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 C3 2007 June Q1
6 marks Moderate -0.3
Find the exact solutions to the equations
  1. \(\ln x + \ln 3 = \ln 6\),
  2. \(\mathrm { e } ^ { x } + 3 \mathrm { e } ^ { - x } = 4\).
Edexcel C3 2007 June Q3
10 marks Moderate -0.3
3. A curve \(C\) has equation $$y = x ^ { 2 } \mathrm { e } ^ { x }$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), using the product rule for differentiation.
  2. Hence find the coordinates of the turning points of \(C\).
  3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
  4. Determine the nature of each turning point of the curve \(C\).
Edexcel C3 2007 June Q4
7 marks Moderate -0.3
4. $$f ( x ) = - x ^ { 3 } + 3 x ^ { 2 } - 1$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) can be rewritten as $$x = \sqrt { } \left( \frac { 1 } { 3 - x } \right)$$
  2. Starting with \(x _ { 1 } = 0.6\), use the iteration $$\left. x _ { n + 1 } = \sqrt { ( } \frac { 1 } { 3 - x _ { n } } \right)$$ to calculate the values of \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving all your answers to 4 decimal places.
  3. Show that \(x = 0.653\) is a root of \(\mathrm { f } ( x ) = 0\) correct to 3 decimal places.
Edexcel C3 2007 June Q5
12 marks Moderate -0.3
5. The functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto \ln ( 2 x - 1 ) , & x \in \mathbb { R } , x > \frac { 1 } { 2 } \\ \mathrm {~g} : x \mapsto \frac { 2 } { x - 3 } , & x \in \mathbb { R } , x \neq 3 \end{array}$$
  1. Find the exact value of fg(4).
  2. Find the inverse function \(\mathrm { f } ^ { - 1 } ( x )\), stating its domain.
  3. Sketch the graph of \(y = | \mathrm { g } ( x ) |\). Indicate clearly the equation of the vertical asymptote and the coordinates of the point at which the graph crosses the \(y\)-axis.
  4. Find the exact values of \(x\) for which \(\left| \frac { 2 } { x - 3 } \right| = 3\).
Edexcel C3 2007 June Q6
11 marks Standard +0.3
  1. Express \(3 \sin x + 2 \cos x\) in the form \(R \sin ( x + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
  2. Hence find the greatest value of \(( 3 \sin x + 2 \cos x ) ^ { 4 }\).
  3. Solve, for \(0 < x < 2 \pi\), the equation $$3 \sin x + 2 \cos x = 1$$ giving your answers to 3 decimal places.
Edexcel C3 2007 June Q7
12 marks Standard +0.3
  1. Prove that $$\frac { \sin \theta } { \cos \theta } + \frac { \cos \theta } { \sin \theta } = 2 \operatorname { cosec } 2 \theta , \quad \theta \neq 90 n ^ { \circ }$$
  2. On the axes on page 20, sketch the graph of \(y = 2 \operatorname { cosec } 2 \theta\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
  3. Solve, for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), the equation $$\frac { \sin \theta } { \cos \theta } + \frac { \cos \theta } { \sin \theta } = 3 ,$$ giving your answers to 1 decimal place. \includegraphics[max width=\textwidth, alt={}, center]{f3c3c777-7808-4d82-a1f4-2dee6674be1e-11_899_1253_315_347}
Edexcel C3 2007 June Q8
7 marks Moderate -0.8
8. The amount of a certain type of drug in the bloodstream \(t\) hours after it has been taken is given by the formula $$x = D \mathrm { e } ^ { - \frac { 1 } { 8 } t } ,$$ where \(x\) is the amount of the drug in the bloodstream in milligrams and \(D\) is the dose given in milligrams. A dose of 10 mg of the drug is given.
  1. Find the amount of the drug in the bloodstream 5 hours after the dose is given. Give your answer in mg to 3 decimal places. A second dose of 10 mg is given after 5 hours.
  2. Show that the amount of the drug in the bloodstream 1 hour after the second dose is 13.549 mg to 3 decimal places. No more doses of the drug are given. At time \(T\) hours after the second dose is given, the amount of the drug in the bloodstream is 3 mg .
