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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] $$5\sqrt{48} + \frac{2+5\sqrt{3}}{5+3\sqrt{3}} - (2\sqrt{3})^3$$
WJEC Unit 1 2022 June Q3
11 marks Moderate -0.8
The line \(L_1\) passes through the points \(A(0, 5)\) and \(B(3, -1)\).
  1. Find the equation of the line \(L_1\). [3]
The line \(L_2\) is perpendicular to \(L_1\) and passes through the origin \(O\).
  1. Write down the equation of \(L_2\). [1]
The lines \(L_1\) and \(L_2\) intersect at the point \(C\).
  1. Calculate the area of triangle \(OAC\). [4]
  2. Find the equation of the line \(L_3\) which is parallel to \(L_1\) and passes through the point \(D(4, 2)\). [2]
  3. The line \(L_3\) intersects the \(y\)-axis at the point \(E\). Find the area of triangle \(ODE\). [1]
WJEC Unit 1 2022 June Q4
4 marks Moderate -0.3
Solve the inequality \(x^2 + 3x - 6 > 4x - 4\). [4]
WJEC Unit 1 2022 June Q5
9 marks Moderate -0.8
The curve \(C_1\) has equation \(y = -x^2 + 2x + 3\) and the curve \(C_2\) has equation \(y = x^2 - x - 6\). The two curves intersect at the points \(A\) and \(B\).
  1. Determine the coordinates of \(A\) and \(B\). [4]
  2. On the same set of axes, sketch the graphs of \(C_1\) and \(C_2\). Clearly label the points where the two curves intersect. [3]
  3. In the diagram drawn in part (b), shade the region satisfying the following inequalities: [2] $$x > 0,$$ $$y < -x^2 + 2x + 3,$$ $$y > x^2 - x - 6.$$
WJEC Unit 1 2022 June Q6
5 marks Standard +0.3
In each of the two statements below, \(x\) and \(y\) are real numbers. One of the statements is true while the other is false. A: \(x^2 + y^2 \geqslant 2xy\), for all real values of \(x\) and \(y\). B: \(x + y \geqslant 2\sqrt{xy}\), for all real values of \(x\) and \(y\).
  1. Identify the statement which is false. Find a counter example to show that this statement is in fact false. [3]
  2. Identify the statement which is true. Give a proof to show that this statement is in fact true. [2]
WJEC Unit 1 2022 June Q7
11 marks Standard +0.3
A circle \(C\) has centre \(A\) and equation \(x^2 + y^2 - 4x - 6y = 3\).
  1. Find the coordinates of \(A\) and the radius of \(C\). [3]
The line \(L\) with equation \(y = x + 5\) intersects \(C\) at the points \(P\) and \(Q\).
  1. Determine the coordinates of \(P\) and \(Q\). [4]
  2. The point \(B\) is on \(PQ\) and is such that \(AB\) is perpendicular to \(PQ\). Find the length of \(PB\). [2]
  3. Show that the area of the smaller segment enclosed by \(C\) and \(L\) is \(4\pi - 8\). [2]
WJEC Unit 1 2022 June Q8
7 marks Easy -1.8
  1. The graph \(G\) shows the relationship between the variables \(y\) and \(x\), where \(y \propto x\). Sketch the graph \(G\). [1]
  2. Mary and Jeff work for a company which pays its employees by hourly rates. Mary's hourly rate is twice Jeff's hourly rate. On a certain day, Jeff worked three times as long as Mary and was paid £120. Calculate Mary's earnings on that day. [3]
  3. Atmospheric pressure, \(P\) units, decreases as the height, \(H\) metres, above sea level increases. The rate of decrease is 12% for every 1000m. At sea level, the pressure \(P\) is 1013 units. Write down the model for \(P\) in terms of \(H\) and find the pressure at the top of Mount Everest, which is 8848m above sea level. [3]
WJEC Unit 1 2022 June Q9
4 marks Standard +0.3
Find the range of values of \(k\) for which the quadratic equation \(x^2 + 2kx + 8k = 0\) has no real roots. [4]
WJEC Unit 1 2022 June Q10
3 marks Moderate -0.8
Showing all your working, solve the equation \(2^x = 53\). Give your answer correct to two decimal places. [3]
WJEC Unit 1 2022 June Q11
15 marks Standard +0.3
The diagram below shows a sketch of the curve \(y = f(x)\), where \(f(x) = 10x + 3x^2 - x^3\). The curve intersects the \(x\)-axis at the origin \(O\) and at the points \(A(-2, 0)\), \(B(5, 0)\). The tangent to the curve at the point \(C(2, 24)\) intersects the \(y\)-axis at the point \(D\). \includegraphics{figure_11}
  1. Find the coordinates of \(D\). [5]
  2. Find the area of the shaded region. [6]
  3. Determine the range of values of \(x\) for which \(f(x)\) is an increasing function. [4]
WJEC Unit 1 2022 June Q12
9 marks Moderate -0.3
  1. Solve the equation \(2x^3 - x^2 - 5x - 2 = 0\). [6]
  2. Find all values of \(\theta\) in the range \(0° < \theta < 180°\) satisfying $$\cos(2\theta - 51°) = 0.891.$$ [3]