Questions AS Paper 1 (363 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
AQA AS Paper 1 Specimen Q1
1 marks
1 The curve \(y = \sqrt { x }\) is translated onto the curve \(y = \sqrt { x + 4 }\)
The translation is described by a vector.
Find this vector.
Circle your answer.
[0pt] [1 mark] $$\left[ \begin{array} { l } 4
0 \end{array} \right] \quad \left[ \begin{array} { c } - 4
0 \end{array} \right] \quad \left[ \begin{array} { l } 0
4 \end{array} \right] \quad \left[ \begin{array} { c } 0
- 4 \end{array} \right]$$
AQA AS Paper 1 Specimen Q2
1 marks
2 Consider the two statements, A and B, below.
A: \(x ^ { 2 } - 6 x + 8 > 0\)
B: \(x > 4\)
Choose the most appropriate option below.
Circle your answer.
[0pt] [1 mark] $$A \Rightarrow B \quad A \Leftarrow B \quad A \Leftrightarrow B$$ There is no connection between \(A\) and B
AQA AS Paper 1 Specimen Q4
8 marks
4
0 \end{array} \right] \quad \left[ \begin{array} { c } - 4
0 \end{array} \right] \quad \left[ \begin{array} { l } 0
4 \end{array} \right] \quad \left[ \begin{array} { c } 0
- 4 \end{array} \right]$$ 2 Consider the two statements, A and B, below.
A: \(x ^ { 2 } - 6 x + 8 > 0\)
B: \(x > 4\)
Choose the most appropriate option below.
Circle your answer.
[0pt] [1 mark] $$A \Rightarrow B \quad A \Leftarrow B \quad A \Leftrightarrow B$$ There is no connection between \(A\) and B 3
  1. Write down the value of \(p\) and the value of \(q\) given that: 3
    1. \(\sqrt { 3 } = 3 ^ { p }\)
      [0pt] [1 mark] 3
  3. (ii) \(\frac { 1 } { 9 } = 3 ^ { q }\)
    [0pt] [1 mark] 3
  4. Find the value of \(x\) for which \(\sqrt { 3 } \times 3 ^ { x } = \frac { 1 } { 9 }\)
    [0pt] [2 marks]
    4 Show that \(\frac { 5 \sqrt { 2 } + 2 } { 3 \sqrt { 2 } + 4 }\) can be expressed in the form \(m + n \sqrt { 2 }\), where \(m\) and \(n\) are integers.
    [0pt] [3 marks]
AQA AS Paper 1 Specimen Q5
2 marks
5 Jessica, a maths student, is asked by her teacher to solve the equation \(\tan x = \sin x\), giving all solutions in the range \(0 ^ { \circ } \leq x \leq 360 ^ { \circ }\) The steps of Jessica's working are shown below. $$\begin{aligned} & \tan x = \sin x
& \text { Step } 1 \Rightarrow \frac { \sin x } { \cos x } = \sin x \quad \text { Write } \tan x \text { as } \frac { \sin x } { \cos x }
& \text { Step } 2 \Rightarrow \sin x = \sin x \cos x \quad \text { Multiply by } \cos x
& \text { Step } 3 \Rightarrow 1 = \cos x \quad \text { Cancel } \sin x
& \Rightarrow \quad x = 0 ^ { \circ } \text { or } 360 ^ { \circ } \end{aligned}$$ The teacher tells Jessica that she has not found all the solutions because of a mistake.
Explain why Jessica's method is not correct.
[0pt] [2 marks]
AQA AS Paper 1 Specimen Q6
4 marks
6 A parallelogram has sides of length 6 cm and 4.5 cm .
The larger interior angles of the parallelogram have size \(\alpha\)
Given that the area of the parallelogram is \(24 \mathrm {~cm} ^ { 2 }\), find the exact value of \(\tan \alpha\)
[0pt] [4 marks]
AQA AS Paper 1 Specimen Q7
4 marks
7 Determine whether the line with equation \(2 x + 3 y + 4 = 0\) is parallel to the line through the points with coordinates \(( 9,4 )\) and \(( 3,8 )\).
