Questions — Edexcel (9685 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 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 Stats 1 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 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 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 C4 Q2
8 marks Moderate -0.3
  1. A curve has the equation
$$4 \cos x + 2 \sin y = 3$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \sin x \sec y\).
  2. Find an equation for the tangent to the curve at the point \(\left( \frac { \pi } { 3 } , \frac { \pi } { 6 } \right)\), giving your answer in the form \(a x + b y = c\), where \(a\) and \(b\) are integers.
Edexcel C4 Q3
9 marks Standard +0.3
3. (a) Express \(\frac { 2 + 20 x } { 1 + 2 x - 8 x ^ { 2 } }\) as a sum of partial fractions.
(b) Hence find the series expansion of \(\frac { 2 + 20 x } { 1 + 2 x - 8 x ^ { 2 } } , | x | < \frac { 1 } { 4 }\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
3. continued
Edexcel C4 Q4
9 marks Standard +0.3
4. The line \(l _ { 1 }\) passes through the points \(P\) and \(Q\) with position vectors ( \(- \mathbf { i } - 8 \mathbf { j } + 3 \mathbf { k }\) ) and ( \(2 \mathbf { i } - 9 \mathbf { j } + \mathbf { k }\) ) respectively, relative to a fixed origin.
  1. Find a vector equation for \(l _ { 1 }\). The line \(l _ { 2 }\) has the equation $$\mathbf { r } = ( 6 \mathbf { i } + a \mathbf { j } + b \mathbf { k } ) + \mu ( \mathbf { i } + 4 \mathbf { j } - \mathbf { k } )$$ and also passes through the point \(Q\).
  2. Find the values of the constants \(a\) and \(b\).
  3. Find, in degrees to 1 decimal place, the acute angle between lines \(l _ { 1 }\) and \(l _ { 2 }\).
    4. continued
Edexcel C4 Q5
10 marks Moderate -0.3
5. At time \(t = 0\), a tank of height 2 metres is completely filled with water. Water then leaks from a hole in the side of the tank such that the depth of water in the tank, \(y\) metres, after \(t\) hours satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} t } = - k \mathrm { e } ^ { - 0.2 t }$$ where \(k\) is a positive constant,
  1. Find an expression for \(y\) in terms of \(k\) and \(t\). Given that two hours after being filled the depth of water in the tank is 1.6 metres,
  2. find the value of \(k\) to 4 significant figures. Given also that the hole in the tank is \(h \mathrm {~cm}\) above the base of the tank,
  3. show that \(h = 79\) to 2 significant figures.
    5. continued
Edexcel C4 Q6
10 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{922f404e-12d5-490b-9c8d-509f3a304c1e-10_438_700_255_518} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with parametric equations $$x = 2 - t ^ { 2 } , \quad y = t ( t + 1 ) , \quad t \geq 0$$
  1. Find the coordinates of the points where the curve meets the coordinate axes.
  2. Find the exact area of the region bounded by the curve and the coordinate axes.
    6. continued
Edexcel C4 Q7
12 marks Standard +0.8
7. (a) Prove that $$\frac { \mathrm { d } } { \mathrm {~d} x } \left( a ^ { x } \right) = a ^ { x } \ln a .$$ A curve has the equation \(y = 4 ^ { x } - 2 ^ { x - 1 } + 1\).
(b) Show that the tangent to the curve at the point where it crosses the \(y\)-axis has the equation $$3 x \ln 2 - 2 y + 3 = 0 .$$ (c) Find the exact coordinates of the stationary point of the curve.
7. continued
Edexcel C4 Q8
13 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{922f404e-12d5-490b-9c8d-509f3a304c1e-14_656_999_146_429} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the curve with equation \(y = \sqrt { \frac { x } { x + 1 } }\).
The shaded region is bounded by the curve, the \(x\)-axis and the line \(x = 3\).
    1. Use the trapezium rule with three strips to find an estimate for the area of the shaded region.
    2. Use the trapezium rule with six strips to find an improved estimate for the area of the shaded region. The shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis.
  2. Show that the volume of the solid formed is \(\pi ( 3 - \ln 4 )\).
    8. continued
    8. continued
Edexcel C4 Q1
8 marks Standard +0.8
  1. A curve has the equation
$$2 x ^ { 2 } + x y - y ^ { 2 } + 18 = 0$$ Find the coordinates of the points where the tangent to the curve is parallel to the \(x\)-axis.
Edexcel C4 Q2
8 marks Standard +0.8
2. Use the substitution \(x = 2 \tan u\) to show that $$\int _ { 0 } ^ { 2 } \frac { x ^ { 2 } } { x ^ { 2 } + 4 } \mathrm {~d} x = \frac { 1 } { 2 } ( 4 - \pi )$$
Edexcel C4 Q3
9 marks Moderate -0.3
  1. (a) Show that \(\left( 1 \frac { 1 } { 24 } \right) ^ { - \frac { 1 } { 2 } } = k \sqrt { 6 }\), where \(k\) is rational.
