Questions C3 (1200 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 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics 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 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics 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
Edexcel C3 Q4
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{909b52e5-2f16-4eab-b691-9d8fcf9bcfd9-3_604_1408_868_269}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x ) , - 1 \leq x \leq 3\). The curve touches the \(x\)-axis at the origin \(O\), crosses the \(x\)-axis at the point \(A ( 2,0 )\) and has a maximum at the point \(B \left( \frac { 4 } { 3 } , 1 \right)\). In separate diagrams, show a sketch of the curve with equation
  1. \(y = \mathrm { f } ( x + 1 )\),
  2. \(y = | \mathrm { f } ( x ) |\),
  3. \(y = \mathrm { f } ( | x | )\),
    marking on each sketch the coordinates of points at which the curve
    1. has a turning point,
    2. meets the \(x\)-axis.
Edexcel C3 Q5
5. (i) Given that \(\sin x = \frac { 3 } { 5 }\), use an appropriate double angle formula to find the exact value of \(\sec 2 x\).
(ii) Prove that $$\cot 2 x + \operatorname { cosec } 2 x \equiv \cot x , \quad \left( x \neq \frac { n \pi } { 2 } , n \in Z \right)$$
Edexcel C3 Q6
  1. The function f is defined by \(\mathrm { f } : x \rightarrow \frac { 3 x - 1 } { x - 3 } , x \in j , x \neq 3\).
    1. Prove that \(\mathrm { f } ^ { - 1 } ( x ) = \mathrm { f } ( x )\) for all \(x \in j , x \neq 3\).
    2. Hence find, in terms of \(k , \operatorname { ff } ( k )\), where \(x \neq 3\).
    \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{909b52e5-2f16-4eab-b691-9d8fcf9bcfd9-5_864_1205_605_242}
    \end{figure} Figure 3 shows a sketch of the one-one function g , defined over the domain \(- 2 \leq x \leq 2\).
  2. Find the value of \(\mathrm { fg } ( - 2 )\).
  3. Sketch the graph of the inverse function \(\mathrm { g } ^ { - 1 }\) and state its domain. The function h is defined by \(\mathrm { h } : x \mapsto 2 \mathrm {~g} ( x - 1 )\).
  4. Sketch the graph of the function h and state its range.
Edexcel C3 Q7
7. (i) (a) Express \(( 12 \cos \theta - 5 \sin \theta )\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\).
(b) Hence solve the equation $$12 \cos \theta - 5 \sin \theta = 4$$ for \(0 < \theta < 90 ^ { \circ }\), giving your answer to 1 decimal place.
(ii) Solve $$8 \cot \theta - 3 \tan \theta = 2$$ for \(0 < \theta < 90 ^ { \circ }\), giving your answer to 1 decimal place.
Edexcel C3 Q8
8. The curve \(C\) has equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = 3 \ln x + \frac { 1 } { x } , \quad x > 0$$ The point \(P\) is a stationary point on \(C\).
  1. Calculate the \(x\)-coordinate of \(P\).
  2. Show that the \(y\)-coordinate of \(P\) may be expressed in the form \(k - k \ln k\), where \(k\) is a constant to be found. The point \(Q\) on \(C\) has \(x\)-coordinate 1 .
  3. Find an equation for the normal to \(C\) at \(Q\). The normal to \(C\) at \(Q\) meets \(C\) again at the point \(R\).
  4. Show that the \(x\)-coordinate of \(R\)
    1. satisfies the equation \(6 \ln x + x + \frac { 2 } { x } - 3 = 0\),
    2. lies between 0.13 and 0.14 .
Edexcel C3 Q1
  1. The curve \(C\) has equation \(y = 2 \mathrm { e } ^ { x } + 3 x ^ { 2 } + 2\). The point \(A\) with coordinates \(( 0,4 )\) lies on \(C\). Find the equation of the tangent to \(C\) at \(A\).
  2. Express \(\frac { x } { ( x + 1 ) ( x + 3 ) } + \frac { x + 12 } { x ^ { 2 } - 9 }\) as a single fraction in its simplest form.
  3. The functions f and g are defined by
$$\begin{aligned} & \mathrm { f } : x \propto x ^ { 2 } - 2 x + 3 , x \in \mathbb { R } , 0 \leq x \leq 4
& \mathrm {~g} : x \propto \lambda x ^ { 2 } + 1 , \text { where } \lambda \text { is a constant, } x \in \mathbb { R } . \end{aligned}$$
  1. Find the range of f.
  2. Given that \(\operatorname { gf } ( 2 ) = 16\), find the value of \(\lambda\).
Edexcel C3 Q4
4. (a) Sketch, on the same set of axes, the graphs of $$y = 2 - \mathrm { e } ^ { - x } \text { and } y = \sqrt { } x$$ [It is not necessary to find the coordinates of any points of intersection with the axes.]
