Questions C3 (1200 questions)

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AQA C3 2014 June Q8
12 marks
8
  1. Show that the expression \(\frac { 1 - \sin x } { \cos x } + \frac { \cos x } { 1 - \sin x }\) can be written as \(2 \sec x\).
    [0pt] [4 marks]
  2. Hence solve the equation $$\frac { 1 - \sin x } { \cos x } + \frac { \cos x } { 1 - \sin x } = \tan ^ { 2 } x - 2$$ giving the values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } \leqslant x < 360 ^ { \circ }\).
    [0pt] [6 marks]
  3. Hence solve the equation $$\frac { 1 - \sin \left( 2 \theta - 30 ^ { \circ } \right) } { \cos \left( 2 \theta - 30 ^ { \circ } \right) } + \frac { \cos \left( 2 \theta - 30 ^ { \circ } \right) } { 1 - \sin \left( 2 \theta - 30 ^ { \circ } \right) } = \tan ^ { 2 } \left( 2 \theta - 30 ^ { \circ } \right) - 2$$ giving the values of \(\theta\) to the nearest degree in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
    [0pt] [2 marks]
    \includegraphics[max width=\textwidth, alt={}]{57412ec0-ad97-4418-8ba8-93f1f7d8aac1-16_1517_1709_1190_153}
    \includegraphics[max width=\textwidth, alt={}]{57412ec0-ad97-4418-8ba8-93f1f7d8aac1-20_2489_1730_221_139}
AQA C3 2016 June Q1
1
  1. Given that \(y = ( 4 x + 1 ) ^ { 3 } \sin 2 x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Given that \(y = \frac { 2 x ^ { 2 } + 3 } { 3 x ^ { 2 } + 4 }\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { p x } { \left( 3 x ^ { 2 } + 4 \right) ^ { 2 } }\), where \(p\) is a constant.
  3. Given that \(y = \ln \left( \frac { 2 x ^ { 2 } + 3 } { 3 x ^ { 2 } + 4 } \right)\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
AQA C3 2016 June Q2
2 The curve with equation \(y = x ^ { x }\), where \(x > 0\), intersects the line \(y = 5\) at a single point, where \(x = \alpha\).
  1. Show that \(\alpha\) lies between 2 and 3 .
  2. Show that the equation \(x ^ { x } = 5\) can be rearranged into the form $$x = \mathrm { e } ^ { \left( \frac { \ln 5 } { x } \right) }$$
  3. Use the iterative formula $$x _ { n + 1 } = \mathrm { e } ^ { \left( \frac { \ln 5 } { x _ { n } } \right) }$$ with \(x _ { 1 } = 2\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.
    1. Use Simpson's rule with 7 ordinates ( 6 strips) to find an approximation to $$\int _ { 0.5 } ^ { 1.7 } \left( 5 - x ^ { x } \right) \mathrm { d } x$$ giving your answer to three significant figures.
    2. Hence find an approximation to \(\int _ { 0.5 } ^ { 1.7 } x ^ { x } \mathrm {~d} x\).
AQA C3 2016 June Q3
5 marks
3 Solve $$x ^ { 2 } \geqslant | 5 x - 6 |$$ [5 marks]
AQA C3 2016 June Q4
6 marks
4
  1. Describe a sequence of two geometrical transformations that maps the graph of \(y = \mathrm { e } ^ { x }\) onto the graph of \(y = \mathrm { e } ^ { 2 x - 5 }\).
  2. The normal to the curve \(y = \mathrm { e } ^ { 2 x - 5 }\) at the point \(P \left( 2 , \mathrm { e } ^ { - 1 } \right)\) intersects the \(x\)-axis at the point \(A\) and the \(y\)-axis at the point \(B\). Show that the area of the triangle \(O A B\) is \(\frac { \left( \mathrm { e } ^ { 2 } + 1 \right) ^ { m } } { \mathrm { e } ^ { n } }\), where \(m\) and \(n\) are integers.
    [0pt] [6 marks]
AQA C3 2016 June Q5
5 The function f is defined by $$\mathrm { f } ( x ) = 16 x - \mathrm { e } ^ { 2 x } , \text { for all real } x$$ The graph of \(y = \mathrm { f } ( x )\) is sketched below.
