Questions - WJEC Unit 3 - A-Level Maths

WJEC Unit 3 2018 June Q1

1 The equation of a curve $C$ is given by the parametric equations $$x = \cos 2 \theta , y = \cos \theta$$ a) Find the Cartesian equation of $C$.
b) Show that the line $x - y + 1 = 0$ meets $C$ at the point $P$, where $\theta = \frac { \pi } { 3 }$, and at the point $Q$, where $\theta = \frac { \pi } { 2 }$. Write down the coordinates of $P$ and $Q$.
c) Determine the equations of the tangents to $C$ at $P$ and $Q$. Write down the coordinates of the point of intersection of the two tangents.

$\mathbf { 1 }$

Prove by contradiction that, for every real number $x$ such that $0 \leqslant x \leqslant \frac { \pi } { 2 }$, $$\sin x + \cos x \geqslant 1$$

1

2

a) Given that $f$ is a function,
i) state the condition for $f ^ { - 1 }$ to exist,
ii) find $f f ^ { - 1 } ( x )$.
b) The functions $g$ and $h$, are given by $$\begin{aligned} & g ( x ) = x ^ { 2 } - 1
& h ( x ) = \mathrm { e } ^ { x } + 1 \end{aligned}$$ i) Suggest a domain for $g$ such that $g ^ { - 1 }$ exists.
ii) Given the domain of $h$ is ( $- \infty , \infty$ ), find an expression for $h ^ { - 1 } ( x )$ and sketch, using the same axes, the graphs of $h ( x )$ and $h ^ { - 1 } ( x )$. Indicate clearly the asymptotes and the points where the graphs cross the coordinate axes.
iii) Determine an expression for $g h ( x )$ in its simplest form.

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3

a) Express $8 \sin \theta - 15 \cos \theta$ in the form $R \sin ( \theta - \alpha )$, where $R$ and $\alpha$ are constants with $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$.
b) Find all values of $\theta$ in the range $0 ^ { \circ } < \theta < 360 ^ { \circ }$ satisfying $$8 \sin \theta - 15 \cos \theta - 7 = 0$$ c) Determine the greatest value and the least value of the expression $$\frac { 1 } { 8 \sin \theta - 15 \cos \theta + 23 }$$

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$\mathbf { 4 }$

Evaluate a) $\int _ { 1 } ^ { 2 } x ^ { 3 } \ln x \mathrm {~d} x$.
b) $\int _ { 0 } ^ { 1 } \frac { 2 + x } { \sqrt { 4 - x ^ { 2 } } } \mathrm {~d} x$.

$\mathbf { 1 }$

$\mathbf { 5 }$

The variable $y$ satisfies the differential equation $$2 \frac { \mathrm {~d} y } { \mathrm {~d} x } = 5 - 2 y , \quad \text { where } x \geqslant 0$$ Given that $y = 1$ when $x = 0$, find an expression for $y$ in terms of $x$.

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$\mathbf { 6 }$

a) Differentiate the following functions with respect to $x$, simplifying your answer wherever possible. i) $e ^ { 3 \tan x }$,
ii) $\frac { \sin 2 x } { x ^ { 2 } }$.
b) A function is defined implicitly by $$3 x ^ { 2 } y + y ^ { 2 } - 5 x = 5$$ Find the equation of the normal at the point (1, 2).

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$\mathbf { 7 }$

By drawing suitable graphs, show that $x - 1 = \cos x$ has only one root. Starting with $x _ { 0 } = 1$, use the Newton-Raphson method to find the value of this root correct to two decimal places.

WJEC Unit 3 2022 June Q1

$\mathbf { 1 }$ & $\mathbf { 0 }$
\hline \end{tabular} \end{center} Solve the equation $$\frac { 6 x ^ { 5 } - 17 x ^ { 4 } - 5 x ^ { 3 } + 6 x ^ { 2 } } { ( 3 x + 2 ) } = 0$$

$\mathbf { 1 }$

a) Express $9 \cos x + 40 \sin x$ in the form $R \cos ( x - \alpha )$, where $R$ and $\alpha$ are constants with $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$. b) Find the maximum possible value of $\frac { 12 } { 9 \cos x + 40 \sin x + 47 }$.

