Questions FP1 AS (80 questions)

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Edexcel FP1 AS 2023 June Q1
  1. (a) Use algebra to determine the values of \(x\) for which
$$\frac { 5 x } { x - 2 } \geqslant 12$$ (b) Hence, given that \(x\) is an integer, deduce the value of \(x\).
Edexcel FP1 AS 2023 June Q2
  1. (a) Use the substitution \(t = \tan \left( \frac { x } { 2 } \right)\) to show that the equation
$$3 \cos x - 2 \sin x = 1$$ can be written in the form $$2 t ^ { 2 } + 2 t - 1 = 0$$ (b) Hence solve, for \(- 180 ^ { \circ } < x < 180 ^ { \circ }\), the equation $$3 \cos x - 2 \sin x = 1$$ giving your answers to one decimal place.
Edexcel FP1 AS 2023 June Q3
  1. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\) where \(c\) is a positive constant.
The line \(l\) has equation \(x - 2 y = c\)
The points \(P\) and \(Q\) are the points of intersection of \(H\) and \(l\)
  1. Determine, in terms of \(c\), the coordinates of \(P\) and the coordinates of \(Q\) The point \(R\) is the midpoint of \(P Q\)
  2. Show that, as \(C\) varies, the coordinates of \(R\) satisfy the equation $$x y = - \frac { c ^ { 2 } } { a }$$ where \(a\) is a constant to be determined.
Edexcel FP1 AS 2023 June Q4
  1. A teacher made a cup of coffee. The temperature \(\theta ^ { \circ } \mathrm { C }\) of the coffee, \(t\) minutes after it was made, is modelled by the differential equation
$$\frac { \mathrm { d } \theta } { \mathrm {~d} t } + 0.05 ( \theta - 20 ) = 0$$ Given that
  • the initial temperature of the coffee was \(95 ^ { \circ } \mathrm { C }\)
  • the coffee can only be safely drunk when its temperature is below \(70 ^ { \circ } \mathrm { C }\)
  • the teacher made the cup of coffee at 1.15 pm
  • the teacher needs to be able to start drinking the coffee by 1.20 pm
    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 estimate whether the teacher will be able to start drinking the coffee at 1.20 pm .
Edexcel FP1 AS 2023 June Q5
  1. The points \(A , B\) and \(C\) are the vertices of a triangle.
Given that
  • \(\overrightarrow { A B } = \left( \begin{array} { l } p
    4
    6 \end{array} \right)\) and \(\overrightarrow { A C } = \left( \begin{array} { l } q
    4
    5 \end{array} \right)\) where \(p\) and \(q\) are constants
  • \(\overrightarrow { A B } \times \overrightarrow { A C }\) is parallel to \(2 \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k }\)
    1. determine the value of \(p\) and the value of \(q\)
    2. Hence, determine the exact area of triangle \(A B C\)
Edexcel FP1 AS 2023 June Q6
  1. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\) where \(a\) is a positive constant.
The point \(P \left( a t ^ { 2 } , 2 a t \right) , t \neq 0\), lies on \(C\)
The normal to \(C\) at \(P\) is parallel to the line with equation \(y = 2 x\)
  1. For the point \(P\), show that \(t = - 2\) The normal to \(C\) at \(P\) intersects \(C\) again when \(x = 9\)
  2. Determine the value of \(a\), giving a reason for your answer.
Edexcel FP1 AS 2024 June Q1
  1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
    1. Sketch the graph of the curve with equation
    $$y = \frac { 1 } { x ^ { 2 } }$$
  2. Solve, using algebra, the inequality $$3 - 2 x ^ { 2 } > \frac { 1 } { x ^ { 2 } }$$
Edexcel FP1 AS 2024 June Q2
  1. An area of woodland contains a mixture of blue and yellow flowers.
A study found that the proportion, \(x\), of blue flowers in the woodland area satisfies the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { x t ( 0.8 - x ) } { x ^ { 2 } + 5 t } \quad t > 0$$ where \(t\) is the number of years since the start of the study.
Given that exactly 3 years after the start of the study half of the flowers in the woodland area were blue,
  1. use one application 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 proportion of blue flowers in the woodland area half a year later.
  2. Deduce from the differential equation the proportion of flowers that will be blue in the long term.
Edexcel FP1 AS 2024 June Q3
  1. Vectors \(\mathbf { u }\) and \(\mathbf { v }\) are given by
$$\mathbf { u } = 5 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k } \quad \text { and } \quad \mathbf { v } = a \mathbf { i } - 6 \mathbf { j } + 2 \mathbf { k }$$ where \(a\) is a constant.
