Questions — Edexcel (9685 questions)

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Edexcel PMT Mocks Q3
9 marks Moderate -0.3
3. The curve \(C\) has equation $$y = 8 \sqrt { x } + \frac { 18 } { \sqrt { x } } - 20 \quad x > 0$$ a. Find
i) \(\frac { d y } { d x }\) ii) \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\) b. Use calculus to find the coordinates of the stationary point of \(C\).
c. Determine whether the stationary point is a maximum or minimum, giving a reason for your answer.
Edexcel PMT Mocks Q4
4 marks Standard +0.3
4. The curve with equation \(y = 2 + \ln ( 4 - x )\) meets the line \(y = x\) at a single point, \(x = \beta\).
a. Show that \(2 < \beta < 3\) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{63d85737-99d4-4916-a479-fe44f77b1505-07_961_1002_296_513} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the graph of \(y = 2 + \ln ( 4 - x )\) and the graph of \(y = x\).
A student uses the iteration formula $$x _ { n + 1 } = 2 + \ln \left( 4 - x _ { n } \right) , \quad n \in N ,$$ in an attempt to find an approximation for \(\beta\).
Using the graph and starting with \(x _ { 1 } = 3\),
b. determine whether the or not this iteration formula can be used to find an approximation for \(\beta\), justifying your answer.
Edexcel PMT Mocks Q5
5 marks Standard +0.3
5. Given that $$y = \frac { 5 \cos \theta } { 4 \cos \theta + 4 \sin \theta } , \quad - \frac { \pi } { 4 } < \theta < \frac { 3 \pi } { 4 }$$ Show that $$\frac { d y } { d \theta } = - \frac { 5 } { 4 ( 1 + \sin 2 \theta ) } , \quad - \frac { \pi } { 4 } < \theta < \frac { 3 \pi } { 4 }$$
Edexcel PMT Mocks Q6
10 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{63d85737-99d4-4916-a479-fe44f77b1505-10_951_1022_306_488} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The circle \(C\) has centre \(A\) with coordinates \(( - 3,1 )\).
The line \(l _ { 1 }\) with equation \(y = - 4 x + 6\), is the tangent to \(C\) at the point \(Q\), as shown in Figure 3.
a. Find the equation of the line \(A Q\) in the form \(a x + b y = c\).
b. Show that the equation of the circle \(C\) is \(( x + 3 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 17\) The line \(l _ { 2 }\) with equation \(y = - 4 x + k , k \neq 6\), is also a tangent to \(C\).
c. Find the value of the constant \(k\).
Edexcel PMT Mocks Q7
7 marks Standard +0.3
7. Given that \(k \in \mathbb { Z } ^ { + }\) a. show that \(\int _ { 2 k } ^ { 3 k } \frac { 6 } { ( 7 k - 2 x ) } \mathrm { d } x\) is independent of \(k\),
b. show that \(\int _ { k } ^ { 2 k } \frac { 2 } { 3 ( 2 x - k ) ^ { 2 } } \mathrm {~d} x\) is inversely proportional to \(k\).
Edexcel PMT Mocks Q8
5 marks Standard +0.3
8. The length of the daylight, \(D ( t )\) in a town in Sweden can be modelled using the equation $$D ( t ) = 12 + 9 \sin \left( \frac { 360 t } { 365 } - 63.435 \right) \quad 0 \leq t \leq 365$$ where \(t\) is the number of days into the year, and the argument of \(\sin x\) is in degrees
a. Find the number of daylight hours after 90 days in that year.
b. Find the values of \(t\) when \(D ( t ) = 17\), giving your answers to the nearest integer. (Solutions based entirely on graphical or numerical methods are not acceptable)
Edexcel PMT Mocks Q9
8 marks Standard +0.8
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{63d85737-99d4-4916-a479-fe44f77b1505-16_871_1017_267_548} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a sketch of the curve with equation \(x ^ { 2 } + y ^ { 3 } - 10 x - 12 y - 5 = 0\) a. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 10 - 2 x } { 3 y ^ { 2 } - 12 }\) At each of the points \(P\) and \(Q\) the tangent to the curve is parallel to the \(y\)-axis.
b. Find the exact coordinates of \(Q\).