Given that \(\mathrm { f } ( x ) = \mathrm { e } ^ { - x } + \sqrt { } x - 2 , x \geq 0\),
(b) explain how your graphs show that the equation \(\mathrm { f } ( x ) = 0\) has only one solution,
(c) show that the solution of \(\mathrm { f } ( x ) = 0\) lies between \(x = 3\) and \(x = 4\). The iterative formula \(x _ { n + 1 } = \left( 2 - \mathrm { e } ^ { - x _ { n } } \right) ^ { 2 }\) is used to solve the equation \(\mathrm { f } ( x ) = 0\).
(d) Taking \(x _ { 0 } = 4\), write down the values of \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), and hence find an approximation to the solution of \(\mathrm { f } ( x ) = 0\), giving your answer to 3 decimal places.
Edexcel C3 Q5
5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{bdb2b50d-0816-4ef2-adfa-aee3afe18582-3_515_739_228_534}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { e } ^ { - x } - 1\).
  1. Copy Fig. 1 and on the same axes sketch the graph of \(y = \frac { 1 } { 2 } | x - 1 |\). Show the coordinates of the points where the graph meets the axes. The \(x\)-coordinate of the point of intersection of the graph is \(\alpha\).
  2. Show that \(x = \alpha\) is a root of the equation \(x + 2 \mathrm { e } ^ { - x } - 3 = 0\).
  3. Show that \(- 1 < \alpha < 0\). The iterative formula \(x _ { \mathrm { n } + 1 } = - \ln \left[ \frac { 1 } { 2 } \left( 3 - x _ { n } \right) \right]\) is used to solve the equation \(x + 2 \mathrm { e } ^ { - x } - 3 = 0\).
  4. Starting with \(x _ { 0 } = - 1\), find the values of \(x _ { 1 }\) and \(x _ { 2 }\).
  5. Show that, to 2 decimal places, \(\alpha = - 0.58\).
Edexcel C3 Q6
6. $$\mathrm { f } ( x ) = x ^ { 2 } - 2 x - 3 , x \in \mathbb { R } , x \geq 1$$
  1. Find the range of f .
  2. Write down the domain and range of \(\mathrm { f } ^ { - 1 }\).
  3. Sketch the graph of \(\mathrm { f } ^ { - 1 }\), indicating clearly the coordinates of any point at which the graph intersects the coordinate axes. Given that \(\mathrm { g } ( x ) = | x - 4 | , x \in \mathbb { R }\),
  4. find an expression for \(\operatorname { gf } ( x )\).
  5. Solve \(\operatorname { gf } ( x ) = 8\).
Edexcel C3 Q7
7. \(\mathrm { f } ( x ) = x + \frac { \mathrm { e } ^ { x } } { 5 } , \quad x \in \mathbb { R }\).
  1. Find \(\mathrm { f } ^ { \prime } ( x )\). The curve \(C\), with equation \(y = \mathrm { f } ( x )\), crosses the \(y\)-axis at the point \(A\).
  2. Find an equation for the tangent to \(C\) at \(A\).
  3. Complete the table, giving the values of \(\sqrt { \left( x + \frac { \mathrm { e } ^ { x } } { 5 } \right) }\) to 2 decimal places.
    \(x\)00.511.52
    \(\sqrt { \left( x + \frac { \mathrm { e } ^ { x } } { 5 } \right) }\)0.450.91
Edexcel C3 Q8
  1. (a) Express \(2 \cos \theta + 5 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
Give the values of \(R\) and \(\alpha\) to 3 significant figures.
(b)Find the maximum and minimum values of \(2 \cos \theta + 5 \sin \theta\) and the smallest possible value of \(\theta\) for which the maximum occurs. The temperature \(T ^ { \circ } \mathrm { C }\), of an unheated building is modelled using the equation $$T = 15 + 2 \cos \left( \frac { \pi t } { 12 } \right) + 5 \sin \left( \frac { \pi t } { 12 } \right) , \quad 0 \leq t < 24$$ where \(t\) hours is the number of hours after 1200 .
(c) Calculate the maximum temperature predicted by this model and the value of \(t\) when this maximum occurs.
(d) Calculate, to the nearest half hour, the times when the temperature is predicted to be \(12 ^ { \circ } \mathrm { C }\).
