Edexcel PMT Mocks (PMT Mocks)

1. Given that \(a\) is a positive constant,
   a. Sketch the graph with equation

$$y = | a - 2 x |$$ Show on your sketch the coordinates of each point at which the graph crosses the \(x\)-axis and \(y\)-axis.
b. Solve the inequality \(| a - 2 x | > x + 2 a\)

2. Solve $$4 ^ { x - 3 } = 6$$ giving your answer in the form \(a + b \log _ { 2 } 3\), where \(a\) and \(b\) are constants to be found.

3. Given that $$y = \frac { 1 } { 3 } x ^ { 3 }$$ use differentiation from first principle to show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 2 }$$

1. A sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 }\) is defined by

$$a _ { n } = \sin ^ { 2 } \left( \frac { n \pi } { 3 } \right)$$ Find the exact values of
a. i) \(a _ { 1 }\)
ii) \(a _ { 2 }\)
iii) \(a _ { 3 }\)
b. Hence find the exact value of $$\sum _ { n = 1 } ^ { 100 } \left\{ n + \sin ^ { 2 } \left( \frac { n \pi } { 3 } \right) \right\}$$

1. The table below shows corresponding values of \(x\) and \(y\) for \(y = \log _ { 3 } ( x )\) The values of \(y\) are given to 2 decimal places as appropriate.

a. Obtain an estimate for \(\int _ { 3 } ^ { 9 } \log _ { 3 } ( x ) \mathrm { d } x\), giving your answer to two decimal places. Use your answer to part (a) and making your method clear, estimate
b. i) \(\int _ { 3 } ^ { 9 } \log _ { 3 } \sqrt { x } \mathrm {~d} x\)
ii) \(\int _ { 3 } ^ { 18 } \log _ { 3 } \left( 9 x ^ { 3 } \right) \mathrm { d } x\)

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation $$f ( x ) = 4 \cos 2 x - 2 x + 1 \quad x > 0$$ and where \(x\) is measured in radians.
The curve crosses the \(x\)-axis at the point \(A\), as shown in figure 1 .
Given that \(x\)-coordinate of \(A\) is \(\alpha\)
a. show that \(\alpha\) lies between 0.7 and 0.8 Given that \(x\)-coordinates of \(B\) and \(C\) are \(\beta\) and \(\gamma\) respectively and they are two smallest values of \(x\) at which local maxima occur
b. find, using calculus, the value of \(\beta\) and the value of \(\gamma\), giving your answers to 3 significant figures.
c. taking \(x _ { 0 } = 0.7\) or 0.8 as a first approximation to \(\alpha\), apply the Newton-Raphson method once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\). Show, your method and give your answer to 2 significant figures.

7. a. Use the binomial theorem to expand $$( 8 - 3 x ) ^ { \frac { 2 } { 3 } }$$ in ascending powers of \(x\), up to and including the term \(x ^ { 3 }\), as a fully simplifying each term. Edward, a student decides to use the expansion with \(x = \frac { 1 } { 3 }\) to find an approximation for \(( 7 ) ^ { \frac { 2 } { 3 } }\). Using the answer to part (a) and without doing any calculations, b. explain clearly whether Edward's approximation will be an overestimate, or, an underestimate.

8. \begin{figure}[h] \end{figure} (5)

9. $$\mathrm { f } ( \theta ) = 4 \cos \theta + 5 \sin \theta \quad \theta \in R$$ a. Express \(\mathrm { f } ( \theta )\) in the form \(R \cos ( \theta - \alpha )\) where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the exact value of \(R\) and give the value of \(\alpha\), in radians, to 3 decimal places. Given that $$\mathrm { g } ( \theta ) = \frac { 135 } { 4 + \mathrm { f } ( \theta ) ^ { 2 } } \quad \theta \in R$$ b.find the range of \(g\).

