Questions P4 - A-Level Maths

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Edexcel P4 2018 Specimen Q4

9 marks Standard +0.3

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4de08317-5fb9-4789-8d57-ccf463224c78-10_899_759_127_621} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the curve $C$ with parametric equations $$x = \sqrt { 3 } \sin 2 t \quad y = 4 \cos ^ { 2 } t \quad 0 \leqslant t \leqslant \pi$$

Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = k \sqrt { 3 } \tan 2 t$, where $k$ is a constant to be found.
Find an equation of the tangent to $C$ at the point where $t = \frac { \pi } { 3 }$ Give your answer in the form $y = a x + b$, where $a$ and $b$ are constants.

Edexcel P4 2018 Specimen Q5

8 marks Standard +0.3

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4de08317-5fb9-4789-8d57-ccf463224c78-14_614_858_303_552} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of part of the curve with equation $y = 4 x - x \mathrm { e } ^ { \frac { 1 } { 2 } x } , x \geqslant 0$ The curve meets the $x$-axis at the origin $O$ and cuts the $x$-axis at the point $A$ .

Find,in terms of $\ln 2$ ,the $x$ coordinate of the point $A$ .
Find $\int x \mathrm { e } ^ { \frac { 1 } { 2 } x } \mathrm {~d} x$ The finite region $R$ ,shown shaded in Figure 2,is bounded by the $x$-axis and the curve with equation $y = 4 x - x \mathrm { e } ^ { \frac { 1 } { 2 } x } , x \geqslant 0$
Find,by integration,the exact value for the area of $R$ . Give your answer in terms of $\ln 2$ \includegraphics[max width=\textwidth, alt={}, center]{4de08317-5fb9-4789-8d57-ccf463224c78-18_2655_1943_114_118}

Edexcel P4 2018 Specimen Q6

4 marks Standard +0.3

6. Prove by contradiction that, if $a , b$ are positive real numbers, then $a + b \geqslant 2 \sqrt { a b }$ \includegraphics[max width=\textwidth, alt={}, center]{4de08317-5fb9-4789-8d57-ccf463224c78-20_2655_1943_114_118}

Edexcel P4 2018 Specimen Q7

5 marks Standard +0.3

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{4de08317-5fb9-4789-8d57-ccf463224c78-21_664_1244_301_351} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Figure 3 shows a sketch of the curve $C$ with parametric equations $$x = 4 \cos \left( t + \frac { \pi } { 6 } \right) \quad y = 2 \sin t \quad 0 \leqslant t \leqslant 2 \pi$$

Show that $$x + y = 2 \sqrt { 3 } \cos t$$
Show that a cartesian equation of $C$ is $$( x + y ) ^ { 2 } + a y ^ { 2 } = b$$ where $a$ and $b$ are integers to be found. \includegraphics[max width=\textwidth, alt={}, center]{4de08317-5fb9-4789-8d57-ccf463224c78-22_2673_1948_107_118}

Edexcel P4 2018 Specimen Q8

11 marks Standard +0.3

8. Water is being heated in a kettle. At time $t$ seconds, the temperature of the water is $\theta ^ { \circ } \mathrm { C }$. The rate of increase of the temperature of the water at time $t$ is modelled by the differential equation $$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = \lambda ( 120 - \theta ) \quad \theta \leqslant 100$$ where $\lambda$ is a positive constant.
Given that $\theta = 20$ when $t = 0$

solve this differential equation to show that $$\theta = 120 - 100 \mathrm { e } ^ { - \lambda t }$$ When the temperature of the water reaches $100 ^ { \circ } \mathrm { C }$, the kettle switches off.
Given that $\lambda = 0.01$, find the time, to the nearest second, when the kettle switches off.
\includegraphics[max width=\textwidth, alt={}]{4de08317-5fb9-4789-8d57-ccf463224c78-26_2642_1833_118_118}