Edexcel C3 Q1
  1. Use the derivatives of \(\sin x\) and \(\cos x\) to prove that the derivative of \(\tan x\) is \(\sec ^ { 2 } x\).
  2. The function f is given by \(\mathrm { f } : x \propto 2 + \frac { 3 } { x + 2 } , x \in \mathbb { R } , x \neq - 2\).
    1. Express \(2 + \frac { 3 } { x + 2 }\) as a single fraction.
    2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
    3. Write down the domain of \(\mathrm { f } ^ { - 1 }\).
    4. (a) Express as a fraction in its simplest form
    $$\frac { 2 } { x - 3 } + \frac { 13 } { x ^ { 2 } + 4 x - 21 }$$
  3. Hence solve $$\frac { 2 } { x - 3 } + \frac { 13 } { x ^ { 2 } + 4 x - 21 } = 1$$
Edexcel C3 Q4
  1. (a) Simplify \(\frac { x ^ { 2 } + 4 x + 3 } { x ^ { 2 } + x }\).
    (b) Find the value of \(x\) for which \(\log _ { 2 } \left( x ^ { 2 } + 4 x + 3 \right) - \log _ { 2 } \left( x ^ { 2 } + x \right) = 4\).
  2. (i) Prove, by counter-example, that the statement
$$\text { " } \sec ( A + B ) \equiv \sec A + \sec B , \text { for all } A \text { and } B \text { " }$$ is false
(ii) Prove that $$\tan \theta + \cot \theta \equiv 2 \operatorname { cosec } 2 \theta , \quad \theta \neq \frac { n \pi } { 2 } , n \in \mathbb { Z }$$
Edexcel C3 Q6
  1. (a) Prove that
$$\frac { 1 - \cos 2 \theta } { \sin 2 \theta } \equiv \tan \theta , \quad \theta \neq \frac { n \pi } { 2 } , \quad n \in \mathbb { Z }$$ (b) Solve, giving exact answers in terms of \(\pi\), $$2 ( 1 - \cos 2 \theta ) = \tan \theta , \quad 0 < \theta < \pi$$
Edexcel C3 Q7
  1. Given that \(y = \log _ { a } x , x > 0\), where \(a\) is a positive constant,
    1. (i) express \(x\) in terms of \(a\) and \(y\),
      (ii) deduce that \(\ln x = y \ln a\).
    2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x \ln a }\).
    The curve \(C\) has equation \(y = \log _ { 10 } x , x > 0\). The point \(A\) on \(C\) has \(x\)-coordinate 10 . Using the result in part (b),
  2. find an equation for the tangent to \(C\) at \(A\). The tangent to \(C\) at \(A\) crosses the \(x\)-axis at the point \(B\).
  3. Find the exact \(x\)-coordinate of \(B\).
Edexcel C3 Q8
8. The curve with equation \(y = \ln 3 x\) crosses the \(x\)-axis at the point \(P ( p , 0 )\).
  1. Sketch the graph of \(y = \ln 3 x\), showing the exact value of \(p\). The normal to the curve at the point \(Q\), with \(x\)-coordinate \(q\), passes through the origin.
  2. Show that \(x = q\) is a solution of the equation \(x ^ { 2 } + \ln 3 x = 0\).
  3. Show that the equation in part (b) can be rearranged in the form \(x = \frac { 1 } { 3 } \mathrm { e } ^ { - x ^ { 2 } }\).
  4. Use the iteration formula \(x _ { n + 1 } = \frac { 1 } { 3 } \mathrm { e } ^ { - x _ { n } ^ { 2 } }\), with \(x _ { 0 } = \frac { 1 } { 3 }\), to find \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\). Hence write down, to 3 decimal places, an approximation for \(q\).
Edexcel C3 Q9
9. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{438fda08-a7c2-409b-afde-17f6f85b5183-5_558_1115_251_306}
\end{figure} Figure 3 shows a sketch of the curve with equation \(y = \mathrm { f } ( x ) , x \geq 0\). The curve meets the coordinate axes at the points \(( 0 , c )\) and \(( d , 0 )\). In separate diagrams sketch the curve with equation
  1. \(y = \mathrm { f } ^ { - 1 } ( x )\),
  2. \(y = 3 \mathrm { f } ( 2 x )\).
    (3) Indicate clearly on each sketch the coordinates, in terms of \(c\) or \(d\), of any point where the curve meets the coordinate axes. Given that f is defined by $$\mathrm { f } : x \mapsto 3 \left( 2 ^ { - x } \right) - 1 , x \in \mathbb { R } , x \geq 0 ,$$
  3. state
    1. the value of \(c\),
    2. the range of \(f\).
