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WJEC Unit 1 2024 June Q8
4 marks Standard +0.3
Prove that \(x - 10 < x^2 - 5x\) for all real values of \(x\). [4]
WJEC Unit 1 2024 June Q9
9 marks Moderate -0.3
  1. Write down the binomial expansion of \((2 - x)^6\) up to and including the term in \(x^2\). [3]
  2. Given that $$(1 + ax)(2 - x)^6 = 64 + bx + 336x^2 + \ldots,$$ find the values of the constants \(a\), \(b\). [6]
WJEC Unit 1 2024 June Q10
6 marks Moderate -0.8
Water is being emptied out of a sink. The depth of water, \(y\)cm, at time \(t\) seconds, may be modelled by $$y = t^2 - 14t + 49 \quad\quad 0 \leqslant t \leqslant 7.$$
  1. Find the value of \(t\) when the depth of water is 25cm. [3]
  2. Find the rate of decrease of the depth of water when \(t = 3\). [3]
WJEC Unit 1 2024 June Q11
4 marks Easy -1.3
  1. Sketch the graph of \(y = 3^x\). Clearly label the coordinates of the point where the graph crosses the \(y\)-axis. [2]
  2. 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. [2]
WJEC Unit 1 2024 June Q12
10 marks Moderate -0.3
A curve C has equation \(y = -x^3 + 12x - 20\).
  1. Find the coordinates of the stationary points of C and determine their nature. [7]
  2. Determine the range of values of \(x\) for which the curve is decreasing. Give your answer in set notation. [3]
WJEC Unit 1 2024 June Q13
8 marks Moderate -0.8
The position vectors of the points A and B, relative to a fixed origin O, are given by $$\mathbf{a} = 4\mathbf{i} + 7\mathbf{j}, \quad\quad \mathbf{b} = \mathbf{i} + 3\mathbf{j},$$ respectively.
  1. Find the vector \(\overrightarrow{AB}\). [2]
  2. Determine the distance between the points A and B. [2]
  3. 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 \(\overrightarrow{CD}\) is parallel to \(\overrightarrow{AB}\). Find the possible position vectors of the point D. [4]
WJEC Unit 1 2024 June Q14
8 marks Moderate -0.3
The diagram below shows a sketch of the curve C with equation \(y = 2 - 3x - 2x^2\) 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{figure_14}
  1. Write down the coordinates of A and B. [2]
  2. Calculate the area enclosed by C and L. [6]
WJEC Unit 1 2024 June Q15
7 marks Standard +0.8
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{figure_15} Find the value of \(x\) at each of the points A, B and C. [7]
WJEC Unit 1 2024 June Q16
10 marks Moderate -0.8
  1. Find the range of values of \(k\) for which the quadratic equation \(x^2 - kx + 4 = 0\) has no real roots. [4]
  2. Determine the coordinates of the points of intersection of the graphs of \(y = x^2 - 3x + 4\) and \(y = x + 16\). [4]
  3. Using the information obtained in parts (a) and (b), sketch the graphs of \(y = x^2 - 3x + 4\) and \(y = x + 16\) on the same set of axes. [2]
WJEC Unit 1 2024 June Q17
7 marks Moderate -0.3
A function \(f\) is defined by \(f(x) = \log_{10}(2 - x)\). Another function \(g\) is defined by \(g(x) = \log_{10}(5 - x)\). The diagram below shows a sketch of the graphs of \(y = f(x)\) and \(y = g(x)\). \includegraphics{figure_17}
  1. The point \((c, 1)\) lies on \(y = f(x)\). Find the value of \(c\). [2]
  2. A point P lies on \(y = f(x)\) and has \(x\)-coordinate \(\alpha\). Another point Q lies on \(y = g(x)\) and also has \(x\)-coordinate \(\alpha\). The distance between P and Q is 1.2 units. Find the value of \(\alpha\), giving your answer correct to three decimal places. [5]
WJEC Unit 1 2024 June Q18
12 marks Standard +0.8
  1. A circle C has centre \((-3, -1)\) and radius \(\sqrt{5}\). Show that the equation of C can be written as \(x^2 + y^2 + 6x + 2y + 5 = 0\). [2]
    1. Find the equations of the tangents to C that pass through the origin O. [6]
    2. Determine the coordinates of the points where the tangents touch the circle. [4]
WJEC Unit 1 Specimen Q1
7 marks Moderate -0.8
The circle \(C\) has centre \(A\) and equation $$x^2 + y^2 - 2x + 6y - 15 = 0.$$
  1. Find the coordinates of \(A\) and the radius of \(C\). [3]
  2. The point \(P\) has coordinates \((4, -7)\) and lies on \(C\). Find the equation of the tangent to \(C\) at \(P\). [4]
WJEC Unit 1 Specimen Q2
6 marks Standard +0.3
Find all values of \(\theta\) between \(0°\) and \(360°\) satisfying $$7 \sin^2 \theta + 1 = 3 \cos^2 \theta - \sin \theta.$$ [6]
WJEC Unit 1 Specimen Q3
6 marks Moderate -0.5
Given that \(y = x^3\), find \(\frac{dy}{dx}\) from first principles. [6]
WJEC Unit 1 Specimen Q4
5 marks Moderate -0.3
The cubic polynomial \(f(x)\) is given by \(f(x) = 2x^3 + ax^2 + bx + c\), where \(a\), \(b\), \(c\) are constants. The graph of \(f(x)\) intersects the \(x\)-axis at the points with coordinates \((-3, 0)\), \((2.5, 0)\) and \((4, 0)\). Find the coordinates of the point where the graph of \(f(x)\) intersects the \(y\)-axis. [5]
WJEC Unit 1 Specimen Q5
12 marks Moderate -0.8
The points \(A(0, 2)\), \(B(-2, 8)\), \(C(20, 12)\) are the vertices of the triangle \(ABC\). The point \(D\) is the mid-point of \(AB\).
