Questions — Edexcel (10514 questions)

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Edexcel PMT Mocks Q2
5 marks Moderate -0.8
2. Relative to a fixed origin \(O\),
the point \(A\) has position vector \(( 3 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } )\) the point \(B\) has position vector ( \(\mathbf { i } + 2 \mathbf { j } - 4 \mathbf { k }\) )
and the point \(C\) has position vector \(( - \mathbf { i } + \mathbf { j } + a \mathbf { k } )\), where \(a\) is a constant and \(a > 0\).
Given that \(| \overrightarrow { B C } | = \sqrt { 41 }\) a. show that \(a = 2\). \(D\) is the point such that \(A B C D\) forms a parallelogram.
b. Find the position vector of \(D\).
Edexcel PMT Mocks Q3
5 marks Moderate -0.3
3. a. "If \(p\) and \(q\) are irrational numbers, where \(p \neq q , q \neq 0\), then \(\frac { p } { q }\) is also irrational." Disprove this statement by means of a counter example.
b. (i) Sketch the graph of \(y = | x | - 2\).
(ii) Explain why \(| x - 2 | \geq | x | - 2\) for all real values of \(x\).
Edexcel PMT Mocks Q4
7 marks Standard +0.3
4.
  1. Show that \(\sum _ { r = 1 } ^ { 20 } \left( 2 ^ { r - 1 } - 3 - 4 r \right) = 1047675\)
  2. A sequence has \(n\)th term \(u _ { n } = \sin \left( 90 n ^ { \circ } \right) n \geq 1\)
    1. Find the order of the sequence.
    2. Find \(\sum _ { r = 1 } ^ { 222 } u _ { r }\)
Edexcel PMT Mocks Q5
5 marks Standard +0.3
5. \(\mathrm { f } ( x ) = \frac { 1 } { 3 } x ^ { 3 } - 4 x - 2\) a. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written in the form \(x = \pm \sqrt { a + \frac { b } { x } }\), and state the values of the integers \(a\) and \(b\). \(\mathrm { f } ( x ) = 0\) has one positive root, \(\alpha\).
The iterative formula \(x _ { n + 1 } = \sqrt { a + \frac { b } { x _ { n } } } , \quad x _ { 0 } = 4\) is used to find an approximation value for \(\alpha\).
b. Calculate the values of \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\) to 4 decimal places.
c. Explain why for this question, the Newton-Raphson method cannot be used with \(x _ { 1 } = 2\).
Edexcel PMT Mocks Q6
7 marks Standard +0.3
6. \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 1\) a. (i) Show that ( \(2 x - 1\) ) is a factor of \(\mathrm { f } ( x )\).
(ii) Express \(\mathrm { f } ( x )\) in the form \(( 2 x - 1 ) ( x + a ) ^ { 2 }\) where \(a\) is an integer. Using the answer to part a) (ii)
b. show that the equation \(2 p ^ { 6 } + 3 p ^ { 4 } - 1\) has exactly two real solutions and state the values of these roots.
c. deduce the number of real solutions, for \(5 \pi \leq \theta \leq 8 \pi\), to the equation $$2 \cos ^ { 3 } \theta + 3 \cos ^ { 2 } \theta - 1 = 0$$
Edexcel PMT Mocks Q7
9 marks Standard +0.3
  1. Solve \(0 \leq \theta \leq 180 ^ { 0 }\), the equation $$4 \cos \theta = \sqrt { 3 } \operatorname { cosec } \theta$$
  2. Solve, for \(0 \leq x \leq 2 \pi\), the equation $$\cos x - \sqrt { 3 } \sin x = \sqrt { 3 }$$
Edexcel PMT Mocks Q8
7 marks Moderate -0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f9dcb521-6aaa-4496-86e8-2dcd07838e10-14_551_1479_388_365} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} In a competition, competitors are going to kick a ball over the barrier walls. The height of the barrier walls are each 9 metres high and 50 cm wide and stand on horizontal ground. The figure 2 is a graph showing the motion of a ball. The ball reaches a maximum height of 12 metres and hits the ground at a point 80 metres from where its kicked.
a. Find a quadratic equation linking \(Y\) with \(x\) that models this situation. The ball pass over the barrier walls.
b. Use your equation to deduce that the ball should be placed about 20 m from the first barrier wall.
Edexcel PMT Mocks Q9
5 marks Standard +0.8
9. Given that \(x\) is measured in radians, prove, from the first principles, that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( \sin x ) = \cos x$$ You may assume the formula for \(\sin ( A \pm B )\) and that as \(h \rightarrow 0 , \frac { \sin h } { h } \rightarrow 1\) and \(\frac { \cos h - 1 } { h } \rightarrow 0\).
