Questions — CAIE (7646 questions)

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CAIE P1 2018 November Q3
5 marks Standard +0.3
\includegraphics{figure_3} The diagram shows part of the curve \(y = x(9 - x^2)\) and the line \(y = 5x\), intersecting at the origin \(O\) and the point \(R\). Point \(P\) lies on the line \(y = 5x\) between \(O\) and \(R\) and the \(x\)-coordinate of \(P\) is \(t\). Point \(Q\) lies on the curve and \(PQ\) is parallel to the \(y\)-axis.
  1. Express the length of \(PQ\) in terms of \(t\), simplifying your answer. [2]
  2. Given that \(t\) can vary, find the maximum value of the length of \(PQ\). [3]
CAIE P1 2018 November Q4
6 marks Moderate -0.8
Functions f and g are defined by $$f : x \mapsto 2 - 3\cos x \text{ for } 0 \leqslant x \leqslant 2\pi,$$ $$g : x \mapsto \frac{1}{2}x \text{ for } 0 \leqslant x \leqslant 2\pi.$$
  1. Solve the equation \(\text{fg}(x) = 1\). [3]
  2. Sketch the graph of \(y = \text{f}(x)\). [3]
CAIE P1 2018 November Q5
7 marks Standard +0.3
The first three terms of an arithmetic progression are \(4\), \(x\) and \(y\) respectively. The first three terms of a geometric progression are \(x\), \(y\) and \(18\) respectively. It is given that both \(x\) and \(y\) are positive.
  1. Find the value of \(x\) and the value of \(y\). [4]
  2. Find the fourth term of each progression. [3]
CAIE P1 2018 November Q6
7 marks Standard +0.3
\includegraphics{figure_6} The diagram shows a triangle \(ABC\) in which \(BC = 20\) cm and angle \(ABC = 90°\). The perpendicular from \(B\) to \(AC\) meets \(AC\) at \(D\) and \(AD = 9\) cm. Angle \(BCA = \theta°\).
  1. By expressing the length of \(BD\) in terms of \(\theta\) in each of the triangles \(ABD\) and \(DBC\), show that \(20\sin^2 \theta = 9\cos \theta\). [4]
  2. Hence, showing all necessary working, calculate \(\theta\). [3]
CAIE P1 2018 November Q7
7 marks Standard +0.3
\includegraphics{figure_7} The diagram shows a solid cylinder standing on a horizontal circular base with centre \(O\) and radius \(4\) units. Points \(A\), \(B\) and \(C\) lie on the circumference of the base such that \(AB\) is a diameter and angle \(BOC = 90°\). Points \(P\), \(Q\) and \(R\) lie on the upper surface of the cylinder vertically above \(A\), \(B\) and \(C\) respectively. The height of the cylinder is \(12\) units. The mid-point of \(CR\) is \(M\) and \(N\) lies on \(BQ\) with \(BN = 4\) units. Unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are parallel to \(OB\) and \(OC\) respectively and the unit vector \(\mathbf{k}\) is vertically upwards. Evaluate \(\overrightarrow{PN} \cdot \overrightarrow{PM}\) and hence find angle \(MPN\). [7]
CAIE P1 2018 November Q8
7 marks Standard +0.3
\includegraphics{figure_8} The diagram shows an isosceles triangle \(ACB\) in which \(AB = BC = 8\) cm and \(AC = 12\) cm. The arc \(XC\) is part of a circle with centre \(A\) and radius \(12\) cm, and the arc \(YC\) is part of a circle with centre \(B\) and radius \(8\) cm. The points \(A\), \(B\), \(X\) and \(Y\) lie on a straight line.
  1. Show that angle \(CBY = 1.445\) radians, correct to \(4\) significant figures. [3]
  2. Find the perimeter of the shaded region. [4]
CAIE P1 2018 November Q9
7 marks Moderate -0.8
The function f is defined by \(\text{f} : x \mapsto 2x^2 - 12x + 7\) for \(x \in \mathbb{R}\).
