Questions FP2 (1157 questions)

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Edexcel FP2 2020 June Q2
  1. Solve the recurrence system
$$\begin{gathered} u _ { 1 } = 1 \quad u _ { 2 } = 4
9 u _ { n + 2 } - 12 u _ { n + 1 } + 4 u _ { n } = 3 n \end{gathered}$$
Edexcel FP2 2020 June Q3
3. $$\mathbf { M } = \left( \begin{array} { r r r } 1 & k & - 2
2 & - 4 & 1
1 & 2 & 3 \end{array} \right)$$ where \(k\) is a constant.
  1. Show that, in terms of \(k\), a characteristic equation for \(\mathbf { M }\) is given by $$\lambda ^ { 3 } - ( 2 k + 13 ) \lambda + 5 ( k + 6 ) = 0$$ Given that \(\operatorname { det } \mathbf { M } = 5\)
    1. find the value of \(k\)
    2. use the Cayley-Hamilton theorem to find the inverse of \(\mathbf { M }\).
Edexcel FP2 2020 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{868aedc8-6afb-4419-ae29-2ecad3461999-12_213_684_257_221} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{868aedc8-6afb-4419-ae29-2ecad3461999-12_193_736_258_1053} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 1 shows a sketch of a design for a road speed bump of width 2.35 metres. The speed bump has a uniform cross-section with vertical ends and its length is 30 cm . A side profile of the speed bump is shown in Figure 2. The curve \(C\) shown in Figure 2 is modelled by the polar equation $$r = 30 \left( 1 - \theta ^ { 2 } \right) \quad 0 \leqslant \theta \leqslant 1$$ The units for \(r\) are centimetres and the initial line lies along the road surface, which is assumed to be horizontal. Once the speed bump has been fixed to the road, the visible surfaces of the speed bump are to be painted. Determine, in \(\mathrm { cm } ^ { 2 }\), the area that is to be painted, according to the model.
Edexcel FP2 2020 June Q5
  1. A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by
$$w = \frac { 1 - 3 z } { z + 2 i } \quad z \neq - 2 i$$ The circle with equation \(| z + \mathrm { i } | = 3\) is mapped by \(T\) onto the circle \(C\).
  1. Show that the equation for \(C\) can be written as $$3 | w + 3 | = | 1 + ( 3 - w ) \mathrm { i } |$$
  2. Hence find
    1. a Cartesian equation for \(C\),
    2. the centre and radius of \(C\).
Edexcel FP2 2020 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{868aedc8-6afb-4419-ae29-2ecad3461999-20_371_328_255_870} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a plane shape made up of a regular hexagon with an equilateral triangle joined to each edge and with alternate equilateral triangles shaded. The symmetries of this shape are the rotations and reflections of the plane that preserve the shape and its shading. The symmetries of the shape can be represented by permutations of the six vertices labelled 1 to 6 in Figure 3. The set of these permutations with the operation of composition form a group, \(G\).
  1. Describe geometrically the symmetry of the shape represented by the permutation $$\left( \begin{array} { l l l l l l } 1 & 2 & 3 & 4 & 5 & 6
    3 & 4 & 5 & 6 & 1 & 2 \end{array} \right)$$
  2. Write down, in similar two-line notation, the remaining elements of the group \(G\).
  3. Explain why each of the following statements is false, making your reasoning clear.
    1. \(G\) has a subgroup of order 4
    2. \(G\) is cyclic. Diagram 1, on page 23, shows an unshaded shape with the same outline as the shape in Figure 3.
  4. Shade the shape in Diagram 1 in such a way that the group of symmetries of the resulting shaded shape is isomorphic to the cyclic group of order 6
    \includegraphics[max width=\textwidth, alt={}]{868aedc8-6afb-4419-ae29-2ecad3461999-23_426_378_1464_845}
    \section*{Diagram 1} \section*{Spare copy of Diagram 1}
    \includegraphics[max width=\textwidth, alt={}]{868aedc8-6afb-4419-ae29-2ecad3461999-23_424_375_2119_845}
    Only use this diagram if you need to redraw your answer to part (d).
