Questions — Edexcel (10514 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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 PURE 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 PURE S1 S2 S3 S4 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 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 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
Sort by: Default | Easiest first | Hardest first
Edexcel PMT Mocks Q13
6 marks Standard +0.3
13. Relative to a fixed origin \(O\)
  • the point \(P\) has position vector \(( 0 , - 1,2 )\)
  • the point \(Q\) has position vector \(( 1,1,5 )\)
  • the point \(R\) has position vector ( \(3,5 , m\) )
    where \(m\) is a constant.
    Given that \(P , Q\) and \(R\) lie on a straight line,
    a. find the value of \(m\)
The line segment \(O Q\) is extended to a point \(T\) so that \(\overrightarrow { R T }\) is parallel to \(\overrightarrow { O P }\) b. Show that \(| \overrightarrow { O T } | = 9 \sqrt { 3 }\).
Edexcel PMT Mocks Q14
9 marks Standard +0.3
14. a. Express \(\frac { 1 } { ( 3 - x ) ( 1 - x ) }\) in partial fractions.
(2) A scientist is studying the mass of a substance in a laboratory.
The mass, \(x\) grams, of a substance at time \(t\) seconds after a chemical reaction starts is modelled by the differential equation $$2 \frac { d x } { d t } = ( 3 - x ) ( 1 - x ) \quad t \geq 0,0 \leq x < 1$$ Given that when \(t = 0 , x = 0\) b. solve the differential equation and show that the solution can be written as $$x = \frac { 3 \left( e ^ { t } - 1 \right) } { 3 e ^ { t } - 1 }$$ c. Find the mass, \(x\) grams, which has formed 2 seconds after the start of the reaction. Give your answer correct to 3 significant figures.
d. Find the limiting value of \(x\) as \(t\) increases.
Edexcel PMT Mocks Q15
9 marks Challenging +1.8
15. The first three terms of a geometric series where \(\theta\) is a constant are $$- 8 \sin \theta , \quad 3 - 2 \cos \theta \quad \text { and } \quad 4 \cot \theta$$ a. Show that \(4 \cos ^ { 2 } \theta + 20 \cos \theta + 9 = 0\) Given that \(\theta\) lies in the interval \(90 ^ { \circ } < \theta < 180 ^ { \circ }\),
b. Find the value of \(\theta\).
c. Hence prove that this series is convergent.
d. Find \(S _ { \infty }\), in the form \(a ( 1 - \sqrt { 3 } )\)
Edexcel PMT Mocks Q16
12 marks Standard +0.3
16. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d37eaba2-0a25-4abf-b2c8-1e08673229fb-26_1241_1130_251_440} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of the curve \(C\) with parametric equations \(x = - 3 + 6 \sin \theta , \quad y = 9 \cos 2 \theta \quad - \frac { \pi } { 2 } \leq \theta \leq \frac { \pi } { 4 }\) where \(\theta\) is a parameter.
a. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\) The line \(l\) is normal to \(C\) at the point \(P\) where \(\theta = \frac { \pi } { 6 }\) b. Show that an equation for \(l\) is $$y = \frac { 1 } { 3 } x + \frac { 9 } { 2 }$$ c. The cartesian equation for the curve \(C\) can be written in the form $$y = a - \frac { 1 } { 2 } ( x + b ) ^ { 2 }$$ where \(a\) and \(b\) are integers to be found. The straight line with equation $$y = \frac { 1 } { 3 } x + k$$ where \(k\) is a constant intersects \(C\) at two distinct points.
d. Find the range of possible values for \(k\).
Edexcel Paper 1 2018 June Q1
3 marks Moderate -0.3
  1. Given that \(\theta\) is small and is measured in radians, use the small angle approximations to find an approximate value of
$$\frac { 1 - \cos 4 \theta } { 2 \theta \sin 3 \theta }$$ (3)
Edexcel Paper 1 2018 June Q2
7 marks Moderate -0.8
  1. A curve \(C\) has equation
$$y = x ^ { 2 } - 2 x - 24 \sqrt { x } , \quad x > 0$$
  1. Find (i) \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) (ii) \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\)
  2. Verify that \(C\) has a stationary point when \(x = 4\)
  3. Determine the nature of this stationary point, giving a reason for your answer.
Edexcel Paper 1 2018 June Q3
4 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b5f50f17-9f1b-4b4c-baf3-e50de5f2ea9c-06_332_348_246_861} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sector \(A O B\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\).
The angle \(A O B\) is \(\theta\) radians.