[0pt] [4 marks]
AQA AS Paper 1 Specimen Q8
6 marks
8
  1. Find the first three terms, in ascending powers of \(x\), of the expansion of \(( 1 - 2 x ) ^ { 10 }\)
    [0pt] [3 marks]
    8
  2. Carly has lost her calculator. She uses the first three terms, in ascending powers of \(x\), of the expansion of \(( 1 - 2 x ) ^ { 10 }\) to evaluate \(0.998 ^ { 10 }\)
    Find Carly's value for \(0.998 ^ { 10 }\) and show that it is correct to five decimal places.
    [0pt] [3 marks]
AQA AS Paper 1 Specimen Q9
5 marks
9
  1. Given that \(\mathrm { f } ( x ) = x ^ { 2 } - 4 x + 2\), find \(\mathrm { f } ( 3 + h )\)
    Express your answer in the form \(h ^ { 2 } + b h + c\), where \(b\) and \(c \in \mathbb { Z }\).
    [0pt] [2 marks]
    L
    9
  2. The curve with equation \(y = x ^ { 2 } - 4 x + 2\) passes through the point \(P ( 3 , - 1 )\) and the point \(Q\) where \(x = 3 + h\). Using differentiation from first principles, find the gradient of the tangent to the curve at the point \(P\).
    [0pt] [3 marks]
AQA AS Paper 1 Specimen Q10
7 marks
10 A student conducts an experiment and records the following data for two variables, \(x\) and \(y\).
\(\boldsymbol { x }\)123456
\(y\)1445130110013003400
\(\log _ { 10 } y\)
The student is told that the relationship between \(x\) and \(y\) can be modelled by an equation of the form \(y = k b ^ { x }\) 10
  1. Plot values of \(\log _ { 10 } y\) against \(x\) on the grid below.
    [0pt] [2 marks]
    \includegraphics[max width=\textwidth, alt={}, center]{3176ee0c-fba2-4878-af3a-c3ac092bbc1f-10_1086_1205_1037_536} 10
  2. State, with a reason, which value of \(y\) is likely to have been recorded incorrectly.
    [0pt] [1 mark] 10
  3. By drawing an appropriate straight line, find the values of \(k\) and \(b\).
    [0pt] [4 marks]
AQA AS Paper 1 Specimen Q11
7 marks
11 Chris claims that, "for any given value of \(x\), the gradient of the curve
\(y = 2 x ^ { 3 } + 6 x ^ { 2 } - 12 x + 3\) is always greater than the gradient of the curve
\(y = 1 + 60 x - 6 x ^ { 2 \prime \prime }\).
Show that Chris is wrong by finding all the values of \(x\) for which his claim is not true.
[0pt] [7 marks]
AQA AS Paper 1 Specimen Q12
9 marks
12 A curve has equation \(y = 6 x \sqrt { x } + \frac { 32 } { x }\) for \(x > 0\)
12
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
    [0pt] [4 marks]
    12
  2. The point \(A\) lies on the curve and has \(x\)-coordinate 4
    Find the coordinates of the point where the tangent to the curve at \(A\) crosses the \(x\)-axis.
    [0pt] [5 marks]
AQA AS Paper 1 Specimen Q13
2 marks
13
  1. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular.
    Find the magnitude of the vector \(- 20 \mathbf { i } + 21 \mathbf { j }\)
    Circle your answer.
    [0pt] [1 mark]
    \(\begin{array} { l l l l } - 1 & 1 & \sqrt { 41 } & 29 \end{array}\) 13
  2. The angle between the vector \(\mathbf { i }\) and the vector \(- 20 \mathbf { i } + 21 \mathbf { j }\) is \(\theta\)
    Which statement about \(\theta\) is true?
    Circle your answer.
