Questions P3 (1243 questions)

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CAIE P3 2018 June Q9
9 marks Standard +0.3
\includegraphics{figure_9} The diagram shows a pyramid \(OABCD\) with a horizontal rectangular base \(OABC\). The sides \(OA\) and \(AB\) have lengths of 8 units and 6 units respectively. The point \(E\) on \(OB\) is such that \(OE = 2\) units. The point \(D\) of the pyramid is 7 units vertically above \(E\). Unit vectors \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\) are parallel to \(OA\), \(OC\) and \(ED\) respectively.
  1. Show that \(\overrightarrow{OE} = 1.6\mathbf{i} + 1.2\mathbf{j}\). [2]
  2. Use a scalar product to find angle \(BDO\). [7]
CAIE P3 2018 June Q10
9 marks Standard +0.3
The one-one function f is defined by \(\mathrm{f}(x) = (x - 2)^2 + 2\) for \(x \geqslant c\), where \(c\) is a constant.
  1. State the smallest possible value of \(c\). [1]
In parts (ii) and (iii) the value of \(c\) is 4.
  1. Find an expression for \(\mathrm{f}^{-1}(x)\) and state the domain of \(\mathrm{f}^{-1}\). [3]
  2. Solve the equation \(\mathrm{f f}(x) = 51\), giving your answer in the form \(a + \sqrt{b}\). [5]
CAIE P3 2018 June Q11
11 marks Standard +0.3
\includegraphics{figure_11} The diagram shows part of the curve \(y = (x + 1)^2 + (x + 1)^{-1}\) and the line \(x = 1\). The point \(A\) is the minimum point on the curve.
  1. Show that the \(x\)-coordinate of \(A\) satisfies the equation \(2(x + 1)^3 = 1\) and find the exact value of \(\frac{\mathrm{d}^2 y}{\mathrm{d}x^2}\) at \(A\). [5]
  2. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360°\) about the \(x\)-axis. [6]
CAIE P3 2013 November Q1
3 marks Moderate -0.8
The equation of a curve is \(y = \frac{1+x}{1+2x}\) for \(x > -\frac{1}{2}\). Show that the gradient of the curve is always negative. [3]
CAIE P3 2013 November Q2
4 marks Standard +0.3
Solve the equation \(2|3^x - 1| = 3^x\), giving your answers correct to 3 significant figures. [4]
CAIE P3 2013 November Q3
5 marks Standard +0.3
Find the exact value of \(\int_1^4 \frac{\ln x}{\sqrt{x}} dx\). [5]
CAIE P3 2013 November Q4
6 marks Standard +0.3
The parametric equations of a curve are $$x = e^{-t}\cos t, \quad y = e^{-t}\sin t.$$ Show that \(\frac{dy}{dx} = \tan(t - \frac{1}{4}\pi)\). [6]
CAIE P3 2013 November Q5
7 marks Standard +0.3
  1. Prove that \(\cot \theta + \tan \theta = 2\cosec 2\theta\). [3]
  2. Hence show that \(\int_{\frac{\pi}{8}}^{\frac{3\pi}{8}} \cosec 2\theta \, d\theta = \frac{1}{2}\ln 3\). [4]
CAIE P3 2013 November Q6
8 marks Challenging +1.2
\includegraphics{figure_6} In the diagram, \(A\) is a point on the circumference of a circle with centre \(O\) and radius \(r\). A circular arc with centre \(A\) meets the circumference at \(B\) and \(C\). The angle \(OAB\) is \(\theta\) radians. The shaded region is bounded by the circumference of the circle and the arc with centre \(A\) joining \(B\) and \(C\). The area of the shaded region is equal to half the area of the circle.
  1. Show that \(\cos 2\theta = \frac{2\sin 2\theta - \pi}{4\theta}\). [5]
  2. Use the iterative formula $$\theta_{n+1} = \frac{1}{2}\cos^{-1}\left(\frac{2\sin 2\theta_n - \pi}{4\theta_n}\right),$$ with initial value \(\theta_1 = 1\), to determine \(\theta\) correct to 2 decimal places, showing the result of each iteration to 4 decimal places. [3]
CAIE P3 2013 November Q7
10 marks Standard +0.3
Let \(f(x) = \frac{2x^2 - 7x - 1}{(x-2)(x^2+3)}\).