Edexcel FP2 2020 June Q7
7. $$I _ { n } = \int \left( 4 - x ^ { 2 } \right) ^ { - n } \mathrm {~d} x \quad n > 0$$
  1. Show that, for \(n > 0\) $$I _ { n + 1 } = \frac { x } { 8 n \left( 4 - x ^ { 2 } \right) ^ { n } } + \frac { 2 n - 1 } { 8 n } I _ { n }$$
  2. Find \(I _ { 2 }\)
Edexcel FP2 2020 June Q8
  1. The four digit number \(n = a b c d\) satisfies the following properties:
    (1) \(n \equiv 3 ( \bmod 7 )\)
    (2) \(n\) is divisible by 9
    (3) the first two digits have the same sum as the last two digits
    (4) the digit \(b\) is smaller than any other digit
    (5) the digit \(c\) is even
    1. Use property (1) to explain why \(6 a + 2 b + 3 c + d \equiv 3 ( \bmod 7 )\)
    2. Use properties (2), (3) and (4) to show that \(a + b = 9\)
    3. Deduce that \(c \equiv 5 ( a - 1 ) ( \bmod 7 )\)
    4. Hence determine the number \(n\), verifying that it is unique. You must make your reasoning clear.
Edexcel FP2 2021 June Q1
  1. In this question you must show detailed reasoning.
Without performing any division, explain why \(n = 20210520\) is divisible by 66
Edexcel FP2 2021 June Q2
  1. A binary operation ★ on the set of non-negative integers, \(\mathbb { Z } _ { 0 } ^ { + }\), is defined by
$$m \star n = | m - n | \quad m , n \in \mathbb { Z } _ { 0 } ^ { + }$$
  1. Explain why \(\mathbb { Z } _ { 0 } ^ { + }\)is closed under the operation
  2. Show that 0 is an identity for \(\left( \mathbb { Z } _ { 0 } ^ { + } , \star \right)\)
  3. Show that all elements of \(\mathbb { Z } _ { 0 } ^ { + }\)have an inverse under ★
  4. Determine if \(\mathbb { Z } _ { 0 } ^ { + }\)forms a group under ★, giving clear justification for your answer.
Edexcel FP2 2021 June Q3
  1. (a) Use the Euclidean Algorithm to find integers \(a\) and \(b\) such that
$$125 a + 87 b = 1$$ (b) Hence write down a multiplicative inverse of 87 modulo 125
(c) Solve the linear congruence $$87 x \equiv 16 ( \bmod 125 )$$
Edexcel FP2 2021 June Q4
  1. Let \(G\) be a group of order \(46 ^ { 46 } + 47 ^ { 47 }\)
Using Fermat's Little Theorem and explaining your reasoning, determine which of the following are possible orders for a subgroup of \(G\)
  1. 11
  2. 21
Edexcel FP2 2021 June Q5
  1. The point \(P\) in the complex plane represents a complex number \(z\) such that
$$| z + 9 | = 4 | z - 12 i |$$ Given that, as \(z\) varies, the locus of \(P\) is a circle,
  1. determine the centre and radius of this circle.
  2. Shade on an Argand diagram the region defined by the set $$\{ z \in \mathbb { C } : | z + 9 | < 4 | z - 12 i | \} \cap \left\{ z \in \mathbb { C } : - \frac { \pi } { 4 } < \arg \left( z - \frac { 3 + 44 i } { 5 } \right) < \frac { \pi } { 4 } \right\}$$
Edexcel FP2 2021 June Q6
  1. A recurrence system is defined by
$$\begin{aligned} u _ { n + 2 } & = 9 ( n + 1 ) ^ { 2 } u _ { n } - 3 u _ { n + 1 } \quad n \geqslant 1
u _ { 1 } & = - 3 , u _ { 2 } = 18 \end{aligned}$$ Prove by induction that, for \(n \in \mathbb { N }\), $$u _ { n } = ( - 3 ) ^ { n } n !$$
Edexcel FP2 2021 June Q7
  1. In this question you must show all stages of your working.
You must not use the integration facility on your calculator. $$I _ { n } = \int t ^ { n } \sqrt { 4 + 5 t ^ { 2 } } \mathrm {~d} t \quad n \geqslant 0$$
  1. Show that, for \(n > 1\) $$I _ { n } = \frac { t ^ { n - 1 } } { 5 ( n + 2 ) } \left( 4 + 5 t ^ { 2 } \right) ^ { \frac { 3 } { 2 } } - \frac { 4 ( n - 1 ) } { 5 ( n + 2 ) } I _ { n - 2 }$$ \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{1241b133-4161-4c04-9b50-067904cc25c2-20_385_394_829_833} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} The curve shown in Figure 1 is defined by the parametric equations $$x = \frac { 1 } { \sqrt { 5 } } t ^ { 5 } \quad y = \frac { 1 } { 2 } t ^ { 4 } \quad 0 \leqslant t \leqslant 1$$ This curve is rotated through \(2 \pi\) radians about the \(x\)-axis to form a hollow open shell.