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$\mathbf { 2 }$

The diagram below shows a sketch of the graph of $y = f ( x )$, where $$f ( x ) = 2 x ^ { 2 } + 12 x + 10 .$$ The graph intersects the $x$-axis at the points $( p , 0 ) , ( q , 0 )$ and the $y$-axis at the point $( 0,10 )$.
\includegraphics[max width=\textwidth, alt={}, center]{72bb1603-edbd-4e2e-bf2b-f33bb667e61b-5_1004_1171_648_440}
a) Write down the value of $f f ( p )$.
b) Determine the values of $p$ and $q$.
c) Express $f ( x )$ in the form $a ( x + b ) ^ { 2 } + c$, where $a , b , c$ are constants whose values are to be found. Write down the coordinates of the minimum point.
d) Explain why $f ^ { - 1 } ( x )$ does not exist.
e) The function $g ( x )$ is defined as $$g ( x ) = f ( x ) \quad \text { for } \quad - 3 \leqslant x < \infty .$$ i) Find an expression for $g ^ { - 1 } ( x )$.
ii) Sketch the graph of $y = g ^ { - 1 } ( x )$, indicating the coordinates of the points where the graph intersects the $x$-axis and the $y$-axis.

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$\mathbf { 3 }$

A function is defined by $f ( x ) = 2 x ^ { 3 } + 3 x - 5$. a) Prove that the graph of $f ( x )$ does not have a stationary point.
b) Show that the graph of $f ( x )$ does have a point of inflection and find the coordinates of the point of inflection.
c) Sketch the graph of $f ( x )$.

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4

Evaluate the integral $\int _ { 0 } ^ { \pi } x ^ { 2 } \sin x \mathrm {~d} x$.

$\mathbf { 1 }$

$\mathbf { 5 }$

A rectangle is inscribed in a semicircle with centre $O$ and radius 4. The point $P ( x , y )$ is the vertex of the rectangle in the first quadrant as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{72bb1603-edbd-4e2e-bf2b-f33bb667e61b-6_553_929_1430_593}
a) Express the area $A$ of the rectangle as a function of $x$.
b) Show that the maximum value of $A$ occurs when $y = x$.

$\mathbf { 1 }$

$\mathbf { 6 }$

The parametric equations of the curve $C$ are $$x = 3 - 4 t + t ^ { 2 } , \quad y = ( 4 - t ) ^ { 2 }$$ a) Find the coordinates of the points where $C$ meets the $y$-axis.
b) Show that the $x$-axis is a tangent to the curve $C$.

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7

a) Prove that $$\cos ( \alpha - \beta ) + \sin ( \alpha + \beta ) \equiv ( \cos \alpha + \sin \alpha ) ( \cos \beta + \sin \beta )$$ b) i) Hence show that $\frac { \cos 3 \theta + \sin 5 \theta } { \cos 4 \theta + \sin 4 \theta }$ can be expressed as $\cos \theta + \sin \theta$.
ii) Explain why $\frac { \cos 3 \theta + \sin 5 \theta } { \cos 4 \theta + \sin 4 \theta } \neq \cos \theta + \sin \theta$ when $\theta = \frac { 3 \pi } { 16 }$.

$\mathbf { 1 }$

$\mathbf { 8 }$

a) Use a suitable substitution to find $$\int \frac { x ^ { 2 } } { ( x + 3 ) ^ { 4 } } \mathrm {~d} x$$ b) Hence evaluate $\int _ { 0 } ^ { 1 } \frac { x ^ { 2 } } { ( x + 3 ) ^ { 4 } } \mathrm {~d} x$. END OF PAPER

WJEC Unit 3 2023 June Q1

1
Two real functions are defined as $$\begin{aligned} & f ( x ) = \frac { 8 } { x - 4 } \quad \text { for } \quad ( - \infty < x < 4 ) \cup ( 4 < x < \infty ) ,
& g ( x ) = ( x - 2 ) ^ { 2 } \quad \text { for } \quad - \infty < x < \infty . \end{aligned}$$ a) i) Find an expression for $f g ( x )$.
ii) Determine the values of $x$ for which $f g ( x )$ does not exist.
b) Find an expression for $f ^ { - 1 } ( x )$.