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Edexcel P4 2018 Specimen Q9

15 marks Standard +0.3

With respect to a fixed origin $O$, the line $l _ { 1 }$ is given by the equation

$$\mathbf { r } = \left( \begin{array} { r } 8 \\ 1 \\ - 3 \end{array} \right) + \mu \left( \begin{array} { r } - 5 \\ 4 \\ 3 \end{array} \right)$$ where $\mu$ is a scalar parameter.
The point $A$ lies on $l _ { 1 }$ where $\mu = 1$

Find the coordinates of $A$. The point $P$ has position vector $\left( \begin{array} { l } 1 \\ 5 \\ 2 \end{array} \right)$ The line $l _ { 2 }$ passes through the point $P$ and is parallel to the line $l _ { 1 }$
Write down a vector equation for the line $l _ { 2 }$
Find the exact value of the distance $A P$. Give your answer in the form $k \sqrt { 2 }$, where $k$ is a constant to be found. The acute angle between $A P$ and $l _ { 2 }$ is $\theta$
Find the value of $\cos \theta$ A point $E$ lies on the line $l _ { 2 }$ Given that $A P = P E$,
find the area of triangle $A P E$,
find the coordinates of the two possible positions of $E$.

Edexcel P4 2021 October Q8

7 marks Standard +0.8

Find $\int x ^ { 2 } \ln x \mathrm {~d} x$ Figure 3 shows a sketch of part of the curve with equation $$y = x \ln x \quad x > 0$$ The region $R$, shown shaded in Figure 3, lies entirely above the $x$-axis and is bounded by the curve, the $x$-axis and the line with equation $x = \mathrm { e }$. This region is rotated through $2 \pi$ radians about the $x$-axis to form a solid of revolution.
Find the exact volume of the solid formed, giving your answer in simplest form. \section*{8. In this question you must show all stages of your working.
In this question you must show all stages of your working.}

Edexcel P4 2024 June Q1

5 marks Standard +0.3

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Find $$\int_0^{\pi/6} x \cos 3x \, dx$$ giving your answer in simplest form. [5]

Edexcel P4 2024 June Q2

6 marks Moderate -0.8

With respect to a fixed origin, $O$, the point $A$ has position vector $$\overrightarrow{OA} = \begin{pmatrix} 7 \\ 2 \\ -5 \end{pmatrix}$$ Given that $$\overrightarrow{AB} = \begin{pmatrix} -2 \\ 4 \\ 3 \end{pmatrix}$$

find the coordinates of the point $B$. [2]

The point $C$ has position vector $$\overrightarrow{OC} = \begin{pmatrix} a \\ 5 \\ -1 \end{pmatrix}$$ where $a$ is a constant. Given that $\overrightarrow{OC}$ is perpendicular to $\overrightarrow{BC}$

find the possible values of $a$. [4]

Edexcel P4 2024 June Q3

7 marks Standard +0.3

The curve $C$ is defined by the equation $$8x^3 - 3y^2 + 2xy = 9$$ Find an equation of the normal to $C$ at the point $(2, 5)$, giving your answer in the form $ax + by + c = 0$, where $a$, $b$ and $c$ are integers. [7]

Edexcel P4 2024 June Q4

6 marks Standard +0.3

\includegraphics{figure_1} Figure 1 shows a sketch of a segment $PQRP$ of a circle with centre $O$ and radius $5$ cm. Given that • angle $PQR$ is $\theta$ radians • $\theta$ is increasing, from $0$ to $\pi$, at a constant rate of $0.1$ radians per second • the area of the segment $PQRP$ is $A$ cm²

show that $$\frac{dA}{d\theta} = K(1 - \cos \theta)$$ where $K$ is a constant to be found. [2]
Find, in cm²s⁻¹, the rate of increase of the area of the segment when $\theta = \frac{\pi}{3}$ [4]

Edexcel P4 2024 June Q5

13 marks Standard +0.3

\includegraphics{figure_2} Figure 2 shows a sketch of the curve defined by the parametric equations $$x = t^2 + 2t \quad y = \frac{2}{t(3-t)} \quad a \leq t \leq b$$ where $a$ and $b$ are constants. The ends of the curve lie on the line with equation $y = 1$