Edexcel PMT Mocks Q10
6 marks Standard +0.3
10. Given that \(y = 8\) at \(x = 1\), solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( 12 x + 9 ) y ^ { \frac { 1 } { 3 } } } { x }$$ Giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).
Edexcel PMT Mocks Q11
7 marks Standard +0.3
11. \(\frac { - 6 x ^ { 2 } + 24 x - 9 } { ( x - 2 ) ( 1 - 3 x ) } \equiv A + \frac { B } { x - 2 } + \frac { C } { 1 - 3 x }\) a. Find the values of the constants \(A , B\) and \(C\).
b. Using part (a), find \(\mathrm { f } ^ { \prime } ( x )\).
c. Prove that \(\mathrm { f } ( x )\) is an increasing function.
Edexcel PMT Mocks Q12
7 marks Standard +0.3
12. a. Prove that $$\frac { \sec ^ { 2 } x - 1 } { \sec ^ { 2 } x } \equiv \sin ^ { 2 } x$$ b. Hence solve, for \(- 360 ^ { \circ } < x < 360 ^ { \circ }\), the equation $$\frac { \sec ^ { 2 } x - 1 } { \sec ^ { 2 } x } = \frac { \cos 2 x } { 2 }$$
Edexcel PMT Mocks Q13
10 marks Standard +0.3
  1. a. Find \(\int \ln x \mathrm {~d} x\)
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f9dcb521-6aaa-4496-86e8-2dcd07838e10-22_919_1139_276_456} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve with equation $$y = \ln x , \quad x > 0$$ The point P lies on \(C\) and has coordinate \(( e , 1 )\).
The line 1 is a normal to \(C\) at \(P\). The line \(l\) cuts the \(x\)-axis at the point \(Q\).
b. Find the exact value of the \(x\)-coordinate of \(Q\). The finite region \(\mathbf { R }\), shown shaded in figure 3, is bounded by the curve, the line \(l\) and the \(x\)-axis.
c. Find the exact area of \(\mathbf { R }\).
Edexcel PMT Mocks Q14
14 marks Standard +0.3
14. A population of ants being studied on an island. The number of ants, \(P\), in the population, is modelled by the equation. $$P = \frac { 900 k e ^ { 0.2 t } } { 1 + k e ^ { 0.2 t } } , \text { where } k \text { is a constant. }$$ Given that there were 360 ants when the study started,
a. show that \(k = \frac { 2 } { 3 }\).
b. Show that \(P = \frac { 1800 } { 2 + 3 e ^ { - 0.2 t } }\). The model predicts an upper limit to the number of ants on the island.
c. State the value of this limit.
d. Find the value of \(t\) when \(P = 520\). Give your answer to one decimal place.
e. i. Show that the rate of growth, \(\frac { \mathrm { d } P } { d t } = \frac { P ( 900 - P ) } { 4500 }\) ii. Hence state the value of \(P\) at which the rate of growth is a maximum.
Edexcel PMT Mocks Q1
5 marks Standard +0.3
  1. Given that \(a\) is a positive constant,
    a. Sketch the graph with equation
$$y = | a - 2 x |$$ Show on your sketch the coordinates of each point at which the graph crosses the \(x\)-axis and \(y\)-axis.
b. Solve the inequality \(| a - 2 x | > x + 2 a\)
Edexcel PMT Mocks Q2
4 marks Moderate -0.8
2. Solve $$4 ^ { x - 3 } = 6$$ giving your answer in the form \(a + b \log _ { 2 } 3\), where \(a\) and \(b\) are constants to be found.
Edexcel PMT Mocks Q3
3 marks Moderate -0.5
3. Given that $$y = \frac { 1 } { 3 } x ^ { 3 }$$ use differentiation from first principle to show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 }$$
Edexcel PMT Mocks Q4
6 marks Standard +0.3
  1. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 }\) is defined by
$$a _ { n } = \sin ^ { 2 } \left( \frac { n \pi } { 3 } \right)$$ Find the exact values of
a. i) \(a _ { 1 }\) ii) \(a _ { 2 }\) iii) \(a _ { 3 }\) b. Hence find the exact value of $$\sum _ { n = 1 } ^ { 100 } \left\{ n + \sin ^ { 2 } \left( \frac { n \pi } { 3 } \right) \right\}$$
Edexcel PMT Mocks Q5
6 marks Standard +0.3
  1. The table below shows corresponding values of \(x\) and \(y\) for \(y = \log _ { 3 } ( x )\) The values of \(y\) are given to 2 decimal places as appropriate.
