Questions - WJEC - A-Level Maths

WJEC Further Unit 1 2018 June Q1

The matrices $\mathbf { A }$ and $\mathbf { B }$ are such that $\mathbf { A } = \left[ \begin{array} { c c } 4 & 2
- 1 & - 3 \end{array} \right]$ and $\mathbf { B } = \left[ \begin{array} { l l } 4 & 2
2 & 1 \end{array} \right]$.
1. Explain why $\mathbf { B }$ has no inverse.
2. 1. Find the inverse of $\mathbf { A }$.
  2. Hence, find the matrix $\mathbf { X }$, where $\mathbf { A X } = \left[ \begin{array} { c } - 4
    1 \end{array} \right]$.
3. Prove, by mathematical induction, that $\sum _ { r = 1 } ^ { n } r ( r + 3 ) = \frac { 1 } { 3 } n ( n + 1 ) ( n + 5 )$ for all positive integers $n$.
4. A cubic equation has roots $\alpha , \beta , \gamma$ such that
$$\alpha + \beta + \gamma = - 9 , \quad \alpha \beta + \beta \gamma + \gamma \alpha = 20 , \quad \alpha \beta \gamma = 0$$

WJEC Further Unit 1 2018 June Q4

4. A complex number is defined by $z = - 3 + 4 \mathrm { i }$.

1. Express $z$ in the form $r ( \cos \theta + \mathrm { i } \sin \theta )$, where $- \pi \leqslant \theta \leqslant \pi$.
2. Express $\bar { z }$, the complex conjugate of $z$, in the form $r ( \cos \theta + \mathrm { i } \sin \theta )$. Another complex number is defined as $w = \sqrt { 5 } ( \cos 2 \cdot 68 + \mathrm { i } \sin 2 \cdot 68 )$.
Express $z w$ in the form $r ( \cos \theta + \mathrm { i } \sin \theta )$.

WJEC Further Unit 1 2018 June Q5

5. (a) Show that $\frac { 2 } { n - 1 } - \frac { 2 } { n + 1 }$ can be expressed as $\frac { 4 } { \left( n ^ { 2 } - 1 \right) }$.
(b) Hence, find an expression for $\sum _ { r = 2 } ^ { n } \frac { 4 } { \left( r ^ { 2 } - 1 \right) }$ in the form $\frac { ( a n + b ) ( n + c ) } { n ( n + 1 ) }$, where $a , b , c$ are integers whose values are to be determined.
(c) Explain why $\sum _ { r = 1 } ^ { 100 } \frac { 4 } { \left( r ^ { 2 } - 1 \right) }$ cannot be calculated.

WJEC Further Unit 1 2018 June Q6

6. (a) Show that $1 - 2 \mathrm { i }$ is a root of the cubic equation $x ^ { 3 } + 5 x ^ { 2 } - 9 x + 35 = 0$.
(b) Find the other two roots of the equation.

WJEC Further Unit 1 2018 June Q7

7. The complex number $z$ is represented by the point $P ( x , y )$ in the Argand diagram and $$| z - 4 - \mathrm { i } | = | z + 2 |$$

Find the equation of the locus of $P$.
Give a geometric interpretation of the locus of $P$.

WJEC Further Unit 1 2018 June Q8

8. The transformation $T$ in the plane consists of a translation in which the point $( x , y )$ is transformed to the point $( x - 1 , y + 1 )$, followed by a reflection in the line $y = x$.

Determine the $3 \times 3$ matrix which represents $T$.
Find the equation of the line of fixed points of $T$.
Find $T ^ { 2 }$ and hence write down $T ^ { - 1 }$.

WJEC Further Unit 1 2018 June Q9

9. The line $L _ { 1 }$ passes through the points $A ( 1,2 , - 3 )$ and $B ( - 2,1,0 )$.

1. Show that the vector equation of $L _ { 1 }$ can be written as $$\mathbf { r } = ( 1 - 3 \lambda ) \mathbf { i } + ( 2 - \lambda ) \mathbf { j } + ( - 3 + 3 \lambda ) \mathbf { k }$$
2. Write down the equation of $L _ { 1 }$ in Cartesian form. The vector equation of the line $L _ { 2 }$ is given by $\mathbf { r } = 2 \mathbf { i } - 4 \mathbf { j } + \mu ( 4 \mathbf { j } + 7 \mathbf { k } )$.
Show that $L _ { 1 }$ and $L _ { 2 }$ do not intersect.
Find a vector in the direction of the common perpendicular to $L _ { 1 }$ and $L _ { 2 }$.

