Questions — WJEC Unit 1 (89 questions)

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WJEC Unit 1 2018 June Q1
Moderate -0.8
Showing all your working, simplify a) \(\frac { 24 \sqrt { a } } { ( \sqrt { a } + 3 ) ^ { 2 } - ( \sqrt { a } - 3 ) ^ { 2 } }\),
b) \(\frac { 3 \sqrt { 7 } + 5 \sqrt { 3 } } { \sqrt { 7 } + \sqrt { 3 } }\).
WJEC Unit 1 2018 June Q2
Standard +0.3
The points \(A\) and \(B\) have coordinates \(( - 1,10 )\) and \(( 5,1 )\) respectively. The straight line \(L\) has equation \(2 x - 3 y + 6 = 0\). a) The line \(L\) intersects the line \(A B\) at the point \(C\). Find the coordinates of \(C\).
b) Determine the ratio in which the line \(L\) divides the line \(A B\).
c) The line \(L\) crosses the \(x\)-axis at the point \(D\). Find the coordinates of \(D\).
d) i) Show that \(L\) is perpendicular to \(A B\).
ii) Calculate the area of the triangle \(A C D\).
WJEC Unit 1 2018 June Q3
Moderate -0.8
Solve the following equation for values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\). $$2 - 3 \cos ^ { 2 } \theta = 2 \sin \theta$$
04
a) Given that \(y = \frac { 5 } { x } + 6 \sqrt [ 3 ] { x }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 8\). b) Find \(\int \left( 5 x ^ { \frac { 3 } { 2 } } + 12 x ^ { - 5 } + 7 \right) \mathrm { d } x\).
05
The diagram below shows a sketch of \(y = f ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{805df86d-3115-4311-982a-263e114a9722-3_659_828_445_639}
a) Sketch the graph of \(y = 4 + f ( x )\), clearly indicating any asymptotes.
b) Sketch the graph of \(y = f ( x - 3 )\), clearly indicating any asymptotes.
□
0 6 \includegraphics[max width=\textwidth, alt={}, center]{805df86d-3115-4311-982a-263e114a9722-3_609_869_1491_619} The sketch shows the curve \(C\) with equation \(y = 14 + 5 x - x ^ { 2 }\) and line \(L\) with equation \(y = x + 2\). The line intersects the curve at the points \(A\) and \(B\).
a) Find the coordinates of \(A\) and \(B\).
b) Calculate the area enclosed by \(L\) and \(C\).
07
Prove that $$\frac { \sin ^ { 3 } \theta + \sin \theta \cos ^ { 2 } \theta } { \cos \theta } \equiv \tan \theta$$
WJEC Unit 1 2018 June Q8
Moderate -0.8
Given that \(( x - 2 )\) and \(( x + 2 )\) are factors of the polynomial \(2 x ^ { 3 } + p x ^ { 2 } + q x - 12\), a) find the values of \(p\) and \(q\),
b) determine the other factor of the polynomial.
WJEC Unit 1 2018 June Q9
5 marks Moderate -0.3
The triangle \(A B C\) is such that \(A C = 16 \mathrm {~cm} , A B = 25 \mathrm {~cm}\) and \(A \widehat { B C } = 32 ^ { \circ }\). Find two possible values for the area of the triangle \(A B C\).
10
a) Use the binomial theorem to expand \(( a + \sqrt { b } ) ^ { 4 }\).
b) Hence, deduce an expression in terms of \(a\) and \(b\) for \(( a + \sqrt { b } ) ^ { 4 } + ( a - \sqrt { b } ) ^ { 4 }\).
11
a) The vectors \(\mathbf { u }\) and \(\mathbf { v }\) are defined by \(\mathbf { u } = 9 \mathbf { i } - 40 \mathbf { j }\) and \(\mathbf { v } = 3 \mathbf { i } - 4 \mathbf { j }\). Determine the range of values for \(\mu\) such that \(\mu | \mathbf { v } | > | \mathbf { u } |\).
b) The point \(A\) has position vector \(11 \mathbf { i } - 4 \mathbf { j }\) and the point \(B\) has position vector \(21 \mathbf { i } + \mathbf { j }\). Determine the position vector of the point \(C\), which lies between \(A\) and \(B\), such that \(A C : C B\) is \(2 : 3\).
12
Find the values of \(m\) for which the equation \(4 x ^ { 2 } + 8 x - 8 = m ( 4 x - 3 )\) has real roots. [5]
WJEC Unit 1 2018 June Q13
Moderate -0.8
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.
14
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.
15
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 \(\pounds 100\).
16
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 \(c\).
17
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 \(4 ^ { x } - 10 \times 2 ^ { x } = - 16\).
WJEC Unit 1 2018 June Q18
Moderate -0.5
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 Q01
6 marks Challenging +1.2
Solve the following equation for values of \(\theta\) between \(0°\) and \(360°\). $$3\tan\theta + 2\cos\theta = 0$$ [6]
WJEC Unit 1 2019 June Q02
7 marks Standard +0.3
Find all the values of \(k\) for which the equation \(x^2 + 2kx + 9k = -4x\) has two distinct real roots. [7]
WJEC Unit 1 2019 June Q03
6 marks Standard +0.3
Use an algebraic method to solve the equation \(12x^3 - 29x^2 + 7x + 6 = 0\). Show all your working. [6]
WJEC Unit 1 2019 June Q04
15 marks Easy -1.3
The line \(L_1\) passes through the points \(A(-1, 3)\) and \(B(2, 9)\). The line \(L_2\) has equation \(2y + x = 25\) and intersects \(L_1\) at the point \(C\). \(L_2\) also intersects the \(x\)-axis at the point \(D\).