$\mathbf { 1 }$

A curve $C$ has equation $f ( x ) = 5 x ^ { 3 } + 2 x ^ { 2 } - 3 x$. a) Find the $x$-coordinate of the point of inflection. State, with a reason, whether the point of inflection is stationary or non-stationary.
b) Determine the range of values of $x$ for which $C$ is concave.

$\mathbf { 1 }$

$\mathbf { 2 }$

The rate of change of a variable $y$ with respect to $x$ is directly proportional to $y$. a) Write down a differential equation satisfied by $y$.
b) When $x = 1$ and $y = 0 \cdot 5$, the rate of change of $y$ with respect to $x$ is 2 . Find $y$ when $x = 3$.

$\mathbf { 1 }$

$\mathbf { 3 }$

The curve $C _ { 1 }$ has parametric equations $x = 3 p + 1 , y = 9 p ^ { 2 }$. The curve $C _ { 2 }$ has parametric equations $x = 4 q , y = 2 q$.
Find the Cartesian coordinates of the points of intersection of $C _ { 1 }$ and $C _ { 2 }$.

1

4

a) Use integration by parts to evaluate $\int _ { 0 } ^ { 1 } ( 3 x - 1 ) \mathrm { e } ^ { 2 x } \mathrm {~d} x$. b) Use the substitution $u = 1 - 2 \cos x$ to find $\int \frac { \sin x } { 1 - 2 \cos x } \mathrm {~d} x$. END OF PAPER

WJEC Unit 3 2024 June Q1

The function $f$ is given by

$$f ( x ) = \frac { 25 x + 32 } { ( 2 x - 5 ) ( x + 1 ) ( x + 2 ) }$$

Express $f ( x )$ in terms of partial fractions.
Show that $\int _ { 1 } ^ { 2 } f ( x ) \mathrm { d } x = - \ln P$, where $P$ is an integer whose value is to be found. \section*{
\includegraphics[max width=\textwidth, alt={}]{e4a4ea5b-7278-4735-925c-265a556ad679-03_65_1597_445_274}
}
Show that the sign of $f ( x )$ changes in the interval $x = 2$ to $x = 3$. Explain why the change of sign method fails to locate a root of the equation $f ( x ) = 0$ in this case.

WJEC Unit 3 2024 June Q2

2. (a) Find all values of $\theta$ in the range $0 ^ { \circ } < \theta < 360 ^ { \circ }$ satisfying $$3 \cot \theta + 4 \operatorname { cosec } ^ { 2 } \theta = 5 .$$ (b) By writing $24 \cos x - 7 \sin x$ in the form $R \cos ( x + \alpha )$, where $R$ and $\alpha$ are constants with $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$, solve the equation $$24 \cos x - 7 \sin x = 16$$ for values of $x$ between $0 ^ { \circ }$ and $360 ^ { \circ }$.

WJEC Unit 3 2024 June Q3

3. The diagram below shows a badge $O D C$. The shape $O A B$ is a sector of a circle centre $O$ and radius $r \mathrm {~cm}$. The shape $O D C$ is a sector of a circle with the same centre $O$. The length $A D$ is 5 cm and angle $A O B$ is $\frac { \pi } { 5 }$ radians. The area of the shaded region, $A B C D$, is $\frac { 13 \pi } { 2 } \mathrm {~cm} ^ { 2 }$.
\includegraphics[max width=\textwidth, alt={}, center]{e4a4ea5b-7278-4735-925c-265a556ad679-06_684_531_571_769}

Determine the value of $r$.
Calculate the perimeter of the shaded region.