\(x\)34.567.59
\(y\)11.371.631.832
a. Obtain an estimate for \(\int _ { 3 } ^ { 9 } \log _ { 3 } ( x ) \mathrm { d } x\), giving your answer to two decimal places. Use your answer to part (a) and making your method clear, estimate
b. i) \(\int _ { 3 } ^ { 9 } \log _ { 3 } \sqrt { x } \mathrm {~d} x\) ii) \(\int _ { 3 } ^ { 18 } \log _ { 3 } \left( 9 x ^ { 3 } \right) \mathrm { d } x\)
Edexcel PMT Mocks Q6
9 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d37eaba2-0a25-4abf-b2c8-1e08673229fb-10_1287_988_278_340} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation $$f ( x ) = 4 \cos 2 x - 2 x + 1 \quad x > 0$$ and where \(x\) is measured in radians.
The curve crosses the \(x\)-axis at the point \(A\), as shown in figure 1 .
Given that \(x\)-coordinate of \(A\) is \(\alpha\) a. show that \(\alpha\) lies between 0.7 and 0.8 Given that \(x\)-coordinates of \(B\) and \(C\) are \(\beta\) and \(\gamma\) respectively and they are two smallest values of \(x\) at which local maxima occur
b. find, using calculus, the value of \(\beta\) and the value of \(\gamma\), giving your answers to 3 significant figures.
c. taking \(x _ { 0 } = 0.7\) or 0.8 as a first approximation to \(\alpha\), apply the Newton-Raphson method once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\). Show, your method and give your answer to 2 significant figures.
Edexcel PMT Mocks Q7
5 marks Standard +0.3
7. a. Use the binomial theorem to expand $$( 8 - 3 x ) ^ { \frac { 2 } { 3 } }$$ in ascending powers of \(x\), up to and including the term \(x ^ { 3 }\), as a fully simplifying each term. Edward, a student decides to use the expansion with \(x = \frac { 1 } { 3 }\) to find an approximation for \(( 7 ) ^ { \frac { 2 } { 3 } }\). Using the answer to part (a) and without doing any calculations, b. explain clearly whether Edward's approximation will be an overestimate, or, an underestimate.
Edexcel PMT Mocks Q8
5 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d37eaba2-0a25-4abf-b2c8-1e08673229fb-14_1090_1205_274_456} \captionsetup{labelformat=empty} \caption{Figure 2
Figure 2 shows a sketch of part of the curve with equation $$y = \frac { 12 x - x ^ { 2 } } { \sqrt { x } } , \quad x > 0$$ The region \(R\), shows shaded in figure 2, is bounded by the curve, the line with equation \(x = 4\), the \(x\)-axis and the line with equation \(x = 8\).
Show that the area of the shaded region \(R\) is \(\frac { 128 } { 5 } ( 3 \sqrt { 2 } - 2 )\).}
\end{figure} (5)
Edexcel PMT Mocks Q9
5 marks Standard +0.8
9. $$\mathrm { f } ( \theta ) = 4 \cos \theta + 5 \sin \theta \quad \theta \in R$$ a. Express \(\mathrm { f } ( \theta )\) in the form \(R \cos ( \theta - \alpha )\) where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the exact value of \(R\) and give the value of \(\alpha\), in radians, to 3 decimal places. Given that $$\mathrm { g } ( \theta ) = \frac { 135 } { 4 + \mathrm { f } ( \theta ) ^ { 2 } } \quad \theta \in R$$ b.find the range of \(g\).
Edexcel PMT Mocks Q10
6 marks Moderate -0.8
10. The functions f and g are defined with their respective domains by $$\begin{array} { l l l } \mathrm { f } ( x ) = 4 - x ^ { 2 } & x \in R & x \geq 0 \\ \mathrm {~g} ( x ) = \frac { 2 } { x + 1 } & x \in R & x \geq 0 \end{array}$$ a. Write down the range of f .
b. Find the value of \(\mathrm { fg } ( 3 )\) c. Find \(\mathrm { g } ^ { - 1 } ( x )\)
Edexcel PMT Mocks Q11
4 marks Standard +0.3
11. Prove, using algebra that $$n ^ { 2 } + 1$$ is not divisible by 4 .
Edexcel PMT Mocks Q12
6 marks Standard +0.3
12. A curve has equation \(y = \frac { 2 x e ^ { x } } { x + k }\) where \(k\) is a positive constant.
i. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { e ^ { x } \left( 2 x ^ { 2 } + 2 k x + 2 k \right) } { ( x + k ) ^ { 2 } }\) ii. Given that the curve has exactly one stationary point find the value of \(k\).