Questions P3 (1243 questions)

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CAIE P3 2017 June Q5
8 marks Standard +0.3
In a certain chemical process a substance \(A\) reacts with and reduces a substance \(B\). The masses of \(A\) and \(B\) at time \(t\) after the start of the process are \(x\) and \(y\) respectively. It is given that \(\frac{dy}{dt} = -0.2xy\) and \(x = \frac{10}{(1 + t)^2}\). At the beginning of the process \(y = 100\).
  1. Form a differential equation in \(y\) and \(t\), and solve this differential equation. [6]
  2. Find the exact value approached by the mass of \(B\) as \(t\) becomes large. State what happens to the mass of \(A\) as \(t\) becomes large. [2]
CAIE P3 2017 June Q6
8 marks Standard +0.3
Throughout this question the use of a calculator is not permitted. The complex number \(2 - \mathrm{i}\) is denoted by \(u\).
  1. It is given that \(u\) is a root of the equation \(x^3 + ax^2 - 3x + b = 0\), where the constants \(a\) and \(b\) are real. Find the values of \(a\) and \(b\). [4]
  2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying both the inequalities \(|z - u| < 1\) and \(|z| < |z + \mathrm{i}|\). [4]
CAIE P3 2017 June Q7
9 marks Standard +0.3
  1. Prove that if \(y = \frac{1}{\cos \theta}\) then \(\frac{dy}{d\theta} = \sec \theta \tan \theta\). [2]
  2. Prove the identity \(\frac{1 + \sin \theta}{1 - \sin \theta} = 2 \sec^2 \theta + 2 \sec \theta \tan \theta - 1\). [3]
  3. Hence find the exact value of \(\int_0^{\frac{\pi}{4}} \frac{1 + \sin \theta}{1 - \sin \theta} d\theta\). [4]
CAIE P3 2017 June Q8
10 marks Standard +0.3
Let \(\mathrm{f}(x) = \frac{5x^2 - 7x + 4}{(3x + 2)(x^2 + 5)}\).
  1. Express \(\mathrm{f}(x)\) in partial fractions. [5]
  2. Hence obtain the expansion of \(\mathrm{f}(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\). [5]
CAIE P3 2017 June Q9
11 marks Standard +0.3
Relative to the origin \(O\), the point \(A\) has position vector given by \(\overrightarrow{OA} = \mathbf{i} + 2\mathbf{j} + 4\mathbf{k}\). The line \(l\) has equation \(\mathbf{r} = 9\mathbf{i} - \mathbf{j} + 8\mathbf{k} + \mu(3\mathbf{i} - \mathbf{j} + 2\mathbf{k})\).
  1. Find the position vector of the foot of the perpendicular from \(A\) to \(l\). Hence find the position vector of the reflection of \(A\) in \(l\). [5]
  2. Find the equation of the plane through the origin which contains \(l\). Give your answer in the form \(ax + by + cz = d\). [3]
  3. Find the exact value of the perpendicular distance of \(A\) from this plane. [3]
CAIE P3 2017 June Q10
11 marks Standard +0.3
\includegraphics{figure_10} The diagram shows the curve \(y = x^2 \cos 2x\) for \(0 \leq x \leq \frac{1}{4}\pi\). The curve has a maximum point at \(M\) where \(x = p\).
  1. Show that \(p\) satisfies the equation \(p = \frac{1}{2} \tan^{-1} \left(\frac{1}{p}\right)\). [3]
  2. Use the iterative formula \(p_{n+1} = \frac{1}{2} \tan^{-1} \left(\frac{1}{p_n}\right)\) to determine the value of \(p\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]
  3. Find, showing all necessary working, the exact area of the region bounded by the curve and the \(x\)-axis. [5]
CAIE P3 2018 June Q1
5 marks Standard +0.3
The coefficient of \(x^2\) in the expansion of \(\left(2 + \frac{x}{2}\right)^6 + (a + x)^5\) is 330. Find the value of the constant \(a\). [5]
CAIE P3 2018 June Q2
5 marks Moderate -0.8
The equation of a curve is \(y = x^2 - 6x + k\), where \(k\) is a constant.