  1. Express \(f(x)\) in partial fractions. [5]
  2. Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\). [5]
CAIE P3 2013 November Q8
10 marks Standard +0.3
Throughout this question the use of a calculator is not permitted.
  1. The complex numbers \(u\) and \(v\) satisfy the equations $$u + 2v = 2i \quad \text{and} \quad iu + v = 3.$$ Solve the equations for \(u\) and \(v\), giving both answers in the form \(x + iy\), where \(x\) and \(y\) are real. [5]
  2. On an Argand diagram, sketch the locus representing complex numbers \(z\) satisfying \(|z + i| = 1\) and the locus representing complex numbers \(w\) satisfying \(\arg(w - 2) = \frac{\pi}{4}\). Find the least value of \(|z - w|\) for points on these loci. [5]
CAIE P3 2013 November Q9
11 marks Standard +0.3
\includegraphics{figure_9} The diagram shows three points \(A\), \(B\) and \(C\) whose position vectors with respect to the origin \(O\) are given by \(\overrightarrow{OA} = \begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}\), \(\overrightarrow{OB} = \begin{pmatrix} 0 \\ 3 \\ 1 \end{pmatrix}\) and \(\overrightarrow{OC} = \begin{pmatrix} 3 \\ 0 \\ 4 \end{pmatrix}\). The point \(D\) lies on \(BC\), between \(B\) and \(C\), and is such that \(CD = 2DB\).
  1. Find the equation of the plane \(ABC\), giving your answer in the form \(ax + by + cz = d\). [6]
  2. Find the position vector of \(D\). [1]
  3. Show that the length of the perpendicular from \(A\) to \(OD\) is \(\frac{1}{3}\sqrt{(65)}\). [4]
CAIE P3 2013 November Q10
11 marks Standard +0.3
\includegraphics{figure_10} A tank containing water is in the form of a cone with vertex \(C\). The axis is vertical and the semi-vertical angle is \(60°\), as shown in the diagram. At time \(t = 0\), the tank is full and the depth of water is \(H\). At this instant, a tap at \(C\) is opened and water begins to flow out. The volume of water in the tank decreases at a rate proportional to \(\sqrt{h}\), where \(h\) is the depth of water at time \(t\). The tank becomes empty when \(t = 60\).
  1. Show that \(h\) and \(t\) satisfy a differential equation of the form $$\frac{dh}{dt} = -Ah^{-\frac{1}{2}},$$ where \(A\) is a positive constant. [4]
  2. Solve the differential equation given in part (i) and obtain an expression for \(t\) in terms of \(h\) and \(H\). [6]
  3. Find the time at which the depth reaches \(\frac{1}{2}H\). [1]
[The volume \(V\) of a cone of vertical height \(h\) and base radius \(r\) is given by \(V = \frac{1}{3}\pi r^2 h\).]
CAIE P3 2017 November Q1
3 marks Moderate -0.8
Find the quotient and remainder when \(x^4\) is divided by \(x^2 + 2x - 1\). [3]
CAIE P3 2017 November Q2
5 marks Moderate -0.8
Two variable quantities \(x\) and \(y\) are believed to satisfy an equation of the form \(y = C(a^x)\), where \(C\) and \(a\) are constants. An experiment produced four pairs of values of \(x\) and \(y\). The table below gives the corresponding values of \(x\) and \(\ln y\).
\(x\)0.91.62.43.2
\(\ln y\)1.71.92.32.6
By plotting \(\ln y\) against \(x\) for these four pairs of values and drawing a suitable straight line, estimate the values of \(C\) and \(a\). Give your answers correct to 2 significant figures. [5] \includegraphics{figure_2}
CAIE P3 2017 November Q3
6 marks Standard +0.3
The equation \(x^3 = 3x + 7\) has one real root, denoted by \(\alpha\).
  1. Show by calculation that \(\alpha\) lies between 2 and 3. [2]
Two iterative formulae, \(A\) and \(B\), derived from this equation are as follows: $$x_{n+1} = (3x_n + 7)^{\frac{1}{3}}, \quad (A)$$ $$x_{n+1} = \frac{x_n^3 - 7}{3}. \quad (B)$$ Each formula is used with initial value \(x_1 = 2.5\).