WJEC Unit 1 2018 June Q1

$\mathbf { 1 }$ & $\mathbf { 3 }$
\hline \end{tabular} \end{center} A curve $C$ has equation $y = x ^ { 3 } - 3 x ^ { 2 }$. a) Find the stationary points of $C$ and determine their nature.
b) Draw a sketch of $C$, clearly indicating the stationary points and the points where the curve crosses the coordinate axes.
c) Without performing the integration, state whether $\int _ { 0 } ^ { 3 } \left( x ^ { 3 } - 3 x ^ { 2 } \right) \mathrm { d } x$ is positive or
negative, giving a reason for your answer.

1

4

In each of the two statements below, $c$ and $d$ are real numbers. One of the statements is true, while the other is false. $$\begin{aligned} & \text { A : } \quad ( 2 c - d ) ^ { 2 } = 4 c ^ { 2 } - d ^ { 2 } , \text { for all values of } c \text { and } d .
& \text { B : } \quad 8 c ^ { 3 } - d ^ { 3 } = ( 2 c - d ) \left( 4 c ^ { 2 } + 2 c d + d ^ { 2 } \right) , \text { for all values of } c \text { and } d . \end{aligned}$$ a) Identify the statement which is false. Show, by counter example, that this statement is in fact false.
b) Identify the statement which is true. Give a proof to show that this statement is in fact true.

1

5

The value of a car, $\pounds V$, may be modelled as a continuous variable. At time $t$ years, the value of the car is given by $V = A \mathrm { e } ^ { k t }$, where $A$ and $k$ are constants. When the car is new, it is worth $\pounds 30000$. When the car is two years old, it is worth $\pounds 20000$. Determine the value of the car when it is six years old, giving your answer correct to the nearest $\pounds 100$.

1

6

The curve $C$ has equation $y = 7 + 13 x - 2 x ^ { 2 }$. The point $P$ lies on $C$ and is such that the tangent to $C$ at $P$ has equation $y = x + c$, where $c$ is a constant. Find the coordinates of $P$ and the value of $c$.

1

7

a) Solve $2 \log _ { 10 } x = 1 + \log _ { 10 } 5 - \log _ { 10 } 2$.
b) Solve $3 = 2 \mathrm { e } ^ { 0 \cdot 5 x }$.
c) Express $4 ^ { x } - 10 \times 2 ^ { x }$ in terms of $y$, where $y = 2 ^ { x }$. Hence solve the equation $4 ^ { x } - 10 \times 2 ^ { x } = - 16$.

$\mathbf { 1 }$

$\mathbf { 8 }$

The coordinates of three points $A , B , C$ are $( 4,6 ) , ( - 3,5 )$ and $( 5 , - 1 )$ respectively. a) Show that $B \widehat { A C }$ is a right angle.
b) A circle passes through all three points $A , B , C$. Determine the equation of the circle.

WJEC Unit 1 2019 June Q1

$\mathbf { 1 }$ & $\mathbf { 1 }$
\hline \end{tabular} \end{center} Two quantities are related by the equation $Q = 1 \cdot 25 P ^ { 3 }$. Explain why the graph of $\log _ { 10 } Q$ against $\log _ { 10 } P$ is a straight line. State the gradient of the straight line and the intercept on the $\log _ { 10 } Q$ axis of the graph.

$\mathbf { 1 }$

$\mathbf { 2 }$

In the binomial expansion of $( 2 - 5 x ) ^ { 8 }$, find a) the number of terms,
b) the $4 ^ { \text {th } }$ term, when the expansion is in ascending powers of $x$,
c) the greatest positive coefficient.