  1. Find the set of values of \(k\) for which the whole of the curve lies above the \(x\)-axis. [2]
  2. Find the value of \(k\) for which the line \(y + 2x = 7\) is a tangent to the curve. [3]
CAIE P3 2018 June Q3
5 marks Moderate -0.8
A company producing salt from sea water changed to a new process. The amount of salt obtained each week increased by 2% of the amount obtained in the preceding week. It is given that in the first week after the change the company obtained 8000 kg of salt.
  1. Find the amount of salt obtained in the 12th week after the change. [3]
  2. Find the total amount of salt obtained in the first 12 weeks after the change. [2]
CAIE P3 2018 June Q4
6 marks Moderate -0.3
The function f is such that \(\mathrm{f}(x) = a + b \cos x\) for \(0 \leqslant x \leqslant 2\pi\). It is given that \(\mathrm{f}\left(\frac{1}{3}\pi\right) = 5\) and \(\mathrm{f}(\pi) = 11\).
  1. Find the values of the constants \(a\) and \(b\). [3]
  2. Find the set of values of \(k\) for which the equation \(\mathrm{f}(x) = k\) has no solution. [3]
CAIE P3 2018 June Q5
6 marks Standard +0.3
\includegraphics{figure_5} The diagram shows a three-dimensional shape. The base \(OAB\) is a horizontal triangle in which angle \(AOB\) is 90°. The side \(OBCD\) is a rectangle and the side \(OAD\) lies in a vertical plane. Unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are parallel to \(OA\) and \(OB\) respectively and the unit vector \(\mathbf{k}\) is vertical. The position vectors of \(A\), \(B\) and \(D\) are given by \(\overrightarrow{OA} = 8\mathbf{i}\), \(\overrightarrow{OB} = 5\mathbf{j}\) and \(\overrightarrow{OD} = 2\mathbf{i} + 4\mathbf{k}\).
  1. Express each of the vectors \(\overrightarrow{DA}\) and \(\overrightarrow{CA}\) in terms of \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\). [2]
  2. Use a scalar product to find angle \(CAD\). [4]
CAIE P3 2018 June Q6
6 marks Standard +0.3
\includegraphics{figure_6} The diagram shows points \(A\) and \(B\) on a circle with centre \(O\) and radius \(r\). The tangents to the circle at \(A\) and \(B\) meet at \(T\). The shaded region is bounded by the minor arc \(AB\) and the lines \(AT\) and \(BT\). Angle \(AOB\) is \(2\theta\) radians.
  1. In the case where the area of the sector \(AOB\) is the same as the area of the shaded region, show that \(\tan \theta = 2\theta\). [3]
  2. In the case where \(r = 8\) cm and the length of the minor arc \(AB\) is 19.2 cm, find the area of the shaded region. [3]
CAIE P3 2018 June Q7
7 marks Moderate -0.8
The function f is defined by \(\mathrm{f} : x \mapsto 7 - 2x^2 - 12x\) for \(x \in \mathbb{R}\).
  1. Express \(7 - 2x^2 - 12x\) in the form \(a - 2(x + b)^2\), where \(a\) and \(b\) are constants. [2]
  2. State the coordinates of the stationary point on the curve \(y = \mathrm{f}(x)\). [1]
The function g is defined by \(\mathrm{g} : x \mapsto 7 - 2x^2 - 12x\) for \(x \geqslant k\).
  1. State the smallest value of \(k\) for which g has an inverse. [1]
  2. For this value of \(k\), find \(\mathrm{g}^{-1}(x)\). [3]
CAIE P3 2018 June Q8
7 marks Standard +0.3
Points \(A\) and \(B\) have coordinates \((h, h)\) and \((4h + 6, 5h)\) respectively. The equation of the perpendicular bisector of \(AB\) is \(3x + 2y = k\). Find the values of the constants \(h\) and \(k\). [7]
CAIE P3 2018 June Q9
8 marks Standard +0.3
A curve is such that \(\frac{\mathrm{d}y}{\mathrm{d}x} = \sqrt{(4x + 1)}\) and \((2, 5)\) is a point on the curve.
