Questions — Edexcel (9685 questions)

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Edexcel C3 Q3
8 marks Challenging +1.2
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
10 marks Standard +0.3
  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
10 marks Standard +0.8
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
12 marks Standard +0.3
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
12 marks Standard +0.3
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
13 marks Standard +0.3
  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
7 marks Moderate -0.3
  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
9 marks Moderate -0.3
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
9 marks Standard +0.3
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
10 marks Standard +0.3
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
13 marks Moderate -0.3
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
13 marks Standard +0.3
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
14 marks Standard +0.3
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
6 marks Moderate -0.3
  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 $$\frac { x + 4 } { 2 x ^ { 2 } + 3 x + 1 } - \frac { 2 } { 2 x + 1 } = \frac { 1 } { 2 } .$$
Edexcel C3 Q2
7 marks Moderate -0.3
  1. (a) Prove, by counter-example, that the statement
$$\text { "cosec } \theta - \sin \theta > 0 \text { for all values of } \theta \text { in the interval } 0 < \theta < \pi \text { " }$$ is false.
(b) Find the values of \(\theta\) in the interval \(0 < \theta < \pi\) such that $$\operatorname { cosec } \theta - \sin \theta = 2$$ giving your answers to 2 decimal places.
Edexcel C3 Q3
8 marks Moderate -0.3
3. Solve each equation, giving your answers in exact form.
  1. \(\quad \ln ( 2 x - 3 ) = 1\)
  2. \(3 \mathrm { e } ^ { y } + 5 \mathrm { e } ^ { - y } = 16\)
Edexcel C3 Q4
8 marks Moderate -0.8
4. Differentiate each of the following with respect to \(x\) and simplify your answers.
  1. \(\quad \ln ( 3 x - 2 )\)
  2. \(\frac { 2 x + 1 } { 1 - x }\)
  3. \(x ^ { \frac { 3 } { 2 } } \mathrm { e } ^ { 2 x }\)
Edexcel C3 Q5
9 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{de511dda-d00f-4881-94c3-9ee643d10f3f-3_529_806_248_408} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve \(y = \mathrm { f } ( x )\) which has a maximum point at ( \(- 3,2\) ) and a minimum point at \(( 2 , - 4 )\).
  1. Showing the coordinates of any stationary points, sketch on separate diagrams the graphs of
    1. \(y = \mathrm { f } ( | x | )\),
    2. \(y = 3 \mathrm { f } ( 2 x )\).
  2. Write down the values of the constants \(a\) and \(b\) such that the curve with equation \(y = a + \mathrm { f } ( x + b )\) has a minimum point at the origin \(O\).
Edexcel C3 Q6
10 marks Standard +0.0
6. The function f is defined by $$\mathrm { f } ( x ) \equiv 4 - \ln 3 x , \quad x \in \mathbb { R } , \quad x > 0 .$$
  1. Solve the equation \(\mathrm { f } ( x ) = 0\).
  2. Sketch the curve \(y = \mathrm { f } ( x )\).
  3. Find an expression for the inverse function, \(\mathrm { f } ^ { - 1 } ( x )\). The function \(g\) is defined by $$\mathrm { g } ( x ) \equiv \mathrm { e } ^ { 2 - x } , \quad x \in \mathbb { R }$$
  4. Show that $$\operatorname { fg } ( x ) = x + a - \ln b$$ where \(a\) and \(b\) are integers to be found.
Edexcel C3 Q7
13 marks Standard +0.3
7. (a) Express \(4 \sin x + 3 \cos x\) in the form \(R \sin ( x + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
(b) State the minimum value of \(4 \sin x + 3 \cos x\) and the smallest positive value of \(x\) for which this minimum value occurs.
(c) Solve the equation $$4 \sin 2 \theta + 3 \cos 2 \theta = 2$$ for \(\theta\) in the interval \(0 \leq \theta \leq \pi\), giving your answers to 2 decimal places.
Edexcel C3 Q8
14 marks Standard +0.8
8. The curve \(C\) has the equation \(y = \sqrt { x } + \mathrm { e } ^ { 1 - 4 x } , x \geq 0\).
  1. Find an equation for the normal to the curve at the point \(\left( \frac { 1 } { 4 } , \frac { 3 } { 2 } \right)\). The curve \(C\) has a stationary point with \(x\)-coordinate \(\alpha\) where \(0.5 < \alpha < 1\).
  2. Show that \(\alpha\) is a solution of the equation $$x = \frac { 1 } { 4 } [ 1 + \ln ( 8 \sqrt { x } ) ]$$
  3. Use the iteration formula $$x _ { n + 1 } = \frac { 1 } { 4 } \left[ 1 + \ln \left( 8 \sqrt { x _ { n } } \right) \right]$$ with \(x _ { 0 } = 1\) to find \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving the value of \(x _ { 4 }\) to 3 decimal places.
  4. Show that your value for \(x _ { 4 }\) is the value of \(\alpha\) correct to 3 decimal places.
  5. Another attempt to find \(\alpha\) is made using the iteration formula $$x _ { n + 1 } = \frac { 1 } { 64 } \mathrm { e } ^ { 8 x _ { n } - 2 }$$ with \(x _ { 0 } = 1\). Describe the outcome of this attempt.
Edexcel C3 Q1
5 marks Standard +0.0
  1. The function f is defined by
$$\mathrm { f } ( x ) \equiv 2 + \ln ( 3 x - 2 ) , \quad x \in \mathbb { R } , \quad x > \frac { 2 } { 3 } .$$
  1. Find the exact value of \(\mathrm { ff } ( 1 )\).
  2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
Edexcel C3 Q2
6 marks Standard +0.3
2. Find, to 2 decimal places, the solutions of the equation $$3 \cot ^ { 2 } x - 4 \operatorname { cosec } x + \operatorname { cosec } ^ { 2 } x = 0$$ in the interval \(0 \leq x \leq 2 \pi\).
Edexcel C3 Q3
8 marks Standard +0.3
3. (a) Given that \(y = \ln x\), find expressions in terms of \(y\) for
  1. \(\quad \log _ { 2 } x\),
  2. \(\ln \frac { x ^ { 2 } } { \mathrm { e } }\).
    (b) Hence, or otherwise, solve the equation $$\log _ { 2 } x = 4 - \ln \frac { x ^ { 2 } } { \mathrm { e } } ,$$ giving your answer to 2 decimal places.
Edexcel C3 Q4
9 marks Challenging +1.2
4. (a) Use the identities for ( \(\sin A + \sin B\) ) and ( \(\cos A + \cos B\) ) to prove that $$\frac { \sin 2 x + \sin 2 y } { \cos 2 x + \cos 2 y } \equiv \tan ( x + y ) .$$ (b) Hence, show that $$\tan 52.5 ^ { \circ } = \sqrt { 6 } - \sqrt { 3 } - \sqrt { 2 } + 2 .$$