Questions — CAIE P1 (1228 questions)

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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]
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]