$\mathbf { 1 }$

$\mathbf { 3 }$

A curve $C$ has equation $y = \frac { 1 } { 9 } x ^ { 3 } - k x + 5$. A point $Q$ lies on $C$ and is such that the tangent to $C$ at $Q$ has gradient - 9 . The $x$-coordinate of $Q$ is 3 . a) Show that $k = 12$.
b) Find the coordinates of each of the stationary points of $C$ and determine their nature.
c) Sketch the curve $C$, clearly labelling the stationary points and the point where the curve crosses the $y$-axis.

$\mathbf { 1 }$

$\mathbf { 4 }$

The diagram below shows a triangle $A B C$ with $A C = 5 \mathrm {~cm} , A B = x \mathrm {~cm} , B C = y \mathrm {~cm}$ and angle $B A C = 120 ^ { \circ }$. The area of the triangle $A B C$ is $14 \mathrm {~cm} ^ { 2 }$. Find the value of $x$ and the value of $y$. Give your answers correct to 2 decimal places.

$\mathbf { 1 }$

$\mathbf { 5 }$

Prove that $f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 13 x - 7$ is an increasing function.

$\mathbf { 1 }$

$\mathbf { 6 }$

The diagram below shows a curve with equation $y = ( x + 2 ) ( x - 2 ) ( x + 1 )$.

\includegraphics[max width=\textwidth, alt={}]{2c33cbe4-b65e-4eae-aa2f-9d1d0f5cb9bd-7_754_743_724_678}

Calculate the total area of the two shaded regions. \section*{END OF PAPER}

WJEC Unit 1 2022 June Q1

$\mathbf { 1 }$ & $\mathbf { 2 }$
\hline \end{tabular} \end{center} a) Solve the equation $2 x ^ { 3 } - x ^ { 2 } - 5 x - 2 = 0$. b) Find all values of $\theta$ in the range $0 ^ { \circ } < \theta < 180 ^ { \circ }$ satisfying $$\cos \left( 2 \theta - 51 ^ { \circ } \right) = 0 \cdot 891$$

$\mathbf { 1 }$

$\mathbf { 3 }$

Find the term which is independent of $x$ in the expansion of $\frac { ( 2 - 3 x ) ^ { 5 } } { x ^ { 3 } }$.

$\mathbf { 1 }$

$\mathbf { 4 }$

A curve $C$ has equation $f ( x ) = 3 x ^ { 3 } - 5 x ^ { 2 } + x - 6$. a) Find the coordinates of the stationary points of $C$ and determine their nature.
b) Without solving the equations, determine the number of distinct real roots for each of the following:
i) $3 x ^ { 3 } - 5 x ^ { 2 } + x + 1 = 0$,
ii) $\quad 6 x ^ { 3 } - 10 x ^ { 2 } + 2 x + 1 = 0$.

$\mathbf { 1 }$

$\mathbf { 5 }$

Solve the simultaneous equations $$\begin{aligned} & 3 \log _ { a } \left( x ^ { 2 } y \right) - \log _ { a } \left( x ^ { 2 } y ^ { 2 } \right) + \log _ { a } \left( \frac { 9 } { x ^ { 2 } y ^ { 2 } } \right) = \log _ { a } 36
& \log _ { a } y - \log _ { a } ( x + 3 ) = 0 \end{aligned}$$

$\mathbf { 1 }$

$\mathbf { 6 }$

The vectors $\mathbf { a }$ and $\mathbf { b }$ are defined by $\mathbf { a } = 2 \mathbf { i } - \mathbf { j }$ and $\mathbf { b } = \mathbf { i } - 3 \mathbf { j }$. a) Find a unit vector in the direction of $\mathbf { a }$.
b) Determine the angle $\mathbf { b }$ makes with the $x$-axis.
c) The vector $\mu \mathbf { a } + \mathbf { b }$ is parallel to $4 \mathbf { i } - 5 \mathbf { j }$.
i) Find the vector $\mu \mathbf { a } + \mathbf { b }$ in terms of $\mu , \mathbf { i }$ and j.
ii) Determine the value of $\mu$.