  4. Find the value of \(d\), giving your answer to 3 decimal places. The function g is defined by $$\mathrm { g } : x \mapsto \log _ { 2 } x , x \in \mathbb { R } , x \geq 1 .$$
  5. Find \(\mathrm { fg } ( x )\), giving your answer in its simplest form.
Edexcel C3 Q1
  1. Given that
$$x = \sec ^ { 2 } y + \tan y ,$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \cos ^ { 2 } y } { 2 \tan y + 1 } .$$
Edexcel C3 Q2
  1. The functions \(f\) and \(g\) are defined by
$$\begin{aligned} & \mathrm { f } : x \rightarrow 3 x - 4 , \quad x \in \mathbb { R } ,
& \mathrm {~g} : x \rightarrow \frac { 2 } { x + 3 } , \quad x \in \mathbb { R } , \quad x \neq - 3 \end{aligned}$$
  1. Evaluate fg(1).
  2. Solve the equation \(\operatorname { gf } ( x ) = 6\).
Edexcel C3 Q3
3. Giving your answers to 2 decimal places, solve the simultaneous equations $$\begin{aligned} & \mathrm { e } ^ { 2 y } - x + 2 = 0
& \ln ( x + 3 ) - 2 y - 1 = 0 \end{aligned}$$
Edexcel C3 Q4
  1. (a) Use the derivatives of \(\sin x\) and \(\cos x\) to prove that
$$\frac { \mathrm { d } } { \mathrm {~d} x } ( \tan x ) = \sec ^ { 2 } x$$ The tangent to the curve \(y = 2 x \tan x\) at the point where \(x = \frac { \pi } { 4 }\) meets the \(y\)-axis at the point \(P\).
(b) Find the \(y\)-coordinate of \(P\) in the form \(k \pi ^ { 2 }\) where \(k\) is a rational constant.
Edexcel C3 Q5
5. (a) Express \(3 \cos x ^ { \circ } + \sin x ^ { \circ }\) in the form \(R \cos ( x - \alpha ) ^ { \circ }\) where \(R > 0\) and \(0 < \alpha < 90\).
(b) Using your answer to part (a), or otherwise, solve the equation $$6 \cos ^ { 2 } x ^ { \circ } + \sin 2 x ^ { \circ } = 0$$ for \(x\) in the interval \(0 \leq x \leq 360\), giving your answers to 1 decimal place where appropriate.
Edexcel C3 Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3db6c0d8-2c8a-47a2-8c98-13fa191320d0-3_727_1006_244_356} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with equation \(y = \mathrm { f } ( x )\). The curve crosses the axes at \(( p , 0 )\) and \(( 0 , q )\) and the lines \(x = 1\) and \(y = 2\) are asymptotes of the curve.
  1. Showing the coordinates of any points of intersection with the axes and the equations of any asymptotes, sketch on separate diagrams the graphs of
    1. \(y = | \mathrm { f } ( x ) |\),
    2. \(y = 2 \mathrm { f } ( x + 1 )\). Given also that $$\mathrm { f } ( x ) \equiv \frac { 2 x - 1 } { x - 1 } , \quad x \in \mathbb { R } , \quad x \neq 1$$
  2. find the values of \(p\) and \(q\),
  3. find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
Edexcel C3 Q7
7. (a) (i) Show that $$\sin ( x + 30 ) ^ { \circ } + \sin ( x - 30 ) ^ { \circ } \equiv a \sin x ^ { \circ }$$ where \(a\) is a constant to be found.
(ii) Hence find the exact value of \(\sin 75 ^ { \circ } + \sin 15 ^ { \circ }\), giving your answer in the form \(b \sqrt { 6 }\).
(b) Solve, for \(0 \leq y \leq 360\), the equation $$2 \cot ^ { 2 } y ^ { \circ } + 5 \operatorname { cosec } y ^ { \circ } + \operatorname { cosec } ^ { 2 } y ^ { \circ } = 0$$
Edexcel C3 Q8
  1. \(f ( x ) = \frac { x ^ { 4 } + x ^ { 3 } - 5 x ^ { 2 } - 9 } { x ^ { 2 } + x - 6 }\).
    1. Using algebraic division, show that
    $$f ( x ) = x ^ { 2 } + A + \frac { B } { x + C }$$ where \(A , B\) and \(C\) are integers to be found.
  2. By sketching two suitable graphs on the same set of axes, show that the equation \(\mathrm { f } ( x ) = 0\) has exactly one real root.
  3. Use the iterative formula $$x _ { n + 1 } = 2 + \frac { 1 } { x _ { n } ^ { 2 } + 1 } ,$$ with a suitable starting value to find the root of the equation \(\mathrm { f } ( x ) = 0\) correct to 3 significant figures and justify the accuracy of your answer.