Edexcel PMT Mocks Q10
9 marks Standard +0.3
10. a. Find \(\int \frac { 1 } { 30 } \cos \frac { \pi } { 6 } t \mathrm {~d} t\). The height above ground, \(X\) metres, of the passenger on a wooden roller coaster can be modelled by the differential equation $$\frac { d \mathrm { X } } { \mathrm {~d} t } = \frac { 1 } { 30 } X \cos \left( \frac { \pi } { 6 } t \right)$$ where \(t\) is the time, in seconds, from the start of the ride.
At time \(t = 0\), the passenger is 6 m above the ground.
b. Show that \(X = k e ^ { \frac { 1 } { 5 \pi } \sin \left( \frac { \pi } { 6 } t \right) }\) where the value of the constant \(k\) should be found.
c. Show that the maximum height of the passenger above the ground is 6.39 m . The passenger reaches the maximum height, for the second time, \(T\) seconds after the start of the ride.
d. Find the value of \(T\).
Edexcel PMT Mocks Q11
10 marks Standard +0.3
11. a. Find the binomial expansion of \(( 4 - x ) ^ { - \frac { 1 } { 2 } }\), up to and including the term in \(x ^ { 2 }\). Given that the binomial expansion of \(\mathrm { f } ( x ) = \sqrt { \frac { 1 + 2 x } { 4 - x } } , | x | < \frac { 1 } { 4 }\), is $$\frac { 1 } { 2 } + \frac { 9 } { 16 } x - A x ^ { 2 } + \cdots$$ b. Show that the value of the constant \(A\) is \(\frac { 45 } { 256 }\) c. By substituting \(x = \frac { 1 } { 4 }\) into the answer for (b) find an approximate for \(\sqrt { 10 }\), giving your answer to 3 decimal places.
Edexcel PMT Mocks Q12
9 marks Moderate -0.8
12. The table shows the average weekly pay of a footballer at a certain club on 1 August 1990 and 1 August 2010.
Year19902010
Average weekly pay\(\pounds 2500\)\(\pounds 50000\)
The average weekly pay of a footballer at this club can be modelled by the equation $$P = A k ^ { t }$$ where \(\pounds P\) is the average weekly pay \(t\) years after 1 August 1990, and \(A\) and \(k\) are constants.
a. i. Write down the value of \(A\).
ii. Show that the value of \(k\) is 1.16159 , correct to five decimal places.
b. With reference to the model, interpret
i. the value of the constant \(A\),
ii. the value of the constant \(k\), Using the model,
c. find the year in which, on 1 August, the average weekly pay of a footballer at this club will first exceed \(\pounds 100000\).
Edexcel PMT Mocks Q13
7 marks Standard +0.3
13. \includegraphics[max width=\textwidth, alt={}, center]{63d85737-99d4-4916-a479-fe44f77b1505-25_679_1043_413_607} Figure 5 shows a sketch of part of the curve with equation \(y = \frac { 6 x } { \sqrt { 3 x + 1 } } , \quad x \geq 0\) The finite region \(\mathbf { R }\), shown shaded in figure 5 is bounded by the curve, the \(x\)-axis and the lines \(x = 2\) and \(x = 5\). Use the substitution \(u = 3 x + 1\) to find the exact area of \(\mathbf { R }\).
(Total for Question 13 is 7 marks)
Edexcel PMT Mocks Q14
10 marks Standard +0.3
14. A curve \(C\) has parametric equations $$x = 1 - \cos t , \quad y = 2 \cos 2 t , \quad 0 \leq t < \pi$$ a. Show that the cartesian equation of the curve can be written as \(y = k ( 1 - x ) ^ { 2 } - 2\) where \(k\) is an integer.
b. i. Sketch the curve C .
ii. Explain briefly why C does not include all points of \(y = k ( 1 - x ) ^ { 2 } - 2 , x \in \mathbb { R }\). The line with equation \(y = k - x\), where \(k\) is a constant, intersects C at two distinct points.
(c) State the range of values of \(k\), writing your answer in set notation.
