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CAIE P1 2010 June Q8
8 marks Moderate -0.3
A solid rectangular block has a square base of side \(x\) cm. The height of the block is \(h\) cm and the total surface area of the block is \(96\) cm\(^2\).
  1. Express \(h\) in terms of \(x\) and show that the volume, \(V\) cm\(^3\), of the block is given by $$V = 24x - \frac{1}{2}x^3.$$ [3]
Given that \(x\) can vary,
  1. find the stationary value of \(V\), [3]
  2. determine whether this stationary value is a maximum or a minimum. [2]
CAIE P1 2010 June Q9
8 marks Standard +0.3
\includegraphics{figure_9} The diagram shows the curve \(y = (x - 2)^2\) and the line \(y + 2x = 7\), which intersect at points \(A\) and \(B\). Find the area of the shaded region. [8]
CAIE P1 2010 June Q10
9 marks Moderate -0.3
The equation of a curve is \(y = \frac{1}{6}(2x - 3)^3 - 4x\).
  1. Find \(\frac{dy}{dx}\). [3]
  2. Find the equation of the tangent to the curve at the point where the curve intersects the \(y\)-axis. [3]
  3. Find the set of values of \(x\) for which \(\frac{1}{6}(2x - 3)^3 - 4x\) is an increasing function of \(x\). [3]
CAIE P1 2010 June Q11
10 marks Moderate -0.3
The function \(f : x \mapsto 4 - 3\sin x\) is defined for the domain \(0 \leq x < 2\pi\).
  1. Solve the equation \(f(x) = 2\). [3]
  2. Sketch the graph of \(y = f(x)\). [2]
  3. Find the set of values of \(k\) for which the equation \(f(x) = k\) has no solution. [2]
The function \(g : x \mapsto 4 - 3\sin x\) is defined for the domain \(\frac{1}{2}\pi \leq x \leq A\).
  1. State the largest value of \(A\) for which \(g\) has an inverse. [1]
  2. For this value of \(A\), find the value of \(g^{-1}(3)\). [2]
CAIE P1 2011 June Q1
3 marks Easy -1.2
Find \(\int \left(x^3 + \frac{1}{x^3}\right) \mathrm{d}x\). [3]
CAIE P1 2011 June Q2
5 marks Moderate -0.8
  1. Find the terms in \(x^2\) and \(x^3\) in the expansion of \(\left(1 - \frac{3}{2}x\right)^6\). [3]
  2. Given that there is no term in \(x^3\) in the expansion of \((k + 2x)\left(1 - \frac{3}{2}x\right)^6\), find the value of the constant \(k\). [2]
CAIE P1 2011 June Q3
5 marks Moderate -0.8
The equation \(x^2 + px + q = 0\), where \(p\) and \(q\) are constants, has roots \(-3\) and \(5\).
  1. Find the values of \(p\) and \(q\). [2]
  2. Using these values of \(p\) and \(q\), find the value of the constant \(r\) for which the equation \(x^2 + px + q + r = 0\) has equal roots. [3]
CAIE P1 2011 June Q4
6 marks Moderate -0.8
A curve has equation \(y = \frac{4}{3x - 4}\) and \(P(2, 2)\) is a point on the curve.
  1. Find the equation of the tangent to the curve at \(P\). [4]
  2. Find the angle that this tangent makes with the \(x\)-axis. [2]
CAIE P1 2011 June Q5
6 marks Moderate -0.3
  1. Prove the identity \(\frac{\cos \theta}{\tan \theta(1 - \sin \theta)} \equiv 1 + \frac{1}{\sin \theta}\). [3]
  2. Hence solve the equation \(\frac{\cos \theta}{\tan \theta(1 - \sin \theta)} = 4\), for \(0° \leq \theta \leq 360°\). [3]
CAIE P1 2011 June Q6
5 marks Moderate -0.3
The function \(f\) is defined by \(f : x \mapsto \frac{x + 3}{2x - 1}\), \(x \in \mathbb{R}\), \(x \neq \frac{1}{2}\).