  1. Show that the equation of the line \(L_1\) is \(y = 2x + 5\). [3]
    1. Find the coordinates of the point \(D\).
    2. Show that \(L_1\) and \(L_2\) are perpendicular.
    3. Determine the coordinates of \(C\). [5]
  3. Find the length of \(CD\). [2]
  4. Calculate the angle \(ADB\). Give your answer in degrees, correct to one decimal place. [5]
WJEC Unit 1 2019 June Q05
3 marks Easy -1.8
Given that \(n\) is an integer such that \(1 \leq n \leq 4\), prove that \(2n^2 + 5\) is a prime number. [3]
WJEC Unit 1 2019 June Q06
5 marks Moderate -0.8
\(OABC\) is a parallelogram with \(O\) as origin. \includegraphics{figure_6} The position vector of \(A\) is \(\mathbf{a}\) and the position vector of \(C\) is \(\mathbf{c}\). The midpoint of \(AB\) is \(D\). The point \(E\) divides the line \(CB\) such that \(CE : EB = 2 : 1\).
  1. Find, in terms of \(\mathbf{a}\) and \(\mathbf{c}\),
    1. the vector \(\overrightarrow{AC}\),
    2. the position vector of \(D\),
    3. the position vector of \(E\). [3]
  2. Determine whether or not \(\overrightarrow{DE}\) is parallel to \(\overrightarrow{AC}\), clearly stating your reason. [2]
WJEC Unit 1 2019 June Q07
6 marks Moderate -0.8
Given that \(a\), \(b\) are integers, simplify the following. Show all your working.
  1. \(\frac{2\sqrt{3} + a}{\sqrt{3} - 1}\) [3]
  2. \(\frac{2\sqrt{6b^2} - \sqrt{27} + \sqrt{192}}{\sqrt{2}}\) [3]
WJEC Unit 1 2019 June Q08
8 marks Standard +0.3
  1. Given that \(y = 2x^2 - 5x\), find \(\frac{dy}{dx}\) from first principles. [5]
  2. Given that \(y = \frac{16}{5}x^4 + \frac{48}{x}\), find the value of \(\frac{dy}{dx}\) when \(x = 16\). [3]
WJEC Unit 1 2019 June Q09
12 marks Moderate -0.3
The points \(A(-2, 4)\) and \(B(6, 10)\) are such that \(AB\) is the diameter of a circle.
  1. Show that the centre of the circle has coordinates \((2, 7)\). [1]
  2. The equation of the circle is \(x^2 + y^2 + ax + by + c = 0\). Determine the values of \(a\), \(b\), \(c\). [3]
A straight line, with equation \(y = x + 6\), passes through the point \(A\) and cuts the circle again at the point \(C\).
  1. Find the coordinates of \(C\). [5]
  2. Calculate the exact area of the triangle \(ABC\). [3]
WJEC Unit 1 2019 June Q10
13 marks Standard +0.3
  1. Solve the following simultaneous equations. $$3^{3x} \times 9^y = 27$$ $$2^{-3x} \times 8^{-y} = \frac{1}{64}$$ [6]
  2. Find the value of \(x\) satisfying the equation $$\log_a 3 + 2\log_a x - \log_a(x - 1) = \log_a(5x + 2).$$ [7]
WJEC Unit 1 2019 June Q11
4 marks Moderate -0.8
Two quantities are related by the equation \(Q = 1.25P^3\). 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. [4]
WJEC Unit 1 2019 June Q12
6 marks Moderate -0.3
In the binomial expansion of \((2 - 5x)^8\), find
  1. the number of terms, [1]
  2. the \(4^{\text{th}}\) term, when the expansion is in ascending powers of \(x\), [2]
  3. the greatest positive coefficient. [3]
WJEC Unit 1 2019 June Q13
11 marks Moderate -0.8
A curve \(C\) has equation \(y = \frac{1}{9}x^3 - kx + 5\). 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 \(3\).
  1. Show that \(k = 12\). [3]
  2. Find the coordinates of each of the stationary points of \(C\) and determine their nature. [6]
  3. Sketch the curve \(C\), clearly labelling the stationary points and the point where the curve crosses the \(y\)-axis. [2]
WJEC Unit 1 2019 June Q14
6 marks Standard +0.3
The diagram below shows a triangle \(ABC\) with \(AC = 5\) cm, \(AB = x\) cm, \(BC = y\) cm and angle \(BAC = 120°\). The area of the triangle \(ABC\) is \(14\) cm\(^2\). \includegraphics{figure_14} Find the value of \(x\) and the value of \(y\). Give your answers correct to 2 decimal places. [6]
WJEC Unit 1 2019 June Q15
4 marks Moderate -0.5
Prove that \(f(x) = x^3 - 6x^2 + 13x - 7\) is an increasing function. [4]
WJEC Unit 1 2019 June Q16
8 marks Standard +0.8
The diagram below shows a curve with equation \(y = (x + 2)(x - 2)(x + 1)\). \includegraphics{figure_16} Calculate the total area of the two shaded regions. [8]
WJEC Unit 1 2022 June Q1
3 marks Easy -1.2
Write down the inverse function of \(y = e^x\). On the same set of axes, sketch the graphs of \(y = e^x\) and its inverse function, clearly labelling the coordinates of the points where the graphs cross the \(x\) and \(y\) axes. [3]
WJEC Unit 1 2022 June Q2
6 marks Moderate -0.3
Showing all your working, simplify the following expression. [6] $$5\sqrt{48} + \frac{2+5\sqrt{3}}{5+3\sqrt{3}} - (2\sqrt{3})^3$$