Edexcel PMT Mocks Q1
4 marks Easy -1.3
  1. The point \(P ( 2 , - 3 )\) lies on the curve with equation \(y = \mathrm { f } ( x )\).
State the coordinates of the image of \(P\) under the transformation represented by the curve
a. \(\quad y = | \mathrm { f } ( x ) |\) b. \(y = \mathrm { f } ( x - 2 )\) c. \(y = 3 \mathrm { f } ( 2 x ) + 2\)
Edexcel PMT Mocks Q2
6 marks Moderate -0.8
2. $$f ( x ) = ( 2 x - 3 ) ( x - k ) - 12$$ where \(k\) is a constant.
a.Write down the value of \(\mathrm { f } ( k )\) When \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ) the remainder is - 5
b. find the value of \(k\).
c. Factorise \(\mathrm { f } ( x )\) completely.
Edexcel PMT Mocks Q3
6 marks Moderate -0.3
3. A circle \(C\) has equation $$x ^ { 2 } - 22 x + y ^ { 2 } + 10 y + 46 = 0$$ a. Find
i. the coordinates of the centre \(A\) of the circle
ii. the radius of the circle. Given that the points \(Q ( 5,3 )\) and \(S\) lie on \(C\) such that the distance \(Q S\) is greatest,
b. find an equation of tangent to \(C\) at \(S\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are constants to be found.
Edexcel PMT Mocks Q4
3 marks Moderate -0.5
4. a. Express \(\lim _ { \mathrm { d } x \rightarrow 0 } \sum _ { 0.2 } ^ { 1.8 } \frac { 1 } { 2 x } \delta x \quad\) as an integral.
b. Hence show that $$\lim _ { \mathrm { d } x \rightarrow 0 } \sum _ { 0.2 } ^ { 1.8 } \frac { 1 } { 2 x } \delta x = \ln k$$ where \(k\) is a constant to be found.
Edexcel PMT Mocks Q5
6 marks Easy -1.2
5. A scientist is studying a population of lizards on an island and uses the linear model \(P = a + b t\) to predict the future population of the lizard where \(P\) is the population and \(t\) is the time in years after the start of the study. Given that
  • The number of population was 900 , exactly 5 years after the start of the study.
  • The number of population was 1200 , exactly 8 years after the start of the study.
    a. find a complete equation for the model.
    b. Sketch the graph of the population for the first 10 years.
    c. Suggest one criticism of this model.
Edexcel PMT Mocks Q6
6 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{48f9a252-61a2-491d-94d0-8470aee96942-07_864_995_299_495} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The figure 1 shows sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\). $$f ( x ) = a x ( x - b ) ^ { 2 } , x \in R$$ where \(a\) and \(b\) are constants.
The curve passes through the origin and touches the \(x\)-axis at the point \(( 3,0 )\).
There is a minimum point at \(( 1 , - 4 )\) and a maximum point at \(( 3,0 )\).
a. Find the equation of \(C\).
b. Deduce the values of \(x\) for which $$\mathrm { f } ^ { \prime } ( x ) > 0$$ Given that the line with equation \(y = k\), where \(k\) is a constant, intersects \(C\) at exactly one point,
c. State the possible values for \(k\).
Edexcel PMT Mocks Q7
5 marks Standard +0.3
7. (i) Given that \(a\) and \(b\) are integers such that $$a + b \text { is odd }$$ Use algebra to prove by contradiction that at least one of \(a\) and \(b\) is odd.
(ii) A student is trying to prove that $$( p + q ) ^ { 2 } < 13 p ^ { 2 } + q ^ { 2 } \quad \text { where } p < 0$$ The student writes: $$\begin{gathered} \qquad \begin{array} { c } p ^ { 2 } + 2 p q + q ^ { 2 } < 13 p ^ { 2 } + q ^ { 2 } \\ 2 p q < 12 p ^ { 2 } \\ \text { so as } p < 0 \quad 2 q < 12 p \\ q < 6 p \end{array} \end{gathered}$$ a. Identify the error made in the proof.
b. Write out the correct solution.
