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CAIE P1 2012 June Q1
4 marks Moderate -0.8
  1. Prove the identity \(\tan^2 \theta - \sin^2 \theta = \tan^2 \theta \sin^2 \theta\). [3]
  2. Use this result to explain why \(\tan \theta > \sin \theta\) for \(0° < \theta < 90°\). [1]
CAIE P1 2012 June Q2
5 marks Moderate -0.8
Relative to an origin \(O\), the position vectors of the points \(A\), \(B\) and \(C\) are given by $$\overrightarrow{OA} = \begin{pmatrix} 2 \\ -1 \\ 4 \end{pmatrix}, \quad \overrightarrow{OB} = \begin{pmatrix} 4 \\ 2 \\ -2 \end{pmatrix} \quad \text{and} \quad \overrightarrow{OC} = \begin{pmatrix} 1 \\ 3 \\ p \end{pmatrix}.$$ Find
  1. the unit vector in the direction of \(\overrightarrow{AB}\), [3]
  2. the value of the constant \(p\) for which angle \(BOC = 90°\). [2]
CAIE P1 2012 June Q3
6 marks Moderate -0.3
The first three terms in the expansion of \((1 - 2x)^2(1 + ax)^6\), in ascending powers of \(x\), are \(1 - x + bx^2\). Find the values of the constants \(a\) and \(b\). [6]
CAIE P1 2012 June Q4
6 marks Moderate -0.8
  1. Solve the equation \(\sin 2x + 3 \cos 2x = 0\) for \(0° \leqslant x \leqslant 360°\). [5]
  2. How many solutions has the equation \(\sin 2x + 3 \cos 2x = 0\) for \(0° \leqslant x \leqslant 1080°\)? [1]
CAIE P1 2012 June Q5
6 marks Standard +0.3
\includegraphics{figure_5} The diagram shows part of the curve \(x = \frac{8}{y^2} - 2\), crossing the \(y\)-axis at the point \(A\). The point \(B (6, 1)\) lies on the curve. The shaded region is bounded by the curve, the \(y\)-axis and the line \(y = 1\). Find the exact volume obtained when this shaded region is rotated through \(360°\) about the \(y\)-axis. [6]
CAIE P1 2012 June Q6
7 marks Moderate -0.3
The first term of an arithmetic progression is 12 and the sum of the first 9 terms is 135.
  1. Find the common difference of the progression. [2]
The first term, the ninth term and the \(n\)th term of this arithmetic progression are the first term, the second term and the third term respectively of a geometric progression.
  1. Find the common ratio of the geometric progression and the value of \(n\). [5]
CAIE P1 2012 June Q7
7 marks Moderate -0.3
The curve \(y = \frac{10}{2x + 1} - 2\) intersects the \(x\)-axis at \(A\). The tangent to the curve at \(A\) intersects the \(y\)-axis at \(C\).
  1. Show that the equation of \(AC\) is \(5y + 4x = 8\). [5]
  2. Find the distance \(AC\). [2]
CAIE P1 2012 June Q8
7 marks Moderate -0.3
\includegraphics{figure_8} In the diagram, \(AB\) is an arc of a circle with centre \(O\) and radius \(r\). The line \(XB\) is a tangent to the circle at \(B\) and \(A\) is the mid-point of \(OX\).
  1. Show that angle \(AOB = \frac{1}{3}\pi\) radians. [2]
Express each of the following in terms of \(r\), \(\pi\) and \(\sqrt{3}\):
  1. the perimeter of the shaded region, [3]
  2. the area of the shaded region. [2]
CAIE P1 2012 June Q9
8 marks Standard +0.3
A curve is such that \(\frac{d^2y}{dx^2} = -4x\). The curve has a maximum point at \((2, 12)\).
  1. Find the equation of the curve. [6]
A point \(P\) moves along the curve in such a way that the \(x\)-coordinate is increasing at 0.05 units per second.
  1. Find the rate at which the \(y\)-coordinate is changing when \(x = 3\), stating whether the \(y\)-coordinate is increasing or decreasing. [2]
CAIE P1 2012 June Q10
9 marks Moderate -0.3
The equation of a line is \(2y + x = k\), where \(k\) is a constant, and the equation of a curve is \(xy = 6\).
  1. In the case where \(k = 8\), the line intersects the curve at the points \(A\) and \(B\). Find the equation of the perpendicular bisector of the line \(AB\). [6]
  2. Find the set of values of \(k\) for which the line \(2y + x = k\) intersects the curve \(xy = 6\) at two distinct points. [3]
CAIE P1 2012 June Q11
10 marks Moderate -0.8
The function \(f\) is such that \(f(x) = 8 - (x - 2)^2\), for \(x \in \mathbb{R}\).
  1. Find the coordinates and the nature of the stationary point on the curve \(y = f(x)\). [3]
The function \(g\) is such that \(g(x) = 8 - (x - 2)^2\), for \(k \leqslant x \leqslant 4\), where \(k\) is a constant.
  1. State the smallest value of \(k\) for which \(g\) has an inverse. [1]
For this value of \(k\),
  1. find an expression for \(g^{-1}(x)\), [3]
  2. sketch, on the same diagram, the graphs of \(y = g(x)\) and \(y = g^{-1}(x)\). [3]
CAIE P1 2015 June Q1
4 marks Easy -1.2
Given that \(\theta\) is an obtuse angle measured in radians and that \(\sin \theta = k\), find, in terms of \(k\), an expression for
  1. \(\cos \theta\), [1]
  2. \(\tan \theta\), [2]
  3. \(\sin(\theta + \pi)\). [1]
CAIE P1 2015 June Q2
5 marks Standard +0.3
\includegraphics{figure_2} The diagram shows the curve \(y = 2x^2\) and the points \(X(-2, 0)\) and \(P(p, 0)\). The point \(Q\) lies on the curve and \(PQ\) is parallel to the \(y\)-axis.