WJEC Unit 3 2024 June Q4

4. A function $f$ is given by $f ( x ) = | 3 x + 4 |$.

Sketch the graph of $y = f ( x )$. Clearly label the coordinates of the point $A$, where the graph meets the $x$-axis, and the coordinates of the point $B$, where the graph cuts the $y$-axis.
On a separate set of axes, sketch the graph of $y = \frac { 1 } { 2 } f ( x ) - 6$, where the points $A$ and $B$ are transformed to the points $A ^ { \prime }$ and $B ^ { \prime }$.
Clearly label the coordinates of the points $A ^ { \prime }$ and $B ^ { \prime }$.

WJEC Unit 3 2024 June Q5

5. Prove by contradiction the following proposition: When $x$ is real and positive, $x + \frac { 81 } { x } \geqslant 18$.
\section*{PLEASE DO NOT WRITE ON THIS PAGE}

WJEC Unit 3 2024 June Q6

(a) Differentiate $\cos x$ from first principles.
(b) Differentiate $\mathrm { e } ^ { 3 x } \sin 4 x$ with respect to $x$.
(c) Find $\int x ^ { 2 } \sin 2 x \mathrm {~d} x$.
Showing all your working, evaluate
(a) $\quad \sum _ { r = 3 } ^ { 50 } ( 4 r + 5 )$,
(b) $\quad \sum _ { r = 2 } ^ { \infty } \left( 540 \times \left( \frac { 1 } { 3 } \right) ^ { r } \right)$.
The function $f$ is defined by

$$f ( x ) = x ^ { 3 } + 4 x ^ { 2 } - 3 x - 1$$ (a) Show that the equation $f ( x ) = 0$ has a root in the interval $[ 0,1 ]$.
(b) Using the Newton-Raphson method with $x _ { 0 } = 0 \cdot 8$,
(i) write down in full the decimal value of $x _ { 1 }$ as given in your calculator,
(ii) determine the value of this root correct to six decimal places.
Explain why the Newton-Raphson method does not work if $x _ { 0 } = \frac { 1 } { 3 }$.

WJEC Unit 3 2024 June Q9

9. The diagram below shows a sketch of the curve $C _ { 1 }$ with equation $y = - x ^ { 2 } + \pi x + 1$ and a sketch of the curve $C _ { 2 }$ with equation $y = \cos 2 x$. The curves intersect at the points where $x = 0$ and $x = \pi$.
\includegraphics[max width=\textwidth, alt={}, center]{e4a4ea5b-7278-4735-925c-265a556ad679-18_817_1173_577_431} Calculate the area of the shaded region enclosed by $C _ { 1 } , C _ { 2 }$ and the $x$-axis. Give your answer in terms of $\pi$.

WJEC Unit 3 2024 June Q10

4 marks

10. The function $f$ has domain $[ 4 , \infty )$ and is defined by $$f ( x ) = \frac { 2 ( 3 x + 1 ) } { x ^ { 2 } - 2 x - 3 } + \frac { x } { x + 1 }$$

Show that $f ( x ) = \frac { x + 2 } { x - 3 }$.
10. The function $f$ has domain $[ 4 , \infty )$ and is defined by $$f ( x ) = \frac { 2 ( 3 x + 1 ) } { x ^ { 2 } - 2 x - 3 } + \frac { x } { x + 1 } .$$
Show that $f ( x ) = \frac { x + 2 } { x - 3 }$. [4]
Determine the range of $f ( x )$.
Find an expression for $f ^ { - 1 } ( x )$ and write down the domain and range of $f ^ { - 1 }$.
Find the value of $x$ when $f ( x ) = f ^ { - 1 } ( x )$.

WJEC Unit 3 2024 June Q11

11. A curve is defined parametrically by $$x = 2 \theta + \sin 2 \theta , \quad y = 1 + \cos 2 \theta$$

Show that the gradient of the curve at the point with parameter $\theta$ is $- \tan \theta$.
Find the equation of the tangent to the curve at the point where $\theta = \frac { \pi } { 4 }$.

