Questions FP2 (1279 questions)

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Edexcel FP2 Specimen Q4
13 marks Challenging +1.8
  1. A group \(G\) contains distinct elements \(a , b\) and \(e\) where \(e\) is the identity element and the group operation is multiplication. Given \(a ^ { 2 } b = b a\), prove \(a b \neq b a\)
  2. The set \(H = \{ 1,2,4,7,8,11,13,14 \}\) forms a group under the operation of multiplication modulo 15
  1. Find the order of each element of \(H\).
  2. Find three subgroups of \(H\) each of order 4, and describe each of these subgroups. The elements of another group \(J\) are the matrices \(\left( \begin{array} { c c } \cos \left( \frac { k \pi } { 4 } \right) & \sin \left( \frac { k \pi } { 4 } \right) \\ - \sin \left( \frac { k \pi } { 4 } \right) & \cos \left( \frac { k \pi } { 4 } \right) \end{array} \right)\) where \(k = 1,2,3,4,5,6,7,8\) and the group operation is matrix multiplication.
  3. Determine whether \(H\) and \(J\) are isomorphic, giving a reason for your answer.
Edexcel FP2 Specimen Q5
12 marks Challenging +1.8
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1c262813-4160-4eda-9a36-e4ba38182c8a-14_480_588_210_740} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} An engineering student makes a miniature arch as part of the design for a piece of coursework. The cross-section of this arch is modelled by the curve with equation $$y = A - \frac { 1 } { 2 } \cosh 2 x , \quad - \ln a \leqslant x \leqslant \ln a$$ where \(a > 1\) and \(A\) is a positive constant. The curve begins and ends on the \(x\)-axis, as shown in Figure 1.
  1. Show that the length of this curve is \(k \left( a ^ { 2 } - \frac { 1 } { a ^ { 2 } } \right)\), stating the value of the constant \(k\). The length of the curved cross-section of the miniature arch is required to be 2 m long.
  2. Find the height of the arch, according to this model, giving your answer to 2 significant figures.
  3. Find also the width of the base of the arch giving your answer to 2 significant figures.
  4. Give the equation of another curve that could be used as a suitable model for the cross-section of an arch, with approximately the same height and width as you found using the first model.
    (You do not need to consider the arc length of your curve)
Edexcel FP2 Specimen Q6
9 marks Challenging +1.2
  1. A curve has equation
$$| z + 6 | = 2 | z - 6 | \quad z \in \mathbb { C }$$
  1. Show that the curve is a circle with equation \(x ^ { 2 } + y ^ { 2 } - 20 x + 36 = 0\)
  2. Sketch the curve on an Argand diagram. The line \(l\) has equation \(a z ^ { * } + a ^ { * } z = 0\), where \(a \in \mathbb { C }\) and \(z \in \mathbb { C }\) Given that the line \(l\) is a tangent to the curve and that \(\arg a = \theta\)
  3. find the possible values of \(\tan \theta\)
Edexcel FP2 Specimen Q7
9 marks Challenging +1.2
7. $$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { n } x \mathrm {~d} x , \quad n \geqslant 0$$
  1. Prove that, for \(n \geqslant 2\), $$n I _ { n } = ( n - 1 ) I _ { n - 2 }$$
  2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{1c262813-4160-4eda-9a36-e4ba38182c8a-22_588_1018_630_520} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} A designer is asked to produce a poster to completely cover the curved surface area of a solid cylinder which has diameter 1 m and height 0.7 m . He uses a large sheet of paper with height 0.7 m and width of \(\pi \mathrm { m }\).
    Figure 2 shows the first stage of the design, where the poster is divided into two sections by a curve. The curve is given by the equation $$y = \sin ^ { 2 } ( 4 x ) - \sin ^ { 10 } ( 4 x )$$ relative to axes taken along the bottom and left hand edge of the paper.
    The region of the poster below the curve is shaded and the region above the curve remains unshaded, as shown in Figure 2. Find the exact area of the poster which is shaded.
