Edexcel PMT Mocks (PMT Mocks)

1. $$y = \sqrt { \left( 2 ^ { x } + x \right) }$$ a. Complete the table below, giving the values of \(y\) to 3 decimal places. (1)
b. Use the trapezium rule with all the values of \(y\) from your table to find an approximation for the value of $$\int _ { 0 } ^ { 1 } \sqrt { \left( 2 ^ { x } + x \right) } \mathrm { d } x$$ giving your answer to 3 significant figures. Using your answer to part (b) and making your method clear, estimate
c. \(\int _ { 0 } ^ { 0.5 } \sqrt { \left( 2 ^ { 2 x } + 2 x \right) } \mathrm { d } x\)

2.
\includegraphics[max width=\textwidth, alt={}, center]{cb92f7b6-2ba5-4703-9595-9ba8570fc52b-04_656_725_283_635} \section*{Figure 1} Figure 1 shows a triangle \(O A C\) where \(O B\) divides \(A C\) in the ratio \(2 : 3\).
Show that \(\mathbf { b } = \frac { 1 } { 5 } ( 3 \mathbf { a } + 2 \mathbf { c } )\)

3. Use the laws of logarithms to solve the equation $$2 + \log _ { 2 } ( 2 x + 1 ) = 2 \log _ { 2 } ( 22 - x )$$

1. In the binomial expansion of \(( 2 - k x ) ^ { 10 }\) where \(k\) is a non-zero positive constant.

The coefficient of \(x ^ { 4 }\) is 256 times the coefficient of \(x ^ { 6 }\).
Find the value of \(k\).

5. a. Given that $$\frac { x ^ { 2 } - 1 } { x + 3 } \equiv x + P + \frac { Q } { x + 3 }$$ find the value of the constant \(P\) and show that \(Q = 8\) \begin{figure}[h] \end{figure} The curve \(C\) has equation \(y = \mathrm { g } ( x )\), where $$\mathrm { g } ( x ) = \frac { x ^ { 2 } - 1 } { x + 3 } \quad x > - 3$$ Figure 3 shows a sketch of the curve \(C\).
The region \(R\), shown shaded in Figure 4, is bounded by \(C\), the \(x\)-axis and the line with equation \(x = 5\).
b. Find the exact area of \(R\), writing your answer in the form \(a \ln 2\), where \(a\) is constant to be found.
(4)

6. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = \frac { 2 x ^ { 2 } - x } { \sqrt { x } } - 2 \ln \left( \frac { x } { 2 } \right) , \quad x > 0$$ The curve has a minimum turning point at \(Q\), as shown in Figure 4.
a. Show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 6 x ^ { 2 } - x - 4 \sqrt { x } } { 2 x \sqrt { x } }\)
b. Show that the \(x\)-coordinate of \(Q\) is the solution of $$x = \sqrt { \frac { x } { 6 } + \frac { 2 \sqrt { x } } { 3 } }$$ To find an approximation for the \(x\)-coordinate of \(Q\), the iteration formula $$x _ { n + 1 } = \sqrt { \frac { x _ { n } } { 6 } + \frac { 2 \sqrt { x _ { n } } } { 3 } }$$ is used.
c. Taking \(x _ { 0 } = 0.8\), find the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\). Give your answers to 3 decimal places.

7. A curve \(C\) has equation \(y = \mathrm { f } ( x )\). Given that

• \(\mathrm { f } ^ { \prime } ( x ) = 18 x ^ { 2 } + 2 a x + b\)
• the \(y\)-intercept of \(C\) is - 48
• the point \(A\), with coordinates \(( - 1,45 )\) lies on \(C\)
  a. Show that \(a - b = 99\)
  b. Find the value of \(a\) and the value of \(b\).
  c. Show that \(( 2 x + 1 )\) is a factor of \(\mathrm { f } ( x )\).