  1. Determine, in terms of \(a\), the vector product \(\mathbf { u } \times \mathbf { v }\) Given that
    • \(\overrightarrow { A B } = 2 \mathbf { u }\)
    • \(\overrightarrow { A C } = \mathbf { v }\)
    • the area of triangle \(A B C\) is 15
    • determine the possible values of \(a\).
Edexcel FP1 AS 2024 June Q4
  1. (a) Given that \(t = \tan \frac { X } { 2 }\) prove that
$$\cos x \equiv \frac { 1 - t ^ { 2 } } { 1 + t ^ { 2 } }$$ (b) Show that the equation $$3 \tan x - 10 \cos x = 10$$ can be written in the form $$( t + 2 ) \left( a t ^ { 2 } + b t + c \right) = 0$$ where \(t = \tan \frac { X } { 2 }\) and \(a , b\) and \(c\) are integers to be determined.
(c) Hence solve, for \(- 180 ^ { \circ } < x < 180 ^ { \circ }\), the equation $$3 \tan x - 10 \cos x = 10$$
Edexcel FP1 AS 2024 June Q5
  1. The parabola \(C\) has equation \(y ^ { 2 } = 16 x\)
The point \(P\) on \(C\) has \(y\) coordinate \(p\), where \(p\) is a positive constant.
  1. Show that an equation of the tangent to \(C\) at \(P\) is given by $$2 p y = 16 x + p ^ { 2 }$$ $$\left[ Y \text { ou may quote without proof that for the general parabola } y ^ { 2 } = 4 a x , \frac { d y } { d x } = \frac { 2 a } { y } \right]$$
  2. Write down the equation of the directrix of \(C\). The line \(l\) is the reflection of the tangent to \(C\) at \(P\) in the directrix of \(C\).
    Given that \(l\) passes through the focus of \(C\),
  3. determine the exact value of \(p\).
Edexcel FP1 AS Specimen Q1
  1. (a) Use the substitution \(\mathrm { t } = \tan \left( \frac { \mathrm { x } } { 2 } \right)\) to show that
$$\sec x - \tan x \equiv \frac { 1 - t } { 1 + t } \quad x \neq ( 2 n + 1 ) \frac { \pi } { 2 } , n \in \mathbb { Z }$$ (b) Use the substitution \(\mathrm { t } = \tan \left( \frac { \mathrm { x } } { 2 } \right)\) and the answer to part (a) to prove that $$\frac { 1 - \sin x } { 1 + \sin x } \equiv ( \sec x - \tan x ) ^ { 2 } \quad x \neq ( 2 n + 1 ) \frac { \pi } { 2 } , n \in \mathbb { Z }$$ \section*{Q uestion 1 continued}
Edexcel FP1 AS Specimen Q2
  1. The value, V hundred pounds, of a particular stock thours after the opening of trading on a given day is modelled by the differential equation
$$\frac { d V } { d t } = \frac { V ^ { 2 } - t } { t ^ { 2 } + t V } \quad 0 < t < 8.5$$ A trader purchases \(\pounds 300\) of the stock one hour after the opening of trading.
Use two iterations of the approximation formula \(\left( \frac { \mathrm { dy } } { \mathrm { dx } } \right) _ { 0 } \approx \frac { \mathrm { y } _ { 1 } - \mathrm { y } _ { 0 } } { \mathrm {~h} }\) to estimate, to the nearest \(\pounds\), the value of the trader's stock half an hour after it was purchased.
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Edexcel FP1 AS Specimen Q3
  1. Use algebra to find the set of values of x for which
$$\frac { 1 } { x } < \frac { x } { x + 2 }$$
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Edexcel FP1 AS Specimen Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ff1fc9b0-6514-44e0-a2a3-46aa6411ce10-08_538_807_251_630} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of a solid sculpture made of glass and concrete. The sculpture is modelled as a parallelepiped. The sculpture is made up of a concrete solid in the shape of a tetrahedron, shown shaded in Figure 1, whose vertices are \(\mathrm { O } ( 0,0,0 ) , \mathrm { A } ( 2,0,0 ) , \mathrm { B } ( 0,3,1 )\) and \(\mathrm { C } ( 1,1,2 )\), where the units are in metres. The rest of the solid parallelepiped is made of glass which is glued to the concrete tetrahedron.