    (b) Expand \(\left( 1 + \frac { 1 } { 2 } x \right) ^ { - \frac { 1 } { 2 } } , | x | < 2\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
    (c) Use your answer to part (b) with \(x = \frac { 1 } { 12 }\) to find an approximate value for \(\sqrt { 6 }\), giving your answer to 5 decimal places.
  2. continued
  3. Relative to a fixed origin, two lines have the equations
$$\mathbf { r } = ( 7 \mathbf { j } - 4 \mathbf { k } ) + s ( 4 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } )$$ and $$\mathbf { r } = ( - 7 \mathbf { i } + \mathbf { j } + 8 \mathbf { k } ) + t ( - 3 \mathbf { i } + 2 \mathbf { k } )$$ where \(s\) and \(t\) are scalar parameters.
(a) Show that the two lines intersect and find the position vector of the point where they meet.
(b) Find, in degrees to 1 decimal place, the acute angle between the lines.
Edexcel C4 Q5
11 marks Standard +0.3
5. A curve has parametric equations $$x = \frac { t } { 2 - t } , \quad y = \frac { 1 } { 1 + t } , \quad - 1 < t < 2$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 1 } { 2 } \left( \frac { 2 - t } { 1 + t } \right) ^ { 2 }\).
  2. Find an equation for the normal to the curve at the point where \(t = 1\).
  3. Show that the cartesian equation of the curve can be written in the form $$y = \frac { 1 + x } { 1 + 3 x }$$
    1. continued
    2. (a) Find \(\int \tan ^ { 2 } x d x\).
    3. Show that
    $$\int \tan x \mathrm {~d} x = \ln | \sec x | + c$$ where \(c\) is an arbitrary constant. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{fe01157a-7617-43d3-900c-8d043bcbe784-10_566_789_648_504} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows part of the curve with equation \(y = x ^ { \frac { 1 } { 2 } } \tan x\).
    The shaded region bounded by the curve, the \(x\)-axis and the line \(x = \frac { \pi } { 3 }\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
  4. Show that the volume of the solid formed is \(\frac { 1 } { 18 } \pi ^ { 2 } ( 6 \sqrt { 3 } - \pi ) - \pi \ln 2\).
Edexcel C4 Q7
17 marks Standard +0.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fe01157a-7617-43d3-900c-8d043bcbe784-12_252_757_267_484} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a hemispherical bowl of radius 5 cm .
The bowl is filled with water but the water leaks from a hole at the base of the bowl. At time \(t\) minutes, the depth of water is \(h \mathrm {~cm}\) and the volume of water in the bowl is \(V \mathrm {~cm} ^ { 3 }\), where $$V = \frac { 1 } { 3 } \pi h ^ { 2 } ( 15 - h ) .$$ In a model it is assumed that the rate at which the volume of water in the bowl decreases is proportional to \(V\).
  1. Show that $$\frac { \mathrm { d } h } { \mathrm {~d} t } = - \frac { k h ( 15 - h ) } { 3 ( 10 - h ) } ,$$ where \(k\) is a positive constant.
  2. Express \(\frac { 3 ( 10 - h ) } { h ( 15 - h ) }\) in partial fractions. Given that when \(t = 0 , h = 5\),
  3. show that $$h ^ { 2 } ( 15 - h ) = 250 \mathrm { e } ^ { - k t }$$ Given also that when \(t = 2 , h = 4\),
  4. find the value of \(k\) to 3 significant figures.
    7. continued
    7. continued
Edexcel C4 Q1
6 marks Moderate -0.3
  1. A curve has the equation
$$x ^ { 2 } + 2 x y ^ { 2 } + y = 4$$ Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
Edexcel C4 Q2
7 marks Standard +0.3
2. Use integration by parts to find $$\int x ^ { 2 } \mathrm { e } ^ { - x } \mathrm {~d} x$$
Edexcel C4 Q3
8 marks Standard +0.3
  1. The first four terms in the series expansion of \(( 1 + a x ) ^ { n }\) in ascending powers of \(x\) are
$$1 - 4 x + 24 x ^ { 2 } + k x ^ { 3 }$$ where \(a , n\) and \(k\) are constants and \(| a x | < 1\).
  1. Find the values of \(a\) and \(n\).
  2. Show that \(k = - 160\).
    3. continued
Edexcel C4 Q4
9 marks Standard +0.3
4. (a) Use the trapezium rule with two intervals of equal width to find an estimate for the value of the integral $$\int _ { 0 } ^ { 3 } e ^ { \cos x } d x$$ giving your answer to 3 significant figures.
(b) Use the trapezium rule with four intervals of equal width to find another estimate for the value of the integral to 3 significant figures.
(c) Given that the true value of the integral lies between the estimates made in parts (a) and (b), comment on the shape of the curve \(y = \mathrm { e } ^ { \cos x }\) in the interval \(0 \leq x \leq 3\) and explain your answer.