  3. Find the value of \(T\).
Edexcel C3 2008 June Q1
6 marks Moderate -0.5
  1. The point \(P\) lies on the curve with equation
$$y = 4 \mathrm { e } ^ { 2 x + 1 }$$ The \(y\)-coordinate of \(P\) is 8 .
  1. Find, in terms of \(\ln 2\), the \(x\)-coordinate of \(P\).
  2. Find the equation of the tangent to the curve at the point \(P\) in the form \(y = a x + b\), where \(a\) and \(b\) are exact constants to be found.
Edexcel C3 2008 June Q2
12 marks Moderate -0.3
2. $$f ( x ) = 5 \cos x + 12 \sin x$$ Given that \(\mathrm { f } ( x ) = R \cos ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\),
  1. find the value of \(R\) and the value of \(\alpha\) to 3 decimal places.
  2. Hence solve the equation $$5 \cos x + 12 \sin x = 6$$ for \(0 \leqslant x < 2 \pi\).
    1. Write down the maximum value of \(5 \cos x + 12 \sin x\).
    2. Find the smallest positive value of \(x\) for which this maximum value occurs.
Edexcel C3 2008 June Q3
12 marks Moderate -0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f47675f8-a2c2-4c4c-b878-ffe15a95c19d-05_623_977_207_479} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the graph of \(y = f ( x ) , x \in \mathbb { R }\).
The graph consists of two line segments that meet at the point \(P\).
The graph cuts the \(y\)-axis at the point \(Q\) and the \(x\)-axis at the points \(( - 3,0 )\) and \(R\). Sketch, on separate diagrams, the graphs of
  1. \(y = | f ( x ) |\),
  2. \(y = \mathrm { f } ( - x )\). Given that \(\mathrm { f } ( x ) = 2 - | x + 1 |\),
  3. find the coordinates of the points \(P , Q\) and \(R\),
  4. solve \(\mathrm { f } ( x ) = \frac { 1 } { 2 } x\).
Edexcel C3 2008 June Q4
12 marks Standard +0.3
4. The function \(f\) is defined by $$f : x \mapsto \frac { 2 ( x - 1 ) } { x ^ { 2 } - 2 x - 3 } - \frac { 1 } { x - 3 } , \quad x > 3$$
  1. Show that \(\mathrm { f } ( x ) = \frac { 1 } { x + 1 } , \quad x > 3\).
  2. Find the range of f.
  3. Find \(\mathrm { f } ^ { - 1 } ( x )\). State the domain of this inverse function. The function \(g\) is defined by $$\mathrm { g } : x \mapsto 2 x ^ { 2 } - 3 , \quad x \in \mathbb { R }$$
  4. Solve \(\mathrm { fg } ( x ) = \frac { 1 } { 8 }\).
Edexcel C3 2008 June Q5
8 marks Standard +0.3
5.
  1. Given that \(\sin ^ { 2 } \theta + \cos ^ { 2 } \theta \equiv 1\), show that \(1 + \cot ^ { 2 } \theta \equiv \operatorname { cosec } ^ { 2 } \theta\).
  2. Solve, for \(0 \leqslant \theta < 180 ^ { \circ }\), the equation $$2 \cot ^ { 2 } \theta - 9 \operatorname { cosec } \theta = 3$$ giving your answers to 1 decimal place.
Edexcel C3 2008 June Q6
14 marks Moderate -0.3
6. (a) Differentiate with respect to \(x\),
  1. \(\mathrm { e } ^ { 3 x } ( \sin x + 2 \cos x )\),
  2. \(x ^ { 3 } \ln ( 5 x + 2 )\). Given that \(y = \frac { 3 x ^ { 2 } + 6 x - 7 } { ( x + 1 ) ^ { 2 } } , \quad x \neq - 1\),
    (b) show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 20 } { ( x + 1 ) ^ { 3 } }\).
    (c) Hence find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and the real values of \(x\) for which \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \frac { 15 } { 4 }\).