  1. Find the surface area of the glued face of the tetrahedron.
  2. Find the volume of glass contained in this parallelepiped.
  3. Give a reason why the volume of concrete predicted by this model may not be an accurate value for the volume of concrete that was used to make the sculpture. \section*{Q uestion 4 continued}
Edexcel FP1 AS Specimen Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ff1fc9b0-6514-44e0-a2a3-46aa6411ce10-10_965_853_212_621} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Diagram not drawn to scale $$\left[ Y \text { ou may quote without proof that for the general parabola } y ^ { 2 } = 4 a x , \frac { d y } { d x } = \frac { 2 a } { y } \right]$$ The parabola C has equation \(\mathrm { y } ^ { 2 } = 16 \mathrm { x }\)
  1. Deduce that the point \(\mathrm { P } \left( 4 \mathrm { p } ^ { 2 } , 8 \mathrm { p } \right)\) is a general point on C . The line I is the tangent to C at the point P .
  2. Show that an equation for I is $$p y = x + 4 p ^ { 2 }$$ The finite region R , shown shaded in Figure 2, is bounded by the line I , the x -axis and the parabola C.
    The line \(I\) intersects the directrix of \(C\) at the point \(B\), where the \(y\) coordinate of \(B\) is \(\frac { 10 } { 3 }\) Given that \(\mathrm { p } > 0\)
  3. show that the area of R is 36 \section*{Q uestion 5 continued}
OCR FP1 AS 2017 December Q1
1 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { c c c } - 3 & 3 & 2
5 & - 4 & - 3
- 1 & 1 & 1 \end{array} \right)\).
  1. Find \(\mathbf { A } ^ { - 1 }\).
  2. Solve the simultaneous equations $$\begin{aligned} - 3 x + 3 y + 2 z & = 12 a
    5 x - 4 y - 3 z & = - 6
    - x + y + z & = 7 \end{aligned}$$ giving your solution in terms of \(a\).
OCR FP1 AS 2017 December Q2
2 The loci \(C _ { 1 }\) and \(C _ { 2 }\) are given by \(| z - ( 3 + 2 \mathrm { i } ) | = 2\) and \(\arg ( z - ( 3 + 2 \mathrm { i } ) ) = \frac { 5 \pi } { 6 }\) respectively.
  1. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on a single Argand diagram.
  2. Find, in surd form, the number represented by the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
  3. Indicate, by shading, the region of the Argand diagram for which $$| z - ( 3 + 2 i ) | \leqslant 2 \text { and } \frac { 5 \pi } { 6 } \leqslant \arg ( z - ( 3 + 2 i ) ) \leqslant \pi$$
OCR FP1 AS 2017 December Q3
3 Two lines, \(l _ { 1 }\) and \(l _ { 2 }\), have the following equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { c } - 11
10
3 \end{array} \right) + \lambda \left( \begin{array} { c } 2
- 2
1 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 5
2
OCR FP1 AS 2017 December Q5
5
2
4 \end{array} \right) + \mu \left( \begin{array} { c } 3
1
- 2 \end{array} \right) \end{aligned}$$ \(P\) is the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).
  1. Find the position vector of \(P\).
  2. Find, correct to 1 decimal place, the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\).
    \(Q\) is a point on \(l _ { 1 }\) which is 12 metres away from \(P . R\) is the point on \(l _ { 2 }\) such that \(Q R\) is perpendicular to \(l _ { 1 }\).
  3. Determine the length \(Q R\). 4 In this question you must show detailed reasoning.
    The distinct numbers \(\omega _ { 1 }\) and \(\omega _ { 2 }\) both satisfy the quadratic equation \(4 x ^ { 2 } + 4 x + 17 = 0\).
  4. Write down the value of \(\omega _ { 1 } \omega _ { 2 }\).
  5. \(A , B\) and \(C\) are the points on an Argand diagram which represent \(\omega _ { 1 } , \omega _ { 2 }\) and \(\omega _ { 1 } \omega _ { 2 }\). Find the area of triangle \(A B C\). 5 In this question you must show detailed reasoning.
    The equation \(x ^ { 3 } + 3 x ^ { 2 } - 2 x + 4 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
  6. Using the identity \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } \equiv ( \alpha + \beta + \gamma ) ^ { 3 } - 3 ( \alpha \beta + \beta \gamma + \gamma \alpha ) ( \alpha + \beta + \gamma ) + 3 \alpha \beta \gamma\) find the value of \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\).
  7. Given that \(\alpha ^ { 3 } \beta ^ { 3 } + \beta ^ { 3 } \gamma ^ { 3 } + \gamma ^ { 3 } \alpha ^ { 3 } = 112\) find a cubic equation whose roots are \(\alpha ^ { 3 } , \beta ^ { 3 }\) and \(\gamma ^ { 3 }\).
OCR FP1 AS 2017 December Q6
6 Prove by induction that \(n ! \geqslant 6 n\) for \(n \geqslant 4\).
OCR FP1 AS 2017 December Q7
7 A transformation is equivalent to a shear parallel to the \(x\)-axis followed by a shear parallel to the \(y\)-axis and is represented by the matrix \(\left( \begin{array} { c c } 1 & s
t & 0 \end{array} \right)\). Find in terms of \(s\) the matrices which represent each of the shears.
