Questions — Edexcel PMT Mocks (92 questions)

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Edexcel PMT Mocks Q1
  1. a. Find the first four terms, in ascending powers of \(x\), of the binomial expansion of
$$\left( \frac { 1 } { 9 } - 2 x \right) ^ { \frac { 1 } { 2 } }$$ giving each coefficient in its simplest form.
b. Explain how you could use \(x = \frac { 1 } { 36 }\) in the expansion to find an approximation for \(\sqrt { 2 }\). There is no need to carry out the calculation.
Edexcel PMT Mocks Q2
2. The curves \(C _ { 1 }\) and \(C _ { 2 }\) have equations $$\begin{aligned} & C _ { 1 } : \quad y = 2 ^ { 3 x + 2 }
& C _ { 2 } : \quad y = 4 ^ { - x } \end{aligned}$$ Show that the \(x\)-coordinate of the point where \(C _ { 1 }\) and \(C _ { 2 }\) intersect is \(\frac { - 2 } { 5 }\).
Edexcel PMT Mocks Q3
3. Relative to a fixed origin,
  • point \(A\) has position vector \(- 2 \mathbf { i } + 4 \mathbf { j } + 7 \mathbf { k }\)
  • point \(B\) has position vector \(- \mathbf { i } + 3 \mathbf { j } + 8 \mathbf { k }\)
  • point \(C\) has position vector \(\mathbf { i } + \mathbf { j } + 4 \mathbf { k }\)
  • point \(D\) has position vector \(- \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }\)
    a. Show that \(\overrightarrow { A B }\) and \(\overrightarrow { C D }\) are parallel and the ratio \(\overrightarrow { A B } : \overrightarrow { C D }\) in its simplest form.
    b. Hence describe the quadrilateral \(A B C D\).
Edexcel PMT Mocks Q4
  1. Ben starts a new company.
  • In year 1 his profits will be \(\pounds 24000\).
  • In year 11 his profit is predicted to be \(\pounds 64000\).
Model \(\boldsymbol { P }\) assumes that his profit will increase by the same amount each year.
a. According to model \(\boldsymbol { P }\), determine Ben's profit in year 5. Model \(\boldsymbol { Q }\) assumes that his profit will increase by the same percentage each year.
b. According to model \(\boldsymbol { Q }\), determine Ben's profit in year 5 . Give your answer to the nearest £10.
Edexcel PMT Mocks Q5
5. The function f is defined by $$\mathrm { f } : x \rightarrow \frac { 2 x - 3 } { x - 1 } \quad x \in R , x \neq 1$$ a. Find \(f ^ { - 1 } ( 3 )\).
b. Show that $$\mathrm { ff } ( x ) = \frac { x + p } { x - 2 } \quad x \in R , \quad x \neq 2$$ where \(p\) is an integer to be found. The function g is defined by $$g : x \rightarrow x ^ { 2 } - 5 x \quad x \in R , 0 \leq x \leq 6$$ c. Find the range of g .
d. Explain why the function g does not have an inverse.
Edexcel PMT Mocks Q6
6. a. Express \(4 \sin x - 5 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\). Give the exact value of \(R\), and give the value of \(\alpha\), in degrees, to 2 decimal places. $$T = \frac { 8400 } { 19 + ( 4 \sin x - 5 \cos x ) ^ { 2 } } , x > 0$$ b. Use your answer to part \(a\) to calculate
i. the minimum value of \(T\).
ii. the smallest value of \(x , x > 0\), at which this minimum value occurs.
Edexcel PMT Mocks Q7
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{802e56f7-5cff-491a-b90b-0759a9b35778-09_928_1093_258_497} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of a curve \(C\) with equation \(y = \mathrm { f } ( x )\) and a straight line \(l\). The curve \(C\) meets \(l\) at the points \(( 2,4 )\) and \(( 6,0 )\) as shown. The shaded region \(R\), shown shaded in Figure 1, is bounded by \(\mathrm { C } , l\) and the \(y\)-axis. Given that \(\mathrm { f } ( x )\) is a quadratic function in \(x\), use inequalities to define region \(R\).
Edexcel PMT Mocks Q8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{802e56f7-5cff-491a-b90b-0759a9b35778-11_1112_1211_280_386} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve \(C\) with the equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \left( 2 x ^ { 2 } - 9 x + 9 \right) e ^ { - x } , \quad x \in R$$ The curve has a minimum turning point at \(A\) and a maximum turning point at \(B\) as shown in the figure above.
a. Find the coordinates of the point where \(C\) crosses the \(y\)-axis.
b. Show that \(\mathrm { f } ^ { \prime } ( x ) = - \left( 2 x ^ { 2 } - 13 x + 18 \right) e ^ { - x }\)
c. Hence find the exact coordinates of the turning points of \(C\). The graph with equation \(y = \mathrm { f } ( x )\) is transformed onto the graph with equation $$y = a \mathrm { f } ( x ) + b , \quad x \geq 0$$ The range of the graph with equation \(y = a \mathrm { f } ( x ) + b\) is \(0 \leq y \leq 9 e ^ { 2 } + 1\)
Given that \(a\) and \(b\) are constants.
d. find the value of \(a\) and the value of \(b\).
