Questions FP1 AS (80 questions)

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Edexcel FP1 AS 2018 June Q1
  1. (a) Use the substitution \(t = \tan \left( \frac { x } { 2 } \right)\) to show that the equation
$$5 \sin x + 12 \cos x = 2$$ can be written in the form $$7 t ^ { 2 } - 5 t - 5 = 0$$ (b) Hence solve, for \(- 180 ^ { \circ } < x < 180 ^ { \circ }\), the equation $$5 \sin x + 12 \cos x = 2$$ giving your answers to one decimal place.
Edexcel FP1 AS 2018 June Q2
  1. The temperature, \(\theta ^ { \circ } \mathrm { C }\), of coffee in a cup, \(t\) minutes after the cup of coffee is put in a room, is modelled by the differential equation
$$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = - k ( \theta - 20 )$$ where \(k\) is a constant.
The coffee has an initial temperature of \(80 ^ { \circ } \mathrm { C }\)
Using \(k = 0.1\)
  1. use two iterations of the approximation formula \(\left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) _ { 0 } = \frac { y _ { 1 } - y _ { 0 } } { h }\) to estimate the temperature of the coffee 3 minutes after it was put in the room. The coffee in a different cup, which also had an initial temperature of \(80 ^ { \circ } \mathrm { C }\) when it was put in the room, cools more slowly.
  2. Use this information to suggest how the value of \(k\) would need to be changed in the model.
    V349 SIHI NI IMIMM ION OCVJYV SIHIL NI LIIIM ION OOVJYV SIHIL NI JIIYM ION OC
Edexcel FP1 AS 2018 June Q3
  1. Use algebra to find the values of \(x\) for which
$$\frac { x } { x ^ { 2 } - 2 x - 3 } \leqslant \frac { 1 } { x + 3 }$$
V349 SIHI NI IMIMM ION OCVJYV SIHIL NI LIIIM ION OOVJYV SIHIL NI JIIYM ION OC
Edexcel FP1 AS 2018 June Q4
  1. A scientist is investigating the properties of a crystal. The crystal is modelled as a tetrahedron whose vertices are \(A ( 12,4 , - 1 ) , B ( 10,15 , - 3 ) , C ( 5,8,5 )\) and \(D ( 2,2 , - 6 )\), where the length of unit is the millimetre. The mass of the crystal is 0.5 grams.
    1. Show that, to one decimal place, the area of the triangular face \(A B C\) is \(52.2 \mathrm {~mm} ^ { 2 }\)
    2. Find the density of the crystal, giving your answer in \(\mathrm { g } \mathrm { cm } ^ { - 3 }\)
    V349 SIHI NI IMIMM ION OCVJYV SIHIL NI LIIIM ION OOVJYV SIHIL NI JIIYM ION OC
Edexcel FP1 AS 2018 June Q5
  1. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\), where \(c\) is a non-zero constant.
The point \(P \left( c p , \frac { c } { p } \right)\), where \(p \neq 0\), lies on \(H\).
  1. Use calculus to show that an equation of the normal to \(H\) at \(P\) is $$p ^ { 3 } x - p y + c \left( 1 - p ^ { 4 } \right) = 0$$ The normal to \(H\) at the point \(P\) meets \(H\) again at the point \(Q\).
  2. Find the coordinates of the midpoint of \(P Q\) in terms of \(c\) and \(p\), simplifying your answer where possible.
Edexcel FP1 AS 2019 June Q1
  1. (a) Write down the \(t\)-formula for \(\sin x\).
    (b) Use the answer to part (a)
    1. to find the exact value of \(\sin x\) when
    $$\tan \left( \frac { x } { 2 } \right) = \sqrt { 2 }$$
  2. to show that $$\cos x = \frac { 1 - t ^ { 2 } } { 1 + t ^ { 2 } }$$ (c) Use the \(t\)-formulae to solve for \(0 < \theta \leqslant 360 ^ { \circ }\) $$7 \sin \theta + 9 \cos \theta + 3 = 0$$ giving your answers to one decimal place.