8. \begin{figure}[h] \end{figure} The curves with equation \(y = 21 - 2 ^ { x }\) meet the curve with equation \(y = 2 ^ { 2 x + 1 }\) at the point \(A\) as shown in Figure 2. Find the exact coordinates of point \(A\).

9. A cup of tea is cooling down in a room. The temperature of tea, \(\theta ^ { \circ } \mathrm { C }\), at time \(t\) minutes after the tea is made, is modelled by the equation $$\theta = A + 70 e ^ { - 0.025 t }$$ where \(A\) is a positive constant.
Given that the initial temperature of the tea is \(85 ^ { \circ } \mathrm { C }\)
a. find the value of \(A\).
b. Find the temperature of the tea 20 minutes after it is made.
c. Find how long it will take the tea to cool down to \(43 ^ { \circ } \mathrm { C }\).
(4)

10. a. Show that $$\sin 3 A \equiv 3 \sin A - 4 \sin ^ { 3 } A$$ b. Hence solve, for \(- \frac { \pi } { 2 } \leq \theta \leq \frac { \pi } { 2 }\) the equation $$1 + \sin 3 \theta = \cos ^ { 2 } \theta$$

1. a. Sketch the graph of the function with equation

$$y = 11 - 2 | 2 - x |$$ Stating the coordinates of the maximum point and any points where the graph cuts the \(y\)-axis.
b. Solve the equation $$4 x = 11 - 2 | 2 - x |$$ A straight line \(l\) has equation \(y = k x + 13\), where \(k\) is a constant.
Given that \(l\) does not meet or intersect \(y = 11 - 2 | 2 - x |\)
c. find the range of possible value of \(k\).

12. \begin{figure}[h] \end{figure} Figure 5 shows part of the curve \(C\) with parametric equations $$x = 2 \cos \theta \quad y = \sin 2 \theta \quad 0 \leq \theta \leq \frac { \pi } { 2 }$$ The region \(R\), shown shaded in figure 5, is bounded by the curve \(C\), the line \(x = \sqrt { 2 }\) and the \(x\)-axis. This shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid revolution.
a. Show that the volume of the solid of revolution formed is given by the integral. $$k \int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 2 } } \sin ^ { 3 } \theta \cos ^ { 2 } \theta \mathrm {~d} \theta$$ where \(k\) is a constant.
\includegraphics[max width=\textwidth, alt={}, center]{cb92f7b6-2ba5-4703-9595-9ba8570fc52b-22_164_1148_54_118}
b. Hence, find the exact value for this volume, giving your answer in the form \(p \pi \sqrt { 2 }\) where \(p\) is a constant.

13. The function \(g\) is defined by $$\mathrm { g } ( x ) = \frac { 2 e ^ { x } - 5 } { e ^ { x } - 4 } \quad x \neq k , x > 0$$ where \(k\) is a constant.
a. Deduce the value of \(k\).
b. Prove that $$\mathrm { g } ^ { \prime } ( x ) < 0$$ For all values of \(x\) in the domain of g .
c. Find the range of values of \(a\) for which $$\mathrm { g } ( a ) > 0$$

1. A circle \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 6 x - 14 y = 40\).

The line \(l\) has equation \(y = x + k\), where \(k\) is a constant.
a. Show that the \(x\)-coordinate of the points where \(C\) and \(l\) intersect are given by the solutions to the equation $$2 x ^ { 2 } + ( 2 k - 20 ) x + k ^ { 2 } - 14 k - 40 = 0$$ b. Hence find the two values of \(k\) for which \(l\) is a tangent to \(C\).

15. An infinite geometric series has first four terms \(1 - 2 x + 4 x ^ { 2 } - 8 x ^ { 3 } + \cdots\). The series is convergent.
a. Find the set of possible values of \(x\) for which the series converges. Given that \(\sum _ { r = 1 } ^ { \infty } ( - 2 x ) ^ { r - 1 } = 8\),
b. calculate the value of \(x\).

16. Prove by contradiction that if \(n ^ { 2 }\) is a multiple of \(3 , n\) is a multiple of 3 .