Edexcel PMT Mocks Q9
9. a. Use the substitution \(t ^ { 2 } = 2 x - 5\) to show that $$\int \frac { 1 } { x + 3 \sqrt { 2 x - 5 } } \mathrm {~d} x = \int \frac { 2 t } { t ^ { 2 } + 6 t + 5 } \mathrm {~d} t$$ b. Hence find the exact value of $$\int _ { 3 } ^ { 27 } \frac { 1 } { x + 3 \sqrt { 2 x - 5 } } \mathrm {~d} x$$
Edexcel PMT Mocks Q10
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{802e56f7-5cff-491a-b90b-0759a9b35778-16_1116_1433_360_420} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Circle \(C _ { 1 }\) has equation \(x ^ { 2 } + y ^ { 2 } = 64\) with centre \(O _ { 1 }\).
Circle \(C _ { 2 }\) has equation \(( x - 6 ) ^ { 2 } + y ^ { 2 } = 100\) with centre \(O _ { 2 }\).
The circles meet at points \(A\) and \(B\) as shown in Figure 3.
a. Show that angle \(A O _ { 2 } B = 1.85\) radians to 3 significant figures.
(3)
b. Find the area of the shaded region, giving your answer correct to 1 decimal place.
Edexcel PMT Mocks Q11
11. In a science experiment, a radio active particle, \(N\), decays over time, \(t\), measured in minutes. The rate of decay of a particle is proportional to the number of particles remaining. Write down a suitable equation for the rate of change of the number of particles, \(N\) in terms of \(t\).
Edexcel PMT Mocks Q12
12. a. Show that $$\sec \theta - \cos \theta = \sin \theta \tan \theta \quad \theta \neq ( \pi n ) ^ { 0 } \quad n \in Z$$ b. Hence, or otherwise, solve for \(0 < x \leq \pi\) $$\sec x - \cos x = \sin x \tan \left( 3 x - \frac { \pi } { 9 } \right)$$
Edexcel PMT Mocks Q13
  1. A sequence \(a _ { 1 } , a _ { 2 } a _ { 3 } , \ldots\) is defined by
$$a _ { n + 1 } = 5 - p a _ { n } \quad n \geq 1$$ where \(p\) is a constant.
Given that
  • \(a _ { 1 } = 4\)
  • the sequence is a periodic sequence of order 2.
    a. Write down an expression for \(a _ { 2 }\) and \(a _ { 3 }\).
    b. Find the value of \(p\).
    c. Find \(\sum _ { r = 1 } ^ { 21 } a _ { r }\)
Edexcel PMT Mocks Q14
  1. A circular stain is growing.
The rate of increase of its radius is inversely proportional to the square of the radius. At time \(t\) seconds the circular stain has radius \(r \mathrm {~cm}\) and area \(A \mathrm {~cm} ^ { 2 }\).
a. Show that \(\frac { \mathrm { d } A } { \mathrm {~d} t } = \frac { k } { \sqrt { A } }\). Given that
  • the initial area of the circular stain is \(0.09 \mathrm {~cm} ^ { 2 }\).
  • after 10 seconds the area of the circular stain is \(0.36 \mathrm {~cm} ^ { 2 }\).
    b. Solve the differential equation to find a complete equation linking \(A\) and \(t\).
Edexcel PMT Mocks Q15
  1. The curve \(C\) has equation
$$y = \frac { 1 } { 2 } x - \frac { 1 } { 4 } \sin 2 x \quad 0 < x < \pi$$ a. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sin ^ { 2 } x\)
b. Find the coordinates of the points of inflection of the curve.
Edexcel PMT Mocks Q16
16. Use algebra to prove that the product of any two consecutive odd numbers is an odd number.
Edexcel PMT Mocks Q1
  1. Given that \(\theta\) is small and is measured in radians, use the small angle approximations to find an approximate value of
$$\frac { \theta \tan 3 \theta } { \cos 2 \theta - 1 }$$
Edexcel PMT Mocks Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{63d85737-99d4-4916-a479-fe44f77b1505-03_442_552_351_721} \captionsetup{labelformat=empty} \caption{Figure 1}
\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.
Edexcel PMT Mocks Q3
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.
Edexcel PMT Mocks Q4
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]
\includegraphics[alt={},max width=\textwidth]{63d85737-99d4-4916-a479-fe44f77b1505-07_961_1002_296_513} \captionsetup{labelformat=empty} \caption{Figure 2}
\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.
Edexcel PMT Mocks Q5
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 $$\frac { d y } { d \theta } = - \frac { 5 } { 4 ( 1 + \sin 2 \theta ) } , \quad - \frac { \pi } { 4 } < \theta < \frac { 3 \pi } { 4 }$$
Edexcel PMT Mocks Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{63d85737-99d4-4916-a479-fe44f77b1505-10_951_1022_306_488} \captionsetup{labelformat=empty} \caption{Figure 3}
\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 \(k\).
Edexcel PMT Mocks Q7
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 \(k\).
Edexcel PMT Mocks Q8
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)
Edexcel PMT Mocks Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{63d85737-99d4-4916-a479-fe44f77b1505-16_871_1017_267_548} \captionsetup{labelformat=empty} \caption{Figure 4}
\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 \(Q\).