Questions (33218 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
WJEC Unit 1 2022 June Q13
4 marks Moderate -0.3
Find the term which is independent of \(x\) in the expansion of \(\frac{(2-3x)^5}{x^3}\). [4]
WJEC Unit 1 2022 June Q14
12 marks Standard +0.3
A curve \(C\) has equation \(f(x) = 3x^3 - 5x^2 + x - 6\).
  1. Find the coordinates of the stationary points of \(C\) and determine their nature. [8]
  2. Without solving the equations, determine the number of distinct real roots for each of the following:
    1. \(3x^3 - 5x^2 + x + 1 = 0\),
    2. \(6x^3 - 10x^2 + 2x + 1 = 0\). [4]
WJEC Unit 1 2022 June Q15
8 marks Challenging +1.2
Solve the simultaneous equations $$3\log_u(x^2y) - \log_u(x^2y^2) + \log_u\left(\frac{9}{x^2y^2}\right) = \log_u 36,$$ $$\log_u y - \log_u(x + 3) = 0.$$ [8]
WJEC Unit 1 2022 June Q16
9 marks Moderate -0.8
The vectors \(\mathbf{a}\) and \(\mathbf{b}\) are defined by \(\mathbf{a} = 2\mathbf{i} - \mathbf{j}\) and \(\mathbf{b} = \mathbf{i} - 3\mathbf{j}\).
  1. Find a unit vector in the direction of \(\mathbf{a}\). [2]
  2. Determine the angle \(\mathbf{b}\) makes with the \(x\)-axis. [2]
  3. The vector \(\mu\mathbf{a} + \mathbf{b}\) is parallel to \(4\mathbf{i} - 5\mathbf{j}\).
    1. Find the vector \(\mu\mathbf{a} + \mathbf{b}\) in terms of \(\mu\), \(\mathbf{i}\) and \(\mathbf{j}\). [1]
    2. Determine the value of \(\mu\). [4]
WJEC Unit 1 2023 June Q1
6 marks Moderate -0.8
  1. Using the binomial theorem, write down and simplify the first three terms in the expansion of \((1 - 3x)^9\) in ascending powers of \(x\). [3]
  2. Hence, by writing \(x = 0.001\) in your expansion in part (a), find an approximate value for \((0.997)^9\). Show all your working and give your answer correct to three decimal places. [3]
WJEC Unit 1 2023 June Q2
7 marks Standard +0.8
Solve the following equation for values of \(\theta\) between \(0°\) and \(360°\). $$3\sin^2 \theta - 5\cos^2 \theta = 2\cos \theta$$ [7]
WJEC Unit 1 2023 June Q3
15 marks Moderate -0.3
The point \(A\) has coordinates \((-2, 5)\) and the point \(B\) has coordinates \((3, 8)\). The point \(C\) lies on the \(x\)-axis such that \(AC\) is perpendicular to \(AB\).
  1. Find the equation of \(AB\). [3]
  2. Show that \(C\) has coordinates \((1, 0)\). [3]
  3. Calculate the area of triangle \(ABC\). [4]
  4. Find the equation of the circle which passes through the points \(A\), \(B\) and \(C\). [5]
WJEC Unit 1 2023 June Q4
10 marks Moderate -0.8
  1. Find the remainder when the polynomial \(3x^3 + 2x^2 + x - 1\) is divided by \((x - 3)\). [3]
  2. The polynomial \(f(x) = 2x^3 - 3x^2 + ax + 6\) is divisible by \((x + 2)\), where \(a\) is a real constant.
    1. Find the value of \(a\). [3]
    2. Showing all your working, solve the equation \(f(x) = 0\). [4]
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]