Questions C3 (1301 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 Q1
8 marks Standard +0.3
  1. Given that \(\sin ^ { 2 } \theta + \cos ^ { 2 } \theta \equiv 1\), show that \(1 + \tan ^ { 2 } \theta \equiv \sec ^ { 2 } \theta\).
  2. Solve, for \(0 \leq \theta < 360 ^ { \circ }\), the equation $$2 \tan ^ { 2 } \theta + \sec \theta = 1 ,$$ giving your answers to 1 decimal place.
Edexcel C3 Q2
10 marks Moderate -0.3
2. (a) Differentiate with respect to \(x\)
  1. \(3 \sin ^ { 2 } x + \sec 2 x\),
  2. \(\{ x + \ln ( 2 x ) \} ^ { 3 }\). Given that \(y = \frac { 5 x ^ { 2 } - 10 x + 9 } { ( x - 1 ) ^ { 2 } } , x \neq 1\),
    (b) show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 8 } { ( x - 1 ) ^ { 3 } }\).
Edexcel C3 Q3
12 marks Standard +0.3
3. The function \(f\) is defined by $$f : x \mapsto \frac { 5 x + 1 } { x ^ { 2 } + x - 2 } - \frac { 3 } { x + 2 } , x > 1$$
  1. Show that \(\mathrm { f } ( x ) = \frac { 2 } { x - 1 } , x > 1\).
  2. Find \(\mathrm { f } ^ { - 1 } ( x )\). The function g is defined by $$\mathrm { g } : x \mapsto x ^ { 2 } + 5 , \quad x \in \mathbb { R } .$$ (b) Solve \(\mathrm { fg } ( x ) = \frac { 1 } { 4 }\).
Edexcel C3 Q4
10 marks Standard +0.3
4. $$\mathrm { f } ( x ) = 3 \mathrm { e } ^ { x } - \frac { 1 } { 2 } \ln x - 2 , \quad x > 0 .$$
  1. Differentiate to find \(\mathrm { f } ^ { \prime } ( x )\). The curve with equation \(y = \mathrm { f } ( x )\) has a turning point at \(P\). The \(x\)-coordinate of \(P\) is \(\alpha\).
  2. Show that \(\alpha = \frac { 1 } { 6 } \mathrm { e } ^ { - \alpha }\). The iterative formula $$x _ { n + 1 } = \frac { 1 } { 6 } \mathrm { e } ^ { - x _ { n } } , \quad x _ { 0 } = 1$$ is used to find an approximate value for \(\alpha\).
  3. Calculate the values of \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to 4 decimal places.
  4. By considering the change of sign of \(\mathrm { f } ^ { \prime } ( x )\) in a suitable interval, prove that \(\alpha = 0.1443\) correct to 4 decimal places.
Edexcel C3 Q5
8 marks Standard +0.3
5.
  1. Using the identity \(\cos ( A + B ) \equiv \cos A \cos B - \sin A \sin B\), prove that $$\cos 2 A \equiv 1 - 2 \sin ^ { 2 } A$$
  2. Show that $$2 \sin 2 \theta - 3 \cos 2 \theta - 3 \sin \theta + 3 \equiv \sin \theta ( 4 \cos \theta + 6 \sin \theta - 3 )$$
  3. Express \(4 \cos \theta + 6 \sin \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
  4. Hence, for \(0 \leq \theta < \pi\), solve $$2 \sin 2 \theta = 3 ( \cos 2 \theta + \sin \theta - 1 )$$ giving your answers in radians to 3 significant figures, where appropriate.
    Hence, for \(0 \leq \theta < \pi\), solve \includegraphics[max width=\textwidth, alt={}]{933ec0b9-3496-455e-9c2c-2612e84f63ff-02_20_26_1509_239} giving your answers in radians to 3 significant figures, where appropriate.
Edexcel C3 Q6
15 marks Standard +0.3
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{933ec0b9-3496-455e-9c2c-2612e84f63ff-02_371_643_338_1852}
\end{figure} Figure 1 shows part of the graph of \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The graph consists of two line segments that meet at the point \(( 1 , a ) , a < 0\). One line meets the \(x\)-axis at \(( 3,0 )\). The other line meets the \(x\)-axis at \(( - 1,0 )\) and the \(y\)-axis at \(( 0 , b ) , b < 0\). In separate diagrams, sketch the graph with equation
  1. \(y = \mathrm { f } ( x + 1 )\),
  2. \(y = \mathrm { f } ( | x | )\). Indicate clearly on each sketch the coordinates of any points of intersection with the axes.
