Questions — CAIE P1 (1228 questions)

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CAIE P1 2024 November Q7
10 marks Standard +0.3
\includegraphics{figure_7} The diagram shows a metal plate \(ABCDEF\) consisting of five parts. The parts \(BCD\) and \(DEF\) are semicircles. The part \(BAFO\) is a sector of a circle with centre \(O\) and radius 20 cm, and \(D\) lies on this circle. The parts \(OBD\) and \(ODF\) are triangles. Angles \(BOD\) and \(DOF\) are both \(\theta\) radians.
  1. Given that \(\theta = 1.2\), find the area of the metal plate. Give your answer correct to 3 significant figures. [5]
  2. Given instead that the area of each semicircle is \(50\pi \text{ cm}^2\), find the exact perimeter of the metal plate. [5]
CAIE P1 2024 November Q8
9 marks Moderate -0.3
  1. Express \(3x^2 - 12x + 14\) in the form \(3(x + a)^2 + b\), where \(a\) and \(b\) are constants to be found. [2]
The function f(x) = \(3x^2 - 12x + 14\) is defined for \(x \geqslant k\), where \(k\) is a constant.
  1. Find the least value of \(k\) for which the function \(\text{f}^{-1}\) exists. [1]
For the rest of this question, you should assume that \(k\) has the value found in part (b).
  1. Find an expression for \(\text{f}^{-1}(x)\). [3]
  2. Hence or otherwise solve the equation \(\text{f f}(x) = 29\). [3]
CAIE P1 2024 November Q9
7 marks Moderate -0.3
\includegraphics{figure_9} The diagram shows the curves with equations \(y = x^3 - 3x + 3\) and \(y = 2x^3 - 4x^2 + 3\).
  1. Find the \(x\)-coordinates of the points of intersection of the curves. [3]
  2. Find the area of the shaded region. [4]
CAIE P1 2024 November Q10
9 marks Standard +0.3
Points \(A\) and \(B\) have coordinates \((4, 3)\) and \((8, -5)\) respectively. A circle with radius 10 passes through the points \(A\) and \(B\).
  1. Show that the centre of the circle lies on the line \(y = \frac{1}{2}x - 4\). [4]
  2. Find the two possible equations of the circle. [5]
CAIE P1 2024 November Q11
12 marks Standard +0.3
The equation of a curve is \(y = kx^{\frac{1}{2}} - 4x^2 + 2\), where \(k\) is a constant.
  1. Find \(\frac{\text{d}y}{\text{d}x}\) and \(\frac{\text{d}^2y}{\text{d}x^2}\) in terms of \(k\). [2]
  2. It is given that \(k = 2\). Find the coordinates of the stationary point and determine its nature. [4]
  3. Points \(A\) and \(B\) on the curve have \(x\)-coordinates 0.25 and 1 respectively. For a different value of \(k\), the tangents to the curve at the points \(A\) and \(B\) meet at a point with \(x\)-coordinate 0.6. Find this value of \(k\). [6]
CAIE P1 2010 June Q1
4 marks Moderate -0.8
  1. Show that the equation $$3(2\sin x - \cos x) = 2(\sin x - 3\cos x)$$ can be written in the form \(\tan x = -\frac{4}{5}\). [2]
  2. Solve the equation \(3(2\sin x - \cos x) = 2(\sin x - 3\cos x)\), for \(0° \leq x \leq 360°\). [2]
CAIE P1 2010 June Q2
4 marks Moderate -0.3
\includegraphics{figure_2} The diagram shows part of the curve \(y = \frac{a}{x}\), where \(a\) is a positive constant. Given that the volume obtained when the shaded region is rotated through \(360°\) about the \(x\)-axis is \(24\pi\), find the value of \(a\). [4]
CAIE P1 2010 June Q3
5 marks Moderate -0.3
The functions f and g are defined for \(x \in \mathbb{R}\) by $$f : x \mapsto 4x - 2x^2,$$ $$g : x \mapsto 5x + 3.$$
  1. Find the range of f. [2]
  2. Find the value of the constant \(k\) for which the equation \(gf(x) = k\) has equal roots. [3]
CAIE P1 2010 June Q4
6 marks Moderate -0.3
\includegraphics{figure_4} In the diagram, \(A\) is the point \((-1, 3)\) and \(B\) is the point \((3, 1)\). The line \(L_1\) passes through \(A\) and is parallel to \(OB\). The line \(L_2\) passes through \(B\) and is perpendicular to \(AB\). The lines \(L_1\) and \(L_2\) meet at \(C\). Find the coordinates of \(C\). [6]
CAIE P1 2010 June Q5
6 marks Moderate -0.8
Relative to an origin \(O\), the position vectors of the points \(A\) and \(B\) are given by $$\overrightarrow{OA} = \begin{pmatrix} -2 \\ 3 \\ 1 \end{pmatrix} \text{ and } \overrightarrow{OB} = \begin{pmatrix} 4 \\ 1 \\ p \end{pmatrix}$$
  1. Find the value of \(p\) for which \(\overrightarrow{OA}\) is perpendicular to \(\overrightarrow{OB}\). [2]
  2. Find the values of \(p\) for which the magnitude of \(\overrightarrow{AB}\) is 7. [4]
CAIE P1 2010 June Q6
7 marks Moderate -0.8
  1. Find the first 3 terms in the expansion of \((1 + ax)^4\) in ascending powers of \(x\). [2]
  2. Given that there is no term in \(x\) in the expansion of \((1 - 2x)(1 + ax)^5\), find the value of the constant \(a\). [2]
  3. For this value of \(a\), find the coefficient of \(x^2\) in the expansion of \((1 - 2x)(1 + ax)^5\). [3]
CAIE P1 2010 June Q7
8 marks Moderate -0.8
  1. Find the sum of all the multiples of 5 between 100 and 300 inclusive. [3]
  2. A geometric progression has a common ratio of \(-\frac{2}{3}\) and the sum of the first 3 terms is 35. Find
    1. the first term of the progression, [3]
    2. the sum to infinity. [2]
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]