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CAIE P1 2016 November Q7
7 marks Moderate -0.3
\includegraphics{figure_2} The diagram shows a triangular pyramid \(ABCD\). It is given that $$\overrightarrow{AB} = 3\mathbf{i} + \mathbf{j} + \mathbf{k}, \quad \overrightarrow{AC} = \mathbf{i} - 2\mathbf{j} - \mathbf{k} \quad \text{and} \quad \overrightarrow{AD} = \mathbf{i} + 4\mathbf{j} - 7\mathbf{k}.$$
  1. Verify, showing all necessary working, that each of the angles \(DAB\), \(DAC\) and \(CAB\) is \(90°\). [3]
  2. Find the exact value of the area of the triangle \(ABC\), and hence find the exact value of the volume of the pyramid. [4]
[The volume \(V\) of a pyramid of base area \(A\) and vertical height \(h\) is given by \(V = \frac{1}{3}Ah\).]
CAIE P1 2016 November Q8
8 marks Moderate -0.3
  1. Express \(4x^2 + 12x + 10\) in the form \((ax + b)^2 + c\), where \(a\), \(b\) and \(c\) are constants. [3]
  2. Functions \(f\) and \(g\) are both defined for \(x > 0\). It is given that \(f(x) = x^2 + 1\) and \(fg(x) = 4x^2 + 12x + 10\). Find \(g(x)\). [1]
  3. Find \((fg)^{-1}(x)\) and give the domain of \((fg)^{-1}\). [4]
CAIE P1 2016 November Q9
8 marks Standard +0.3
  1. Two convergent geometric progressions, \(P\) and \(Q\), have the same sum to infinity. The first and second terms of \(P\) are \(6\) and \(6r\) respectively. The first and second terms of \(Q\) are \(12\) and \(-12r\) respectively. Find the value of the common sum to infinity. [3]
  2. The first term of an arithmetic progression is \(\cos\theta\) and the second term is \(\cos\theta + \sin^2\theta\), where \(0 \leq \theta \leq \pi\). The sum of the first \(13\) terms is \(52\). Find the possible values of \(\theta\). [5]
CAIE P1 2016 November Q10
12 marks Standard +0.3
A curve is such that \(\frac{dy}{dx} = \frac{2}{a}x^{-\frac{1}{2}} + ax^{-\frac{3}{2}}\), where \(a\) is a positive constant. The point \(A(a^2, 3)\) lies on the curve. Find, in terms of \(a\),
  1. the equation of the tangent to the curve at \(A\), simplifying your answer, [3]
  2. the equation of the curve. [4]
It is now given that \(B(16, 8)\) also lies on the curve.
  1. Find the value of \(a\) and, using this value, find the distance \(AB\). [5]
CAIE P1 2016 November Q11
12 marks Standard +0.3
A curve has equation \(y = (kx - 3)^{-1} + (kx - 3)\), where \(k\) is a non-zero constant.
  1. Find the \(x\)-coordinates of the stationary points in terms of \(k\), and determine the nature of each stationary point. Justify your answers. [7]
  1. \includegraphics{figure_3} The diagram shows part of the curve for the case when \(k = 1\). Showing all necessary working, find the volume obtained when the region between the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 2\), shown shaded in the diagram, is rotated through \(360°\) about the \(x\)-axis. [5]
CAIE P1 2018 November Q1
4 marks Standard +0.3
Find the coefficient of \(\frac{1}{x^2}\) in the expansion of \(\left(3x + \frac{2}{3x^2}\right)^7\). [4]
CAIE P1 2018 November Q2
4 marks Moderate -0.8
Showing all necessary working, find \(\int_1^4 \left(\sqrt{x} + \frac{2}{\sqrt{x}}\right) \text{d}x\). [4]
CAIE P1 2018 November Q3
5 marks Standard +0.3
\includegraphics{figure_3} The diagram shows part of the curve \(y = x(9 - x^2)\) and the line \(y = 5x\), intersecting at the origin \(O\) and the point \(R\). Point \(P\) lies on the line \(y = 5x\) between \(O\) and \(R\) and the \(x\)-coordinate of \(P\) is \(t\). Point \(Q\) lies on the curve and \(PQ\) is parallel to the \(y\)-axis.