WJEC Unit 3 2024 June Q12

12. (a) Given that $\theta$ is small, show that $2 \cos \theta + \sin \theta - 1 \approx 1 + \theta - \theta ^ { 2 }$.
(b) Hence, when $\theta$ is small, show that $$\frac { 1 } { 2 \cos \theta + \sin \theta - 1 } \approx 1 + a \theta + b \theta ^ { 2 }$$ where $a , b$ are constants to be found.
\includegraphics[max width=\textwidth, alt={}, center]{e4a4ea5b-7278-4735-925c-265a556ad679-25_2382_1837_278_178}

WJEC Unit 3 2024 June Q14

14. (a) Given that $y = \frac { 1 + \ln x } { x }$, show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - \ln x } { x ^ { 2 } }$.
(b) Hence, solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { x ^ { 2 } t } { \ln x } ,$$ given that $t = 3$ when $x = 1$.
Give your answer in the form $t ^ { 2 } = g ( x )$, where $g$ is a function of $x$.

WJEC Unit 3 2024 June Q15

15. Robert wants to deposit $\pounds P$ into a savings account. He has a choice of two accounts.

Account $A$ offers an annual compound interest rate of $1 \%$.
Account $B$ offers an interest rate of $5 \%$ for the first year and an annual compound interest rate of $0.6 \%$ for each subsequent year.

After $n$ years, account $A$ is more profitable than account $B$. Find the smallest value of $n$.
Additional page, if required. Write the question number(s) in the left-hand margin. \section*{PLEASE DO NOT WRITE ON THIS PAGE} \section*{PLEASE DO NOT WRITE ON THIS PAGE}

WJEC Unit 3 Specimen Q1

Find a small positive value of $x$ which is an approximate solution of the equation.

$$\cos x - 4 \sin x = x ^ { 2 }$$

WJEC Unit 3 Specimen Q2

Air is pumped into a spherical balloon at the rate of $250 \mathrm {~cm} ^ { 3 }$ per second. When the radius of the balloon is 15 cm , calculate the rate at which the radius is increasing, giving your answer to three decimal places
(a) Sketch the graph of $y = x ^ { 2 } + 6 x + 13$, identifying the stationary point.
(b) The function $f$ is defined by $f ( x ) = x ^ { 2 } + 6 x + 13$ with domain $( a , b )$.
1. Explain why $f ^ { - 1 }$ does not exist when $a = - 10$ and $b = 10$.
2. Write down a value of $a$ and a value of $b$ for which the inverse of $f$ does exist and derive an expression for $f ^ { - 1 } ( x )$.
3. (a) Expand $( 1 - x ) ^ { - \frac { 1 } { 2 } }$ in ascending power of $x$ as far as the term in $x ^ { 2 }$. State the range of $x$ for which the expansion is valid.
  (b) By taking $x = \frac { 1 } { 10 }$, find an approximation for $\sqrt { 10 }$ in the form $\frac { a } { b }$, where $a$ and $b$ are to be determined.
4. Aled decides to invest $\pounds 1000$ in a savings scheme on the first day of each year. The scheme pays 8\% compound interest per annum, and interest is added on the last day of each year. The amount of savings, in pounds, at the end of the third year is given by
$$1000 \times 1 \cdot 08 + 1000 \times 1 \cdot 08 ^ { 2 } + 1000 \times 1 \cdot 08 ^ { 3 }$$ Calculate, to the nearest pound, the amount of savings at the end of thirty years.

WJEC Unit 3 Specimen Q6

6. The lengths of the sides of a fifteen-sided plane figure form an arithmetic sequence. The perimeter of the figure is 270 cm and the length of the largest side is eight times that of the smallest side. Find the length of the smallest side.

WJEC Unit 3 Specimen Q7

7. The curve $y = a x ^ { 4 } + b x ^ { 3 } + 18 x ^ { 2 }$ has a point of inflection at $( 1,11 )$.

Show that $2 a + b + 6 = 0$.
Find the values of the constants $a$ and $b$ and show that the curve has another point of inflection at $( 3,27 )$.
Sketch the curve, identifying all the stationary points including their nature.