  2. Show that the external surface area of the shell is given by $$\pi \int _ { 0 } ^ { 1 } t ^ { 7 } \sqrt { 4 + 5 t ^ { 2 } } \mathrm {~d} t$$ Using the results in parts (a) and (b) and making each step of your working clear,
  3. determine the value of the external surface area of the shell, giving your answer to 3 significant figures.
Edexcel FP2 2021 June Q8
8. $$\mathbf { A } = \left( \begin{array} { r r r } 5 & - 2 & 5
0 & 3 & p
- 6 & 6 & - 4 \end{array} \right) \quad \text { where } p \text { is a constant }$$ Given that \(\left( \begin{array} { r } 2
1
- 2 \end{array} \right)\) is an eigenvector for \(\mathbf { A }\)
    1. determine the eigenvalue corresponding to this eigenvector
    2. hence show that \(p = 2\)
    3. determine the remaining eigenvalues and corresponding eigenvectors of \(\mathbf { A }\)
  2. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } = \mathbf { P D P } ^ { - 1 }\)
    1. Solve the differential equation \(\dot { u } = k u\), where \(k\) is a constant. With respect to a fixed origin \(O\), the velocity of a particle moving through space is modelled by $$\left( \begin{array} { c } \dot { x }
      \dot { y }
      \dot { z } \end{array} \right) = \mathbf { A } \left( \begin{array} { l } x
      y
      z \end{array} \right)$$ By considering \(\left( \begin{array} { c } u
      v
      w \end{array} \right) = \mathbf { P } ^ { - 1 } \left( \begin{array} { c } x
      y
      z \end{array} \right)\) so that \(\left( \begin{array} { c } \dot { u }
      \dot { v }
      \dot { w } \end{array} \right) = \mathbf { P } ^ { - 1 } \left( \begin{array} { c } \dot { x }
      \dot { y }
      \dot { z } \end{array} \right)\)
    2. determine a general solution for the displacement of the particle.
Edexcel FP2 2022 June Q1
  1. The group \(\mathrm { S } _ { 4 }\) is the set of all possible permutations that can be performed on the four numbers 1, 2, 3 and 4, under the operation of composition.
For the group \(\mathrm { S } _ { 4 }\)
  1. write down the identity element,
  2. write down the inverse of the element \(a\), where $$a = \left( \begin{array} { l l l l } 1 & 2 & 3 & 4
    3 & 4 & 2 & 1 \end{array} \right)$$
  3. demonstrate that the operation of composition is associative using the following elements $$a = \left( \begin{array} { l l l l } 1 & 2 & 3 & 4
    3 & 4 & 2 & 1 \end{array} \right) \quad b = \left( \begin{array} { l l l l } 1 & 2 & 3 & 4
    2 & 4 & 3 & 1 \end{array} \right) \quad \text { and } c = \left( \begin{array} { l l l l } 1 & 2 & 3 & 4
    4 & 1 & 2 & 3 \end{array} \right)$$
  4. Explain why it is possible for the group \(\mathrm { S } _ { 4 }\) to have a subgroup of order 4 You do not need to find such a subgroup.
Edexcel FP2 2022 June Q2
  1. Matrix \(\mathbf { M }\) is given by
$$\mathbf { M } = \left( \begin{array} { r r r } 1 & 0 & a
- 3 & b & 1
0 & 1 & a \end{array} \right)$$ where \(a\) and \(b\) are integers, such that \(a < b\)
Given that the characteristic equation for \(\mathbf { M }\) is $$\lambda ^ { 3 } - 7 \lambda ^ { 2 } + 13 \lambda + c = 0$$ where \(c\) is a constant,
  1. determine the values of \(a , b\) and \(c\).
  2. Hence, using the Cayley-Hamilton theorem, determine the matrix \(\mathbf { M } ^ { - 1 }\)
Edexcel FP2 2022 June Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9516df6d-0e85-45d8-afb0-281c80450159-08_321_615_294_726} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} There are three lily pads on a pond. A frog hops repeatedly from one lily pad to another.
The frog starts on lily pad A, as shown in Figure 1.
In a model, the frog hops from its position on one lily pad to either of the other two lily pads with equal probability. Let \(p _ { n }\) be the probability that the frog is on lily pad A after \(n\) hops.
  1. Explain, with reference to the model, why \(p _ { 1 } = 0\) The probability \(p _ { n }\) satisfies the recurrence relation $$p _ { n + 1 } = \frac { 1 } { 2 } \left( 1 - p _ { n } \right) \quad n \geqslant 1 \quad \text { where } p _ { 1 } = 0$$
  2. Prove by induction that, for \(n \geqslant 1\) $$p _ { n } = \frac { 2 } { 3 } \left( - \frac { 1 } { 2 } \right) ^ { n } + \frac { 1 } { 3 }$$
  3. Use the result in part (b) to explain why, in the long term, the probability that the frog is on lily pad A is \(\frac { 1 } { 3 }\)
Edexcel FP2 2022 June Q4
  1. (a) Use the Euclidean algorithm to show that 124 and 17 are relatively prime (coprime).
