { \circ } q = r\) and \(q ^ { \circ } p = s\). By reference to the group axioms, prove that \(\{ p , q , r , s \}\) is not a group under \({ } ^ { \circ }\). Option 5: Markov chains \section*{This question requires the use of a calculator with the ability to handle matrices.}
OCR MEI FP3 2016 June Q5
24 marks Challenging +1.2
5 Each day that Adam is at work he carries out one of three tasks A, B or C. Each task takes a whole day. Adam chooses the task to carry out on each day according to the following set of three rules.
  1. If, on any given day, he has worked on task A then the next day he will choose task A with probability 0.75 , and tasks B and C with equal probability.
  2. If, on any given day, he has worked on task B then the next day he will choose task B or task C with equal probability but will never choose task A .
  3. If, on any given day, he has worked on task C then the next day he will choose task A with probability \(p\) and tasks B and C with equal probability.
    1. Write down the transition matrix.
    2. Over a long period Adam carries out the tasks \(\mathrm { A } , \mathrm { B }\) and C with equal frequency. Find the value of \(p\).
    3. On day 1 Adam chooses task A . Find the probability that he also chooses task A on day 5 .
    Adam decides to change rule 3 as follows.
    If, on any given day, he has worked on task C then the next day he will choose tasks \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) with probabilities \(0.4,0.3,0.3\) respectively.
  4. On day 1 Adam chooses task A. Find the probability that he chooses the same task on day 7 as he did on day 4 .
  5. On a particular day, Adam chooses task A. Find the expected number of consecutive further days on which he will choose A. Adam changes all three rules again as follows.
    • If he works on A one day then on the next day he chooses C .
    • If he works on B one day then on the next day he chooses A or C each with probability 0.5.
    • If he works on C one day then on the next day he chooses A or B each with probability 0.5 .
    • Find the long term probabilities for each task.
AQA FP3 2008 January Q1
6 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 $$\mathrm { f } ( x , y ) = x ^ { 2 } - y ^ { 2 }$$ and $$y ( 2 ) = 1$$
  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 \(y ( 2.1 )\).
  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), to obtain an approximation to \(y ( 2.2 )\).
AQA FP3 2008 January Q2
9 marks Challenging +1.2
2 The diagram shows a sketch of part of the curve \(C\) whose polar equation is \(r = 1 + \tan \theta\). The point \(O\) is the pole. \includegraphics[max width=\textwidth, alt={}, center]{0c177d90-02ae-4e91-bc9d-d0c7051799b8-3_561_629_406_772} The points \(P\) and \(Q\) on the curve are given by \(\theta = 0\) and \(\theta = \frac { \pi } { 3 }\) respectively.
  1. Show that the area of the region bounded by the curve \(C\) and the lines \(O P\) and \(O Q\) is $$\frac { 1 } { 2 } \sqrt { 3 } + \ln 2$$ (6 marks)
  2. Hence find the area of the shaded region bounded by the line \(P Q\) and the arc \(P Q\) of \(C\).
AQA FP3 2008 January Q3
10 marks Standard +0.3
3
  1. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 5$$
  2. Hence express \(y\) in terms of \(x\), given that \(y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3\) when \(x = 0\).
AQA FP3 2008 January Q4
7 marks Standard +0.3
4
  1. Explain why \(\int _ { 1 } ^ { \infty } x \mathrm { e } ^ { - 3 x } \mathrm {~d} x\) is an improper integral.
  2. Find \(\int x \mathrm { e } ^ { - 3 x } \mathrm {~d} x\).
  3. Hence evaluate \(\int _ { 1 } ^ { \infty } x \mathrm { e } ^ { - 3 x } \mathrm {~d} x\), showing the limiting process used.
AQA FP3 2008 January Q5
9 marks Standard +0.3
5 By using an integrating factor, find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 4 x } { x ^ { 2 } + 1 } y = x$$ given that \(y = 1\) when \(x = 0\). Give your answer in the form \(y = \mathrm { f } ( x )\).
AQA FP3 2008 January Q6
8 marks Standard +0.8
6 A curve \(C\) has polar equation $$r ^ { 2 } \sin 2 \theta = 8$$
  1. Find the cartesian equation of \(C\) in the form \(y = \mathrm { f } ( x )\).
  2. Sketch the curve \(C\).
  3. The line with polar equation \(r = 2 \sec \theta\) intersects \(C\) at the point \(A\). Find the polar coordinates of \(A\).
