Questions — WJEC (504 questions)

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WJEC Unit 1 2023 June Q5
7 marks Moderate -0.8
Simplify the expression \(\sqrt[3]{512a^7} - \frac{a^{\frac{7}{2}} \times a^{-\frac{1}{3}}}{a^6}\). [4]
WJEC Unit 1 2023 June Q6
7 marks Standard +0.3
The diagram below shows a triangle \(ABC\). \includegraphics{figure_6} Given that \(AB = 3\), \(BC = 2\sqrt{5}\), \(AC = 4 + \sqrt{3}\), find the value of \(\cos ABC\). Show all your working and give your answer in the form \(\frac{(a - b\sqrt{3})}{6\sqrt{5}}\), where \(a\), \(b\) are integers. [7]
WJEC Unit 1 2023 June Q7
13 marks Moderate -0.3
The curve \(C\) has equation \(y = 2x^2 + 5x - 12\) and the line \(L\) has equation \(y = mx - 14\), where \(m\) is a real constant.
  1. Given that \(L\) is a tangent to \(C\),
    1. show that \(m\) satisfies the equation $$m^2 - 10m + 9 = 0,$$ [5]
    2. find the coordinates of the two possible points of contact of \(C\) and \(L\). [6]
  2. Given instead that \(L\) intersects \(C\) at two distinct points, find the range of values of \(m\). [2]
WJEC Unit 1 2023 June Q8
3 marks Easy -1.8
Show, by counter example, that the following statement is false. "For all positive integer values of \(n\), \(n^2 + 1\) is a prime number." [3]
WJEC Unit 1 2023 June Q9
11 marks Moderate -0.3
  1. Given that \(y = x^2 - 3x\), find \(\frac{dy}{dx}\) from first principles. [5]
  2. The function \(f\) is defined by \(f(x) = 4x^{\frac{3}{2}} + \frac{6}{\sqrt{x}}\) for \(x > 0\).
    1. Find \(f'(x)\). [2]
    2. When \(x > k\), \(f(x)\) is an increasing function. Determine the least possible value of \(k\). Give your answer correct to two decimal places. [4]
WJEC Unit 1 2023 June Q10
11 marks Moderate -0.3
Solve the following equations for values of \(x\).
  1. \(\ln(2x + 5) = 3\) [2]
  2. \(5^{2x+1} = 14\) [3]
  3. \(3\log_7(2x) - \log_7(8x^2) + \log_7 x = \log_3 81\) [6]
WJEC Unit 1 2023 June Q11
7 marks Moderate -0.8
The function \(f\) is defined by \(f(x) = \frac{8}{x^2}\).
  1. Sketch the graph of \(y = f(x)\). [2]
  2. 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. [3]
  3. 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 \(\frac{8}{(x-2)^2} = \frac{8}{x}\). [2]
WJEC Unit 1 2023 June Q12
8 marks Moderate -0.8
The position vectors of the points \(A\) and \(B\), relative to a fixed origin \(O\), are given by $$\mathbf{a} = -3\mathbf{i} + 4\mathbf{j}, \quad \mathbf{b} = 5\mathbf{i} + 8\mathbf{j},$$ respectively.
  1. Find the vector \(\overrightarrow{AB}\). [2]
    1. Find a unit vector in the direction of \(\mathbf{a}\). [2]
    2. The point \(C\) is such that the vector \(\overrightarrow{OC}\) is in the direction of \(\mathbf{a}\). Given that the length of \(\overrightarrow{OC}\) is 7 units, write down the position vector of \(C\). [1]
  3. Calculate the angle \(AOB\). [3]
WJEC Unit 1 2023 June Q13
12 marks Standard +0.3
  1. Find \(\int \left(4x^{-\frac{2}{3}} + 5x^3 + 7\right) dx\). [3]
  2. The diagram below shows the graph of \(y = x(x + 6)(x - 3)\). \includegraphics{figure_13} Calculate the total area of the regions enclosed by the graph and the \(x\)-axis. [9]
WJEC Unit 1 2023 June Q14
6 marks Moderate -0.8
  1. 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 \(x\). [1]
  2. 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 \(Y = Ae^{-kt}\), where \(A\) and \(k\) are constants. A patient was given a dose of the drug that resulted in an initial concentration of 5 units.
    1. After 4 hours, the concentration had dropped to 1.25 units. Show that \(k = 0.3466\), correct to four decimal places. [2]
    2. 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? [3]
WJEC Unit 1 2024 June Q1
4 marks Moderate -0.8
Given that \(y = 12\sqrt{x} - \frac{27}{x} + 4\), find the value of \(\frac{dy}{dx}\) when \(x = 9\). [4]
WJEC Unit 1 2024 June Q2
3 marks Moderate -0.8
Find all values of \(\theta\) in the range \(0° < \theta < 180°\) that satisfy the equation $$2\sin 2\theta = 1.$$ [3]
WJEC Unit 1 2024 June Q3
3 marks Easy -1.2
Find \(\int\left(5x^4 + 3x^{-2} - 2\right)dx\). [3]
WJEC Unit 1 2024 June Q4
3 marks Easy -1.2
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. [3]
WJEC Unit 1 2024 June Q5
4 marks Moderate -0.5
A triangle \(ABC\) has sides \(AB = 6\)cm, \(BC = 11\)cm and \(AC = 13\)cm. Calculate the area of the triangle. [4]
WJEC Unit 1 2024 June Q6
7 marks Moderate -0.8
  1. Find the exact value of \(x\) that satisfies the equation $$\frac{7x^{\frac{5}{4}}}{x^{\frac{1}{2}}} = \sqrt{147}.$$ [4]
  2. Show that \(\frac{(8x-18)}{(2\sqrt{x}-3)}\), where \(x \neq \frac{9}{4}\), may be written as \(2(2\sqrt{x}+3)\). [3]
WJEC Unit 1 2024 June Q7
11 marks Easy -1.2
  1. The line \(L_1\) passes through the points \(A(-3, 0)\) and \(B(1, 4)\). Determine the equation of \(L_1\). [3]
  2. The line \(L_2\) has equation \(y = 3x - 3\).
    1. Given that \(L_1\) and \(L_2\) intersect at the point C, find the coordinates of C.
    2. The line \(L_2\) crosses the \(x\)-axis at the point D. Show that the coordinates of D are \((1, 0)\). [4]
  3. Calculate the area of triangle \(ACD\). [2]
  4. Determine the angle \(ACD\). [2]
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]