4. continued
Edexcel C4 Q5
9 marks Standard +0.3
5. A straight road passes through villages at the points \(A\) and \(B\) with position vectors ( \(9 \mathbf { i } - 8 \mathbf { j } + 2 \mathbf { k }\) ) and ( \(4 \mathbf { j } + \mathbf { k }\) ) respectively, relative to a fixed origin. The road ends at a junction at the point \(C\) with another straight road which lies along the line with equation $$\mathbf { r } = ( 2 \mathbf { i } + 16 \mathbf { j } - \mathbf { k } ) + \mu ( - 5 \mathbf { i } + 3 \mathbf { j } ) ,$$ where \(\mu\) is a scalar parameter.
  1. Find the position vector of \(C\). Given that 1 unit on each coordinate axis represents 200 metres,
  2. find the distance, in kilometres, from the village at \(A\) to the junction at \(C\).
    5. continued
Edexcel C4 Q6
10 marks Standard +0.3
6. A small town had a population of 9000 in the year 2001. In a model, it is assumed that the population of the town, \(P\), at time \(t\) years after 2001 satisfies the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = 0.05 P \mathrm { e } ^ { - 0.05 t }$$
  1. Show that, according to the model, the population of the town in 2011 will be 13300 to 3 significant figures.
  2. Find the value which the population of the town will approach in the long term, according to the model.
    6. continued
Edexcel C4 Q7
11 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4232f6a1-00ff-4e88-b5f4-1abf3d4742c4-12_560_911_146_456} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with parametric equations $$x = t ^ { 3 } + 1 , \quad y = \frac { 2 } { t } , \quad t > 0 .$$ The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 2\) and \(x = 9\).
  1. Find the area of the shaded region.
  2. Show that the volume of the solid formed when the shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis is \(12 \pi\).
  3. Find a cartesian equation for the curve in the form \(y = \mathrm { f } ( x )\).
    7. continued
Edexcel C4 Q8
15 marks Standard +0.3
8. (a) Show that the substitution \(u = \sin x\) transforms the integral $$\int \frac { 6 } { \cos x ( 2 - \sin x ) } d x$$ into the integral $$\int \frac { 6 } { \left( 1 - u ^ { 2 } \right) ( 2 - u ) } \mathrm { d } u .$$ (b) Express \(\frac { 6 } { \left( 1 - u ^ { 2 } \right) ( 2 - u ) }\) in partial fractions.
(c) Hence, evaluate $$\int _ { 0 } ^ { \frac { \pi } { 6 } } \frac { 6 } { \cos x ( 2 - \sin x ) } d x$$ giving your answer in the form \(a \ln 2 + b \ln 3\), where \(a\) and \(b\) are integers.
8. continued
8. continued
Edexcel C4 Q1
5 marks Moderate -0.3
  1. (a) Expand \(( 1 + 4 x ) ^ { \frac { 3 } { 2 } }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
    (b) State the set of values of \(x\) for which your expansion is valid.
  2. Use the substitution \(u = 1 + \sin x\) to find the value of
$$\int _ { 0 } ^ { \frac { \pi } { 2 } } \cos x ( 1 + \sin x ) ^ { 3 } d x$$
Edexcel C4 Q3
8 marks Moderate -0.3
  1. (a) Express \(\frac { x + 11 } { ( x + 4 ) ( x - 3 ) }\) as a sum of partial fractions.
    (b) Evaluate
$$\int _ { 0 } ^ { 2 } \frac { x + 11 } { ( x + 4 ) ( x - 3 ) } d x$$ giving your answer in the form \(\ln k\), where \(k\) is an exact simplified fraction. (5)
3. continued
Edexcel C4 Q4
8 marks Standard +0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4d4aa72e-d781-409d-8401-ccb4241bb12f-06_588_886_255_513} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with equation \(y = 2 \sin x + \operatorname { cosec } x , 0 < x < \pi\). The shaded region bounded by the curve, the \(x\)-axis and the lines \(x = \frac { \pi } { 6 }\) and \(x = \frac { \pi } { 2 }\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Show that the volume of the solid formed is \(\frac { 1 } { 2 } \pi ( 4 \pi + 3 \sqrt { 3 } )\).
4. continued
Edexcel C4 Q5
8 marks Standard +0.3
5. A curve has the equation $$x ^ { 2 } - 3 x y - y ^ { 2 } = 12$$
  1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
  2. Find an equation for the tangent to the curve at the point \(( 2 , - 2 )\).
    5. continued
Edexcel C4 Q6
10 marks Standard +0.8
6. Relative to a fixed origin, \(O\), the points \(A\) and \(B\) have position vectors \(\left( \begin{array} { c } 1 \\ 5 \\ - 1 \end{array} \right)\) and \(\left( \begin{array} { c } 6 \\ 3 \\ - 6 \end{array} \right)\) respectively. Find, in exact, simplified form,
  1. the cosine of \(\angle A O B\),
  2. the area of triangle \(O A B\),
  3. the shortest distance from \(A\) to the line \(O B\).
    6. continued