WJEC Unit 3 Specimen Q8

8. (a) Integrate

$\quad \mathrm { e } ^ { - 3 x + 5 }$
$x ^ { 2 } \ln x$
(b) Use an appropriate substitution to show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { x ^ { 2 } } { \sqrt { 1 - x ^ { 2 } } } \mathrm {~d} x = \frac { \pi } { 12 } - \frac { \sqrt { 3 } } { 8 }$$

WJEC Unit 3 Specimen Q9

9.

\includegraphics[max width=\textwidth, alt={}]{59529f86-2ec7-4578-a68b-c4e3d8a4fc80-3_617_1084_1382_495}

The diagram above shows a sketch of the curves $y = x ^ { 2 } + 4$ and $y = 12 - x ^ { 2 }$.
Find the area of the region bounded by the two curves.

WJEC Unit 3 Specimen Q10

10. The equation $$1 + 5 x - x ^ { 4 } = 0$$ has a positive root $\alpha$.

Show that $\alpha$ lies between 1 and 2 .
Use the iterative sequence based on the arrangement $$x = \sqrt [ 4 ] { 1 + 5 x }$$ with starting value 1.5 to find $\alpha$ correct to two decimal places.
Use the Newton-Raphson method to find $\alpha$ correct to six decimal places.

WJEC Unit 3 Specimen Q11

11. (a) The curve $C$ is given by the equation $$x ^ { 4 } + x ^ { 2 } y + y ^ { 2 } = 13$$ Find the value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ at the point ( $- 1,3$ ).
(b) Show that the equation of the normal to the curve $y ^ { 2 } = 4 x$ at the point $P \left( p ^ { 2 } , 2 p \right)$ is $$y + p x = 2 p + p ^ { 3 }$$ Given that $p \neq 0$ and that the normal at $P$ cuts the $x$-axis at $B ( b , 0 )$, show that $b > 2$.

WJEC Unit 3 Specimen Q12

12. (a) Differentiate $\cos x$ from first principles.
(b) Differentiate the following with respect to $x$, simplifying your answer as far as possible.

$\frac { 3 x ^ { 2 } } { x ^ { 3 } + 1 }$
$x ^ { 3 } \tan 3 x$

WJEC Unit 3 Specimen Q13

13. (a) Solve the equation $$\operatorname { cosec } ^ { 2 } x + \cot ^ { 2 } x = 5$$ for $0 ^ { \circ } \leq x \leq 360 ^ { \circ }$.
(b) (i) Express $4 \sin \theta + 3 \cos \theta$ in the form $R \sin ( \theta + \alpha )$, where $R > 0$ and $0 ^ { \circ } \leq \alpha \leq 90 ^ { \circ }$.
(ii) Solve the equation $$4 \sin \theta + 3 \cos \theta = 2$$ for $0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }$, giving your answer correct to the nearest degree

Questions — WJEC Unit 3 (28 questions)

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WJEC Unit 3 2018 June Q1

WJEC Unit 3 2022 June Q1

WJEC Unit 3 2023 June Q1

WJEC Unit 3 2024 June Q1

WJEC Unit 3 2024 June Q2

WJEC Unit 3 2024 June Q3

WJEC Unit 3 2024 June Q4

WJEC Unit 3 2024 June Q5

WJEC Unit 3 2024 June Q6

WJEC Unit 3 2024 June Q9

WJEC Unit 3 2024 June Q10

WJEC Unit 3 2024 June Q11

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WJEC Unit 3 2024 June Q14

WJEC Unit 3 2024 June Q15

WJEC Unit 3 Specimen Q1

WJEC Unit 3 Specimen Q2

WJEC Unit 3 Specimen Q6

WJEC Unit 3 Specimen Q7

WJEC Unit 3 Specimen Q8

WJEC Unit 3 Specimen Q9

WJEC Unit 3 Specimen Q10

WJEC Unit 3 Specimen Q11

WJEC Unit 3 Specimen Q12

WJEC Unit 3 Specimen Q13