  1. Show that \(f f(x) = x\). [3]
  2. Hence, or otherwise, obtain an expression for \(f^{-1}(x)\). [2]
CAIE P1 2011 June Q7
7 marks Moderate -0.3
The line \(L_1\) passes through the points \(A(2, 5)\) and \(B(10, 9)\). The line \(L_2\) is parallel to \(L_1\) and passes through the origin. The point \(C\) lies on \(L_2\) such that \(AC\) is perpendicular to \(L_2\). Find
  1. the coordinates of \(C\), [5]
  2. the distance \(AC\). [2]
CAIE P1 2011 June Q8
8 marks Moderate -0.3
Relative to the origin \(O\), the position vectors of the points \(A\), \(B\) and \(C\) are given by $$\overrightarrow{OA} = \begin{pmatrix} 2 \\ 3 \\ 5 \end{pmatrix}, \quad \overrightarrow{OB} = \begin{pmatrix} 4 \\ 2 \\ 3 \end{pmatrix} \quad \text{and} \quad \overrightarrow{OC} = \begin{pmatrix} 10 \\ 0 \\ 6 \end{pmatrix}.$$
  1. Find angle \(ABC\). [6]
The point \(D\) is such that \(ABCD\) is a parallelogram.
  1. Find the position vector of \(D\). [2]
CAIE P1 2011 June Q9
8 marks Moderate -0.3
The function \(f\) is such that \(f(x) = 3 - 4\cos^k x\), for \(0 \leq x \leq \pi\), where \(k\) is a constant.
  1. In the case where \(k = 2\),
    1. find the range of \(f\), [2]
    2. find the exact solutions of the equation \(f(x) = 1\). [3]
  2. In the case where \(k = 1\),
    1. sketch the graph of \(y = f(x)\), [2]
    2. state, with a reason, whether \(f\) has an inverse. [1]
CAIE P1 2011 June Q10
11 marks Standard +0.3
  1. A circle is divided into 6 sectors in such a way that the angles of the sectors are in arithmetic progression. The angle of the largest sector is 4 times the angle of the smallest sector. Given that the radius of the circle is 5 cm, find the perimeter of the smallest sector. [6]
  2. The first, second and third terms of a geometric progression are \(2k + 3\), \(k + 6\) and \(k\), respectively. Given that all the terms of the geometric progression are positive, calculate
    1. the value of the constant \(k\), [3]
    2. the sum to infinity of the progression. [2]
CAIE P1 2011 June Q11
11 marks Standard +0.3
\includegraphics{figure_11} The diagram shows part of the curve \(y = 4\sqrt{x} - x\). The curve has a maximum point at \(M\) and meets the \(x\)-axis at \(O\) and \(A\).
  1. Find the coordinates of \(A\) and \(M\). [5]
  2. Find the volume obtained when the shaded region is rotated through \(360°\) about the \(x\)-axis, giving your answer in terms of \(\pi\). [6]
CAIE P1 2012 June Q1
4 marks Moderate -0.3
\includegraphics{figure_1} The diagram shows the region enclosed by the curve \(y = \frac{6}{2x - 3}\), the \(x\)-axis and the lines \(x = 2\) and \(x = 3\). Find, in terms of \(\pi\), the volume obtained when this region is rotated through \(360°\) about the \(x\)-axis. [4]
CAIE P1 2012 June Q2
5 marks Moderate -0.8
The equation of a curve is \(y = 4\sqrt{x} + \frac{2}{\sqrt{x}}\).