\includegraphics[max width=\textwidth, alt={}, center]{bf427498-f1ee-4167-a6f2-ddaa2ff5ef81-12_789_1349_534_347}
  1. Find the range of f.
  2. The composite function fg is defined by $$\operatorname { fg } ( x ) = \frac { 16 } { x } - \mathrm { e } ^ { \frac { 2 } { x } } , \text { for real } x , x \neq 0$$ Find an expression for \(\operatorname { gg } ( x )\).
AQA C3 2016 June Q6
7 marks
6
  1. Use integration by parts to find \(\int \frac { \ln ( 3 x ) } { x ^ { 2 } } \mathrm {~d} x\).
  2. The region bounded by the curve \(y = \frac { \ln ( 3 x ) } { x }\), the \(x\)-axis from \(\frac { 1 } { 3 }\) to 1 , and the line \(x = 1\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid. Find the exact value of the volume of the solid generated.
    [0pt] [7 marks]
AQA C3 2016 June Q7
6 marks
7
  1. By writing \(\sec x = ( \cos x ) ^ { - 1 }\), use the chain rule to show that, if \(y = \sec x\), then $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x \tan x$$
  2. The function f is defined by $$\mathrm { f } ( x ) = 2 \tan x - 3 \sec x , \text { for } 0 < x < \frac { \pi } { 2 }$$ Find the value of the \(y\)-coordinate of the stationary point of the graph of \(y = \mathrm { f } ( x )\), giving your answer in the form \(p \sqrt { q }\), where \(p\) and \(q\) are integers.
    [0pt] [6 marks]
AQA C3 2016 June Q8
8 Use the substitution \(u = 4 x - 1\) to find the exact value of $$\int _ { \frac { 1 } { 4 } } ^ { \frac { 1 } { 2 } } ( 5 - 2 x ) ( 4 x - 1 ) ^ { \frac { 1 } { 3 } } \mathrm {~d} x$$
\includegraphics[max width=\textwidth, alt={}]{bf427498-f1ee-4167-a6f2-ddaa2ff5ef81-18_2104_1712_603_153}
AQA C3 2016 June Q9
3 marks
9
  1. It is given that \(\sec x - \tan x = - 5\).
    1. Show that \(\sec x + \tan x = - 0.2\).
    2. Hence find the exact value of \(\cos x\).
  2. Hence solve the equation $$\sec \left( 2 x - 70 ^ { \circ } \right) - \tan \left( 2 x - 70 ^ { \circ } \right) = - 5$$ giving all values of \(x\), to one decimal place, in the interval \(- 90 ^ { \circ } \leqslant x \leqslant 90 ^ { \circ }\).
    [0pt] [3 marks] \section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}
Edexcel C3 Q1
  1. Express as a single fraction in its simplest form
$$\frac { x ^ { 2 } - 8 x + 15 } { x ^ { 2 } - 9 } \times \frac { 2 x ^ { 2 } + 6 x } { ( x - 5 ) ^ { 2 } }$$
Edexcel C3 Q2
  1. The root of the equation \(\mathrm { f } ( x ) = 0\), where
$$f ( x ) = x + \ln 2 x - 4$$ is to be estimated using the iterative formula \(x _ { n + 1 } = 4 - \ln 2 x _ { n }\), with \(x _ { 0 } = 2.4\).
  1. Showing your values of \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\), obtain the value, to 3 decimal places, of the root.
  2. By considering the change of sign of \(\mathrm { f } ( x )\) in a suitable interval, justify the accuracy of your answer to part (a).
Edexcel C3 Q3
3. The function \(f\) is defined by $$f : x \text { a } | 2 x - a | , \quad x \in ^ { \circ }$$ where \(a\) is a positive constant.
  1. Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of the points where the graph cuts the axes.
  2. On a separate diagram, sketch the graph of \(y = \mathrm { f } ( 2 x )\), showing the coordinates of the points where the graph cuts the axes.
  3. Given that a solution of the equation \(\mathrm { f } ( x ) = \frac { 1 } { 2 } x\) is \(x = 4\), find the two possible values of \(a\).