Find the value of $a$ and the value of $b$ [2]

The region $R$, shown shaded in Figure 2, is bounded by the curve and the line with equation $y = 1$

Show that the area of region $R$ is given by $$M - k \int_a^b \frac{t+1}{t(3-t)} dt$$ where $M$ and $k$ are constants to be found. [5]
1. Write $\frac{t+1}{t(3-t)}$ in partial fractions.
2. Use algebraic integration to find the exact area of $R$, giving your answer in simplest form. [6]

Edexcel P4 2024 June Q6

10 marks Standard +0.3

With respect to a fixed origin $O$, the line $l_1$ is given by the equation $$\mathbf{r} = \mathbf{i} + 2\mathbf{j} + 5\mathbf{k} + \lambda(8\mathbf{i} - \mathbf{j} + 4\mathbf{k})$$ where $\lambda$ is a scalar parameter. The point $A$ lies on $l_1$ Given that $|\overrightarrow{OA}| = 5\sqrt{10}$

show that at $A$ the parameter $\lambda$ satisfies $$81\lambda^2 + 52\lambda - 220 = 0$$ [3]

Hence

1. show that one possible position vector for $A$ is $-15\mathbf{i} + 4\mathbf{j} - 3\mathbf{k}$
2. find the other possible position vector for $A$. [3]

The line $l_2$ is parallel to $l_1$ and passes through $O$. Given that • $\overrightarrow{OA} = -15\mathbf{i} + 4\mathbf{j} - 3\mathbf{k}$ • point $B$ lies on $l_2$ where $|\overrightarrow{OB}| = 4\sqrt{10}$

find the area of triangle $OAB$, giving your answer to one decimal place. [4]

Edexcel P4 2024 June Q7

11 marks Standard +0.3

The current, $x$ amps, at time $t$ seconds after a switch is closed in a particular electric circuit is modelled by the equation $$\frac{dx}{dt} = k - 3x$$ where $k$ is a constant. Initially there is zero current in the circuit.

Solve the differential equation to find an equation, in terms of $k$, for the current in the circuit at time $t$ seconds. Give your answer in the form $x = f(t)$. [6]

Given that in the long term the current in the circuit approaches $7$ amps,

find the value of $k$. [2]
Hence find the time in seconds it takes for the current to reach $5$ amps, giving your answer to $2$ significant figures. [3]

Edexcel P4 2024 June Q8

8 marks Standard +0.8

$$f(x) = (8 - 3x)^{\frac{4}{3}} \quad 0 < x < \frac{8}{3}$$

Show that the binomial expansion of $f(x)$ in ascending powers of $x$ up to and including the term in $x^3$ is $$A - 8x + \frac{x^2}{2} + Bx^3 + ...$$ where $A$ and $B$ are constants to be found. [4]
Use proof by contradiction to prove that the curve with equation $$y = 8 + 8x - \frac{15}{2}x^2$$ does not intersect the curve with equation $$y = A - 8x + \frac{x^2}{2} + Bx^3 \quad 0 < x < \frac{8}{3}$$ where $A$ and $B$ are the constants found in part (a). (Solutions relying on calculator technology are not acceptable.) [4]

Edexcel P4 2024 June Q9

9 marks Challenging +1.2

\includegraphics{figure_3} The curve $C$, shown in Figure 3, has equation $$y = \frac{x^{-\frac{1}{4}}}{\sqrt{1+x}\left(\arctan\sqrt{x}\right)}$$ The region $R$, shown shaded in Figure 3, is bounded by $C$, the line with equation $x = 3$, the $x$-axis and the line with equation $x = \frac{1}{3}$ The region $R$ is rotated through $360°$ about the $x$-axis to form a solid. Using the substitution $\tan u = \sqrt{x}$

show that the volume $V$ of the solid formed is given by $$k \int_a^b \frac{1}{u^2} du$$ where $k$, $a$ and $b$ are constants to be found. [6]
Hence, using algebraic integration, find the value of $V$ in simplest form. [3]