    Given that \(\mathrm { f } ( x ) = | x - 1 | - 2\), find
  3. the value of \(a\) and the value of \(b\),
  4. the value of \(x\) for which \(\mathrm { f } ( x ) = 5 x\).
Edexcel C3 Q7
11 marks Standard +0.8
7. A particular species of orchid is being studied. The population \(p\) at time \(t\) years after the study started is assumed to be $$p = \frac { 2800 a \mathrm { e } ^ { 0.2 t } } { 1 + a \mathrm { e } ^ { 0.2 t } } , \text { where } a \text { is a constant. }$$ Given that there were 300 orchids when the study started,
  1. show that \(a = 0.12\),
  2. use the equation with \(a = 0.12\) to predict the number of years before the population of orchids reaches 1850 .
  3. Show that \(p = \frac { 336 } { 0.12 + \mathrm { e } ^ { - 0.2 t } }\).
  4. Hence show that the population cannot exceed 2800.
Edexcel C3 Q8
13 marks Moderate -0.8
8. The functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } : x \mapsto 2 x + \ln 2 , & x \in \mathbb { R } , \\ \mathrm {~g} : x \mapsto \mathrm { e } ^ { 2 x } , & x \in \mathbb { R } . \end{array}$$
  1. Prove that the composite function gf is $$\operatorname { gf } : x \mapsto 4 \mathrm { e } ^ { 4 x } , \quad x \in \mathbb { R }$$
  2. Sketch the curve with equation \(y = \operatorname { gf } ( x )\), and show the coordinates of the point where the curve cuts the \(y\)-axis.
  3. Write down the range of gf .
  4. Find the value of \(x\) for which \(\frac { \mathrm { d } } { \mathrm { d } x } [ \operatorname { gf } ( x ) ] = 3\), giving your answer to 3 significant figures.
Edexcel C3 Q9
9 marks Moderate -0.3
9. (i) Find the exact solutions to the equations
  1. \(\ln ( 3 x - 7 ) = 5\),
  2. \(3 ^ { x } \mathrm { e } ^ { 7 x + 2 } = 15\).
    (ii) The functions f and g are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } + 3 , & x \in \mathbb { R } , \\ \mathrm {~g} ( x ) = \ln ( x - 1 ) , & x \in \mathbb { R } , \quad x > 1 . \end{array}$$
    1. Find \(\mathrm { f } ^ { - 1 }\) and state its domain.
    2. Find fg and state its range.
Edexcel C3 Q6
Standard +0.3
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{9527d80b-a2a2-442f-9e32-d8768fbbd01a-009_458_876_285_539}
\end{figure} Figure 1 shows part of the graph of \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The graph consists of two line segments that meet at the point \(( 1 , a ) , a < 0\). One line meets the \(x\)-axis at \(( 3,0 )\). The other line meets the \(x\)-axis at \(( - 1,0 )\) and the \(y\)-axis at \(( 0 , b ) , b < 0\). In separate diagrams, sketch the graph with equation
  1. \(y = \mathrm { f } ( x + 1 )\),
  2. \(y = \mathrm { f } ( | x | )\). Indicate clearly on each sketch the coordinates of any points of intersection with the axes. Given that \(\mathrm { f } ( x ) = | x - 1 | - 2\), find
  3. the value of \(a\) and the value of \(b\),
  4. the value of \(x\) for which \(\mathrm { f } ( x ) = 5 x\).
AQA C3 Q2
Moderate -0.3
2 Use Simpson's rule with 5 ordinates ( 4 strips) to find an approximation to $$\int _ { 1 } ^ { 3 } \frac { 1 } { \sqrt { 1 + x ^ { 3 } } } \mathrm {~d} x$$ giving your answer to three significant figures.
AQA C3 Q5
Standard +0.3
5 The diagram shows part of the graph of \(y = \mathrm { e } ^ { 2 x } - 9\). The graph cuts the coordinate axes at ( \(0 , a\) ) and ( \(b , 0\) ). \includegraphics[max width=\textwidth, alt={}, center]{9aac4ee4-2435-4315-a87d-fe9fa8e15665-004_817_908_479_550}
  1. State the value of \(a\), and show that \(b = \ln 3\).
  2. Show that \(y ^ { 2 } = \mathrm { e } ^ { 4 x } - 18 \mathrm { e } ^ { 2 x } + 81\).
  3. The shaded region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the volume of the solid formed, giving your answer in the form \(\pi ( p \ln 3 + q )\), where \(p\) and \(q\) are integers.
  4. Sketch the curve with equation \(y = \left| \mathrm { e } ^ { 2 x } - 9 \right|\) for \(x \geqslant 0\).
AQA C3 Q6
Moderate -0.5
6 [Figure 1, printed on the insert, is provided for use in this question.]