Edexcel C3 2008 June Q7
11 marks Standard +0.2
7. $$f ( x ) = 3 x ^ { 3 } - 2 x - 6$$
  1. Show that \(\mathrm { f } ( x ) = 0\) has a root, \(\alpha\), between \(x = 1.4\) and \(x = 1.45\)
  2. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as $$x = \sqrt { } \left( \frac { 2 } { x } + \frac { 2 } { 3 } \right) , \quad x \neq 0$$
  3. Starting with \(x _ { 0 } = 1.43\), use the iteration $$x _ { \mathrm { n } + 1 } = \sqrt { } \left( \frac { 2 } { x _ { \mathrm { n } } } + \frac { 2 } { 3 } \right)$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 4 decimal places.
  4. By choosing a suitable interval, show that \(\alpha = 1.435\) is correct to 3 decimal places.
Edexcel C3 2009 June Q1
6 marks Moderate -0.3
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bcb0c693-66ae-4b97-99f8-b10fb9396886-02_579_1240_251_383} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows part of the curve with equation \(y = - x ^ { 3 } + 2 x ^ { 2 } + 2\), which intersects the \(x\)-axis at the point \(A\) where \(x = \alpha\). To find an approximation to \(\alpha\), the iterative formula $$x _ { n + 1 } = \frac { 2 } { \left( x _ { n } \right) ^ { 2 } } + 2$$ is used.
  1. Taking \(x _ { 0 } = 2.5\), find the values of \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\). Give your answers to 3 decimal places where appropriate.
  2. Show that \(\alpha = 2.359\) correct to 3 decimal places.
Edexcel C3 2009 June Q2
8 marks Standard +0.3
2.
  1. Use the identity \(\cos ^ { 2 } \theta + \sin ^ { 2 } \theta = 1\) to prove that \(\tan ^ { 2 } \theta = \sec ^ { 2 } \theta - 1\).
  2. Solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), the equation $$2 \tan ^ { 2 } \theta + 4 \sec \theta + \sec ^ { 2 } \theta = 2$$
Edexcel C3 2009 June Q3
8 marks Moderate -0.8
  1. Rabbits were introduced onto an island. The number of rabbits, \(P , t\) years after they were introduced is modelled by the equation
$$P = 80 \mathrm { e } ^ { \frac { 1 } { 5 } t } , \quad t \in \mathbb { R } , t \geqslant 0$$
  1. Write down the number of rabbits that were introduced to the island.
  2. Find the number of years it would take for the number of rabbits to first exceed 1000.
  3. Find \(\frac { \mathrm { d } P } { \mathrm {~d} t }\).
  4. Find \(P\) when \(\frac { \mathrm { d } P } { \mathrm {~d} t } = 50\).
Edexcel C3 2009 June Q4
13 marks Moderate -0.3
4. (i) Differentiate with respect to \(x\)
  1. \(x ^ { 2 } \cos 3 x\)
  2. \(\frac { \ln \left( x ^ { 2 } + 1 \right) } { x ^ { 2 } + 1 }\) (ii) A curve \(C\) has the equation $$y = \sqrt { } ( 4 x + 1 ) , \quad x > - \frac { 1 } { 4 } , \quad y > 0$$ The point \(P\) on the curve has \(x\)-coordinate 2 . Find an equation of the tangent to \(C\) at \(P\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C3 2009 June Q5
10 marks Moderate -0.3
5.
[diagram]
Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\).
The curve meets the coordinate axes at the points \(A ( 0,1 - k )\) and \(B \left( \frac { 1 } { 2 } \ln k , 0 \right)\), where \(k\) is a constant and \(k > 1\), as shown in Figure 2. On separate diagrams, sketch the curve with equation
  1. \(y = | f ( x ) |\),
  2. \(y = \mathrm { f } ^ { - 1 } ( x )\). Show on each sketch the coordinates, in terms of \(k\), of each point at which the curve meets or cuts the axes. Given that \(\mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } - k\),
  3. state the range of f ,
  4. find \(\mathrm { f } ^ { - 1 } ( x )\),
  5. write down the domain of \(\mathrm { f } ^ { - 1 }\).