OCR FP1 AS 2017 December Q10
10
3 \end{array} \right) + \lambda \left( \begin{array} { c } 2
- 2
1 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 5
2
4 \end{array} \right) + \mu \left( \begin{array} { c } 3
1
- 2 \end{array} \right) \end{aligned}$$ \(P\) is the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).
  1. Find the position vector of \(P\).
  2. Find, correct to 1 decimal place, the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\).
    \(Q\) is a point on \(l _ { 1 }\) which is 12 metres away from \(P . R\) is the point on \(l _ { 2 }\) such that \(Q R\) is perpendicular to \(l _ { 1 }\).
  3. Determine the length \(Q R\). 4 In this question you must show detailed reasoning.
    The distinct numbers \(\omega _ { 1 }\) and \(\omega _ { 2 }\) both satisfy the quadratic equation \(4 x ^ { 2 } + 4 x + 17 = 0\).
  4. Write down the value of \(\omega _ { 1 } \omega _ { 2 }\).
  5. \(A , B\) and \(C\) are the points on an Argand diagram which represent \(\omega _ { 1 } , \omega _ { 2 }\) and \(\omega _ { 1 } \omega _ { 2 }\). Find the area of triangle \(A B C\). 5 In this question you must show detailed reasoning.
    The equation \(x ^ { 3 } + 3 x ^ { 2 } - 2 x + 4 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
  6. Using the identity \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } \equiv ( \alpha + \beta + \gamma ) ^ { 3 } - 3 ( \alpha \beta + \beta \gamma + \gamma \alpha ) ( \alpha + \beta + \gamma ) + 3 \alpha \beta \gamma\) find the value of \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\).
  7. Given that \(\alpha ^ { 3 } \beta ^ { 3 } + \beta ^ { 3 } \gamma ^ { 3 } + \gamma ^ { 3 } \alpha ^ { 3 } = 112\) find a cubic equation whose roots are \(\alpha ^ { 3 } , \beta ^ { 3 }\) and \(\gamma ^ { 3 }\). 6 Prove by induction that \(n ! \geqslant 6 n\) for \(n \geqslant 4\). 7 A transformation is equivalent to a shear parallel to the \(x\)-axis followed by a shear parallel to the \(y\)-axis and is represented by the matrix \(\left( \begin{array} { c c } 1 & s
    t & 0 \end{array} \right)\). Find in terms of \(s\) the matrices which represent each of the shears. 8
  8. (a) Find, in terms of \(x\), a vector which is perpendicular to the vectors \(\left( \begin{array} { c } x - 2
    5
    1 \end{array} \right)\) and \(\left( \begin{array} { c } x
    6
    2 \end{array} \right)\).
    (b) Find the shortest possible vector of the form \(\left( \begin{array} { l } 1
    a
    b \end{array} \right)\) which is perpendicular to the vectors \(\left( \begin{array} { c } x - 2
    5
    1 \end{array} \right)\) and \(\left( \begin{array} { c } x
    6
    2 \end{array} \right)\).
  9. Vector \(\mathbf { v }\) is perpendicular to both \(\left( \begin{array} { c } - 1
    1
    1 \end{array} \right)\) and \(\left( \begin{array} { c } 1
    p
    p ^ { 2 } \end{array} \right)\) where \(p\) is a real number. Show that it is impossible for \(\mathbf { v }\) to be perpendicular to the vector \(\left( \begin{array} { c } 1
    1
    p - 1 \end{array} \right)\). \section*{OCR} Oxford Cambridge and RSA
OCR FP1 AS 2018 March Q1
1
  1. The complex number 3-4i is denoted by \(z _ { 1 }\). Write \(z _ { 1 }\) in modulus-argument form, giving your angle in radians to 3 significant figures.
  2. The complex number \(z _ { 2 }\) has modulus 6 and argument - 2.5 radians. Express \(z _ { 1 } z _ { 2 }\) in modulus-argument form with the angle in radians correct to 3 significant figures.
OCR FP1 AS 2018 March Q2
2 In this question you must show detailed reasoning.
The quadratic equation \(3 x ^ { 2 } - 7 x + 5 = 0\) has roots \(\alpha\) and \(\beta\).
  1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
  2. Hence find the values of the following expressions.
    (a) \(\frac { 1 } { \alpha } + \frac { 1 } { \beta }\)
    (b) \(\alpha ^ { 2 } + \beta ^ { 2 }\)
    \(3 \quad l _ { 1 }\) and \(l _ { 2 }\) are two intersecting straight lines with the following equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { c }