Edexcel PMT Mocks (PMT Mocks)

1. The figure 1 shows part of the graph of \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { a x + 4 } { x - b } , \quad x > 2\)

\begin{figure}[h] \end{figure} a. State the values of \(a\) and \(b\).
b. State the range of f.
c. Find \(\mathrm { f } ^ { - 1 } ( x )\), stating its domain.

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 \(D\).

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 \(x\).

4. (a) Show that \(\sum _ { r = 1 } ^ { 20 } \left( 2 ^ { r - 1 } - 3 - 4 r \right) = 1047675\)
(b) 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 \(\sum _ { r = 1 } ^ { 222 } u _ { r }\)

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 \(x _ { 1 } = 2\).

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 $$2 \cos ^ { 3 } \theta + 3 \cos ^ { 2 } \theta - 1 = 0$$

1. (i) Solve \(0 \leq \theta \leq 180 ^ { 0 }\), the equation

$$4 \cos \theta = \sqrt { 3 } \operatorname { cosec } \theta$$ (ii) Solve, for \(0 \leq x \leq 2 \pi\), the equation $$\cos x - \sqrt { 3 } \sin x = \sqrt { 3 }$$

8. \begin{figure}[h] \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.

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 \(\frac { \cos h - 1 } { h } \rightarrow 0\).

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 \(y ^ { 2 } = \mathrm { f } ( x )\).

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.

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 $$\frac { \sec ^ { 2 } x - 1 } { \sec ^ { 2 } x } = \frac { \cos 2 x } { 2 }$$

1. a. Find \(\int \ln x \mathrm {~d} x\)

\begin{figure}[h] \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 \(\mathbf { R }\).

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.