Edexcel FP2 Specimen Q8
9 marks Standard +0.8
  1. A staircase has \(n\) steps. A tourist moves from the bottom (step zero) to the top (step \(n\) ). At each move up the staircase she can go up either one step or two steps, and her overall climb up the staircase is a combination of such moves.
If \(u _ { n }\) is the number of ways that the tourist can climb up a staircase with \(n\) steps,
  1. explain why \(u _ { n }\) satisfies the recurrence relation $$u _ { n } = u _ { n - 1 } + u _ { n - 2 } , \text { with } u _ { 1 } = 1 \text { and } u _ { 2 } = 2$$
  2. Find the number of ways in which she can climb up a staircase when there are eight steps. A staircase at a certain tourist attraction has 400 steps.
  3. Show that the number of ways in which she could climb up to the top of this staircase is given by $$\frac { 1 } { \sqrt { 5 } } \left[ \left( \frac { 1 + \sqrt { 5 } } { 2 } \right) ^ { 401 } - \left( \frac { 1 - \sqrt { 5 } } { 2 } \right) ^ { 401 } \right]$$
OCR MEI FP2 2009 January Q2
18 marks Standard +0.3
  1. Write down the modulus and argument of the complex number \(\mathrm { e } ^ { \mathrm { j } \pi / 3 }\).
  2. The triangle OAB in an Argand diagram is equilateral. O is the origin; A corresponds to the complex number \(a = \sqrt { 2 } ( 1 + \mathrm { j } ) ; \mathrm { B }\) corresponds to the complex number \(b\). Show A and the two possible positions for B in a sketch. Express \(a\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\). Find the two possibilities for \(b\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\).
  3. Given that \(z _ { 1 } = \sqrt { 2 } \mathrm { e } ^ { \mathrm { j } \pi / 3 }\), show that \(z _ { 1 } ^ { 6 } = 8\). Write down, in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\), the other five complex numbers \(z\) such that \(z ^ { 6 } = 8\). Sketch all six complex numbers in a new Argand diagram. Let \(w = z _ { 1 } \mathrm { e } ^ { - \mathrm { j } \pi / 12 }\).
  4. Find \(w\) in the form \(x + \mathrm { j } y\), and mark this complex number on your Argand diagram.
  5. Find \(w ^ { 6 }\), expressing your answer in as simple a form as possible.
OCR MEI FP2 2013 January Q2
18 marks Challenging +1.3
    1. Show that $$1 + \mathrm { e } ^ { \mathrm { j } 2 \theta } = 2 \cos \theta ( \cos \theta + \mathrm { j } \sin \theta )$$
    2. The series \(C\) and \(S\) are defined as follows. $$\begin{aligned} & C = 1 + \binom { n } { 1 } \cos 2 \theta + \binom { n } { 2 } \cos 4 \theta + \ldots + \cos 2 n \theta \\ & S = \binom { n } { 1 } \sin 2 \theta + \binom { n } { 2 } \sin 4 \theta + \ldots + \sin 2 n \theta \end{aligned}$$ By considering \(C + \mathrm { j } S\), show that $$C = 2 ^ { n } \cos ^ { n } \theta \cos n \theta$$ and find a corresponding expression for \(S\).
    1. Express \(\mathrm { e } ^ { \mathrm { j } 2 \pi / 3 }\) in the form \(x + \mathrm { j } y\), where the real numbers \(x\) and \(y\) should be given exactly.
    2. An equilateral triangle in the Argand diagram has its centre at the origin. One vertex of the triangle is at the point representing \(2 + 4 \mathrm { j }\). Obtain the complex numbers representing the other two vertices, giving your answers in the form \(x + \mathrm { j } y\), where the real numbers \(x\) and \(y\) should be given exactly.
    3. Show that the length of a side of the triangle is \(2 \sqrt { 15 }\).
CAIE FP2 2018 June Q5
12 marks Challenging +1.2
  1. Show that the moment of inertia of the object about the axis \(l\) is \(180 M a ^ { 2 }\).
  2. Show that small oscillations of the object about the axis \(l\) are approximately simple harmonic, and state the period.