Edexcel C3 Q4
4. Prove that $$\frac { 1 - \tan ^ { 2 } \theta } { 1 + \tan ^ { 2 } \theta } \equiv \cos 2 \theta$$ \section*{EDEXCEL CORE MATHEMATICS PRACTICE PAPER 1}
Edexcel C3 Q5
  1. Express \(\frac { 3 } { x ^ { 2 } + 2 x } + \frac { x - 4 } { x ^ { 2 } - 4 }\) as a single fraction in its simplest form.
  2. The function f , defined for \(x \in ^ { \circ } , x > 0\), is such that
$$\mathrm { f } ^ { \prime } ( x ) = x ^ { 2 } - 2 + \frac { 1 } { x ^ { 2 } }$$
  1. Find the value of \(\mathrm { f } ^ { \prime \prime } ( x )\) at \(x = 4\).
  2. Given that \(\mathrm { f } ( 3 ) = 0\), find \(\mathrm { f } ( x )\).
  3. Prove that f is an increasing function.
Edexcel C3 Q7
7. \(\quad \mathrm { f } ( x ) = \frac { 2 } { x - 1 } - \frac { 6 } { ( x - 1 ) ( 2 x + 1 ) } , x > 1\)
  1. Prove that \(\mathrm { f } ( x ) = \frac { 4 } { 2 x + 1 }\).
  2. Find the range of f.
  3. Find \(\mathrm { f } ^ { - 1 } ( x )\).
  4. Find the range of \(\mathrm { f } ^ { - 1 } ( x )\).
Edexcel C3 Q8
8. The function f is given by $$f : x \text { a } \ln ( 3 x - 6 ) , \quad x \in ^ { \circ } , \quad x > 2$$
  1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
  2. Write down the domain of \(\mathrm { f } ^ { - 1 }\) and the range of \(\mathrm { f } ^ { - 1 }\).
  3. Find, to 3 significant figures, the value of \(x\) for which \(\mathrm { f } ( x ) = 3\). The function g is given by $$g : x \text { a } \ln | 3 x - 6 | , \quad x \in ^ { \circ } , \quad x \neq 2$$
  4. Sketch the graph of \(y = \mathrm { g } ( x )\).
  5. Find the exact coordinates of all the points at which the graph of \(y = \mathrm { g } ( x )\) meets the coordinate axes.
Edexcel C3 Q1
  1. The function \(f\) is given by
$$\mathrm { f } : x \propto \frac { x } { x ^ { 2 } - 1 } - \frac { 1 } { x + 1 } , x > 1$$
  1. Show that \(\mathrm { f } ( x ) = \frac { 1 } { ( x - 1 ) ( x + 1 ) }\).
  2. Find the range of f. The function \(g\) is given by $$\mathrm { g } : x \propto \frac { 2 } { x } , x > 0$$
  3. Solve \(\operatorname { gf } ( x ) = 70\).
Edexcel C3 Q3
3. The function f is even and has domain \(\mathbb { R }\). For \(x \geq 0 , \mathrm { f } ( x ) = x ^ { 2 } - 4 a x\), where \(a\) is a positive constant.
  1. In the space below, sketch the curve with equation \(y = \mathrm { f } ( x )\), showing the coordinates of all the points at which the curve meets the axes.
  2. Find, in terms of \(a\), the value of \(\mathrm { f } ( 2 a )\) and the value of \(\mathrm { f } ( - 2 a )\). Given that \(a = 3\),
  3. use algebra to find the values of \(x\) for which \(\mathrm { f } ( x ) = 45\).
Edexcel C3 Q4
4. $$\mathrm { f } ( x ) = x ^ { 3 } + x ^ { 2 } - 4 x - 1$$ The equation \(\mathrm { f } ( x ) = 0\) has only one positive root, \(\alpha\).
  1. Show that \(\mathrm { f } ( x ) = 0\) can be rearranged as $$x = \sqrt { \left( \frac { 4 x + 1 } { x + 1 } \right) } , x \neq - 1$$ The iterative formula \(x _ { n + 1 } = \sqrt { \left( \frac { 4 x _ { n } + 1 } { x _ { n } + 1 } \right) }\) is used to find an approximation to \(\alpha\).