    [0pt] [1 mark] $$0 ^ { \circ } < \theta < 45 ^ { \circ } \quad 45 ^ { \circ } < \theta < 90 ^ { \circ } \quad 90 ^ { \circ } < \theta < 135 ^ { \circ } \quad 135 ^ { \circ } < \theta < 180 ^ { \circ }$$
AQA AS Paper 1 Specimen Q14
3 marks
14 In this question use \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
A man of mass 80 kg is travelling in a lift.
The lift is rising vertically.
\includegraphics[max width=\textwidth, alt={}, center]{3176ee0c-fba2-4878-af3a-c3ac092bbc1f-15_529_332_525_808} The lift decelerates at a rate of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Find the magnitude of the force exerted on the man by the lift.
[0pt] [3 marks]
AQA AS Paper 1 Specimen Q15
5 marks
15 The graph shows how the speed of a cyclist varies during a timed section of length 120 metres along a straight track.
\includegraphics[max width=\textwidth, alt={}, center]{3176ee0c-fba2-4878-af3a-c3ac092bbc1f-16_877_1338_481_463} 15
  1. Find the acceleration of the cyclist during the first 10 seconds.
    [0pt] [1 mark] 15
  2. After the first 15 seconds, the cyclist travels at a constant speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for a further \(T\) seconds to complete the 120 -metre section.
    Calculate the value of \(T\).
    [0pt] [4 marks]
AQA AS Paper 1 Specimen Q16
8 marks
16 A particle, of mass 400 grams, is initially at rest at the point \(O\).
The particle starts to move in a straight line so that its velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at time \(t\) seconds is given by $$v = 6 t ^ { 2 } - 12 t ^ { 3 } \text { for } t > 0$$ 16
  1. Find an expression, in terms of \(t\), for the force acting on the particle.
    [0pt] [3 marks] 16
  2. Find the time when the particle next passes through \(O\).
    [0pt] [5 marks] In this question use \(\boldsymbol { g } = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    A van of mass 1300 kg and a crate of mass 300 kg are connected by a light inextensible rope.
    The rope passes over a light smooth pulley, as shown in the diagram.
    The rope between the pulley and the van is horizontal.
    \includegraphics[max width=\textwidth, alt={}, center]{3176ee0c-fba2-4878-af3a-c3ac092bbc1f-20_515_766_685_607} Initially, the van is at rest and the crate rests on the lower level. The rope is taut.
    The van moves away from the pulley to lift the crate from the lower level.
    The van's engine produces a constant driving force of 5000 N .
    A constant resistance force of magnitude 780 N acts on the van.
    Assume there is no resistance force acting on the crate.
AQA AS Paper 1 Specimen Q17
5 marks
17
    1. Draw a diagram to show the forces acting on the crate while it is being lifted. 17
  2. (ii) Draw a diagram to show the forces acting on the van while the crate is being lifted. [1 mark] 17
  3. Show that the acceleration of the van is \(0.80 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
    [0pt] [4 marks]
    17
  4. Find the tension in the rope.
    17
  5. Suggest how the assumption of a constant resistance force could be refined to produce a better model.
Edexcel AS Paper 1 2018 June Q1
  1. Find
$$\int \left( \frac { 2 } { 3 } x ^ { 3 } - 6 \sqrt { x } + 1 \right) \mathrm { d } x$$ giving your answer in its simplest form.
Edexcel AS Paper 1 2018 June Q2
  1. (i) Show that \(x ^ { 2 } - 8 x + 17 > 0\) for all real values of \(x\)
    (ii) "If I add 3 to a number and square the sum, the result is greater than the square of the original number."
State, giving a reason, if the above statement is always true, sometimes true or never true.
Edexcel AS Paper 1 2018 June Q3
  1. Given that the point \(A\) has position vector \(4 \mathbf { i } - 5 \mathbf { j }\) and the point \(B\) has position vector \(- 5 \mathbf { i } - 2 \mathbf { j }\), (a) find the vector \(\overrightarrow { A B }\),
    (b) find \(| \overrightarrow { A B } |\).
Give your answer as a simplified surd.