Edexcel FP1 AS 2019 June Q2
  1. A student was set the following problem.
Use algebra to find the set of values of \(x\) for which $$\frac { x } { x - 24 } > \frac { 1 } { x + 11 }$$ The student's attempt at a solution is written below. $$\begin{gathered} x ( x - 24 ) ( x + 11 ) ^ { 2 } > ( x + 11 ) ( x - 24 ) ^ { 2 }
x ( x - 24 ) ( x + 11 ) ^ { 2 } - ( x + 11 ) ( x - 24 ) ^ { 2 } > 0
( x - 24 ) ( x + 11 ) [ x ( x + 11 ) - x - 24 ] > 0
( x - 24 ) ( x + 11 ) \left[ x ^ { 2 } + 10 x - 24 \right] > 0
( x - 24 ) ( x + 11 ) ( x + 12 ) ( x - 2 ) > 0
x = 24 , x = - 11 , x = - 12 , x = 2
\{ x \in \mathbb { R } : - 12 < x < - 11 \} \cup \{ x \in \mathbb { R } : 2 < x < 24 \} \end{gathered}$$ Line 3 There are errors in the student's solution.
  1. Identify the error made
    1. in line 3
    2. in line 7
  2. Find a correct solution to this problem.
Edexcel FP1 AS 2019 June Q3
  1. Julie decides to start a business breeding rabbits to sell as pets.
Initially she buys 20 rabbits. After \(t\) years the number of rabbits, \(R\), is modelled by the differential equation $$\frac { \mathrm { d } R } { \mathrm {~d} t } = 2 R + 4 \sin t \quad t > 0$$ Julie needs to have at least 40 rabbits before she can start to sell them.
Use two iterations of the approximation formula $$\left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) _ { n } \approx \frac { y _ { n + 1 } - y _ { n } } { h }$$ to find out if, according to the model, Julie will be able to start selling rabbits after 4 months.
Edexcel FP1 AS 2019 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6b9c61ac-23ec-4346-933f-cf00a2e63695-08_435_807_285_630} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of a solid doorstop made of wood. The doorstop is modelled as a tetrahedron. Relative to a fixed origin \(O\), the vertices of the tetrahedron are \(A ( 2,1,4 )\), \(B ( 6,1,2 ) , C ( 4,10,3 )\) and \(D ( 5,8 , d )\), where \(d\) is a positive constant and the units are in centimetres.
  1. Find the area of the triangle \(A B C\). Given that the volume of the doorstop is \(21 \mathrm {~cm} ^ { 3 }\)
  2. find the value of the constant \(d\).
Edexcel FP1 AS 2019 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6b9c61ac-23ec-4346-933f-cf00a2e63695-12_744_697_294_683} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the rectangular hyperbola \(H\) with equation $$x y = c ^ { 2 } \quad x > 0$$ where \(c\) is a positive constant.
The point \(P \left( c t , \frac { c } { t } \right)\) lies on \(H\).
The line \(l\) is the tangent to \(H\) at the point \(P\).
The line \(l\) crosses the \(x\)-axis at the point \(A\) and crosses the \(y\)-axis at the point \(B\).
The region \(R\), shown shaded in Figure 2, is bounded by the \(x\)-axis, the \(y\)-axis and the line \(l\). Given that the length \(O B\) is twice the length of \(O A\), where \(O\) is the origin, and that the area of \(R\) is 32 , find the exact coordinates of the point \(P\).
Edexcel FP1 AS 2020 June Q1
  1. The variables \(x\) and \(y\) satisfy the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 y ^ { 2 } - x - 1$$ where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3\) and \(y = 0\) at \(x = 0\)
Use the approximations $$\left( \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } \right) _ { n } \approx \frac { \left( y _ { n + 1 } - 2 y _ { n } + y _ { n - 1 } \right) } { h ^ { 2 } } \text { and } \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) _ { n } \approx \frac { \left( y _ { n + 1 } - y _ { n - 1 } \right) } { 2 h }$$ with \(h = 0.1\) to find an estimate for the value of \(y\) at \(x = 0.2\)
Edexcel FP1 AS 2020 June Q2
  1. Use algebra to determine the values of \(x\) for which
$$\frac { x + 1 } { 2 x ^ { 2 } + 5 x - 3 } > \frac { x } { 4 x ^ { 2 } - 1 }$$
Edexcel FP1 AS 2020 June Q3
    1. Use the substitution \(t = \tan \left( \frac { x } { 2 } \right)\) to prove that
$$\cot x + \tan \left( \frac { x } { 2 } \right) = \operatorname { cosec } x \quad x \neq n \pi , n \in \mathbb { Z }$$ (ii) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1e5324f5-a9bc-4041-bfbb-cb940417ea63-08_389_455_573_877} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} An engineer models the vertical height above the ground of the tip of one blade of a wind turbine, shown in Figure 1. The ground is assumed to be horizontal. The vertical height of the tip of the blade above the ground, \(H\) metres, at time \(x\) seconds after the wind turbine has reached its constant operating speed, is modelled by the equation $$H = 90 - 30 \cos ( 120 x ) ^ { \circ } - 40 \sin ( 120 x ) ^ { \circ }$$
  1. Show that \(H = 60\) when \(x = 0\) Using the substitution \(t = \tan ( 60 x ) ^ { \circ }\)
  2. show that equation (I) can be rewritten as $$H = \frac { 120 t ^ { 2 } - 80 t + 60 } { 1 + t ^ { 2 } }$$
  3. Hence find, according to the model, the value of \(x\) when the tip of the blade is 100 m above the ground for the first time after the wind turbine has reached its constant operating speed.
