Questions — CAIE P1 (1228 questions)

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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]
CAIE P1 2015 June Q5
5 marks Moderate -0.3
  1. Prove the identity \(\frac{\sin \theta - \cos \theta}{\sin \theta + \cos \theta} \equiv \frac{\tan \theta - 1}{\tan \theta + 1}\). [1]
  2. Hence solve the equation \(\frac{\sin \theta - \cos \theta}{\sin \theta + \cos \theta} = \frac{\tan \theta}{6}\), for \(0° \leqslant \theta \leqslant 180°\). [4]
CAIE P1 2015 June Q6
6 marks Moderate -0.3
A tourist attraction in a city centre is a big vertical wheel on which passengers can ride. The wheel turns in such a way that the height, \(h\) m, of a passenger above the ground is given by the formula \(h = 60(1 - \cos kt)\). In this formula, \(k\) is a constant, \(t\) is the time in minutes that has elapsed since the passenger started the ride at ground level and \(kt\) is measured in radians.
  1. Find the greatest height of the passenger above the ground. [1]
One complete revolution of the wheel takes 30 minutes.
  1. Show that \(k = \frac{\pi}{15}\pi\). [2]
  2. Find the time for which the passenger is above a height of 90 m. [3]
CAIE P1 2015 June Q7
7 marks Moderate -0.3
The point \(C\) lies on the perpendicular bisector of the line joining the points \(A(4, 6)\) and \(B(10, 2)\). \(C\) also lies on the line parallel to \(AB\) through \((3, 11)\).
  1. Find the equation of the perpendicular bisector of \(AB\). [4]
  2. Calculate the coordinates of \(C\). [3]
CAIE P1 2015 June Q8
9 marks Moderate -0.8
  1. The first, second and last terms in an arithmetic progression are 56, 53 and \(-22\) respectively. Find the sum of all the terms in the progression. [4]
  2. The first, second and third terms of a geometric progression are \(2k + 6\), \(2k\) and \(k + 2\) respectively, where \(k\) is a positive constant.
    1. Find the value of \(k\). [3]
    2. Find the sum to infinity of the progression. [2]
CAIE P1 2015 June Q9
9 marks Moderate -0.3
Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow{OA} = 2\mathbf{i} + 4\mathbf{j} + 4\mathbf{k} \quad \text{and} \quad \overrightarrow{OB} = 3\mathbf{i} + \mathbf{j} + 4\mathbf{k}.$$
  1. Use a vector method to find angle \(AOB\). [4]
The point \(C\) is such that \(\overrightarrow{AB} = \overrightarrow{BC}\).
  1. Find the unit vector in the direction of \(\overrightarrow{OC}\). [4]
  2. Show that triangle \(OAC\) is isosceles. [1]
CAIE P1 2015 June Q10
10 marks Standard +0.3
The equation of a curve is \(y = \frac{A}{2x - 1}\).
  1. Find, showing all necessary working, the volume obtained when the region bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 2\) is rotated through \(360°\) about the \(x\)-axis. [4]
  2. Given that the line \(2y = x + c\) is a normal to the curve, find the possible values of the constant \(c\). [6]
CAIE P1 2015 June Q11
12 marks Moderate -0.3
The function f is defined by \(\mathrm{f} : x \mapsto 2x^2 - 6x + 5\) for \(x \in \mathbb{R}\).
  1. Find the set of values of \(p\) for which the equation \(\mathrm{f}(x) = p\) has no real roots. [3]
The function g is defined by \(\mathrm{g} : x \mapsto 2x^2 - 6x + 5\) for \(0 \leqslant x \leqslant 4\).
  1. Express \(\mathrm{g}(x)\) in the form \(a(x + b)^2 + c\), where \(a\), \(b\) and \(c\) are constants. [3]
  2. Find the range of g. [2]
The function h is defined by \(\mathrm{h} : x \mapsto 2x^2 - 6x + 5\) for \(k \leqslant x \leqslant 4\), where \(k\) is a constant.
  1. State the smallest value of \(k\) for which h has an inverse. [1]
  2. For this value of \(k\), find an expression for \(\mathrm{h}^{-1}(x)\). [3]
CAIE P1 2017 June Q1
3 marks Moderate -0.5
The coefficients of \(x\) and \(x^2\) in the expansion of \((2 + ax)^7\) are equal. Find the value of the non-zero constant \(a\). [3]
CAIE P1 2017 June Q2
4 marks Standard +0.3
The common ratio of a geometric progression is \(r\). The first term of the progression is \((r^2 - 3r + 2)\) and the sum to infinity is \(S\).
  1. Show that \(S = 2 - r\). [2]
  2. Find the set of possible values that \(S\) can take. [2]
CAIE P1 2017 June Q3
4 marks Moderate -0.8
Find the coordinates of the points of intersection of the curve \(y = x^{\frac{2}{3}} - 1\) with the curve \(y = x^{\frac{1}{3}} + 1\). [4]
CAIE P1 2017 June Q4
6 marks Moderate -0.8
Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow{OA} = \begin{pmatrix} 5 \\ 1 \\ 3 \end{pmatrix} \text{ and } \overrightarrow{OB} = \begin{pmatrix} 5 \\ 4 \\ -3 \end{pmatrix}$$ The point \(P\) lies on \(AB\) and is such that \(\overrightarrow{AP} = \frac{3}{4}\overrightarrow{AB}\).
  1. Find the position vector of \(P\). [3]
  2. Find the distance \(OP\). [1]
  3. Determine whether \(OP\) is perpendicular to \(AB\). Justify your answer. [2]
CAIE P1 2017 June Q5
6 marks Moderate -0.3
  1. Show that the equation \(\frac{2 \sin \theta + \cos \theta}{\sin \theta + \cos \theta} = 2 \tan \theta\) may be expressed as \(\cos^2 \theta = 2 \sin^2 \theta\). [3]
  2. Hence solve the equation \(\frac{2 \sin \theta + \cos \theta}{\sin \theta + \cos \theta} = 2 \tan \theta\) for \(0° < \theta < 180°\). [3]
CAIE P1 2017 June Q6
6 marks Standard +0.3
The line \(3y + x = 25\) is a normal to the curve \(y = x^2 - 5x + k\). Find the value of the constant \(k\). [6]