Edexcel P4 2022 October Q1

3 marks Moderate -0.8

A curve $C$ has parametric equations $$x = \frac{t}{t-3}, \quad y = \frac{1}{t} + 2, \quad t \in \mathbb{R}, \quad t > 3$$ Show that all points on $C$ lie on the curve with Cartesian equation $$y = \frac{ax - 1}{bx}$$ where $a$ and $b$ are constants to be found. [3]

Edexcel P4 2022 October Q2

7 marks Moderate -0.3

Express $\frac{3x}{(2x-1)(x-2)}$ in partial fraction form. [3]
Hence show that $$\int_5^{25} \frac{3x}{(2x-1)(x-2)} \, dx = \ln k$$ where $k$ is a fully simplified fraction to be found. (Solutions relying entirely on calculator technology are not acceptable.) [4]

Edexcel P4 2022 October Q3

5 marks Standard +0.3

\includegraphics{figure_1} Figure 1 shows a sketch of triangle $PQR$. Given that • $\overrightarrow{PQ} = 2\mathbf{i} - 3\mathbf{j} + 4\mathbf{k}$ • $\overrightarrow{PR} = 8\mathbf{i} - 5\mathbf{j} + 3\mathbf{k}$

Find $\overrightarrow{RQ}$ [2]
Find the size of angle $PQR$, in degrees, to three significant figures. [3]

Edexcel P4 2022 October Q4

8 marks Standard +0.3

$$g(x) = \frac{1}{\sqrt{4-x^2}}$$

Find, in ascending powers of $x$, the first four non-zero terms of the binomial expansion of $g(x)$. Give each coefficient in simplest form. [5]
State the range of values of $x$ for which this expansion is valid. [1]
Use the expansion from part (a) to find a fully simplified rational approximation for $\sqrt{3}$ Show your working and make your method clear. [2]

Edexcel P4 2022 October Q5

6 marks Challenging +1.2

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. \includegraphics{figure_2} Figure 2 shows a sketch of part of the curve with equation $$y = \frac{12\sqrt{x}}{(2x^2 + 3)^3}$$ The region $R$, shown shaded in Figure 2, is bounded by the curve, the line with equation $x = \frac{1}{\sqrt{2}}$, the $x$-axis and the line with equation $x = k$. This region is rotated through $360°$ about the $x$-axis to form a solid of revolution. Given that the volume of this solid is $\frac{713\pi}{648}$, use algebraic integration to find the exact value of the constant $k$. [6]

Edexcel P4 2022 October Q6

8 marks Standard +0.3

\includegraphics{figure_3} Figure 3 shows a sketch of the curve $C$ with parametric equations $$x = 1 + 3\tan t, \quad y = 2\cos 2t, \quad -\frac{\pi}{6} \leq t \leq \frac{\pi}{3}$$ The curve crosses the $x$-axis at point $P$, as shown in Figure 3.

Find the equation of the tangent to $C$ at $P$, writing your answer in the form $y = mx + c$, where $m$ and $c$ are constants to be found. [5]

The curve $C$ has equation $y = f(x)$, where $f$ is a function with domain $\left[k, 1 + 3\sqrt{3}\right]$

Find the exact value of the constant $k$. [1]
Find the range of $f$. [2]

Edexcel P4 2022 October Q7

12 marks Standard +0.3

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

Use the substitution $u = e^x - 3$ to show that $$\int_{\ln 5}^{\ln 7} \frac{4e^{3x}}{e^x - 3} \, dx = a + b \ln 2$$ where $a$ and $b$ are constants to be found. [7]
Show, by integration, that $$\int 3e^x \cos 2x \, dx = pe^x \sin 2x + qe^x \cos 2x + c$$ where $p$ and $q$ are constants to be found and $c$ is an arbitrary constant. [5]