Edexcel FP1 AS 2020 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1e5324f5-a9bc-4041-bfbb-cb940417ea63-12_611_608_274_715} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the parabola \(C\) with equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant. The point \(S\) is the focus of \(C\) and the point \(P \left( a p ^ { 2 } \right.\), 2ap) lies on \(C\) where \(p > 0\)
  1. Write down the coordinates of \(S\).
  2. Write down the length of SP in terms of \(a\) and \(p\). The point \(Q \left( a q ^ { 2 } , 2 a q \right)\), where \(p \neq q\), also lies on \(C\).
    The point \(M\) is the midpoint of \(P Q\).
    Given that \(p q = - 1\)
  3. prove that, as \(P\) varies, the locus of \(M\) has equation $$y ^ { 2 } = 2 a ( x - a )$$
Edexcel FP1 AS 2020 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1e5324f5-a9bc-4041-bfbb-cb940417ea63-16_360_773_255_646} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a solid display stand with parallel triangular faces \(A B C\) and \(D E F\). Triangle \(D E F\) is similar to triangle \(A B C\). With respect to a fixed origin \(O\), the points \(A , B\) and \(C\) have coordinates ( \(3 , - 3,1\) ), ( \(- 5,3,3\) ) and ( \(1,7,5\) ) respectively and the points \(D , E\) and \(F\) have coordinates ( \(2 , - 1,8\) ), ( \(- 2,2,9\) ) and ( \(1,4,10\) ) respectively. The units are in centimetres.
  1. Show that the area of the triangular face \(D E F\) is \(\frac { 1 } { 2 } \sqrt { 339 } \mathrm {~cm} ^ { 2 }\)
  2. Find, in \(\mathrm { cm } ^ { 3 }\), the exact volume of the display stand.
Edexcel FP1 AS 2021 June Q1
  1. Use algebra to determine the values of \(x\) for which
$$x ( x - 1 ) > \frac { x - 1 } { x }$$ giving your answer in set notation.
Edexcel FP1 AS 2021 June Q2
  1. The variables \(x\) and \(y\) satisfy the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 15 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 3 y ^ { 2 } = 2 x$$ where \(y = 1\) at \(x = 0\) and where \(y = 2\) at \(x = 0.1\)
Use the approximations $$\left( \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } \right) _ { n } \approx \frac { \left( y _ { n + 1 } - 2 y _ { n } + y _ { n - 1 } \right) } { h ^ { 2 } } \text { and } \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) _ { n } \approx \frac { \left( y _ { n + 1 } - y _ { n - 1 } \right) } { 2 h }$$ with \(h = 0.1\) to find an estimate for the value of \(y\) when \(x = 0.3\)
Edexcel FP1 AS 2021 June Q3
  1. On a particular day, the depth of water in a river estuary at a specific location is modelled by the equation
$$D = 2 \sin \left( \frac { x } { 3 } \right) + 3 \cos \left( \frac { x } { 3 } \right) + 6 \quad 0 \leqslant x \leqslant 7 \pi$$ where the depth of water is \(D\) metres at time \(x\) hours after midnight on that day.
  1. Write down the depth of water at midnight, according to the model. Using the substitution \(t = \tan \left( \frac { x } { 6 } \right)\)
  2. show that equation (I) can be re-written as $$D = \frac { 3 t ^ { 2 } + 4 t + 9 } { 1 + t ^ { 2 } }$$
  3. Hence determine, according to the model, the time after midnight when the depth of water is 5 metres for the first time. Give your answer to the nearest minute.