CAIE FP2 2019 June Q4
11 marks Challenging +1.3
  1. Find the moment of inertia of the object, consisting of the rod and two spheres, about \(L\).
    The object is pivoted at \(A\) so that it can rotate freely about \(L\). The object is released from rest with the rod making an angle of \(60 ^ { \circ }\) to the downward vertical. The greatest angular speed attained by the object in the subsequent motion is \(\frac { 9 } { 20 } \sqrt { } \left( \frac { g } { a } \right)\).
  2. Find the value of \(k\).
CAIE FP2 2017 November Q5
12 marks Challenging +1.3
  1. Show that the moment of inertia of the system, consisting of frame and small object, about an axis through \(O\) perpendicular to the plane of the frame, is \(\frac { 169 } { 3 } m a ^ { 2 }\).
  2. Show that small oscillations of the system about this axis are approximately simple harmonic and state their period.
CAIE FP2 2017 Specimen Q3
11 marks Standard +0.8
  1. Find the value of \(k\).
  2. The particle \(P\) is released from rest at a point between \(A\) and \(B\) where both strings are taut. Show that \(P\) performs simple harmonic motion and state the period of the motion.
  3. In the case where \(P\) is released from rest at a distance \(0.2 a \mathrm {~m}\) from \(M\), the speed of \(P\) is \(0.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(P\) is \(0.05 a \mathrm {~m}\) from \(M\). Find the value of \(a\).
OCR MEI FP2 2006 June Q5
18 marks Challenging +1.2
5 A curve has parametric equations $$x = \theta - k \sin \theta , \quad y = 1 - \cos \theta ,$$ where \(k\) is a positive constant.
  1. For the case \(k = 1\), use your graphical calculator to sketch the curve. Describe its main features.
  2. Sketch the curve for a value of \(k\) between 0 and 1 . Describe briefly how the main features differ from those for the case \(k = 1\).
  3. For the case \(k = 2\) :
    (A) sketch the curve;
    (B) find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\);
    (C) show that the width of each loop, measured parallel to the \(x\)-axis, is $$2 \sqrt { 3 } - \frac { 2 \pi } { 3 }$$
  4. Use your calculator to find, correct to one decimal place, the value of \(k\) for which successive loops just touch each other.
AQA FP2 2006 January Q1
6 marks Standard +0.3
1
  1. Show that $$\frac { 1 } { r ^ { 2 } } - \frac { 1 } { ( r + 1 ) ^ { 2 } } = \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } }$$
  2. Hence find the sum of the first \(n\) terms of the series $$\frac { 3 } { 1 ^ { 2 } \times 2 ^ { 2 } } + \frac { 5 } { 2 ^ { 2 } \times 3 ^ { 2 } } + \frac { 7 } { 3 ^ { 2 } \times 4 ^ { 2 } } + \ldots$$
AQA FP2 2006 January Q2
10 marks Standard +0.3
2 The cubic equation $$x ^ { 3 } + p x ^ { 2 } + q x + r = 0$$ where \(p , q\) and \(r\) are real, has roots \(\alpha , \beta\) and \(\gamma\).
  1. Given that $$\alpha + \beta + \gamma = 4 \quad \text { and } \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 20$$ find the values of \(p\) and \(q\).
  2. Given further that one root is \(3 + \mathrm { i }\), find the value of \(r\).
AQA FP2 2006 January Q3
12 marks Moderate -0.3
3 The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by $$z _ { 1 } = \frac { 1 + \mathrm { i } } { 1 - \mathrm { i } } \quad \text { and } \quad z _ { 2 } = \frac { 1 } { 2 } + \frac { \sqrt { 3 } } { 2 } \mathrm { i }$$