WJEC Unit 1 2023 June Q1

$\mathbf { 1 }$ & $\mathbf { 0 }$
\hline \end{tabular} \end{center} Solve the following equations for values of $x$. a) $\quad \ln ( 2 x + 5 ) = 3$
b) $\quad 5 ^ { 2 x + 1 } = 14$
c) $\quad 3 \log _ { 7 } ( 2 x ) - \log _ { 7 } \left( 8 x ^ { 2 } \right) + \log _ { 7 } x = \log _ { 3 } 81$

$\mathbf { 1 }$

The function $f$ is defined by $f ( x ) = \frac { 8 } { x ^ { 2 } }$. a) Sketch the graph of $y = f ( x )$.
b) On a separate set of axes, sketch the graph of $y = f ( x - 2 )$. Indicate the vertical asymptote and the point where the curve crosses the $y$-axis.
c) Sketch the graphs of $y = \frac { 8 } { x }$ and $y = \frac { 8 } { ( x - 2 ) ^ { 2 } }$ on the same set of axes. Hence state the number of roots of the equation $\frac { 8 } { ( x - 2 ) ^ { 2 } } = \frac { 8 } { x }$.

$\mathbf { 1 }$

$\mathbf { 2 }$

The position vectors of the points $A$ and $B$, relative to a fixed origin $O$, are given by $$\mathbf { a } = - 3 \mathbf { i } + 4 \mathbf { j } , \quad \mathbf { b } = 5 \mathbf { i } + 8 \mathbf { j }$$ respectively.
a) Find the vector $\mathbf { A B }$.
b) i) Find a unit vector in the direction of $\mathbf { a }$.
ii) The point $C$ is such that the vector $\mathbf { O C }$ is in the direction of $\mathbf { a }$. Given that the length of $\mathbf { O C }$ is 7 units, write down the position vector of $C$.
c) Calculate the angle $A O B$. \section*{

$\mathbf { 1 }$

$\mathbf { 3 }$

a) Find $\int \left( 4 x ^ { - \frac { 2 } { 3 } } + 5 x ^ { 3 } + 7 \right) \mathrm { d } x$.} b) The diagram below shows the graph of $y = x ( x + 6 ) ( x - 3 )$.
\includegraphics[max width=\textwidth, alt={}, center]{631084a7-d827-401a-af0b-bbe1860dc027-7_614_1107_641_470} Calculate the total area of the regions enclosed by the graph and the $x$-axis. 1 4 a) Two variables, $x$ and $y$, are such that the rate of change of $y$ with respect to $x$ is proportional to $y$. State a model which may be appropriate for $y$ in terms of $x$.
b) The concentration, $Y$ units, of a certain drug in a patient's body decreases exponentially with respect to time. At time $t$ hours the concentration can be modelled by $Y = A \mathrm { e } ^ { - k t }$, where $A$ and $k$ are constants. A patient was given a dose of the drug that resulted in an initial concentration of 5 units.
i) After 4 hours, the concentration had dropped to 1.25 units. Show that $k = 0 \cdot 3466$, correct to four decimal places.
ii) The minimum effective concentration of the drug is 0.6 units. How much longer would it take for the drug concentration to drop to the minimum effective level?

WJEC Unit 1 2024 June Q1

Given that $y = 12 \sqrt { x } - \frac { 27 } { x } + 4$, find the value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ when $x = 9$.
Find all values of $\theta$ in the range $0 ^ { \circ } < \theta < 180 ^ { \circ }$ that satisfy the equation

$$2 \sin 2 \theta = 1$$

WJEC Unit 1 2024 June Q3

3. Find $\int \left( 5 x ^ { \frac { 1 } { 4 } } + 3 x ^ { - 2 } - 2 \right) \mathrm { d } x$.

WJEC Unit 1 2024 June Q4

4. Given that $n$ is an integer such that $1 \leqslant n \leqslant 6$, use proof by exhaustion to show that $n ^ { 2 } - 2$ is not divisible by 3 .

WJEC Unit 1 2024 June Q5

5. A triangle $A B C$ has sides $A B = 6 \mathrm {~cm} , B C = 11 \mathrm {~cm}$ and $A C = 13 \mathrm {~cm}$. Calculate the area of the triangle.

WJEC Unit 1 2024 June Q6

6. (a) Find the exact value of $x$ that satisfies the equation $$\frac { 7 x ^ { \frac { 5 } { 4 } } } { x ^ { \frac { 1 } { 2 } } } = \sqrt { 147 }$$ (b) Show that $\frac { ( 8 x - 18 ) } { ( 2 \sqrt { x } - 3 ) }$, where $x \neq \frac { 9 } { 4 }$, may be written as $2 ( 2 \sqrt { x } + 3 )$.

