Edexcel PMT Mocks (PMT Mocks)

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. \(y = 3 \mathrm { f } ( 2 x ) + 2\)

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.

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.

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.

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.

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

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.

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

9. \begin{figure}[h] \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 \(X A P B\).

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] \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.

11. \begin{figure}[h] \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 \(P\).

12. \begin{figure}[h] \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 \(\frac { e ^ { 6 } } { 72 } ( 35 + 24 \ln 2 )\).

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.

14. Given that $$2 \cos ( x + 60 ) ^ { 0 } = \sin ( x - 30 ) ^ { 0 }$$ a. Show, without using a calculator, that $$\tan x = \frac { \sqrt { 3 } } { 3 }$$ b. Hence solve, for \(0 \leq \theta < 360 ^ { 0 }\) $$2 \cos ( 2 \theta + 90 ) ^ { 0 } = \sin ( 2 \theta ) ^ { 0 }$$

15. \begin{figure}[h] \end{figure} Figure 7 shows an open tank for storing water, \(A B C D E F\). The sides \(A C D F\) and \(A B E F\) are rectangles. The faces \(A B C\) and \(F E D\) are sectors of a circle with radius \(A B\) and \(F E\) respectively.

• \(A B = F E = r \mathrm {~cm}\)
• \(A F = B E = C D = l \mathrm {~cm}\)
• angle \(B A C =\) angle \(E F D = 0.9\) radians

Given that the volume of the tank is \(360 \mathrm {~cm} ^ { 3 }\)
a. show that the surface area of the tank, \(S \mathrm {~cm} ^ { 2 }\), is given by $$S = 0.9 r ^ { 2 } + \frac { 1600 } { r }$$ (4) Given that \(r\) can vary
b. use calculus to find the value of \(r\) for which \(S\) is stationary.
c. Find, to 3 significant figures the minimum value of \(S\).

16. \begin{figure}[h] \end{figure} Figure 8 shows a sketch of the curve with parametric equations $$x = 4 \cos t \quad y = 2 \sin 2 t \quad 0 \leq t \leq \frac { \pi } { 2 }$$ where \(t\) is a parameter.
The finite region \(R\) is enclosed by the curve \(C\), the \(x\)-axis and the line \(x = 2\), as shown in Figure 7.
a. Show that the area of \(R\) is given by $$\int _ { \frac { \pi } { 3 } } ^ { \frac { \pi } { 2 } } 16 \sin ^ { 2 } t \cos t \mathrm {~d} t$$ b. Hence, using algebraic integration, find the exact area of \(R\), giving in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are constants to be determined.