  1. Express the length of \(PQ\) in terms of \(t\), simplifying your answer. [2]
  2. Given that \(t\) can vary, find the maximum value of the length of \(PQ\). [3]
CAIE P1 2018 November Q4
6 marks Moderate -0.8
Functions f and g are defined by $$f : x \mapsto 2 - 3\cos x \text{ for } 0 \leqslant x \leqslant 2\pi,$$ $$g : x \mapsto \frac{1}{2}x \text{ for } 0 \leqslant x \leqslant 2\pi.$$
  1. Solve the equation \(\text{fg}(x) = 1\). [3]
  2. Sketch the graph of \(y = \text{f}(x)\). [3]
CAIE P1 2018 November Q5
7 marks Standard +0.3
The first three terms of an arithmetic progression are \(4\), \(x\) and \(y\) respectively. The first three terms of a geometric progression are \(x\), \(y\) and \(18\) respectively. It is given that both \(x\) and \(y\) are positive.
  1. Find the value of \(x\) and the value of \(y\). [4]
  2. Find the fourth term of each progression. [3]
CAIE P1 2018 November Q6
7 marks Standard +0.3
\includegraphics{figure_6} The diagram shows a triangle \(ABC\) in which \(BC = 20\) cm and angle \(ABC = 90°\). The perpendicular from \(B\) to \(AC\) meets \(AC\) at \(D\) and \(AD = 9\) cm. Angle \(BCA = \theta°\).
  1. By expressing the length of \(BD\) in terms of \(\theta\) in each of the triangles \(ABD\) and \(DBC\), show that \(20\sin^2 \theta = 9\cos \theta\). [4]
  2. Hence, showing all necessary working, calculate \(\theta\). [3]
CAIE P1 2018 November Q7
7 marks Standard +0.3
\includegraphics{figure_7} The diagram shows a solid cylinder standing on a horizontal circular base with centre \(O\) and radius \(4\) units. Points \(A\), \(B\) and \(C\) lie on the circumference of the base such that \(AB\) is a diameter and angle \(BOC = 90°\). Points \(P\), \(Q\) and \(R\) lie on the upper surface of the cylinder vertically above \(A\), \(B\) and \(C\) respectively. The height of the cylinder is \(12\) units. The mid-point of \(CR\) is \(M\) and \(N\) lies on \(BQ\) with \(BN = 4\) units. Unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are parallel to \(OB\) and \(OC\) respectively and the unit vector \(\mathbf{k}\) is vertically upwards. Evaluate \(\overrightarrow{PN} \cdot \overrightarrow{PM}\) and hence find angle \(MPN\). [7]
CAIE P1 2018 November Q8
7 marks Standard +0.3
\includegraphics{figure_8} The diagram shows an isosceles triangle \(ACB\) in which \(AB = BC = 8\) cm and \(AC = 12\) cm. The arc \(XC\) is part of a circle with centre \(A\) and radius \(12\) cm, and the arc \(YC\) is part of a circle with centre \(B\) and radius \(8\) cm. The points \(A\), \(B\), \(X\) and \(Y\) lie on a straight line.
  1. Show that angle \(CBY = 1.445\) radians, correct to \(4\) significant figures. [3]
  2. Find the perimeter of the shaded region. [4]
CAIE P1 2018 November Q9
7 marks Moderate -0.8
The function f is defined by \(\text{f} : x \mapsto 2x^2 - 12x + 7\) for \(x \in \mathbb{R}\).
  1. Express \(2x^2 - 12x + 7\) in the form \(2(x + a)^2 + b\), where \(a\) and \(b\) are constants. [2]
  2. State the range of f. [1]
The function g is defined by \(\text{g} : x \mapsto 2x^2 - 12x + 7\) for \(x \leqslant k\).
  1. State the largest value of \(k\) for which g has an inverse. [1]
  2. Given that g has an inverse, find an expression for \(\text{g}^{-1}(x)\). [3]
CAIE P1 2018 November Q10
9 marks Moderate -0.3
The equation of a curve is \(y = 2x + \frac{12}{x}\) and the equation of a line is \(y + x = k\), where \(k\) is a constant.
  1. Find the set of values of \(k\) for which the line does not meet the curve. [3]
In the case where \(k = 15\), the curve intersects the line at points \(A\) and \(B\).