Edexcel AS Paper 1 2018 June Q4
  1. The line \(l _ { 1 }\) has equation \(4 y - 3 x = 10\)
The line \(l _ { 2 }\) passes through the points \(( 5 , - 1 )\) and \(( - 1,8 )\).
Determine, giving full reasons for your answer, whether lines \(l _ { 1 }\) and \(l _ { 2 }\) are parallel, perpendicular or neither.
Edexcel AS Paper 1 2018 June Q5
  1. A student's attempt to solve the equation \(2 \log _ { 2 } x - \log _ { 2 } \sqrt { x } = 3\) is shown below.
$$\begin{aligned} & 2 \log _ { 2 } x - \log _ { 2 } \sqrt { x } = 3
& 2 \log _ { 2 } \left( \frac { x } { \sqrt { x } } \right) = 3
& 2 \log _ { 2 } ( \sqrt { x } ) = 3
& \log _ { 2 } x = 3
& x = 3 ^ { 2 } = 9 \end{aligned}$$ using the subtraction law for logs simplifying using the power law for logs using the definition of a log
  1. Identify two errors made by this student, giving a brief explanation of each.
  2. Write out the correct solution.
Edexcel AS Paper 1 2018 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f7935caa-6626-4ba8-87ef-e9bb59e1ac3e-12_599_1084_292_486} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A company makes a particular type of children's toy.
The annual profit made by the company is modelled by the equation $$P = 100 - 6.25 ( x - 9 ) ^ { 2 }$$ where \(P\) is the profit measured in thousands of pounds and \(x\) is the selling price of the toy in pounds. A sketch of \(P\) against \(x\) is shown in Figure 1.
Using the model,
  1. explain why \(\pounds 15\) is not a sensible selling price for the toy. Given that the company made an annual profit of more than \(\pounds 80000\)
  2. find, according to the model, the least possible selling price for the toy. The company wishes to maximise its annual profit.
    State, according to the model,
    1. the maximum possible annual profit,
    2. the selling price of the toy that maximises the annual profit.
Edexcel AS Paper 1 2018 June Q7
  1. In a triangle \(A B C\), side \(A B\) has length 10 cm , side \(A C\) has length 5 cm , and angle \(B A C = \theta\) where \(\theta\) is measured in degrees. The area of triangle \(A B C\) is \(15 \mathrm {~cm} ^ { 2 }\)
    1. Find the two possible values of \(\cos \theta\)
    Given that \(B C\) is the longest side of the triangle,
  2. find the exact length of \(B C\).
Edexcel AS Paper 1 2018 June Q8
  1. A lorry is driven between London and Newcastle.
In a simple model, the cost of the journey \(\pounds C\) when the lorry is driven at a steady speed of \(v\) kilometres per hour is $$C = \frac { 1500 } { v } + \frac { 2 v } { 11 } + 60$$
  1. Find, according to this model,
    1. the value of \(v\) that minimises the cost of the journey,
    2. the minimum cost of the journey.
      (Solutions based entirely on graphical or numerical methods are not acceptable.)
  2. Prove by using \(\frac { \mathrm { d } ^ { 2 } C } { \mathrm {~d} v ^ { 2 } }\) that the cost is minimised at the speed found in (a)(i).
  3. State one limitation of this model.
Edexcel AS Paper 1 2018 June Q9
9. $$g ( x ) = 4 x ^ { 3 } - 12 x ^ { 2 } - 15 x + 50$$
  1. Use the factor theorem to show that \(( x + 2 )\) is a factor of \(\mathrm { g } ( x )\).
  2. Hence show that \(\mathrm { g } ( x )\) can be written in the form \(\mathrm { g } ( x ) = ( x + 2 ) ( a x + b ) ^ { 2 }\), where \(a\) and \(b\) are integers to be found. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f7935caa-6626-4ba8-87ef-e9bb59e1ac3e-22_517_807_607_621} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { g } ( x )\)
  3. Use your answer to part (b), and the sketch, to deduce the values of \(x\) for which
    1. \(\mathrm { g } ( x ) \leqslant 0\)
    2. \(\mathrm { g } ( 2 x ) = 0\)