Edexcel FP1 AS 2021 June Q4
  1. With respect to a fixed origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors given by
$$\overrightarrow { O A } = 18 \mathbf { i } - 14 \mathbf { j } - 2 \mathbf { k } \quad \overrightarrow { O B } = - 7 \mathbf { i } - 5 \mathbf { j } + 3 \mathbf { k } \quad \overrightarrow { O C } = - 2 \mathbf { i } - 9 \mathbf { j } - 6 \mathbf { k }$$ The points \(O , A , B\) and \(C\) form the vertices of a tetrahedron.
  1. Show that the area of the triangular face \(A B C\) of the tetrahedron is 108 to 3 significant figures.
  2. Find the volume of the tetrahedron. An oak wood block is made in the shape of the tetrahedron, with centimetres taken for the units. The density of oak is \(0.85 \mathrm {~g} \mathrm {~cm} ^ { - 3 }\)
  3. Determine the mass of the block, giving your answer in kg.
Edexcel FP1 AS 2021 June Q5
  1. The point \(P \left( a p ^ { 2 } , 2 a p \right)\), where \(a\) is a positive constant, lies on the parabola with equation
$$y ^ { 2 } = 4 a x$$ The normal to the parabola at \(P\) meets the parabola again at the point \(Q \left( a q ^ { 2 } , 2 a q \right)\)
  1. Show that $$q = \frac { - p ^ { 2 } - 2 } { p }$$
  2. Hence show that $$P Q ^ { 2 } = \frac { k a ^ { 2 } } { p ^ { 4 } } \left( p ^ { 2 } + 1 \right) ^ { n }$$ where \(k\) and \(n\) are integers to be determined.
Edexcel FP1 AS 2022 June Q1
  1. Use algebra to find the set of values of \(x\) for which
$$x \geqslant \frac { 2 x + 15 } { 2 x + 3 }$$
Edexcel FP1 AS 2022 June Q2
  1. A population of deer was introduced onto an island.
The number of deer, \(P\), on the island at time \(t\) years following their introduction is modelled by the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { P } { 5000 } \left( 1000 - \frac { P ( t + 1 ) } { 6 t + 5 } \right) \quad t > 0$$ It was estimated that there were 540 deer on the island six months after they were introduced.
Use two applications of the approximation formula \(\left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) _ { n } \approx \frac { y _ { n + 1 } - y _ { n } } { h }\) to estimate the number of deer on the island 10 months after they were introduced.
Edexcel FP1 AS 2022 June Q3
  1. (a) Use \(t = \tan \frac { \theta } { 2 }\) to show that, where both sides are defined
$$\frac { 29 - 21 \sec \theta } { 20 - 21 \tan \theta } \equiv \frac { 5 t + 2 } { 2 t + 5 }$$ (b) Hence, again using \(t = \tan \frac { \theta } { 2 }\), prove that, where both sides are defined $$\frac { 20 + 21 \tan \theta } { 29 + 21 \sec \theta } \equiv \frac { 29 - 21 \sec \theta } { 20 - 21 \tan \theta }$$
Edexcel FP1 AS 2022 June Q4
  1. The parabola \(C\) has equation \(y ^ { 2 } = 10 x\)
The point \(F\) is the focus of \(C\).
  1. Write down the coordinates of \(F\). The point \(P\) on \(C\) has \(y\) coordinate \(q\), where \(q > 0\)
  2. Show that an equation for the tangent to \(C\) at \(P\) is given by $$10 x - 2 q y + q ^ { 2 } = 0$$ The tangent to \(C\) at \(P\) intersects the directrix of \(C\) at the point \(A\).
    The point \(B\) lies on the directrix such that \(P B\) is parallel to the \(x\)-axis.
  3. Show that the point of intersection of the diagonals of quadrilateral \(P B A F\) always lies on the \(y\)-axis.
Edexcel FP1 AS 2022 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1be33445-e669-49af-a97e-a8ae84d63463-12_762_1129_246_468} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The points \(A ( 3,2 , - 4 ) , B ( 9 , - 4,2 ) , C ( - 6 , - 10,8 )\) and \(D ( - 4 , - 5,10 )\) are the vertices of a tetrahedron. The plane with equation \(z = 0\) cuts the tetrahedron into two pieces, one on each side of the plane. The edges \(A B , A C\) and \(A D\) of the tetrahedron intersect the plane at the points \(M , N\) and \(P\) respectively, as shown in Figure 1. Determine
  1. the coordinates of the points \(M , N\) and \(P\),
  2. the area of triangle \(M N P\),
  3. the exact volume of the solid \(B C D P N M\).