WJEC Unit 1 2024 June Q7

7. (a) The line $L _ { 1 }$ passes through the points $A ( - 3,0 )$ and $B ( 1,4 )$. Determine the equation of $L _ { 1 }$.
(b) The line $L _ { 2 }$ has equation $y = 3 x - 3$.

Given that $L _ { 1 }$ and $L _ { 2 }$ intersect at the point $C$, find the coordinates of $C$.
The line $L _ { 2 }$ crosses the $x$-axis at the point $D$. Show that the coordinates of $D$ are $( 1,0 )$.
(c) Calculate the area of triangle $A C D$.
(d) Determine the angle $A C D$.

WJEC Unit 1 2024 June Q8

8. Prove that $x - 10 < x ^ { 2 } - 5 x$ for all real values of $x$.

WJEC Unit 1 2024 June Q9

9. (a) Write down the binomial expansion of $( 2 - x ) ^ { 6 }$ up to and including the term in $x ^ { 2 }$.
(b) Given that $$( 1 + a x ) ( 2 - x ) ^ { 6 } \equiv 64 + b x + 336 x ^ { 2 } + \ldots$$ find the values of the constants $a , b$.

WJEC Unit 1 2024 June Q10

10. Water is being emptied out of a sink. The depth of water, $y \mathrm {~cm}$, at time $t$ seconds, may be modelled by $$y = t ^ { 2 } - 14 t + 49 \quad 0 \leqslant t \leqslant 7$$

Find the value of $t$ when the depth of water is 25 cm .
Find the rate of decrease of the depth of water when $t = 3$.

WJEC Unit 1 2024 June Q11

11. (a) Sketch the graph of $y = 3 ^ { x }$. Clearly label the coordinates of the point where the graph crosses the $y$-axis.
(b) On the same set of axes, sketch the graph of $y = 3 ^ { ( x + 1 ) }$, clearly labelling the coordinates of the point where the graph crosses the $y$-axis.

WJEC Unit 1 2024 June Q12

12. A curve $C$ has equation $y = - x ^ { 3 } + 12 x - 20$.

Find the coordinates of the stationary points of $C$ and determine their nature.
Determine the range of values of $x$ for which the curve is decreasing. Give your answer in set notation.

WJEC Unit 1 2024 June Q13

13. The position vectors of the points $A$ and $B$, relative to a fixed origin $O$, are given by $$\mathbf { a } = 4 \mathbf { i } + 7 \mathbf { j } , \quad \mathbf { b } = \mathbf { i } + 3 \mathbf { j }$$ respectively.

Find the vector $\mathbf { A B }$.
Determine the distance between the points $A$ and $B$.
The position vector of the point $C$ is given by $\mathbf { c } = - 2 \mathbf { i } + 5 \mathbf { j }$. The point $D$ is such that the distance between $C$ and $D$ is equal to the distance between $A$ and $B$, and $C D$ is parallel to $A B$. Find the possible position vectors of the point $D$.

WJEC Unit 1 2024 June Q14

6 marks

14. The diagram below shows a sketch of the curve $C$ with equation $y = 2 - 3 x - 2 x ^ { 2 }$ and the line $L$ with equation $y = x + 2$. The curve and the line intersect the coordinate axes at the points $A$ and $B$.
\includegraphics[max width=\textwidth, alt={}, center]{9bb29d6e-2dbb-4212-b3e0-45e7b12c0c43-18_775_970_589_543}

Write down the coordinates of $A$ and $B$.

(b) Calculate the area enclosed by $C$ and $L$.
[6]
Examiner only

WJEC Unit 1 2024 June Q15

The diagram shows a sketch of part of the curve with equation $y = 2 \sin x + 3 \cos ^ { 2 } x - 3$. The curve crosses the $x$-axis at the points $O , A , B$ and $C$.
\includegraphics[max width=\textwidth, alt={}, center]{9bb29d6e-2dbb-4212-b3e0-45e7b12c0c43-20_620_1009_516_520}

Find the value of $x$ at each of the points $A , B$ and $C$.

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