Edexcel C3 2009 June Q6
12 marks Standard +0.3
  1. Use the identity \(\cos ( A + B ) = \cos A \cos B - \sin A \sin B\), to show that $$\cos 2 A = 1 - 2 \sin ^ { 2 } A$$ The curves \(C _ { 1 }\) and \(C _ { 2 }\) have equations $$\begin{aligned} & C _ { 1 } : \quad y = 3 \sin 2 x \\ & C _ { 2 } : \quad y = 4 \sin ^ { 2 } x - 2 \cos 2 x \end{aligned}$$
  2. Show that the \(x\)-coordinates of the points where \(C _ { 1 }\) and \(C _ { 2 }\) intersect satisfy the equation $$4 \cos 2 x + 3 \sin 2 x = 2$$
  3. Express \(4 \cos 2 x + 3 \sin 2 x\) in the form \(R \cos ( 2 x - \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) to 2 decimal places.
  4. Hence find, for \(0 \leqslant x < 180 ^ { \circ }\), all the solutions of $$4 \cos 2 x + 3 \sin 2 x = 2$$ giving your answers to 1 decimal place.
Edexcel C3 2009 June Q7
12 marks Standard +0.3
7. The function f is defined by $$\mathrm { f } ( x ) = 1 - \frac { 2 } { ( x + 4 ) } + \frac { x - 8 } { ( x - 2 ) ( x + 4 ) } , \quad x \in \mathbb { R } , x \neq - 4 , x \neq 2$$
  1. Show that \(\mathrm { f } ( x ) = \frac { x - 3 } { x - 2 }\) The function g is defined by $$\mathrm { g } ( x ) = \frac { \mathrm { e } ^ { x } - 3 } { \mathrm { e } ^ { x } - 2 } , \quad x \in \mathbb { R } , x \neq \ln 2$$
  2. Differentiate \(\mathrm { g } ( x )\) to show that \(\mathrm { g } ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { x } } { \left( \mathrm { e } ^ { x } - 2 \right) ^ { 2 } }\)
  3. Find the exact values of \(x\) for which \(\mathrm { g } ^ { \prime } ( x ) = 1\)
Edexcel C3 2009 June Q8
6 marks Standard +0.3
8.
  1. Write down \(\sin 2 x\) in terms of \(\sin x\) and \(\cos x\).
  2. Find, for \(0 < x < \pi\), all the solutions of the equation $$\operatorname { cosec } x - 8 \cos x = 0$$ giving your answers to 2 decimal places.
Edexcel C3 2010 June Q1
5 marks Moderate -0.3
  1. Show that $$\frac { \sin 2 \theta } { 1 + \cos 2 \theta } = \tan \theta$$
  2. Hence find, for \(- 180 ^ { \circ } \leqslant \theta < 180 ^ { \circ }\), all the solutions of $$\frac { 2 \sin 2 \theta } { 1 + \cos 2 \theta } = 1$$ Give your answers to 1 decimal place.
Edexcel C3 2010 June Q2
7 marks Moderate -0.3
2. A curve \(C\) has equation $$y = \frac { 3 } { ( 5 - 3 x ) ^ { 2 } } , \quad x \neq \frac { 5 } { 3 }$$ The point \(P\) on \(C\) has \(x\)-coordinate 2. Find an equation of the normal to \(C\) at \(P\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C3 2010 June Q3
9 marks Moderate -0.3
3. \(\mathrm { f } ( x ) = 4 \operatorname { cosec } x - 4 x + 1\), where \(x\) is in radians.
  1. Show that there is a root \(\alpha\) of \(\mathrm { f } ( x ) = 0\) in the interval [1.2,1.3].
  2. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written in the form $$x = \frac { 1 } { \sin x } + \frac { 1 } { 4 }$$
  3. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { \sin x _ { n } } + \frac { 1 } { 4 } , \quad x _ { 0 } = 1.25$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 4 decimal places.
  4. By considering the change of sign of \(\mathrm { f } ( x )\) in a suitable interval, verify that \(\alpha = 1.291\) correct to 3 decimal places.