AQA FP3 2008 January Q7
15 marks Challenging +1.2
7
    1. Write down the expansion of \(\ln ( 1 + 2 x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\).
    2. State the range of values of \(x\) for which this expansion is valid.
    1. Given that \(y = \ln \cos x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\).
    2. Find the value of \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\) when \(x = 0\).
    3. Hence, by using Maclaurin's theorem, show that the first two non-zero terms in the expansion, in ascending powers of \(x\), of \(\ln \cos x\) are $$- \frac { x ^ { 2 } } { 2 } - \frac { x ^ { 4 } } { 12 }$$
  3. Find $$\lim _ { x \rightarrow 0 } \left[ \frac { x \ln ( 1 + 2 x ) } { x ^ { 2 } - \ln \cos x } \right]$$
AQA FP3 2008 January Q8
11 marks Standard +0.8
8
  1. Given that \(x = \mathrm { e } ^ { t }\) and that \(y\) is a function of \(x\), show that:
    1. \(x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { \mathrm { d } y } { \mathrm {~d} t }\);
    2. \(\quad x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} t }\).
  2. Hence find the general solution of the differential equation $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 6 y = 0$$
AQA FP3 2009 January Q1
8 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 $$\mathrm { f } ( x , y ) = \frac { x ^ { 2 } + y ^ { 2 } } { x + y }$$ and $$y ( 1 ) = 3$$
  1. Use the Euler formula $$y _ { r + 1 } = y _ { r } + h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$ with \(h = 0.2\), to obtain an approximation to \(y ( 1.2 )\).
  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 \(h = 0.2\), to obtain an approximation to \(y ( 1.2 )\), giving your answer to four decimal places.
AQA FP3 2009 January Q2
7 marks Standard +0.3
2
  1. Show that \(\frac { 1 } { x ^ { 2 } }\) is an integrating factor for the first-order differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } - \frac { 2 } { x } y = x$$
  2. Hence find the general solution of this differential equation, giving your answer in the form \(y = \mathrm { f } ( x )\).
AQA FP3 2009 January Q3
6 marks Challenging +1.2
3 The diagram shows a sketch of a loop, the pole \(O\) and the initial line. \includegraphics[max width=\textwidth, alt={}, center]{f4fdffc7-5647-4462-a983-1564d4e76a4d-3_305_553_383_740} The polar equation of the loop is $$r = ( 2 + \cos \theta ) \sqrt { \sin \theta } , \quad 0 \leqslant \theta \leqslant \pi$$ Find the area enclosed by the loop.
AQA FP3 2009 January Q4
6 marks Standard +0.8
4
  1. Use integration by parts to show that \(\int \ln x \mathrm {~d} x = x \ln x - x + c\), where \(c\) is an arbitrary constant.
  2. Hence evaluate \(\int _ { 0 } ^ { 1 } \ln x \mathrm {~d} x\), showing the limiting process used.
AQA FP3 2009 January Q5
13 marks Standard +0.8
5 The diagram shows a sketch of a curve \(C\), the pole \(O\) and the initial line. \includegraphics[max width=\textwidth, alt={}, center]{f4fdffc7-5647-4462-a983-1564d4e76a4d-3_301_668_1644_689} The curve \(C\) has polar equation $$r = \frac { 2 } { 3 + 2 \cos \theta } , \quad 0 \leqslant \theta \leqslant 2 \pi$$
  1. Verify that the point \(L\) with polar coordinates ( \(2 , \pi\) ) lies on \(C\).
  2. The circle with polar equation \(r = 1\) intersects \(C\) at the points \(M\) and \(N\).
    1. Find the polar coordinates of \(M\) and \(N\).
    2. Find the area of triangle \(L M N\).
  3. Find a cartesian equation of \(C\), giving your answer in the form \(9 y ^ { 2 } = \mathrm { f } ( x )\).
AQA FP3 2009 January Q6
16 marks Standard +0.8
6 The function f is defined by \(\mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } ( 1 + 3 x ) ^ { - \frac { 2 } { 3 } }\).
    1. Use the series expansion for \(\mathrm { e } ^ { x }\) to write down the first four terms in the series expansion of \(\mathrm { e } ^ { 2 x }\).
    2. Use the binomial series expansion of \(( 1 + 3 x ) ^ { - \frac { 2 } { 3 } }\) and your answer to part (a)(i) to show that the first three non-zero terms in the series expansion of \(\mathrm { f } ( x )\) are \(1 + 3 x ^ { 2 } - 6 x ^ { 3 }\).