  1. Find the coordinates of \(A\) and \(B\). [3]
  2. Find the equation of the perpendicular bisector of the line joining \(A\) and \(B\). [3]
CAIE P1 2018 November Q11
12 marks Moderate -0.3
\includegraphics{figure_11} The diagram shows part of the curve \(y = 3\sqrt{(4x + 1)} - 2x\). The curve crosses the \(y\)-axis at \(A\) and the stationary point on the curve is \(M\).
  1. Obtain expressions for \(\frac{\text{d}y}{\text{d}x}\) and \(\int y \text{d}x\). [5]
  2. Find the coordinates of \(M\). [3]
  3. Find, showing all necessary working, the area of the shaded region. [4]
Edexcel P1 2018 Specimen Q1
6 marks Easy -1.2
Given that \(y = 4x^3 - \frac{5}{x^2}\), \(x \neq 0\), find in their simplest form
  1. \(\frac{dy}{dx}\). [3]
  2. \(\int y \, dx\) [3]
Edexcel P1 2018 Specimen Q2
5 marks Easy -1.2
  1. Given that \(3^{-1.5} = a\sqrt{3}\) find the exact value of \(a\) [2]
  2. Simplify fully \(\frac{(2x^{\frac{1}{2}})^3}{4x^2}\) [3]
Edexcel P1 2018 Specimen Q3
6 marks Moderate -0.3
Solve the simultaneous equations $$y + 4x + 1 = 0$$ $$y^2 + 5x^2 + 2y = 0$$ [6]
Edexcel P1 2018 Specimen Q4
5 marks Standard +0.3
The straight line with equation \(y = 4x + c\), where \(c\) is a constant, is a tangent to the curve with equation \(y = 2x^2 + 8x + 3\) Calculate the value of \(c\) [5]
Edexcel P1 2018 Specimen Q5
8 marks Moderate -0.8
  1. On the same axes, sketch the graphs of \(y = x + 2\) and \(y = x^2 - x - 6\) showing the coordinates of all points at which each graph crosses the coordinate axes. [4]
  2. On your sketch, show, by shading, the region \(R\) defined by the inequalities $$y < x + 2 \text{ and } y > x^2 - x - 6$$ [1]
  3. Hence, or otherwise, find the set of values of \(x\) for which \(x^2 - 2x - 8 < 0\) [3]
Edexcel P1 2018 Specimen Q6
7 marks Moderate -0.3
\includegraphics{figure_1} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \text{f}(x)\) The curve \(C\) passes through the origin and through \((6, 0)\) The curve \(C\) has a minimum at the point \((3, -1)\) On separate diagrams, sketch the curve with equation
  1. \(y = \text{f}(2x)\) [3]
  2. \(y = \text{f}(x + p)\), where \(p\) is a constant and \(0 < p < 3\) [4]
On each diagram show the coordinates of any points where the curve intersects the \(x\)-axis and of any minimum or maximum points.
Edexcel P1 2018 Specimen Q7
5 marks Moderate -0.8
A curve with equation \(y = \text{f}(x)\) passes through the point \((4, 25)\) Given that $$\text{f}'(x) = \frac{3}{8}x^2 - 10x^{-\frac{1}{2}} + 1, \quad x > 0$$ find \(\text{f}(x)\), simplifying each term. [5]
Edexcel P1 2018 Specimen Q8
10 marks Moderate -0.3
\includegraphics{figure_2} The line \(l_1\) shown in Figure 2 has equation \(2x + 3y = 26\) The line \(l_2\) passes through the origin \(O\) and is perpendicular to \(l_1\)
  1. Find an equation for the line \(l_2\) [4]
The line \(l_1\) intersects the line \(l_1\) at the point \(C\). Line \(l_1\) crosses the \(y\)-axis at the point \(B\) as shown in Figure 2.
  1. Find the area of triangle \(OBC\). Give your answer in the form \(\frac{a}{b}\), where \(a\) and \(b\) are integers to be found. [6]
Edexcel P1 2018 Specimen Q9
11 marks Standard +0.3
\includegraphics{figure_3} A sketch of part of the curve \(C\) with equation $$y = 20 - 4x - \frac{18}{x}, \quad x > 0$$ is shown in Figure 3. Point \(A\) lies on \(C\) and has \(x\) coordinate equal to 2
  1. Show that the equation of the normal to \(C\) at \(A\) is \(y = -2x + 7\). [6]
The normal to \(C\) at \(A\) meets \(C\) again at the point \(B\), as shown in Figure 3.
  1. Use algebra to find the coordinates of \(B\). [5]