  1. Express \(2x^2 - 12x + 7\) in the form \(2(x + a)^2 + b\), where \(a\) and \(b\) are constants. [2]
  2. State the range of f. [1]
The function g is defined by \(\text{g} : x \mapsto 2x^2 - 12x + 7\) for \(x \leqslant k\).
  1. State the largest value of \(k\) for which g has an inverse. [1]
  2. Given that g has an inverse, find an expression for \(\text{g}^{-1}(x)\). [3]
CAIE P1 2018 November Q10
9 marks Moderate -0.3
The equation of a curve is \(y = 2x + \frac{12}{x}\) and the equation of a line is \(y + x = k\), where \(k\) is a constant.
  1. Find the set of values of \(k\) for which the line does not meet the curve. [3]
In the case where \(k = 15\), the curve intersects the line at points \(A\) and \(B\).
  1. Find the coordinates of \(A\) and \(B\). [3]
  2. Find the equation of the perpendicular bisector of the line joining \(A\) and \(B\). [3]
CAIE P1 2018 November Q11
12 marks Moderate -0.3
\includegraphics{figure_11} The diagram shows part of the curve \(y = 3\sqrt{(4x + 1)} - 2x\). The curve crosses the \(y\)-axis at \(A\) and the stationary point on the curve is \(M\).
  1. Obtain expressions for \(\frac{\text{d}y}{\text{d}x}\) and \(\int y \text{d}x\). [5]
  2. Find the coordinates of \(M\). [3]
  3. Find, showing all necessary working, the area of the shaded region. [4]
CAIE P2 2024 June Q1
4 marks Standard +0.3
Solve the inequality \(|5x + 7| > |2x - 3|\). [4]
CAIE P2 2024 June Q2
4 marks Standard +0.3
Use logarithms to solve the equation \(6^{2x-1} = 5e^{3x+2}\). Give your answer correct to 4 significant figures. [4]
CAIE P2 2024 June Q3
8 marks Moderate -0.3
\includegraphics{figure_3} The diagram shows the curve with equation \(y = 8e^{-x} - e^{2x}\). The curve crosses the y-axis at the point A and the x-axis at the point B. The shaded region is bounded by the curve and the two axes.
  1. Find the gradient of the curve at A. [3]
  2. Show that the x-coordinate of B is \(\ln 2\) and hence find the area of the shaded region. [5]
CAIE P2 2024 June Q4
7 marks Standard +0.3
A curve is defined by the parametric equations $$x = 4\cos^2 t, \quad y = \sqrt{3}\sin 2t,$$ for values of \(t\) such that \(0 < t < \frac{1}{2}\pi\). Find the equation of the normal to the curve at the point for which \(t = \frac{1}{6}\pi\). Give your answer in the form \(ax + by + c = 0\) where \(a\), \(b\) and \(c\) are integers. [7]
CAIE P2 2024 June Q5
8 marks Standard +0.3
The polynomial \(p(x)\) is defined by \(p(x) = 9x^3 + 18x^2 + 5x + 4\).
  1. Find the quotient when \(p(x)\) is divided by \((3x + 2)\), and show that the remainder is 6. [3]
  2. Find the value of \(\int_0^2 \frac{p(x)}{3x + 2} \, dx\), giving your answer in the form \(a + \ln b\) where \(a\) and \(b\) are integers. [5]
CAIE P2 2024 June Q6
9 marks Standard +0.3
\includegraphics{figure_6} The diagram shows the curve with equation \(y = \frac{\ln(2x + 1)}{x + 3}\). The curve has a maximum point M.