  1. Show that \(z _ { 1 } = \mathrm { i }\).
  2. Show that \(\left| z _ { 1 } \right| = \left| z _ { 2 } \right|\).
  3. Express both \(z _ { 1 }\) and \(z _ { 2 }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
  4. Draw an Argand diagram to show the points representing \(z _ { 1 } , z _ { 2 }\) and \(z _ { 1 } + z _ { 2 }\).
  5. Use your Argand diagram to show that $$\tan \frac { 5 } { 12 } \pi = 2 + \sqrt { 3 }$$
AQA FP2 2006 January Q4
9 marks Challenging +1.2
4
  1. Prove by induction that $$2 + ( 3 \times 2 ) + \left( 4 \times 2 ^ { 2 } \right) + \ldots + ( n + 1 ) 2 ^ { n - 1 } = n 2 ^ { n }$$ for all integers \(n \geqslant 1\).
  2. Show that $$\sum _ { r = n + 1 } ^ { 2 n } ( r + 1 ) 2 ^ { r - 1 } = n 2 ^ { n } \left( 2 ^ { n + 1 } - 1 \right)$$
AQA FP2 2006 January Q5
9 marks Standard +0.8
5 The complex number \(z\) satisfies the relation $$| z + 4 - 4 i | = 4$$
  1. Sketch, on an Argand diagram, the locus of \(z\).
  2. Show that the greatest value of \(| z |\) is \(4 ( \sqrt { 2 } + 1 )\).
  3. Find the value of \(z\) for which $$\arg ( z + 4 - 4 \mathrm { i } ) = \frac { 1 } { 6 } \pi$$ Give your answer in the form \(a + \mathrm { i } b\).
AQA FP2 2006 January Q6
12 marks Challenging +1.2
6 It is given that \(z = \mathrm { e } ^ { \mathrm { i } \theta }\).
    1. Show that $$z + \frac { 1 } { z } = 2 \cos \theta$$ (2 marks)
    2. Find a similar expression for $$z ^ { 2 } + \frac { 1 } { z ^ { 2 } }$$ (2 marks)
    3. Hence show that $$z ^ { 2 } - z + 2 - \frac { 1 } { z } + \frac { 1 } { z ^ { 2 } } = 4 \cos ^ { 2 } \theta - 2 \cos \theta$$ (3 marks)
  2. Hence solve the quartic equation $$z ^ { 4 } - z ^ { 3 } + 2 z ^ { 2 } - z + 1 = 0$$ giving the roots in the form \(a + \mathrm { i } b\).
AQA FP2 2006 January Q7
17 marks Challenging +1.2
7
  1. Use the definitions $$\sinh \theta = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \theta } - \mathrm { e } ^ { - \theta } \right) \quad \text { and } \quad \cosh \theta = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \theta } + \mathrm { e } ^ { - \theta } \right)$$ to show that:
    1. \(2 \sinh \theta \cosh \theta = \sinh 2 \theta\);
    2. \(\cosh ^ { 2 } \theta + \sinh ^ { 2 } \theta = \cosh 2 \theta\).
  2. A curve is given parametrically by $$x = \cosh ^ { 3 } \theta , \quad y = \sinh ^ { 3 } \theta$$
    1. Show that $$\left( \frac { \mathrm { d } x } { \mathrm {~d} \theta } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} \theta } \right) ^ { 2 } = \frac { 9 } { 4 } \sinh ^ { 2 } 2 \theta \cosh 2 \theta$$
    2. Show that the length of the arc of the curve from the point where \(\theta = 0\) to the point where \(\theta = 1\) is $$\frac { 1 } { 2 } \left[ ( \cosh 2 ) ^ { \frac { 3 } { 2 } } - 1 \right]$$
AQA FP2 2007 January Q1
7 marks Standard +0.3
1
  1. Given that $$4 \cosh ^ { 2 } x = 7 \sinh x + 1$$ find the two possible values of \(\sinh x\).
  2. Hence obtain the two possible values of \(x\), giving your answers in the form \(\ln p\).
AQA FP2 2007 January Q2
8 marks Standard +0.3
2
  1. Sketch on one diagram:
    1. the locus of points satisfying \(| z - 4 + 2 \mathrm { i } | = 2\);
    2. the locus of points satisfying \(| z | = | z - 3 - 2 \mathrm { i } |\).