10. The functions f and g are defined with their respective domains by $$\begin{array} { l l l } \mathrm { f } ( x ) = 4 - x ^ { 2 } & x \in R & x \geq 0
\mathrm {~g} ( x ) = \frac { 2 } { x + 1 } & x \in R & x \geq 0 \end{array}$$ a. Write down the range of f .
b. Find the value of \(\mathrm { fg } ( 3 )\)
c. Find \(\mathrm { g } ^ { - 1 } ( x )\)

11. Prove, using algebra that $$n ^ { 2 } + 1$$ is not divisible by 4 .

12. A curve has equation \(y = \frac { 2 x e ^ { x } } { x + k }\) where \(k\) is a positive constant.
i. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { e ^ { x } \left( 2 x ^ { 2 } + 2 k x + 2 k \right) } { ( x + k ) ^ { 2 } }\)
ii. Given that the curve has exactly one stationary point find the value of \(k\).

13. Relative to a fixed origin \(O\)

• the point \(P\) has position vector \(( 0 , - 1,2 )\)
• the point \(Q\) has position vector \(( 1,1,5 )\)
• the point \(R\) has position vector ( \(3,5 , m\) )
  where \(m\) is a constant.
  Given that \(P , Q\) and \(R\) lie on a straight line,
  a. find the value of \(m\)

The line segment \(O Q\) is extended to a point \(T\) so that \(\overrightarrow { R T }\) is parallel to \(\overrightarrow { O P }\)
b. Show that \(| \overrightarrow { O T } | = 9 \sqrt { 3 }\).

14. a. Express \(\frac { 1 } { ( 3 - x ) ( 1 - x ) }\) in partial fractions.
(2) A scientist is studying the mass of a substance in a laboratory.
The mass, \(x\) grams, of a substance at time \(t\) seconds after a chemical reaction starts is modelled by the differential equation $$2 \frac { d x } { d t } = ( 3 - x ) ( 1 - x ) \quad t \geq 0,0 \leq x < 1$$ Given that when \(t = 0 , x = 0\)
b. solve the differential equation and show that the solution can be written as $$x = \frac { 3 \left( e ^ { t } - 1 \right) } { 3 e ^ { t } - 1 }$$ c. Find the mass, \(x\) grams, which has formed 2 seconds after the start of the reaction. Give your answer correct to 3 significant figures.
d. Find the limiting value of \(x\) as \(t\) increases.

15. The first three terms of a geometric series where \(\theta\) is a constant are $$- 8 \sin \theta , \quad 3 - 2 \cos \theta \quad \text { and } \quad 4 \cot \theta$$ a. Show that \(4 \cos ^ { 2 } \theta + 20 \cos \theta + 9 = 0\) Given that \(\theta\) lies in the interval \(90 ^ { \circ } < \theta < 180 ^ { \circ }\),
b. Find the value of \(\theta\).
c. Hence prove that this series is convergent.
d. Find \(S _ { \infty }\), in the form \(a ( 1 - \sqrt { 3 } )\)

16. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve \(C\) with parametric equations \(x = - 3 + 6 \sin \theta , \quad y = 9 \cos 2 \theta \quad - \frac { \pi } { 2 } \leq \theta \leq \frac { \pi } { 4 }\) where \(\theta\) is a parameter.
a. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(\theta\) The line \(l\) is normal to \(C\) at the point \(P\) where \(\theta = \frac { \pi } { 6 }\)
b. Show that an equation for \(l\) is $$y = \frac { 1 } { 3 } x + \frac { 9 } { 2 }$$ c. The cartesian equation for the curve \(C\) can be written in the form $$y = a - \frac { 1 } { 2 } ( x + b ) ^ { 2 }$$ where \(a\) and \(b\) are integers to be found. The straight line with equation $$y = \frac { 1 } { 3 } x + k$$ where \(k\) is a constant intersects \(C\) at two distinct points.
d. Find the range of possible values for \(k\).