Questions — Edexcel (9685 questions)

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Edexcel C3 Q5
10 marks Standard +0.3
5. $$f ( x ) = 3 - \frac { x - 1 } { x - 3 } + \frac { x + 11 } { 2 x ^ { 2 } - 5 x - 3 } , \quad x \in \mathbb { R } , \quad x < - 1$$
  1. Show that $$f ( x ) = \frac { 4 x - 1 } { 2 x + 1 }$$
  2. Find an equation for the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where \(x = - 2\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Edexcel C3 Q6
11 marks Standard +0.3
6. A curve has the equation \(y = \mathrm { e } ^ { 3 x } \cos 2 x\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \mathrm { e } ^ { 3 x } ( 5 \cos 2 x - 12 \sin 2 x )\). The curve has a stationary point in the interval \([ 0,1 ]\).
  3. Find the \(x\)-coordinate of the stationary point to 3 significant figures.
  4. Determine whether the stationary point is a maximum or minimum point and justify your answer.
Edexcel C3 Q7
12 marks Standard +0.8
7. (a) Sketch on the same diagram the graphs of \(y = 4 a ^ { 2 } - x ^ { 2 }\) and \(y = | 2 x - a |\), where \(a\) is a positive constant. Show, in terms of \(a\), the coordinates of any points where each graph meets the coordinate axes.
(b) Find the exact solutions of the equation $$4 - x ^ { 2 } = | 2 x - 1 |$$
Edexcel C3 Q8
14 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d17a1b86-d758-4470-834a-b32a41f90c89-4_478_937_251_450} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with equation \(y = 2 x - 3 \ln ( 2 x + 5 )\) and the normal to the curve at the point \(P ( - 2 , - 4 )\).
  1. Find an equation for the normal to the curve at \(P\). The normal to the curve at \(P\) intersects the curve again at the point \(Q\) with \(x\)-coordinate \(q\).
  2. Show that \(1 < q < 2\).
  3. Show that \(q\) is a solution of the equation $$x = \frac { 12 } { 7 } \ln ( 2 x + 5 ) - 2 .$$
  4. Use the iterative formula $$x _ { n + 1 } = \frac { 12 } { 7 } \ln \left( 2 x _ { n } + 5 \right) - 2 ,$$ with \(x _ { 0 } = 1.5\), to find the value of \(q\) to 3 significant figures and justify the accuracy of your answer.
Edexcel C3 Q2
10 marks Standard +0.3
2. (a) Prove that, for \(\cos x \neq 0\), $$\sin 2 x - \tan x \equiv \tan x \cos 2 x .$$ (b) Hence, or otherwise, solve the equation $$\sin 2 x - \tan x = 2 \cos 2 x ,$$ for \(x\) in the interval \(0 \leq x \leq 180 ^ { \circ }\).
Edexcel C3 Q3
11 marks Standard +0.8
3. $$f ( x ) = x ^ { 2 } + 5 x - 2 \sec x , \quad x \in \mathbb { R } , \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 } .$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root in the interval [1,1.5]. A more accurate estimate of this root is to be found using iterations of the form $$x _ { n + 1 } = \arccos \mathrm { g } \left( x _ { n } \right) .$$
  2. Find a suitable form for \(\mathrm { g } ( x )\) and use this formula with \(x _ { 0 } = 1.25\) to find \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\). Give the value of \(x _ { 4 }\) to 3 decimal places. The curve \(y = \mathrm { f } ( x )\) has a stationary point at \(P\).
  3. Show that the \(x\)-coordinate of \(P\) is 1.0535 correct to 5 significant figures.
Edexcel C3 Q4
11 marks Moderate -0.3
4. (a) Differentiate each of the following with respect to \(x\) and simplify your answers.
  1. \(\sqrt { 1 - \cos x }\)
  2. \(x ^ { 3 } \ln x\) (b) Given that $$x = \frac { y + 1 } { 3 - 2 y } ,$$ find and simplify an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(y\).
