Questions — Edexcel C3 (377 questions)

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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.
Edexcel C3 Q1
  1. (a) Simplify
$$\frac { x ^ { 2 } + 7 x + 12 } { 2 x ^ { 2 } + 9 x + 4 }$$ (b) Solve the equation $$\ln \left( x ^ { 2 } + 7 x + 12 \right) - 1 = \ln \left( 2 x ^ { 2 } + 9 x + 4 \right)$$ giving your answer in terms of e.
Edexcel C3 Q2
2. A curve has the equation \(y = \sqrt { 3 x + 11 }\). The point \(P\) on the curve has \(x\)-coordinate 3 .
  1. Show that the tangent to the curve at \(P\) has the equation $$3 x - 4 \sqrt { 5 } y + 31 = 0$$ The normal to the curve at \(P\) crosses the \(y\)-axis at \(Q\).
  2. Find the \(y\)-coordinate of \(Q\) in the form \(k \sqrt { 5 }\).
Edexcel C3 Q3
3. (a) Use the identities for \(\sin ( A + B )\) and \(\sin ( A - B )\) to prove that $$\sin P + \sin Q \equiv 2 \sin \frac { P + Q } { 2 } \cos \frac { P - Q } { 2 } \text {. }$$ (b) Find, in terms of \(\pi\), the solutions of the equation $$\sin 5 x + \sin x = 0$$ for \(x\) in the interval \(0 \leq x < \pi\).
Edexcel C3 Q4
4. The curve with equation \(y = x ^ { \frac { 5 } { 2 } } \ln \frac { x } { 4 } , x > 0\) crosses the \(x\)-axis at the point \(P\).
  1. Write down the coordinates of \(P\). The normal to the curve at \(P\) crosses the \(y\)-axis at the point \(Q\).
  2. Find the area of triangle \(O P Q\) where \(O\) is the origin. The curve has a stationary point at \(R\).
  3. Find the \(x\)-coordinate of \(R\) in exact form.
Edexcel C3 Q5
5. $$\mathrm { f } ( x ) \equiv 2 x ^ { 2 } + 4 x + 2 , \quad x \in \mathbb { R } , \quad x \geq - 1 .$$
  1. Express \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\).
  2. Describe fully two transformations that would map the graph of \(y = x ^ { 2 } , x \geq 0\) onto the graph of \(y = \mathrm { f } ( x )\).
  3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.
  4. Sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) on the same diagram and state the relationship between them.
Edexcel C3 Q6
6. $$\mathrm { f } ( x ) = \mathrm { e } ^ { 3 x + 1 } - 2 , \quad x \in \mathbb { R } .$$
  1. State the range of f . The curve \(y = \mathrm { f } ( x )\) meets the \(y\)-axis at the point \(P\) and the \(x\)-axis at the point \(Q\).
  2. Find the exact coordinates of \(P\) and \(Q\).
  3. Show that the tangent to the curve at \(P\) has the equation $$y = 3 \mathrm { e } x + \mathrm { e } - 2 .$$
  4. Find to 3 significant figures the \(x\)-coordinate of the point where the tangent to the curve at \(P\) meets the tangent to the curve at \(Q\).
Edexcel C3 Q7
7. (a) Solve the equation $$\pi - 3 \arccos \theta = 0$$ (b) Sketch on the same diagram the curves \(y = \arccos ( x - 1 ) , 0 \leq x \leq 2\) and \(y = \sqrt { x + 2 } , x \geq - 2\). Given that \(\alpha\) is the root of the equation $$\arccos ( x - 1 ) = \sqrt { x + 2 }$$ (c) show that \(0 < \alpha < 1\),
(d) use the iterative formula $$x _ { n + 1 } = 1 + \cos \sqrt { x _ { n } + 2 }$$ with \(x _ { 0 } = 1\) to find \(\alpha\) correct to 3 decimal places. END
Edexcel C3 Q1
  1. (a) Express
$$\frac { x + 4 } { 2 x ^ { 2 } + 3 x + 1 } - \frac { 2 } { 2 x + 1 }$$ as a single fraction in its simplest form.
(b) Hence, find the values of \(x\) such that $$\frac { x + 4 } { 2 x ^ { 2 } + 3 x + 1 } - \frac { 2 } { 2 x + 1 } = \frac { 1 } { 2 } .$$