  1. Obtain an expression for \(\frac{dy}{dx}\). [3]
  2. A point is moving along the curve in such a way that the \(x\)-coordinate is increasing at a constant rate of \(0.12\) units per second. Find the rate of change of the \(y\)-coordinate when \(x = 4\). [2]
CAIE P1 2012 June Q3
5 marks Standard +0.3
The coefficient of \(x^3\) in the expansion of \((a + x)^5 + (2 - x)^6\) is \(90\). Find the value of the positive constant \(a\). [5]
CAIE P1 2012 June Q4
6 marks Moderate -0.8
The point \(A\) has coordinates \((-1, -5)\) and the point \(B\) has coordinates \((7, 1)\). The perpendicular bisector of \(AB\) meets the \(x\)-axis at \(C\) and the \(y\)-axis at \(D\). Calculate the length of \(CD\). [6]
CAIE P1 2012 June Q5
6 marks Moderate -0.3
  1. Prove the identity \(\tan x + \frac{1}{\tan x} = \frac{1}{\sin x \cos x}\). [2]
  2. Solve the equation \(\frac{2}{\sin x \cos x} = 1 + 3 \tan x\), for \(0° \leqslant x \leqslant 180°\). [4]
CAIE P1 2012 June Q6
8 marks Moderate -0.3
\includegraphics{figure_6} The diagram shows a metal plate made by removing a segment from a circle with centre \(O\) and radius \(8\) cm. The line \(AB\) is a chord of the circle and angle \(AOB = 2.4\) radians. Find
  1. the length of \(AB\), [2]
  2. the perimeter of the plate, [3]
  3. the area of the plate. [3]
CAIE P1 2012 June Q7
8 marks Moderate -0.8
  1. In an arithmetic progression, the sum of the first \(n\) terms, denoted by \(S_n\), is given by $$S_n = n^2 + 8n.$$ Find the first term and the common difference. [3]
  2. In a geometric progression, the second term is \(9\) less than the first term. The sum of the second and third terms is \(30\). Given that all the terms of the progression are positive, find the first term. [5]
CAIE P1 2012 June Q8
10 marks Moderate -0.3
  1. Find the angle between the vectors \(\mathbf{3i} - \mathbf{4k}\) and \(\mathbf{2i} + \mathbf{3j} - \mathbf{6k}\). [4]
The vector \(\overrightarrow{OA}\) has a magnitude of \(15\) units and is in the same direction as the vector \(\mathbf{3i} - \mathbf{4k}\). The vector \(\overrightarrow{OB}\) has a magnitude of \(14\) units and is in the same direction as the vector \(\mathbf{2i} + \mathbf{3j} - \mathbf{6k}\).
  1. Express \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\) in terms of \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\). [3]
  2. Find the unit vector in the direction of \(\overrightarrow{AB}\). [3]
CAIE P1 2012 June Q9
11 marks Standard +0.3
\includegraphics{figure_9} The diagram shows part of the curve \(y = -x^2 + 8x - 10\) which passes through the points \(A\) and \(B\). The curve has a maximum point at \(A\) and the gradient of the line \(BA\) is \(2\).
  1. Find the coordinates of \(A\) and \(B\). [7]
  2. Find \(\int y \, dx\) and hence evaluate the area of the shaded region. [4]
CAIE P1 2012 June Q10
12 marks Standard +0.3
Functions \(f\) and \(g\) are defined by $$f : x \mapsto 2x + 5 \quad \text{for } x \in \mathbb{R},$$ $$g : x \mapsto \frac{8}{x - 3} \quad \text{for } x \in \mathbb{R}, x \neq 3.$$
  1. Obtain expressions, in terms of \(x\), for \(f^{-1}(x)\) and \(g^{-1}(x)\), stating the value of \(x\) for which \(g^{-1}(x)\) is not defined. [4]
  2. Sketch the graphs of \(y = f(x)\) and \(y = f^{-1}(x)\) on the same diagram, making clear the relationship between the two graphs. [3]
  3. Given that the equation \(fg(x) = 5 - kx\), where \(k\) is a constant, has no solutions, find the set of possible values of \(k\). [5]