Edexcel C3 Q5
12 marks Moderate -0.3
5. (a) Express \(\sqrt { 3 } \sin \theta + \cos \theta\) in the form \(R \sin ( \theta + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
(b) State the maximum value of \(\sqrt { 3 } \sin \theta + \cos \theta\) and the smallest positive value of \(\theta\) for which this maximum value occurs.
(c) Solve the equation $$\sqrt { 3 } \sin \theta + \cos \theta + \sqrt { 3 } = 0 ,$$ for \(\theta\) in the interval \(- \pi \leq \theta \leq \pi\), giving your answers in terms of \(\pi\).
Edexcel C3 Q6
13 marks Standard +0.3
6. The function f is defined by $$\mathrm { f } ( x ) \equiv 3 - x ^ { 2 } , \quad x \in \mathbb { R } , \quad x \geq 0 .$$
  1. State the range of f.
  2. Sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) on the same diagram.
  3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain. The function g is defined by $$\mathrm { g } ( x ) \equiv \frac { 8 } { 3 - x } , \quad x \in \mathbb { R } , \quad x \neq 3 .$$
  4. Evaluate \(\mathrm { fg } ( - 3 )\).
  5. Solve the equation $$\mathrm { f } ^ { - 1 } ( x ) = \mathrm { g } ( x ) .$$
Edexcel C3 Q7
13 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{795e472b-ad43-432a-a7cf-457b0f5e66f5-4_499_1107_242_415} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a graph of the temperature of a room, \(T ^ { \circ } \mathrm { C }\), at time \(t\) minutes.
The temperature is controlled by a thermostat such that when the temperature falls to \(12 ^ { \circ } \mathrm { C }\), a heater is turned on until the temperature reaches \(18 ^ { \circ } \mathrm { C }\). The room then cools until the temperature again falls to \(12 ^ { \circ } \mathrm { C }\). For \(t\) in the interval \(10 \leq t \leq 60\), \(T\) is given by $$T = 5 + A \mathrm { e } ^ { - k t } ,$$ where \(A\) and \(k\) are constants.
Given that \(T = 18\) when \(t = 10\) and that \(T = 12\) when \(t = 60\),
  1. show that \(k = 0.0124\) to 3 significant figures and find the value of \(A\),
  2. find the rate at which the temperature of the room is decreasing when \(t = 20\). The temperature again reaches \(18 ^ { \circ } \mathrm { C }\) when \(t = 70\) and the graph for \(70 \leq t \leq 120\) is a translation of the graph for \(10 \leq t \leq 60\).
  3. Find the value of the constant \(B\) such that for \(70 \leq t \leq 120\) $$T = 5 + B \mathrm { e } ^ { - k t } .$$
Edexcel C3 Q1
6 marks Standard +0.8
  1. Solve the equation
$$3 \operatorname { cosec } \theta ^ { \circ } + 8 \cos \theta ^ { \circ } = 0$$ for \(\theta\) in the interval \(0 \leq \theta \leq 180\), giving your answers to 1 decimal place.
Edexcel C3 Q2
7 marks Standard +0.3
2. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \rightarrow 1 - a x , \quad x \in \mathbb { R } , \\ & \mathrm {~g} : x \rightarrow x ^ { 2 } + 2 a x + 2 , \quad x \in \mathbb { R } , \end{aligned}$$ where \(a\) is a constant.
  1. Find the range of g in terms of \(a\). Given that \(\operatorname { gf } ( 3 ) = 7\),
  2. find the two possible values of \(a\).
Edexcel C3 Q3
8 marks Moderate -0.8
3. (a) Solve the equation $$\ln ( 3 x + 1 ) = 2$$ giving your answer in terms of e.
(b) Prove, by counter-example, that the statement $$\text { "ln } \left( 3 x ^ { 2 } + 5 x + 3 \right) \geq 0 \text { for all real values of } x \text { " }$$ is false.