  1. Express the area, \(A\), of triangle \(XPQ\) in terms of \(p\). [2]
The point \(P\) moves along the \(x\)-axis at a constant rate of 0.02 units per second and \(Q\) moves along the curve so that \(PQ\) remains parallel to the \(y\)-axis.
  1. Find the rate at which \(A\) is increasing when \(p = 2\). [3]
CAIE P1 2015 June Q3
7 marks Easy -1.2
  1. Find the first three terms, in ascending powers of \(x\), in the expansion of
    1. \((1 - x)^6\), [2]
    2. \((1 + 2x)^6\). [2]
  2. Hence find the coefficient of \(x^2\) in the expansion of \([(1 - x)(1 + 2x)]^6\). [3]
CAIE P1 2015 June Q4
7 marks Moderate -0.3
Relative to the origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow{OA} = \begin{pmatrix} 3 \\ 0 \\ -4 \end{pmatrix} \text{ and } \overrightarrow{OB} = \begin{pmatrix} 6 \\ -3 \\ 2 \end{pmatrix}.$$
  1. Find the cosine of angle \(AOB\). [3]
The position vector of \(C\) is given by \(\overrightarrow{OC} = \begin{pmatrix} k \\ -2k \\ 2k - 3 \end{pmatrix}\).
  1. Given that \(AB\) and \(OC\) have the same length, find the possible values of \(k\). [4]
CAIE P1 2015 June Q5
7 marks Moderate -0.3
A piece of wire of length 24 cm is bent to form the perimeter of a sector of a circle of radius \(r\) cm.
  1. Show that the area of the sector, \(A\) cm\(^2\), is given by \(A = 12r - r^2\). [3]
  2. Express \(A\) in the form \(a - (r - b)^2\), where \(a\) and \(b\) are constants. [2]
  3. Given that \(r\) can vary, state the greatest value of \(A\) and find the corresponding angle of the sector. [2]
CAIE P1 2015 June Q6
7 marks Standard +0.3
The line with gradient \(-2\) passing through the point \(P(3t, 2t)\) intersects the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).
  1. Find the area of triangle \(AOB\) in terms of \(t\). [3]
The line through \(P\) perpendicular to \(AB\) intersects the \(x\)-axis at \(C\).
  1. Show that the mid-point of \(PC\) lies on the line \(y = x\). [4]
CAIE P1 2015 June Q7
8 marks Moderate -0.3
  1. The third and fourth terms of a geometric progression are \(\frac{1}{4}\) and \(\frac{2}{9}\) respectively. Find the sum to infinity of the progression. [4]
  2. A circle is divided into 5 sectors in such a way that the angles of the sectors are in arithmetic progression. Given that the angle of the largest sector is 4 times the angle of the smallest sector, find the angle of the largest sector. [4]
CAIE P1 2015 June Q8
9 marks Moderate -0.3
The function \(\text{f} : x \mapsto 5 + 3\cos(\frac{1}{3}x)\) is defined for \(0 \leqslant x \leqslant 2\pi\).
  1. Solve the equation \(\text{f}(x) = 7\), giving your answer correct to 2 decimal places. [3]
  2. Sketch the graph of \(y = \text{f}(x)\). [2]
  3. Explain why \(\text{f}\) has an inverse. [1]
  4. Obtain an expression for \(\text{f}^{-1}(x)\). [3]
CAIE P1 2015 June Q9
10 marks Standard +0.3
The equation of a curve is \(y = x^3 + px^2\), where \(p\) is a positive constant.
  1. Show that the origin is a stationary point on the curve and find the coordinates of the other stationary point in terms of \(p\). [4]
  2. Find the nature of each of the stationary points. [3]
Another curve has equation \(y = x^3 + px^2 + px\).
  1. Find the set of values of \(p\) for which this curve has no stationary points. [3]
CAIE P1 2015 June Q10
11 marks Standard +0.3
\includegraphics{figure_10} The diagram shows part of the curve \(y = \frac{8}{\sqrt{(3x + 4)}}\). The curve intersects the \(y\)-axis at \(A(0, 4)\). The normal to the curve at \(A\) intersects the line \(x = 4\) at the point \(B\).
  1. Find the coordinates of \(B\). [5]
  2. Show, with all necessary working, that the areas of the regions marked \(P\) and \(Q\) are equal. [6]
CAIE P1 2015 June Q1
3 marks Easy -1.2
The function f is such that \(\mathrm{f}'(x) = 5 - 2x^2\) and \((3, 5)\) is a point on the curve \(y = \mathrm{f}(x)\). Find \(\mathrm{f}(x)\). [3]
CAIE P1 2015 June Q2
4 marks Moderate -0.8
\includegraphics{figure_2} In the diagram, \(AYB\) is a semicircle with \(AB\) as diameter and \(OAXB\) is a sector of a circle with centre \(O\) and radius \(r\). Angle \(AOB = 2\theta\) radians. Find an expression, in terms of \(r\) and \(\theta\), for the area of the shaded region. [4]
CAIE P1 2015 June Q3
5 marks Moderate -0.8
  1. Find the coefficients of \(x^2\) and \(x^3\) in the expansion of \((2 - x)^6\). [3]
  2. Find the coefficient of \(x^3\) in the expansion of \((3x + 1)(2 - x)^6\). [2]
CAIE P1 2015 June Q4
5 marks Moderate -0.3
Variables \(u\), \(x\) and \(y\) are such that \(u = 2x(y - x)\) and \(x + 3y = 12\). Express \(u\) in terms of \(x\) and hence find the stationary value of \(u\). [5]