  2. Shade on your sketch the region in which
    both $$| z - 4 + 2 i | \leqslant 2$$ and $$| z | \leqslant | z - 3 - 2 \mathrm { i } |$$
AQA FP2 2007 January Q3
7 marks Standard +0.3
3 The cubic equation $$z ^ { 3 } + 2 ( 1 - \mathrm { i } ) z ^ { 2 } + 32 ( 1 + \mathrm { i } ) = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).
  1. It is given that \(\alpha\) is of the form \(k \mathrm { i }\), where \(k\) is real. By substituting \(z = k \mathrm { i }\) into the equation, show that \(k = 4\).
  2. Given that \(\beta = - 4\), find the value of \(\gamma\).
AQA FP2 2007 January Q4
18 marks Challenging +1.8
4
  1. Given that \(y = \operatorname { sech } t\), show that:
    1. \(\frac { \mathrm { d } y } { \mathrm {~d} t } = - \operatorname { sech } t \tanh t\);
    2. \(\left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = \operatorname { sech } ^ { 2 } t - \operatorname { sech } ^ { 4 } t\).
  2. The diagram shows a sketch of part of the curve given parametrically by $$x = t - \tanh t \quad y = \operatorname { sech } t$$
    \includegraphics[max width=\textwidth, alt={}]{1891766e-7744-49ac-82b6-7e51cb63b381-3_424_625_863_703}
    The curve meets the \(y\)-axis at the point \(K\), and \(P ( x , y )\) is a general point on the curve. The arc length \(K P\) is denoted by \(s\). Show that:
    1. \(\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = \tanh ^ { 2 } t\);
    2. \(s = \ln \cosh t\);
    3. \(y = \mathrm { e } ^ { - s }\).
  3. The arc \(K P\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Show that the surface area generated is $$2 \pi \left( 1 - \mathrm { e } ^ { - S } \right)$$ (4 marks)
AQA FP2 2007 January Q5
14 marks Standard +0.8
5
  1. Prove by induction that, if \(n\) is a positive integer, $$( \cos \theta + \mathrm { i } \sin \theta ) ^ { n } = \cos n \theta + \mathrm { i } \sin n \theta$$
  2. Find the value of \(\left( \cos \frac { \pi } { 6 } + \mathrm { i } \sin \frac { \pi } { 6 } \right) ^ { 6 }\).
  3. Show that $$( \cos \theta + \mathrm { i } \sin \theta ) ( 1 + \cos \theta - \mathrm { i } \sin \theta ) = 1 + \cos \theta + \mathrm { i } \sin \theta$$
  4. Hence show that $$\left( 1 + \cos \frac { \pi } { 6 } + i \sin \frac { \pi } { 6 } \right) ^ { 6 } + \left( 1 + \cos \frac { \pi } { 6 } - i \sin \frac { \pi } { 6 } \right) ^ { 6 } = 0$$
AQA FP2 2007 January Q6
12 marks Standard +0.8
6
  1. Find the three roots of \(z ^ { 3 } = 1\), giving the non-real roots in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\), where \(- \pi < \theta \leqslant \pi\).
  2. Given that \(\omega\) is one of the non-real roots of \(z ^ { 3 } = 1\), show that $$1 + \omega + \omega ^ { 2 } = 0$$
  3. By using the result in part (b), or otherwise, show that:
    1. \(\frac { \omega } { \omega + 1 } = - \frac { 1 } { \omega }\);
    2. \(\frac { \omega ^ { 2 } } { \omega ^ { 2 } + 1 } = - \omega\);
    3. \(\left( \frac { \omega } { \omega + 1 } \right) ^ { k } + \left( \frac { \omega ^ { 2 } } { \omega ^ { 2 } + 1 } \right) ^ { k } = ( - 1 ) ^ { k } 2 \cos \frac { 2 } { 3 } k \pi\), where \(k\) is an integer.