  1. Find an expression for \(\frac{dy}{dx}\). [2]
  2. Show that the x-coordinate of M satisfies the equation \(x = \frac{x + 3}{\ln(2x + 1)} - 0.5\). [2]
  3. Show by calculation that the x-coordinate of M lies between 2.5 and 3.0. [2]
  4. Use an iterative formula based on the equation in part (b) to find the x-coordinate of M correct to 4 significant figures. Give the result of each iteration to 6 significant figures. [3]
CAIE P2 2024 June Q7
10 marks Standard +0.3
  1. Prove that \(2\sin\theta\cosec 2\theta \equiv \sec\theta\). [2]
  2. Solve the equation \(\tan^2\theta + 7\sin\theta\cosec 2\theta = 8\) for \(-\pi < \theta < \pi\). [5]
  3. Find \(\int 8\sin^2\frac{1}{2}x\cosec^2 x \, dx\). [3]
CAIE P2 2023 March Q1
4 marks Standard +0.3
Find the exact value of \(\int_0^{\frac{\pi}{4}} 2 \tan^2(\frac{1}{2}x) \, dx\). [4]
CAIE P2 2023 March Q2
5 marks Standard +0.8
Solve the equation \(\tan(\theta - 60°) = 3 \cot \theta\) for \(-90° < \theta < 90°\). [5]
CAIE P2 2023 March Q3
8 marks Moderate -0.3
The polynomial \(p(x)\) is defined by $$p(x) = ax^3 - ax^2 + ax + b,$$ where \(a\) and \(b\) are constants. It is given that \((x + 2)\) is a factor of \(p(x)\), and that the remainder is 35 when \(p(x)\) is divided by \((x - 3)\).
  1. Find the values of \(a\) and \(b\). [5]
  2. Hence factorise \(p(x)\) and show that the equation \(p(x) = 0\) has exactly one real root. [3]
CAIE P2 2023 March Q4
7 marks Moderate -0.3
  1. Sketch, on the same diagram, the graphs of \(y = |2x - 1|\) and \(y = 3x - 3\). [2]
  2. Solve the inequality \(|2x - 1| < 3x - 3\). [3]
  3. Find the smallest integer \(N\) satisfying the inequality \(|2 \ln N - 1| < 3 \ln N - 3\). [2]
CAIE P2 2023 March Q5
8 marks Standard +0.3
It is given that \(\int_1^a \left(\frac{4}{1 + 2x} + \frac{3}{x}\right) dx = \ln 10\), where \(a\) is a constant greater than 1.
  1. Show that \(a = \sqrt{90(1 + 2a)^{-2}}\). [5]
  2. Use an iterative formula, based on the equation in (a), to find the value of \(a\) correct to 3 significant figures. Use an initial value of 1.7 and give the result of each iteration to 5 significant figures. [3]
CAIE P2 2023 March Q6
8 marks Standard +0.3
\includegraphics{figure_6} The diagram shows the curve with equation \(y = \frac{4e^{2x} + 9}{e^x + 2}\). The curve has a minimum point \(M\) and crosses the \(y\)-axis at the point \(P\).
  1. Find the exact value of the gradient of the curve at \(P\). [4]
  2. Find the exact coordinates of \(M\). [4]
CAIE P2 2023 March Q7
10 marks Standard +0.3
\includegraphics{figure_7} The diagram shows the curve with parametric equations $$x = k \tan t, \quad y = 3 \sin 2t - 4 \sin t,$$ for \(0 < t < \frac{1}{2}\pi\). It is given that \(k\) is a positive constant. The curve crosses the \(x\)-axis at the point \(P\).
  1. Find the value of \(\cos t\) at \(P\), giving your answer as an exact fraction. [3]
  2. Express \(\frac{dy}{dx}\) in terms of \(k\) and \(\cos t\). [4]
  3. Given that the normal to the curve at \(P\) has gradient \(\frac{9}{10}\), find the value of \(k\), giving your answer as an exact fraction. [3]
CAIE P2 2024 March Q1
4 marks Moderate -0.5
Use logarithms to solve the equation \(3^{4t+3} = 5^{2t+7}\). Give your answer correct to 3 significant figures. [4]
CAIE P2 2024 March Q2
4 marks Moderate -0.3
  1. Sketch the graph of \(y = |3x - 7|\), stating the coordinates of the points where the graph meets the axes. [2]
  2. Hence find the set of values of the constant \(k\) for which the equation \(|3x - 7| = k(x - 4)\) has exactly two real roots. [2]