Questions Unit 1 (89 questions)

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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
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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]
WJEC Unit 1 Specimen Q15
8 marks Moderate -0.8
The size \(N\) of the population of a small island at time \(t\) years may be modelled by \(N = Ae^{kt}\), where \(A\) and \(k\) are constants. It is known that \(N = 100\) when \(t = 2\) and that \(N = 160\) when \(t = 12\).
  1. Interpret the constant \(A\) in the context of the question. [1]
  2. Show that \(k = 0.047\), correct to three decimal places. [4]
  3. Find the size of the population when \(t = 20\). [3]
WJEC Unit 1 Specimen Q16
5 marks Standard +0.3
Find the range of values of \(x\) for which the function $$f(x) = x^3 - 5x^2 - 8x + 13$$ is an increasing function. [5]
WJEC Unit 1 Specimen Q17
12 marks Standard +0.3
\includegraphics{figure_17} The diagram above shows a sketch of the curve \(y = 3x - x^2\). The curve intersects the \(x\)-axis at the origin and at the point \(A\). The tangent to the curve at the point \(B(2, 2)\) intersects the \(x\)-axis at the point \(C\).
  1. Find the equation of the tangent to the curve at \(B\). [4]
  2. Find the area of the shaded region. [8]
WJEC Unit 1 Specimen Q18
7 marks Moderate -0.8
  1. The vectors \(\mathbf{u}\) and \(\mathbf{v}\) are defined by \(\mathbf{u} = 2\mathbf{i} - 3\mathbf{j}\), \(\mathbf{v} = -4\mathbf{i} + 5\mathbf{j}\).
    1. Find the vector \(4\mathbf{u} - 3\mathbf{v}\).
    2. The vectors \(\mathbf{u}\) and \(\mathbf{v}\) are the position vectors of the points \(U\) and \(V\), respectively. Find the length of the line \(UV\). [4]
  2. Two villages \(A\) and \(B\) are 40 km apart on a long straight road passing through a desert. The position vectors of \(A\) and \(B\) are denoted by \(\mathbf{a}\) and \(\mathbf{b}\), respectively.
    1. Village \(C\) lies on the road between \(A\) and \(B\) at a distance 4 km from \(B\). Find the position vector of \(C\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).
    2. Village \(D\) has position vector \(\frac{2}{9}\mathbf{a} + \frac{5}{9}\mathbf{b}\). Explain why village \(D\) cannot possibly be on the straight road passing through \(A\) and \(B\). [3]