Edexcel PMT Mocks (PMT Mocks)

1. Given that \(\theta\) is small and is measured in radians, use the small angle approximations to find an approximate value of

$$\frac { \theta \tan 3 \theta } { \cos 2 \theta - 1 }$$

2. \begin{figure}[h] \end{figure} Figure 1 shows a sector \(P O Q\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\).
The angle \(P O Q\) is 0.5 radians.
The area of the sector is \(9 \mathrm {~cm} ^ { 2 }\).
Show that the perimeter of the sector is \(k\) times the length of the arc, where \(k\) is an integer.

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.

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] \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.

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 $$\frac { d y } { d \theta } = - \frac { 5 } { 4 ( 1 + \sin 2 \theta ) } , \quad - \frac { \pi } { 4 } < \theta < \frac { 3 \pi } { 4 }$$

6. \begin{figure}[h] \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 \(k\).

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 \(k\).

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)

9. \begin{figure}[h] \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 \(Q\).

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 \(T\).

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.

12. The table shows the average weekly pay of a footballer at a certain club on 1 August 1990 and 1 August 2010. 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 \(\pounds 100000\).

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)

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.