Questions Unit 1 (89 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 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]
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]