  1. Show that one of these formulae produces a sequence which fails to converge, and use the other formula to calculate \(\alpha\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [4]
CAIE P3 2017 November Q4
7 marks Standard +0.3
  1. Prove the identity \(\tan(45° + x) + \tan(45° - x) = 2 \sec 2x\). [4]
  2. Hence sketch the graph of \(y = \tan(45° + x) + \tan(45° - x)\) for \(0° \leqslant x \leqslant 90°\). [3]
CAIE P3 2017 November Q5
8 marks Standard +0.3
The equation of a curve is \(2x^4 + xy^3 + y^4 = 10\).
  1. Show that \(\frac{dy}{dx} = -\frac{8x^3 + y^3}{3xy^2 + 4y^3}\). [4]
  2. Hence show that there are two points on the curve at which the tangent is parallel to the \(x\)-axis and find the coordinates of these points. [4]
CAIE P3 2017 November Q6
8 marks Standard +0.3
The variables \(x\) and \(y\) satisfy the differential equation $$\frac{dy}{dx} = 4 \cos^2 y \tan x,$$ for \(0 \leqslant x < \frac{1}{2}\pi\), and \(x = 0\) when \(y = \frac{1}{4}\pi\). Solve this differential equation and find the value of \(x\) when \(y = \frac{1}{8}\pi\). [8]
CAIE P3 2017 November Q7
9 marks Standard +0.3
  1. The complex number \(u\) is given by \(u = 8 - 15\text{i}\). Showing all necessary working, find the two square roots of \(u\). Give answers in the form \(a + ib\), where the numbers \(a\) and \(b\) are real and exact. [5]
  2. On an Argand diagram, shade the region whose points represent complex numbers satisfying both the inequalities \(|z - 2 - \text{i}| \leqslant 2\) and \(0 \leqslant \arg(z - \text{i}) \leqslant \frac{1}{4}\pi\). [4]
CAIE P3 2017 November Q8
9 marks Standard +0.3
Let \(\text{f}(x) = \frac{4x^2 + 9x - 8}{(x + 2)(2x - 1)}\).
  1. Express \(\text{f}(x)\) in the form \(A + \frac{B}{x + 2} + \frac{C}{2x - 1}\). [4]
  2. Hence show that \(\int_1^4 \text{f}(x) \, dx = 6 + \frac{1}{2} \ln\left(\frac{16}{7}\right)\). [5]
CAIE P3 2017 November Q9
9 marks Standard +0.8
\includegraphics{figure_9} The diagram shows the curve \(y = (1 + x^2)\text{e}^{-\frac{3x}{4}}\) for \(x \geqslant 0\). The shaded region \(R\) is enclosed by the curve, the \(x\)-axis and the lines \(x = 0\) and \(x = 2\).
  1. Find the exact values of the \(x\)-coordinates of the stationary points of the curve. [4]
  2. Show that the exact value of the area of \(R\) is \(18 - \frac{42}{\text{e}}\). [5]
CAIE P3 2017 November Q10
11 marks Standard +0.3
The equations of two lines \(l\) and \(m\) are \(\mathbf{r} = 3\mathbf{i} - \mathbf{j} - 2\mathbf{k} + \lambda(-\mathbf{i} + \mathbf{j} + 4\mathbf{k})\) and \(\mathbf{r} = 4\mathbf{i} + 4\mathbf{j} - 3\mathbf{k} + \mu(2\mathbf{i} + \mathbf{j} - 2\mathbf{k})\) respectively.
  1. Show that the lines do not intersect. [3]
  2. Calculate the acute angle between the directions of the lines. [3]
  3. Find the equation of the plane which passes through the point \((3, -2, -1)\) and which is parallel to both \(l\) and \(m\). Give your answer in the form \(ax + by + cz = d\). [5]
CAIE P3 2018 November Q1
4 marks Moderate -0.3
Solve the inequality \(3|2x - 1| > |x + 4|\). [4]
CAIE P3 2018 November Q2
4 marks Moderate -0.3
Showing all necessary working, solve the equation \(\sin(\theta - 30°) + \cos \theta = 2 \sin \theta\), for \(0° < \theta < 180°\). [4]