Edexcel C3 Q4
8 marks Standard +0.3
4. A curve has the equation \(x = y \sqrt { 1 - 2 y }\).
  1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \sqrt { 1 - 2 y } } { 1 - 3 y } .$$ The point \(A\) on the curve has \(y\)-coordinate - 1 .
  2. Show that the equation of tangent to the curve at \(A\) can be written in the form $$\sqrt { 3 } x + p y + q = 0$$ where \(p\) and \(q\) are integers to be found.
Edexcel C3 Q5
10 marks Standard +0.3
5. (a) Sketch the graph of \(y = 2 + \sec \left( x - \frac { \pi } { 6 } \right)\) for \(x\) in the interval \(0 \leq x \leq 2 \pi\). Show on your sketch the coordinates of any turning points and the equations of any asymptotes.
(b) Find, in terms of \(\pi\), the \(x\)-coordinates of the points where the graph crosses the \(x\)-axis.
Edexcel C3 Q6
11 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1db60b49-1373-43d4-a74d-dfe8f9a952df-3_559_992_712_477} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve \(y = \mathrm { f } ( x )\) which has a minimum point at \(\left( - \frac { 3 } { 2 } , 0 \right)\), a maximum point at \(( 3,6 )\) and crosses the \(y\)-axis at \(( 0,4 )\). Sketch each of the following graphs on separate diagrams. In each case, show the coordinates of any turning points and of any points where the graph meets the coordinate axes.
  1. \(y = \mathrm { f } ( | x | )\)
  2. \(y = 2 + \mathrm { f } ( x + 3 )\)
  3. \(\quad y = \frac { 1 } { 2 } \mathrm { f } ( - x )\)
Edexcel C3 Q7
12 marks Standard +0.3
7. $$f ( x ) = 1 + \frac { 4 x } { 2 x - 5 } - \frac { 15 } { 2 x ^ { 2 } - 7 x + 5 } , \quad x \in \mathbb { R } , \quad x < 1$$
  1. Show that $$f ( x ) = \frac { 3 x + 2 } { x - 1 }$$
  2. Find an expression for the inverse function \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.
  3. Solve the equation \(\mathrm { f } ( x ) = 2\).
Edexcel C3 Q8
13 marks Standard +0.3
8. A curve has the equation \(y = x ^ { 2 } - \sqrt { 4 + \ln x }\).
  1. Show that the tangent to the curve at the point where \(x = 1\) has the equation $$7 x - 4 y = 11$$ The curve has a stationary point with \(x\)-coordinate \(\alpha\).
  2. Show that \(0.3 < \alpha < 0.4\)
  3. Show that \(\alpha\) is a solution of the equation $$x = \frac { 1 } { 2 } ( 4 + \ln x ) ^ { - \frac { 1 } { 4 } }$$
  4. Use the iteration formula $$x _ { n + 1 } = \frac { 1 } { 2 } \left( 4 + \ln x _ { n } \right) ^ { - \frac { 1 } { 4 } }$$ with \(x _ { 0 } = 0.35\), to find \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to 5 decimal places. END
Edexcel C3 Q1
7 marks Standard +0.3
  1. A curve has the equation \(y = ( 3 x - 5 ) ^ { 3 }\).
    1. Find an equation for the tangent to the curve at the point \(P ( 2,1 )\).
    The tangent to the curve at the point \(Q\) is parallel to the tangent at \(P\).
  2. Find the coordinates of \(Q\).
Edexcel C3 Q2
9 marks Standard +0.3
2. (a) Use the identities for \(\cos ( A + B )\) and \(\cos ( A - B )\) to prove that $$2 \cos A \cos B \equiv \cos ( A + B ) + \cos ( A - B ) .$$ (b) Hence, or otherwise, find in terms of \(\pi\) the solutions of the equation $$2 \cos \left( x + \frac { \pi } { 2 } \right) = \sec \left( x + \frac { \pi } { 6 } \right) ,$$ for \(x\) in the interval \(0 \leq x \leq \pi\).