The area of the sector \(A O B\) is \(11 \mathrm {~cm} ^ { 2 }\) Given that the perimeter of the sector is 4 times the length of the arc \(A B\), find the exact value of \(r\).
Edexcel Paper 1 2018 June Q4
4 marks Standard +0.3
The curve with equation \(y = 2 \ln ( 8 - x )\) meets the line \(y = x\) at a single point, \(x = \alpha\).
  1. Show that \(3 < \alpha < 4\) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b5f50f17-9f1b-4b4c-baf3-e50de5f2ea9c-08_666_1061_445_502} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows the graph of \(y = 2 \ln ( 8 - x )\) and the graph of \(y = x\).
    A student uses the iteration formula $$x _ { n + 1 } = 2 \ln \left( 8 - x _ { n } \right) , \quad n \in \mathbb { N }$$ in an attempt to find an approximation for \(\alpha\).
    Using the graph and starting with \(x _ { 1 } = 4\)
  2. determine whether or not this iteration formula can be used to find an approximation for \(\alpha\), justifying your answer.
Edexcel Paper 1 2018 June Q5
5 marks Standard +0.8
  1. Given that
$$y = \frac { 3 \sin \theta } { 2 \sin \theta + 2 \cos \theta } \quad - \frac { \pi } { 4 } < \theta < \frac { 3 \pi } { 4 }$$ show that $$\frac { d y } { d \theta } = \frac { A } { 1 + \sin 2 \theta } \quad - \frac { \pi } { 4 } < \theta < \frac { 3 \pi } { 4 }$$ where \(A\) is a rational constant to be found.
Edexcel Paper 1 2018 June Q6
10 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b5f50f17-9f1b-4b4c-baf3-e50de5f2ea9c-12_549_592_244_731} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Not to scale The circle \(C\) has centre \(A\) with coordinates (7,5).
The line \(l\), with equation \(y = 2 x + 1\), is the tangent to \(C\) at the point \(P\), as shown in Figure 3 .
  1. Show that an equation of the line \(P A\) is \(2 y + x = 17\)
  2. Find an equation for \(C\). The line with equation \(y = 2 x + k , \quad k \neq 1\) is also a tangent to \(C\).
  3. Find the value of the constant \(k\).
Edexcel Paper 1 2018 June Q7
7 marks Standard +0.3
Given that \(k \in \mathbb { Z } ^ { + }\)
  1. show that \(\int _ { k } ^ { 3 k } \frac { 2 } { ( 3 x - k ) } \mathrm { d } x\) is independent of \(k\),
  2. show that \(\int _ { k } ^ { 2 k } \frac { 2 } { ( 2 x - k ) ^ { 2 } } \mathrm {~d} x\) is inversely proportional to \(k\).
Edexcel Paper 1 2018 June Q8
5 marks Moderate -0.3
  1. The depth of water, \(D\) metres, in a harbour on a particular day is modelled by the formula
$$D = 5 + 2 \sin ( 30 t ) ^ { \circ } \quad 0 \leqslant t < 24$$ where \(t\) is the number of hours after midnight. A boat enters the harbour at 6:30 am and it takes 2 hours to load its cargo. The boat requires the depth of water to be at least 3.8 metres before it can leave the harbour.
  1. Find the depth of the water in the harbour when the boat enters the harbour.
  2. Find, to the nearest minute, the earliest time the boat can leave the harbour. (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel Paper 1 2018 June Q9
10 marks Standard +0.8
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b5f50f17-9f1b-4b4c-baf3-e50de5f2ea9c-22_537_748_242_662} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of the curve with equation \(x ^ { 2 } - 2 x y + 3 y ^ { 2 } = 50\)
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y - x } { 3 y - x }\) The curve is used to model the shape of a cycle track with both \(x\) and \(y\) measured in km .
    The points \(P\) and \(Q\) represent points that are furthest west and furthest east of the origin \(O\), as shown in Figure 4. Using part (a),
  2. find the exact coordinates of the point \(P\).
  3. Explain briefly how to find the coordinates of the point that is furthest north of the origin \(O\). (You do not need to carry out this calculation).
Edexcel Paper 1 2018 June Q10
8 marks Standard +0.3
  1. The height above ground, \(H\) metres, of a passenger on a roller coaster can be modelled by the differential equation
$$\frac { \mathrm { d } H } { \mathrm {~d} t } = \frac { H \cos ( 0.25 t ) } { 40 }$$ where \(t\) is the time, in seconds, from the start of the ride. Given that the passenger is 5 m above the ground at the start of the ride,
  1. show that \(H = 5 \mathrm { e } ^ { 0.1 \sin ( 0.25 t ) }\)
  2. State the maximum height of the passenger above the ground. The passenger reaches the maximum height, for the second time, \(T\) seconds after the start of the ride.