  1. Find the equation of the curve. [4]
  2. A point \(P\) moves along the curve in such a way that the \(y\)-coordinate is increasing at a constant rate of 0.06 units per second. Find the rate of change of the \(x\)-coordinate when \(P\) passes through \((2, 5)\). [2]
  3. Show that \(\frac{\mathrm{d}^2y}{\mathrm{d}x^2} \times \frac{\mathrm{d}y}{\mathrm{d}x}\) is constant. [2]
CAIE P3 2018 June Q10
8 marks Moderate -0.3
  1. Solve the equation \(2 \cos x + 3 \sin x = 0\), for \(0° \leqslant x \leqslant 360°\). [3]
  2. Sketch, on the same diagram, the graphs of \(y = 2 \cos x\) and \(y = -3 \sin x\) for \(0° \leqslant x \leqslant 360°\). [3]
  3. Use your answers to parts (i) and (ii) to find the set of values of \(x\) for \(0° \leqslant x \leqslant 360°\) for which \(2 \cos x + 3 \sin x > 0\). [2]
CAIE P3 2018 June Q11
12 marks Standard +0.8
\includegraphics{figure_11} The diagram shows part of the curve \(y = \frac{x}{2} + \frac{6}{x}\). The line \(y = 4\) intersects the curve at the points \(P\) and \(Q\).
  1. Show that the tangents to the curve at \(P\) and \(Q\) meet at a point on the line \(y = x\). [6]
  2. Find, showing all necessary working, the volume obtained when the shaded region is rotated through 360° about the \(x\)-axis. Give your answer in terms of \(\pi\). [6]
CAIE P3 2018 June Q1
3 marks Easy -1.2
Express \(3x^2 - 12x + 7\) in the form \(a(x + b)^2 + c\), where \(a\), \(b\) and \(c\) are constants. [3]
CAIE P3 2018 June Q2
3 marks Easy -1.2
Find the coefficient of \(\frac{1}{x}\) in the expansion of \(\left(x - \frac{2}{x}\right)^5\). [3]
CAIE P3 2018 June Q3
5 marks Standard +0.3
The common ratio of a geometric progression is 0.99. Express the sum of the first 100 terms as a percentage of the sum to infinity, giving your answer correct to 2 significant figures. [5]
CAIE P3 2018 June Q4
6 marks Moderate -0.3
A curve with equation \(y = \mathrm{f}(x)\) passes through the point \(A(3, 1)\) and crosses the \(y\)-axis at \(B\). It is given that \(\mathrm{f}'(x) = (3x - 1)^{-\frac{1}{3}}\). Find the \(y\)-coordinate of \(B\). [6]
CAIE P3 2018 June Q5
5 marks Standard +0.3
\includegraphics{figure_5} The diagram shows a triangle \(OAB\) in which angle \(OAB = 90°\) and \(OA = 5\) cm. The arc \(AC\) is part of a circle with centre \(O\). The arc has length 6 cm and it meets \(OB\) at \(C\). Find the area of the shaded region. [5]
CAIE P3 2018 June Q6
7 marks Moderate -0.3
The coordinates of points \(A\) and \(B\) are \((-3k - 1, k + 3)\) and \((k + 3, 3k + 5)\) respectively, where \(k\) is a constant \((k \neq -1)\).
  1. Find and simplify the gradient of \(AB\), showing that it is independent of \(k\). [2]
  2. Find and simplify the equation of the perpendicular bisector of \(AB\). [5]
CAIE P3 2018 June Q7
9 marks Moderate -0.3
    1. Express \(\frac{\tan^2 \theta - 1}{\tan^2 \theta + 1}\) in the form \(a \sin^2 \theta + b\), where \(a\) and \(b\) are constants to be found. [3]
    2. Hence, or otherwise, and showing all necessary working, solve the equation $$\frac{\tan^2 \theta - 1}{\tan^2 \theta + 1} = \frac{1}{4}$$ for \(-90° \leqslant \theta \leqslant 0°\). [2]
  2. \includegraphics{figure_7b} The diagram shows the graphs of \(y = \sin x\) and \(y = 2 \cos x\) for \(-\pi \leqslant x \leqslant \pi\). The graphs intersect at the points \(A\) and \(B\).
    1. Find the \(x\)-coordinate of \(A\). [2]
    2. Find the \(y\)-coordinate of \(B\). [2]
CAIE P3 2018 June Q8
8 marks Moderate -0.3
  1. The tangent to the curve \(y = x^3 - 9x^2 + 24x - 12\) at a point \(A\) is parallel to the line \(y = 2 - 3x\). Find the equation of the tangent at \(A\). [6]
  2. The function f is defined by \(\mathrm{f}(x) = x^3 - 9x^2 + 24x - 12\) for \(x > k\), where \(k\) is a constant. Find the smallest value of \(k\) for f to be an increasing function. [2]