Edexcel C3 Q3
10 marks Moderate -0.8
3. Differentiate each of the following with respect to \(x\) and simplify your answers.
  1. \(\quad \ln ( \cos x )\)
  2. \(x ^ { 2 } \sin 3 x\)
  3. \(\frac { 6 } { \sqrt { 2 x - 7 } }\)
Edexcel C3 Q4
10 marks Standard +0.3
4. (a) Express \(2 \sin x ^ { \circ } - 3 \cos x ^ { \circ }\) in the form \(R \sin ( x - \alpha ) ^ { \circ }\) where \(R > 0\) and \(0 < \alpha < 90\).
(b) Show that the equation $$\operatorname { cosec } x ^ { \circ } + 3 \cot x ^ { \circ } = 2$$ can be written in the form $$2 \sin x ^ { \circ } - 3 \cos x ^ { \circ } = 1 .$$ (c) Solve the equation $$\operatorname { cosec } x ^ { \circ } + 3 \cot x ^ { \circ } = 2 ,$$ for \(x\) in the interval \(0 \leq x \leq 360\), giving your answers to 1 decimal place.
Edexcel C3 Q5
12 marks Standard +0.3
5. (a) Show that \(( 2 x + 3 )\) is a factor of \(\left( 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 \right)\).
(b) Hence, simplify $$\frac { 2 x ^ { 2 } + x - 3 } { 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 } .$$ (c) Find the coordinates of the stationary points of the curve with equation $$y = \frac { 2 x ^ { 2 } + x - 3 } { 2 x ^ { 3 } - x ^ { 2 } + 4 x + 15 } .$$
Edexcel C3 Q6
13 marks Moderate -0.3
  1. The population in thousands, \(P\), of a town at time \(t\) years after \(1 ^ { \text {st } }\) January 1980 is modelled by the formula
$$P = 30 + 50 \mathrm { e } ^ { 0.002 t }$$ Use this model to estimate
  1. the population of the town on \(1 { } ^ { \text {st } }\) January 2010,
  2. the year in which the population first exceeds 84000 . The population in thousands, \(Q\), of another town is modelled by the formula $$Q = 26 + 50 \mathrm { e } ^ { 0.003 t }$$
  3. Show that the value of \(t\) when \(P = Q\) is a solution of the equation $$t = 1000 \ln \left( 1 + 0.08 \mathrm { e } ^ { - 0.002 t } \right) .$$
  4. Use the iteration formula $$t _ { n + 1 } = 1000 \ln \left( 1 + 0.08 \mathrm { e } ^ { - 0.002 t _ { n } } \right)$$ with \(t _ { 0 } = 50\) to find \(t _ { 1 } , t _ { 2 }\) and \(t _ { 3 }\) and hence, the year in which the populations of these two towns will be equal according to these models.
Edexcel C3 Q7
14 marks Standard +0.2
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a36989df-555f-4727-b6c6-e66362380011-4_481_808_248_424} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the graph of \(y = \mathrm { f } ( x )\) which meets the coordinate axes at the points \(( a , 0 )\) and \(( 0 , b )\), where \(a\) and \(b\) are constants.
  1. Showing, in terms of \(a\) and \(b\), the coordinates of any points of intersection with the axes, sketch on separate diagrams the graphs of
    1. \(\quad y = \mathrm { f } ^ { - 1 } ( x )\),
    2. \(y = 2 \mathrm { f } ( 3 x )\). Given that $$\mathrm { f } ( x ) = 2 - \sqrt { x + 9 } , \quad x \in \mathbb { R } , \quad x \geq - 9 ,$$
  2. find the values of \(a\) and \(b\),
  3. find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.