  2. Taking \(x _ { 1 } = 1\), find, to 2 decimal places, the values of \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\).
  3. By choosing values of \(x\) in a suitable interval, prove that \(\alpha = 1.70\), correct to 2 decimal places.
  4. Write down a value of \(x _ { 1 }\) for which the iteration formula \(x _ { n + 1 } = \sqrt { \left( \frac { 4 x _ { n } + 1 } { x _ { n } + 1 } \right) }\) does not produce a valid value for \(x _ { 2 }\). Justify your answer.
Edexcel C3 Q5
5. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \alpha \quad | x - a | + a , x \in \mathbb { R }
& \mathrm {~g} : x \alpha \quad 4 x + a , \quad x \in \mathbb { R } \end{aligned}$$ where \(a\) is a positive constant.
  1. On the same diagram, sketch the graphs of f and g , showing clearly the coordinates of any points at which your graphs meet the axes.
  2. Use algebra to find, in terms of \(a\), the coordinates of the point at which the graphs of f and g intersect.
  3. Find an expression for \(\mathrm { fg } ( x )\).
  4. Solve, for \(x\) in terms of \(a\), the equation $$\mathrm { fg } ( x ) = 3 a$$ \section*{6.}
Edexcel C3 Q6
6. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{79c1c4df-3812-4295-b01d-4724eda3457d-4_656_791_315_386}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = 10 + \ln ( 3 x ) - \frac { 1 } { 2 } \mathrm { e } ^ { x } , 0.1 \leq x \leq 3.3 .$$ Given that \(\mathrm { f } ( k ) = 0\),
  1. show, by calculation, that \(3.1 < k < 3.2\).
  2. Find \(\mathrm { f } ^ { \prime } ( x )\). The tangent to the graph at \(x = 1\) intersects the \(y\)-axis at the point \(P\).
    1. Find an equation of this tangent.
    2. Find the exact \(y\)-coordinate of \(P\), giving your answer in the form \(a + \ln b\).
Edexcel C3 Q7
7. (a) Express \(\sin x + \sqrt { 3 } \cos x\) in the form \(R \sin ( x + \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\).
(b) Show that the equation \(\sec x + \sqrt { 3 } \operatorname { cosec } x = 4\) can be written in the form $$\sin x + \sqrt { 3 } \cos x = 2 \sin 2 x$$ (c) Deduce from parts (a) and (b) that \(\sec x + \sqrt { 3 } \operatorname { cosec } x = 4\) can be written in the form $$\sin 2 x - \sin \left( x + 60 ^ { \circ } \right) = 0$$ END
Edexcel C3 Q1
\begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{909b52e5-2f16-4eab-b691-9d8fcf9bcfd9-2_679_1189_516_520}
\end{figure} Figure 2 shows part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = 0.5 \mathrm { e } ^ { x } - x ^ { 2 }$$ The curve \(C\) cuts the \(y\)-axis at \(A\) and there is a minimum at the point \(B\).
  1. Find an equation of the tangent to \(C\) at \(A\). The \(x\)-coordinate of \(B\) is approximately 2.15 . A more exact estimate is to be made of this coordinate using iterations \(x _ { n + 1 } = \ln \mathrm { g } \left( x _ { n } \right)\).
  2. Show that a possible form for \(\mathrm { g } ( x )\) is \(\mathrm { g } ( x ) = 4 x\).
  3. Using \(x _ { n + 1 } = \ln 4 x _ { n }\), with \(x _ { 0 } = 2.15\), calculate \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\). Give the value of \(x _ { 3 }\) to 4 decimal places.
Edexcel C3 Q3
3. (a) Sketch the graph of \(y = | 2 x + a | , a > 0\), showing the coordinates of the points where the graph meets the coordinate axes.
(b) On the same axes, sketch the graph of \(y = \frac { 1 } { x }\).
(c) Explain how your graphs show that there is only one solution of the equation $$x | 2 x + a | - 1 = 0$$ (d) Find, using algebra, the value of \(x\) for which \(x | 2 x + 1 | - 1 = 0\).