    (b) Hence solve the equation
$$124 x + 17 y = 10$$ (c) Solve the congruence equation $$124 x \equiv 6 \bmod 17$$
Edexcel FP2 2022 June Q5
  1. The locus of points \(z\) satisfies
$$| z + a \mathrm { i } | = 3 | z - a |$$ where \(a\) is an integer.
The locus is a circle with its centre in the third quadrant and radius \(\frac { 3 } { 2 } \sqrt { 2 }\)
Determine
  1. the value of \(a\),
  2. the coordinates of the centre of the circle. \footnotetext{Question 5 continued }
Edexcel FP2 2022 June Q6
6. (a) Determine the general solution of the recurrence relation $$u _ { n } = 2 u _ { n - 1 } - u _ { n - 2 } + 2 ^ { n } \quad n \geqslant 2$$ (b) Hence solve this recurrence relation given that \(u _ { 0 } = 2 u _ { 1 }\) and \(u _ { 4 } = 3 u _ { 2 }\)
Edexcel FP2 2022 June Q7
    1. The polynomial \(\mathrm { F } ( x )\) is a quartic such that
$$\mathrm { F } ( x ) = p x ^ { 4 } + q x ^ { 3 } + 2 x ^ { 2 } + r x + s$$ where \(p , q , r\) and \(s\) are distinct constants.
Determine the number of possible quartics given that
  1. the constants \(p , q , r\) and \(s\) belong to the set \(\{ - 4 , - 2,1,3,5 \}\)
  2. the constants \(p , q , r\) and \(s\) belong to the set \(\{ - 4 , - 2,0,1,3,5 \}\)
    (ii) A 3-digit positive integer \(N = a b c\) has the following properties
    • \(N\) is divisible by 11
    • the sum of the digits of \(N\) is even
    • \(N \equiv 8 \bmod 9\)
    • Use the first two properties to show that
    $$a - b + c = 0$$
  3. Hence determine all possible integers \(N\), showing all your working and reasoning.
Edexcel FP2 2022 June Q8
  1. The locus of points \(z = x + \mathrm { i } y\) that satisfy
$$\arg \left( \frac { z - 8 - 5 i } { z - 2 - 5 i } \right) = \frac { \pi } { 3 }$$ is an arc of a circle \(C\).
  1. On an Argand diagram sketch the locus of \(z\).
  2. Explain why the centre of \(C\) has \(x\) coordinate 5
  3. Determine the radius of \(C\).
  4. Determine the \(y\) coordinate of the centre of \(C\).
Edexcel FP2 2022 June Q9
9. $$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { n } 2 x d x$$
  1. Prove that for \(n \geqslant 2\) $$I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }$$
  2. Hence determine the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 2 } } 64 \sin ^ { 5 } x \cos ^ { 5 } x d x$$
Edexcel FP2 2022 June Q10
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9516df6d-0e85-45d8-afb0-281c80450159-28_387_474_340_324} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9516df6d-0e85-45d8-afb0-281c80450159-28_448_716_315_1023} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 2 shows a picture of a plant pot.
The plant pot has
  • a flat circular base of radius 10 cm
  • a height of 15 cm
Figure 3 shows a sketch of the curve \(C\) with parametric equations $$x = 10 + 15 t - 5 t ^ { 3 } \quad y = 15 t ^ { 2 } \quad 0 \leqslant t \leqslant 1$$ The curved inner surface of the plant pot is modelled by the surface of revolution formed by rotating curve \(C\) through \(2 \pi\) radians about the \(y\)-axis.
  1. Show that, according to the model, the area of the curved inner surface of the plant pot is given by $$150 \pi \int _ { 0 } ^ { 1 } \left( 2 + 3 t + 2 t ^ { 2 } + 2 t ^ { 3 } - t ^ { 5 } \right) \mathrm { d } t$$
  2. Determine, according to the model, the total area of the inner surface of the plant pot. Each plant pot will be painted with one coat of paint, both inside and outside. The paint in one tin will cover an area of \(12 \mathrm {~m} ^ { 2 }\)
  3. Use the answer to part (b) to estimate how many plant pots can be painted using one tin of paint.
  4. Give a reason why the model might not give an accurate answer to part (c).