Questions — Edexcel (9685 questions)

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Edexcel C2 Q2
4 marks Easy -1.2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{05006f1f-ebf0-4d70-9dbb-68221c09043e-2_510_842_534_513} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with equation \(y = \sqrt { 4 x - 1 }\). Use the trapezium rule with five equally-spaced ordinates to estimate the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 3\).
Edexcel C2 Q3
7 marks Moderate -0.3
3. (a) Given that \(y = \log _ { 2 } x\), find expressions in terms of \(y\) for
  1. \(\quad \log _ { 2 } \left( \frac { x } { 2 } \right)\),
  2. \(\log _ { 2 } ( \sqrt { x } )\).
    (b) Hence, or otherwise, solve the equation $$2 \log _ { 2 } \left( \frac { x } { 2 } \right) + \log _ { 2 } ( \sqrt { x } ) = 8$$
Edexcel C2 Q4
9 marks Moderate -0.8
4. $$f ( x ) = 2 - x - x ^ { 3 }$$
  1. Show that \(\mathrm { f } ( x )\) is decreasing for all values of \(x\).
  2. Verify that the point \(( 1,0 )\) lies on the curve \(y = \mathrm { f } ( x )\).
  3. Find the area of the region bounded by the curve \(y = \mathrm { f } ( x )\) and the coordinate axes.
Edexcel C2 Q5
9 marks Standard +0.3
5. Figure 2 Figure 2 shows triangle \(P Q R\) in which \(P Q = 7\) and \(P R = 3 \sqrt { 5 }\).
Given that \(\sin ( \angle Q P R ) = \frac { 2 } { 3 }\) and that \(\angle Q P R\) is acute,
  1. find the exact value of \(\cos ( \angle Q P R )\) in its simplest form,
  2. show that \(Q R = 2 \sqrt { 6 }\),
  3. find \(\angle P Q R\) in degrees to 1 decimal place.
Edexcel C2 Q6
10 marks Moderate -0.3
6. The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 2 x ^ { 3 } + x ^ { 2 } + a x + b ,$$ where \(a\) and \(b\) are constants.
Given that when \(\mathrm { p } ( x )\) is divided by \(( x + 2 )\) there is a remainder of 20 ,
  1. find an expression for \(b\) in terms of \(a\). Given also that \(( x + 3 )\) is a factor of \(\mathrm { p } ( x )\),
  2. find the values of \(a\) and \(b\),
  3. fully factorise \(\mathrm { p } ( x )\).
Edexcel C2 Q7
10 marks Standard +0.3
7. (a) Find, to 2 decimal places, the values of \(x\) in the interval \(0 \leq x < 2 \pi\) for which $$\tan \left( x + \frac { \pi } { 4 } \right) = 3 .$$ (b) Find, in terms of \(\pi\), the values of \(y\) in the interval \(0 \leq y < 2 \pi\) for which $$2 \sin y = \tan y .$$
Edexcel C2 Q8
11 marks Moderate -0.3
  1. The point \(A\) has coordinates ( 4,6 ).
Given that \(O A\), where \(O\) is the origin, is a diameter of circle \(C\),
  1. find an equation for \(C\). Circle \(C\) crosses the \(x\)-axis at \(O\) and at the point \(B\).
  2. Find the coordinates of \(B\).
  3. Find an equation for the tangent to \(C\) at \(B\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
Edexcel C2 Q9
11 marks Standard +0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{05006f1f-ebf0-4d70-9dbb-68221c09043e-4_325_662_1345_520} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows part of a design being produced by a computer program.
The program draws a series of circles with each one touching the previous one and such that their centres lie on a horizontal straight line. The radii of the circles form a geometric sequence with first term 1 mm and second term 1.5 mm . The width of the design is \(w\) as shown.
  1. Find the radius of the fourth circle to be drawn.
  2. Show that when eight circles have been drawn, \(w = 98.5 \mathrm {~mm}\) to 3 significant figures.
  3. Find the total area of the design in square centimetres when ten circles have been drawn.
Edexcel C2 Q1
4 marks Easy -1.2
  1. A geometric series has first term 75 and second term - 15 .
    1. Find the common ratio of the series.
    2. Find the sum to infinity of the series.
    3. A circle has the equation
    $$x ^ { 2 } + y ^ { 2 } + 8 x - 4 y + k = 0 ,$$ where \(k\) is a constant.