  3. Find the value of \(T\).
Edexcel Paper 1 2018 June Q11
10 marks Standard +0.3
  1. Use binomial expansions to show that \(\sqrt { \frac { 1 + 4 x } { 1 - x } } \approx 1 + \frac { 5 } { 2 } x - \frac { 5 } { 8 } x ^ { 2 }\) A student substitutes \(x = \frac { 1 } { 2 }\) into both sides of the approximation shown in part (a) in an attempt to find an approximation to \(\sqrt { 6 }\)
  2. Give a reason why the student should not use \(x = \frac { 1 } { 2 }\)
  3. Substitute \(x = \frac { 1 } { 11 }\) into $$\sqrt { \frac { 1 + 4 x } { 1 - x } } = 1 + \frac { 5 } { 2 } x - \frac { 5 } { 8 } x ^ { 2 }$$ to obtain an approximation to \(\sqrt { 6 }\). Give your answer as a fraction in its simplest form.
Edexcel Paper 1 2018 June Q12
10 marks Standard +0.3
  1. The value, \(\pounds V\), of a vintage car \(t\) years after it was first valued on 1 st January 2001, is modelled by the equation
$$V = A p ^ { t } \quad \text { where } A \text { and } p \text { are constants }$$ Given that the value of the car was \(\pounds 32000\) on 1st January 2005 and \(\pounds 50000\) on 1st January 2012
    1. find \(p\) to 4 decimal places,
    2. show that \(A\) is approximately 24800
  2. With reference to the model, interpret
    1. the value of the constant \(A\),
    2. the value of the constant \(p\). Using the model,
  3. find the year during which the value of the car first exceeds \(\pounds 100000\)
Edexcel Paper 1 2018 June Q13
7 marks Standard +0.3
  1. Show that
$$\int _ { 0 } ^ { 2 } 2 x \sqrt { x + 2 } \mathrm {~d} x = \frac { 32 } { 15 } ( 2 + \sqrt { 2 } )$$
Edexcel Paper 1 2018 June Q14
10 marks Standard +0.3
  1. A curve \(C\) has parametric equations
$$x = 3 + 2 \sin t , \quad y = 4 + 2 \cos 2 t , \quad 0 \leqslant t < 2 \pi$$
  1. Show that all points on \(C\) satisfy \(y = 6 - ( x - 3 ) ^ { 2 }\)
    1. Sketch the curve \(C\).
    2. Explain briefly why \(C\) does not include all points of \(y = 6 - ( x - 3 ) ^ { 2 } , \quad x \in \mathbb { R }\) The line with equation \(x + y = k\), where \(k\) is a constant, intersects \(C\) at two distinct points.
  3. State the range of values of \(k\), writing your answer in set notation.
Edexcel Paper 1 2019 June Q2
5 marks Moderate -0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{91a2f26a-add2-4b58-997d-2ae229548217-04_670_1447_212_333} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a plot of part of the curve with equation \(y = \cos x\) where \(x\) is measured in radians. Diagram 1, on the opposite page, is a copy of Figure 1.
  1. Use Diagram 1 to show why the equation $$\cos x - 2 x - \frac { 1 } { 2 } = 0$$ has only one real root, giving a reason for your answer. Given that the root of the equation is \(\alpha\), and that \(\alpha\) is small,
  2. use the small angle approximation for \(\cos x\) to estimate the value of \(\alpha\) to 3 decimal places.
    \includegraphics[max width=\textwidth, alt={}]{91a2f26a-add2-4b58-997d-2ae229548217-05_664_1452_246_333}
    \section*{Diagram 1}
Edexcel Paper 1 2019 June Q3
5 marks Moderate -0.3
3. $$y = \frac { 5 x ^ { 2 } + 10 x } { ( x + 1 ) ^ { 2 } } \quad x \neq - 1$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { A } { ( x + 1 ) ^ { n } }\) where \(A\) and \(n\) are constants to be found.