The curve \(y = x ^ { 3 } + 4 x - 3\) intersects the \(x\)-axis at the point \(A\) where \(x = \alpha\).
  1. Show that \(\alpha\) lies between 0.5 and 1.0.
  2. Show that the equation \(x ^ { 3 } + 4 x - 3 = 0\) can be rearranged into the form \(x = \frac { 3 - x ^ { 3 } } { 4 }\).
    (1 mark)
    1. Use the iteration \(x _ { n + 1 } = \frac { 3 - x _ { n } { } ^ { 3 } } { 4 }\) with \(x _ { 1 } = 0.5\) to find \(x _ { 3 }\), giving your answer to two decimal places.
    2. The sketch on Figure 1 shows parts of the graphs of \(y = \frac { 3 - x ^ { 3 } } { 4 }\) and \(y = x\), and the position of \(x _ { 1 }\). On Figure 1, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of \(x _ { 2 }\) and \(x _ { 3 }\) on the \(x\)-axis.
      (3 marks)
AQA C3 Q7
Standard +0.3
7
  1. The sketch shows the graph of \(y = \sin ^ { - 1 } x\). \includegraphics[max width=\textwidth, alt={}, center]{9aac4ee4-2435-4315-a87d-fe9fa8e15665-006_819_824_456_591} Write down the coordinates of the points \(P\) and \(Q\), the end-points of the graph.
  2. Sketch the graph of \(y = - \sin ^ { - 1 } ( x - 1 )\).
AQA C3 Q8
Moderate -0.3
8 The functions \(f\) and \(g\) are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = x ^ { 2 } & \text { for all real values of } x \\ \mathrm {~g} ( x ) = \frac { 1 } { x + 2 } & \text { for real values of } x , \quad x \neq - 2 \end{array}$$
  1. State the range of f.
    1. Find fg(x).
    2. Solve the equation \(\operatorname { fg } ( x ) = 4\).
    1. Explain why the function f does not have an inverse.
    2. The inverse of g is \(\mathrm { g } ^ { - 1 }\). Find \(\mathrm { g } ^ { - 1 } ( x )\).
AQA C3 Q9
Standard +0.3
9
  1. Given that \(y = x ^ { - 2 } \ln x\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 - 2 \ln x } { x ^ { 3 } }\).
  2. Using integration by parts, find \(\int x ^ { - 2 } \ln x \mathrm {~d} x\).
  3. The sketch shows the graph of \(y = x ^ { - 2 } \ln x\). \includegraphics[max width=\textwidth, alt={}, center]{9aac4ee4-2435-4315-a87d-fe9fa8e15665-007_593_1034_696_543}
    1. Using the answer to part (a), find, in terms of e, the \(x\)-coordinate of the stationary point \(A\).
    2. The region \(R\) is bounded by the curve, the \(x\)-axis and the line \(x = 5\). Using your answer to part (b), show that the area of \(R\) is $$\frac { 1 } { 5 } ( 4 - \ln 5 )$$
AQA C3 Q10
Standard +0.3
10
    1. By writing \(\ln x\) as \(( \ln x ) \times 1\), use integration by parts to find \(\int \ln x \mathrm {~d} x\).
    2. Find \(\int ( \ln x ) ^ { 2 } \mathrm {~d} x\).
  2. Use the substitution \(u = \sqrt { x }\) to find the exact value of $$\int _ { 1 } ^ { 4 } \frac { 1 } { x + \sqrt { x } } \mathrm {~d} x$$ (7 marks)
AQA C3 2006 January Q1
5 marks Moderate -0.8
1
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = \tan 3 x\).
    (2 marks)
  2. Given that \(y = \frac { 3 x + 1 } { 2 x + 1 }\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { ( 2 x + 1 ) ^ { 2 } }\).
    (3 marks)
AQA C3 2006 January Q2
4 marks Moderate -0.5
2 Use Simpson's rule with 5 ordinates (4 strips) to find an approximation to $$\int _ { 1 } ^ { 3 } \frac { 1 } { \sqrt { 1 + x ^ { 3 } } } \mathrm {~d} x$$ giving your answer to three significant figures.