    1. Given that \(y = \ln ( 1 + 2 \sin x )\), find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
    2. By using Maclaurin's theorem, show that, for small values of \(x\), $$\ln ( 1 + 2 \sin x ) \approx 2 x - 2 x ^ { 2 }$$
  3. Find $$\lim _ { x \rightarrow 0 } \frac { 1 - \mathrm { f } ( x ) } { x \ln ( 1 + 2 \sin x ) }$$
AQA FP3 2009 January Q7
19 marks Challenging +1.2
7
  1. Given that \(x = \mathrm { e } ^ { t }\) and that \(y\) is a function of \(x\), show that $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} t }$$
  2. Hence show that the substitution \(x = \mathrm { e } ^ { t }\) transforms the differential equation $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 10$$ into $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 5 \frac { \mathrm {~d} y } { \mathrm {~d} t } = 10$$
  3. Find the general solution of the differential equation \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 5 \frac { \mathrm {~d} y } { \mathrm {~d} t } = 10\).
  4. Hence solve the differential equation \(x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 10\), given that \(y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 8\) when \(x = 1\).
AQA FP3 2010 January Q1
8 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 $$\mathrm { f } ( x , y ) = x \ln ( 2 x + y )$$ and $$y ( 3 ) = 2$$
  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 \(y ( 3.1 )\), 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 \(y ( 3.1 )\), giving your answer to four decimal places.
AQA FP3 2010 January Q2
8 marks Standard +0.3
2
  1. Given that \(y = \ln ( 4 + 3 x )\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  2. Hence, by using Maclaurin's theorem, find the first three terms in the expansion, in ascending powers of \(x\), of \(\ln ( 4 + 3 x )\).
  3. Write down the first three terms in the expansion, in ascending powers of \(x\), of \(\ln ( 4 - 3 x )\).
  4. Show that, for small values of \(x\), $$\ln \left( \frac { 4 + 3 x } { 4 - 3 x } \right) \approx \frac { 3 } { 2 } x$$
AQA FP3 2010 January Q3
9 marks Standard +0.8
3
  1. A differential equation is given by $$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } = 3 x$$ Show that the substitution $$u = \frac { \mathrm { d } y } { \mathrm {~d} x }$$ transforms this differential equation into $$\frac { \mathrm { d } u } { \mathrm {~d} x } + \frac { 2 } { x } u = 3$$
  2. Find the general solution of $$\frac { \mathrm { d } u } { \mathrm {~d} x } + \frac { 2 } { x } u = 3$$ giving your answer in the form \(u = \mathrm { f } ( x )\).
  3. Hence find the general solution of the differential equation $$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } = 3 x$$ giving your answer in the form \(y = \mathrm { g } ( x )\).
AQA FP3 2010 January Q4
5 marks Standard +0.8
4
  1. Write down the expansion of \(\sin 3 x\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\).
  2. Find $$\lim _ { x \rightarrow 0 } \left[ \frac { 3 x \cos 2 x - \sin 3 x } { 5 x ^ { 3 } } \right]$$
AQA FP3 2010 January Q5
12 marks Challenging +1.2
5 It is given that \(y\) satisfies the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 3 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 2 \mathrm { e } ^ { - 2 x }$$
  1. Find the value of the constant \(p\) for which \(y = p x \mathrm { e } ^ { - 2 x }\) is a particular integral of the given differential equation.
  2. Solve the differential equation, expressing \(y\) in terms of \(x\), given that \(y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 0\).
AQA FP3 2010 January Q6
9 marks Challenging +1.2
6
  1. Explain why \(\int _ { 1 } ^ { \infty } \frac { \ln x ^ { 2 } } { x ^ { 3 } } \mathrm {~d} x\) is an improper integral.
    1. Show that the substitution \(y = \frac { 1 } { x }\) transforms \(\int \frac { \ln x ^ { 2 } } { x ^ { 3 } } \mathrm {~d} x\) into \(\int 2 y \ln y \mathrm {~d} y\).
    2. Evaluate \(\int _ { 0 } ^ { 1 } 2 y \ln y \mathrm {~d} y\), showing the limiting process used.
    3. Hence write down the value of \(\int _ { 1 } ^ { \infty } \frac { \ln x ^ { 2 } } { x ^ { 3 } } \mathrm {~d} x\).