  2. Hence deduce the range of values for \(x\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } < 0\)
Edexcel Paper 1 2019 June Q4
6 marks Standard +0.3
  1. Find the first three terms, in ascending powers of \(x\), of the binomial expansion of $$\frac { 1 } { \sqrt { 4 - x } }$$ giving each coefficient in its simplest form. The expansion can be used to find an approximation to \(\sqrt { 2 }\) Possible values of \(x\) that could be substituted into this expansion are:
    • \(x = - 14\) because \(\frac { 1 } { \sqrt { 4 - x } } = \frac { 1 } { \sqrt { 18 } } = \frac { \sqrt { 2 } } { 6 }\)
    • \(x = 2\) because \(\frac { 1 } { \sqrt { 4 - x } } = \frac { 1 } { \sqrt { 2 } } = \frac { \sqrt { 2 } } { 2 }\)
    • \(x = - \frac { 1 } { 2 }\) because \(\frac { 1 } { \sqrt { 4 - x } } = \frac { 1 } { \sqrt { \frac { 9 } { 2 } } } = \frac { \sqrt { 2 } } { 3 }\)
    • Without evaluating your expansion,
      1. state, giving a reason, which of the three values of \(x\) should not be used
      2. state, giving a reason, which of the three values of \(x\) would lead to the most accurate approximation to \(\sqrt { 2 }\)
Edexcel Paper 1 2019 June Q5
10 marks Moderate -0.8
5. $$\mathrm { f } ( x ) = 2 x ^ { 2 } + 4 x + 9 \quad x \in \mathbb { R }$$
  1. Write \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a\), \(b\) and \(c\) are integers to be found.
  2. Sketch the curve with equation \(y = \mathrm { f } ( x )\) showing any points of intersection with the coordinate axes and the coordinates of any turning point.
    1. Describe fully the transformation that maps the curve with equation \(y = \mathrm { f } ( x )\) onto the curve with equation \(y = \mathrm { g } ( x )\) where $$\mathrm { g } ( x ) = 2 ( x - 2 ) ^ { 2 } + 4 x - 3 \quad x \in \mathbb { R }$$
    2. Find the range of the function $$\mathrm { h } ( x ) = \frac { 21 } { 2 x ^ { 2 } + 4 x + 9 } \quad x \in \mathbb { R }$$
Edexcel Paper 1 2019 June Q6
8 marks Standard +0.3
  1. Solve, for \(- 180 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\), the equation $$5 \sin 2 \theta = 9 \tan \theta$$ giving your answers, where necessary, to one decimal place.
    [0pt] [Solutions based entirely on graphical or numerical methods are not acceptable.]
  2. Deduce the smallest positive solution to the equation $$5 \sin \left( 2 x - 50 ^ { \circ } \right) = 9 \tan \left( x - 25 ^ { \circ } \right)$$
Edexcel Paper 1 2019 June Q7
7 marks Moderate -0.3
  1. In a simple model, the value, \(\pounds V\), of a car depends on its age, \(t\), in years.
The following information is available for \(\operatorname { car } A\)
  • its value when new is \(\pounds 20000\)
  • its value after one year is \(\pounds 16000\)
    1. Use an exponential model to form, for car \(A\), a possible equation linking \(V\) with \(t\).
The value of car \(A\) is monitored over a 10-year period.
Its value after 10 years is \(\pounds 2000\)
  • Evaluate the reliability of your model in light of this information. The following information is available for car \(B\)
    • it has the same value, when new, as car \(A\)
    • its value depreciates more slowly than that of \(\operatorname { car } A\)
    • Explain how you would adapt the equation found in (a) so that it could be used to model the value of car \(B\).
    •
    Edexcel Paper 1 2019 June Q8
    10 marks Moderate -0.3
    8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{91a2f26a-add2-4b58-997d-2ae229548217-22_812_958_244_555} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = x ( x + 2 ) ( x - 4 )\).
    The region \(R _ { 1 }\) shown shaded in Figure 2 is bounded by the curve and the negative \(x\)-axis.
    1. Show that the exact area of \(R _ { 1 }\) is \(\frac { 20 } { 3 }\) The region \(R _ { 2 }\) also shown shaded in Figure 2 is bounded by the curve, the positive \(x\)-axis and the line with equation \(x = b\), where \(b\) is a positive constant and \(0 < b < 4\) Given that the area of \(R _ { 1 }\) is equal to the area of \(R _ { 2 }\)
    2. verify that \(b\) satisfies the equation $$( b + 2 ) ^ { 2 } \left( 3 b ^ { 2 } - 20 b + 20 \right) = 0$$ The roots of the equation \(3 b ^ { 2 } - 20 b + 20 = 0\) are 1.225 and 5.442 to 3 decimal places. The value of \(b\) is therefore 1.225 to 3 decimal places.
    3. Explain, with the aid of a diagram, the significance of the root 5.442