  2. Find the coordinates of the centre of the circle. Given that the \(x\)-axis is a tangent to the circle,
  3. find the value of \(k\).
Edexcel C2 Q3
6 marks Standard +0.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a7285542-c32a-45b1-921b-d528676ad6b5-2_586_513_1219_593} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a circle of radius \(r\) and centre \(O\) in which \(A D\) is a diameter.
The points \(B\) and \(C\) lie on the circle such that \(O B\) and \(O C\) are arcs of circles of radius \(r\) with centres \(A\) and \(D\) respectively. Show that the area of the shaded region \(O B C\) is \(\frac { 1 } { 6 } r ^ { 2 } ( 3 \sqrt { 3 } - \pi )\).
Edexcel C2 Q4
6 marks Standard +0.8
4. (a) Sketch on the same diagram the graphs of \(y = \sin 2 x\) and \(y = \tan \frac { x } { 2 }\) for \(x\) in the interval \(0 \leq x \leq 360 ^ { \circ }\).
(b) Hence state how many solutions exist to the equation $$\sin 2 x = \tan \frac { x } { 2 }$$ for \(x\) in the interval \(0 \leq x \leq 360 ^ { \circ }\) and give a reason for your answer.
Edexcel C2 Q5
7 marks Moderate -0.3
5. (a) Find the value of \(a\) such that $$\log _ { a } 27 = 3 + \log _ { a } 8$$ (b) Solve the equation $$2 ^ { x + 3 } = 6 ^ { x - 1 }$$ giving your answer to 3 significant figures.
Edexcel C2 Q6
9 marks Moderate -0.8
6. (a) Expand \(( 2 + x ) ^ { 4 }\) in ascending powers of \(x\), simplifying each coefficient.
(b) Find the integers \(A , B\) and \(C\) such that $$( 2 + x ) ^ { 4 } + ( 2 - x ) ^ { 4 } \equiv A + B x ^ { 2 } + C x ^ { 4 }$$ (c) Find the real values of \(x\) for which $$( 2 + x ) ^ { 4 } + ( 2 - x ) ^ { 4 } = 136$$
Edexcel C2 Q7
11 marks Standard +0.3
7. $$f ( x ) = 2 x ^ { 3 } - 5 x ^ { 2 } + x + 2$$
  1. Show that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Fully factorise \(\mathrm { f } ( x )\).
  3. Solve the equation \(\mathrm { f } ( x ) = 0\).
  4. Find the values of \(\theta\) in the interval \(0 \leq \theta \leq 2 \pi\) for which $$2 \sin ^ { 3 } \theta - 5 \sin ^ { 2 } \theta + \sin \theta + 2 = 0$$ giving your answers in terms of \(\pi\).
Edexcel C2 Q8
13 marks Moderate -0.3
8. The curve \(C\) has the equation $$y = 3 - x ^ { \frac { 1 } { 2 } } - 2 x ^ { - \frac { 1 } { 2 } } , \quad x > 0 .$$
  1. Find the coordinates of the points where \(C\) crosses the \(x\)-axis.
  2. Find the exact coordinates of the stationary point of \(C\).
  3. Determine the nature of the stationary point.
  4. Sketch the curve \(C\).
Edexcel C2 Q9
14 marks Standard +0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a7285542-c32a-45b1-921b-d528676ad6b5-4_620_872_895_504} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the curve \(C\) with equation \(y = 3 x - 4 \sqrt { x } + 2\) and the tangent to \(C\) at the point \(A\). Given that \(A\) has \(x\)-coordinate 4,
  1. show that the tangent to \(C\) at \(A\) has the equation \(y = 2 x - 2\). The shaded region is bounded by \(C\), the tangent to \(C\) at \(A\) and the positive coordinate axes.
  2. Find the area of the shaded region.
Edexcel C3 Q1
4 marks Moderate -0.8
  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
6 marks Standard +0.3
  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
8 marks Standard +0.3
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
6 marks Moderate -0.3
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
7 marks Moderate -0.8
  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
10 marks Moderate -0.3
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
13 marks Moderate -0.3
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
9 marks Standard +0.3
  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
9 marks Moderate -0.3
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 \(\mathrm { f } ( x ) = 45\).