AQA C3 2006 January Q3
10 marks Standard +0.3
3
    1. Given that \(\mathrm { f } ( x ) = x ^ { 4 } + 2 x\), find \(\mathrm { f } ^ { \prime } ( x )\).
    2. Hence, or otherwise, find \(\int \frac { 2 x ^ { 3 } + 1 } { x ^ { 4 } + 2 x } \mathrm {~d} x\).
    1. Use the substitution \(u = 2 x + 1\) to show that $$\int x \sqrt { 2 x + 1 } \mathrm {~d} x = \frac { 1 } { 4 } \int \left( u ^ { \frac { 3 } { 2 } } - u ^ { \frac { 1 } { 2 } } \right) \mathrm { d } u$$
    2. Hence show that \(\int _ { 0 } ^ { 4 } x \sqrt { 2 x + 1 } \mathrm {~d} x = 19.9\) correct to three significant figures.
AQA C3 2006 January Q4
7 marks Moderate -0.3
4 It is given that \(2 \operatorname { cosec } ^ { 2 } x = 5 - 5 \cot x\).
  1. Show that the equation \(2 \operatorname { cosec } ^ { 2 } x = 5 - 5 \cot x\) can be written in the form $$2 \cot ^ { 2 } x + 5 \cot x - 3 = 0$$
  2. Hence show that \(\tan x = 2\) or \(\tan x = - \frac { 1 } { 3 }\).
  3. Hence, or otherwise, solve the equation \(2 \operatorname { cosec } ^ { 2 } x = 5 - 5 \cot x\), giving all values of \(x\) in radians to one decimal place in the interval \(- \pi < x \leqslant \pi\).
AQA C3 2006 January Q5
12 marks Standard +0.3
5 The diagram shows part of the graph of \(y = \mathrm { e } ^ { 2 x } - 9\). The graph cuts the coordinate axes at \(( 0 , a )\) and \(( b , 0 )\). \includegraphics[max width=\textwidth, alt={}, center]{908f530c-076d-47b1-90dd-38dbfe44f898-03_826_924_477_541}
  1. State the value of \(a\), and show that \(b = \ln 3\).
  2. Show that \(y ^ { 2 } = \mathrm { e } ^ { 4 x } - 18 \mathrm { e } ^ { 2 x } + 81\).
  3. The shaded region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the volume of the solid formed, giving your answer in the form \(\pi ( p \ln 3 + q )\), where \(p\) and \(q\) are integers.
  4. Sketch the curve with equation \(y = \left| \mathrm { e } ^ { 2 x } - 9 \right|\) for \(x \geqslant 0\).
AQA C3 2006 January Q6
12 marks Moderate -0.3
6 [Figure 1, printed on the insert, is provided for use in this question.]
The curve \(y = x ^ { 3 } + 4 x - 3\) intersects the \(x\)-axis at the point \(A\) where \(x = \alpha\).
  1. Show that \(\alpha\) lies between 0.5 and 1.0.
  2. Show that the equation \(x ^ { 3 } + 4 x - 3 = 0\) can be rearranged into the form \(x = \frac { 3 - x ^ { 3 } } { 4 }\).
    (1 mark)
    1. Use the iteration \(x _ { n + 1 } = \frac { 3 - x _ { n } { } ^ { 3 } } { 4 }\) with \(x _ { 1 } = 0.5\) to find \(x _ { 3 }\), giving your answer to two decimal places.
      (3 marks)
    2. The sketch on Figure 1 shows parts of the graphs of \(y = \frac { 3 - x ^ { 3 } } { 4 }\) and \(y = x\), and the position of \(x _ { 1 }\). On Figure 1, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of \(x _ { 2 }\) and \(x _ { 3 }\) on the \(x\)-axis.
      (3 marks)
AQA C3 2006 January Q7
5 marks Standard +0.3
7
  1. The sketch shows the graph of \(y = \sin ^ { - 1 } x\). \includegraphics[max width=\textwidth, alt={}, center]{908f530c-076d-47b1-90dd-38dbfe44f898-05_835_834_447_587} Write down the coordinates of the points \(P\) and \(Q\), the end-points of the graph.
  2. Sketch the graph of \(y = - \sin ^ { - 1 } ( x - 1 )\).
AQA C3 2006 January Q8
10 marks Moderate -0.8
8 The functions \(f\) and \(g\) are defined with their respective domains by $$\begin{array} { l l } \mathrm { f } ( x ) = x ^ { 2 } & \text { for all real values of } x \\ \mathrm {~g} ( x ) = \frac { 1 } { x + 2 } & \text { for real values of } x , x \neq - 2 \end{array}$$
  1. State the range of f.
    1. Find fg(x).
    2. Solve the equation \(\operatorname { fg } ( x ) = 4\).
    1. Explain why the function f does not have an inverse.
    2. The inverse of g is \(\mathrm { g } ^ { - 1 }\). Find \(\mathrm { g } ^ { - 1 } ( x )\).