Edexcel PMT Mocks Q8
8 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{48f9a252-61a2-491d-94d0-8470aee96942-10_689_1011_294_486} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where \(x \in R\), \(x > 0\) $$\mathrm { f } ( x ) = ( 0.5 x - 8 ) \ln ( x + 1 ) \quad 0 \leq x \leq A$$ a. Find the value of \(A\).
b. Find \(\mathrm { f } ^ { \prime } ( x )\) The curve has a minimum turning point at \(B\).
c. Show that the \(x\)-coordinate of \(B\) is a solution of the equation $$x = \frac { 17 } { \ln ( x + 1 ) + 1 } - 1$$ d. Use the iteration formula $$x _ { n + 1 } = \frac { 17 } { \ln \left( x _ { n } + 1 \right) + 1 } - 1$$ with \(x _ { 0 } = 5\) to find the values of \(x _ { 1 }\) and the value of \(x _ { 6 }\) giving your answers to three decimal places.
Edexcel PMT Mocks Q9
8 marks Standard +0.8
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{48f9a252-61a2-491d-94d0-8470aee96942-12_451_519_328_717} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of a parallelogram \(X A P B\).
Given that \(\overrightarrow { O X } = \left( \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right)\) $$\begin{aligned} & \overrightarrow { O A } = \left( \begin{array} { l } 0 \\ 4 \\ 1 \end{array} \right) \\ & \overrightarrow { O B } = \left( \begin{array} { l } 3 \\ 3 \\ 1 \end{array} \right) \end{aligned}$$ a. Find the coordinates of the point \(P\).
b. Show that \(X A P B\) is a rhombus.
c. Find the exact area of the rhombus \(X A P B\).
Edexcel PMT Mocks Q10
8 marks Standard +0.3
10. The figure 4 shows the curves \(\mathrm { f } ( x ) = A - B e ^ { - 0.5 x }\) and \(\mathrm { g } ( x ) = 26 + e ^ { 0.5 x }\) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{48f9a252-61a2-491d-94d0-8470aee96942-14_718_1152_347_340} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Given that \(\mathrm { f } ( x )\) passes through \(( 0,8 )\) and has an horizontal asymptote \(y = 48\) a. Find the values of \(A\) and \(B\) for \(\mathrm { f } ( x )\) (3)
b. State the range of \(\mathrm { g } ( x )\) (1) The curves \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) meet at the points \(C\) and \(D\) c. Find the \(x\)-coordinates of the intersection points \(C\) and \(D\), in the form \(\ln k\), where \(k\) is an integer.
Edexcel PMT Mocks Q11
7 marks Moderate -0.3
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{48f9a252-61a2-491d-94d0-8470aee96942-16_1123_1031_280_511} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} The figure 5 shows part of the curves \(C _ { 1 }\) and \(C _ { 2 }\) with equations $$\begin{array} { c c } C _ { 1 } : y = x ^ { 3 } - 2 x ^ { 2 } & x > 0 \\ C _ { 2 } : y = 9 - \frac { 5 } { 2 } ( x - 3 ) ^ { 2 } & x > 0 \end{array}$$ The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the points \(P\) and \(Q\).
a. Verify that the point \(Q\) has coordinates \(( 3,9 )\) b. Use algebra to find the coordinates of the point \(P\).
Edexcel PMT Mocks Q12
5 marks Standard +0.8
12. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{48f9a252-61a2-491d-94d0-8470aee96942-18_1038_1271_244_440} \captionsetup{labelformat=empty} \caption{Figure 6}
\end{figure} The figure 6 shows a sketch of the curve with equation $$y = x ^ { 2 } \ln 2 x$$ The finite region \(R\), shown shaded in figure 5, is bounded by the line with equation \(x = \frac { e ^ { 2 } } { 2 }\), the curve \(C\), the line with equation \(x = e ^ { 2 }\) and the \(x\)-axis.
Show that the exact value of the area of region \(R\) is \(\frac { e ^ { 6 } } { 72 } ( 35 + 24 \ln 2 )\).
Edexcel PMT Mocks Q13
4 marks Moderate -0.3
13. A construction company had a 30 -year programme to build new houses in Newtown. They began in the year 1991 (Year 1) and finished in 2020 (Year 30).
The company built 120 houses in year 1, 140 in year 2, 160 houses in year 3 and so on, so that the number of houses they built form an arithmetic sequence.
A total of 8400 new houses were built in year \(n\).
a. Show that $$n ^ { 2 } + 11 n - 840 = 0$$ b. Solve the equation $$n ^ { 2 } + 11 n - 840 = 0$$ and hence find in which year 8400 new houses were built.