  1. Show that \(CD\) is perpendicular to \(AB\). [6]
  2. Find the exact value of \(\tan CAB\). [5]
  3. Write down the geometrical name for the triangle \(ABC\). [1]
WJEC Unit 1 Specimen Q6
5 marks Standard +0.3
In each of the two statements below, \(c\) and \(d\) are real numbers. One of the statements is true while the other is false. A: Given that \((2c + 1)^2 = (2d + 1)^2\), then \(c = d\). B: Given that \((2c + 1)^3 = (2d + 1)^3\), then \(c = d\).
  1. Identify the statement which is false. Find a counter example to show that this statement is in fact false.
  2. Identify the statement which is true. Give a proof to show that this statement is in fact true. [5]
WJEC Unit 1 Specimen Q7
5 marks Moderate -0.8
Figure 1 shows a sketch of the graph of \(y = f(x)\). The graph has a minimum point at \((-3, -4)\) and intersects the \(x\)-axis at the points \((-8, 0)\) and \((2, 0)\). \includegraphics{figure_1}
  1. Sketch the graph of \(y = f(x + 3)\), indicating the coordinates of the stationary point and the coordinates of the points of intersection of the graph with the \(x\)-axis. [3]
  2. Figure 2 shows a sketch of the graph having one of the following equations with an appropriate value of either \(p\), \(q\) or \(r\). \(y = f(px)\), where \(p\) is a constant \(y = f(x) + q\), where \(q\) is a constant \(y = rf(x)\), where \(r\) is a constant \includegraphics{figure_2} Write down the equation of the graph sketched in Figure 2, together with the value of the corresponding constant. [2]
WJEC Unit 1 Specimen Q8
6 marks Moderate -0.3
The circle \(C\) has radius 5 and its centre is the origin. The point \(T\) has coordinates \((11, 0)\). The tangents from \(T\) to the circle \(C\) touch \(C\) at the points \(R\) and \(S\).
  1. Write down the geometrical name for the quadrilateral \(ORTS\). [1]
  2. Find the exact value of the area of the quadrilateral \(ORTS\). Give your answer in its simplest form. [5]
WJEC Unit 1 Specimen Q9
7 marks Standard +0.8
The quadratic equation \(4x^2 - 12x + m = 0\), where \(m\) is a positive constant, has two distinct real roots. Show that the quadratic equation \(3x^2 + mx + 7 = 0\) has no real roots. [7]
WJEC Unit 1 Specimen Q10
8 marks Standard +0.8
  1. Use the binomial theorem to express \(\left(\sqrt{3} - \sqrt{2}\right)^5\) in the form \(a\sqrt{3} + b\sqrt{2}\), where \(a\), \(b\) are integers whose values are to be found. [5]
  2. Given that \(\left(\sqrt{3} - \sqrt{2}\right)^5 \approx 0\), use your answer to part (a) to find an approximate value for \(\sqrt{6}\) in the form \(\frac{c}{d}\), where \(c\) and \(d\) are positive integers whose values are to be found. [3]
WJEC Unit 1 Specimen Q11
3 marks Moderate -0.8
\includegraphics{figure_11} The diagram shows a sketch of the curve \(y = 6 + 4x - x^2\) and the line \(y = x + 2\). The point \(P\) has coordinates \((a, b)\). Write down the three inequalities involving \(a\) and \(b\) which are such that the point \(P\) will be strictly contained within the shaded area above, if and only if, all three inequalities are satisfied. [3]
WJEC Unit 1 Specimen Q12
3 marks Easy -1.8
Prove that $$\log_a a \times \log_a 19 = \log_a 19$$ whatever the value of the positive constant \(a\). [3]
WJEC Unit 1 Specimen Q13
7 marks Standard +0.3
In triangle \(ABC\), \(BC = 12\) cm and \(\cos ABC = \frac{2}{3}\). The length of \(AC\) is 2 cm greater than the length of \(AB\).
  1. Find the lengths of \(AB\) and \(AC\). [4]
  2. Find the exact value of \(\sin BAC\). Give your answer in its simplest form. [3]
WJEC Unit 1 Specimen Q14
8 marks Standard +0.3
The diagram below shows a closed box in the form of a cuboid, which is such that the length of its base is twice the width of its base. The volume of the box is 9000 cm³. The total surface area of the box is denoted by \(S\) cm². \includegraphics{figure_14}
  1. Show that \(S = 4x^2 + \frac{27000}{x}\), where \(x\) cm denotes the width of the base. [3]
  2. Find the minimum value of \(S\), showing that the value you have found is a minimum value. [5]