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CAIE P1 2019 March Q5
7 marks Moderate -0.8
Two vectors, \(\mathbf{u}\) and \(\mathbf{v}\), are such that $$\mathbf{u} = \begin{pmatrix} q \\ 1 \\ 6 \end{pmatrix} \quad \text{and} \quad \mathbf{v} = \begin{pmatrix} 8 \\ q - 1 \\ q^2 - 7 \end{pmatrix},$$ where \(q\) is a constant.
  1. Find the values of \(q\) for which \(\mathbf{u}\) is perpendicular to \(\mathbf{v}\). [3]
  2. Find the angle between \(\mathbf{u}\) and \(\mathbf{v}\) when \(q = 0\). [4]
CAIE P1 2019 March Q6
7 marks Moderate -0.3
  1. The first and second terms of a geometric progression are \(p\) and \(2p\) respectively, where \(p\) is a positive constant. The sum of the first \(n\) terms is greater than \(1000p\). Show that \(2^n > 1001\). [2]
  2. In another case, \(p\) and \(2p\) are the first and second terms respectively of an arithmetic progression. The \(n\)th term is \(336\) and the sum of the first \(n\) terms is \(7224\). Write down two equations in \(n\) and \(p\) and hence find the values of \(n\) and \(p\). [5]
CAIE P1 2019 March Q7
8 marks Standard +0.3
  1. Solve the equation \(3\sin^2 2\theta + 8\cos 2\theta = 0\) for \(0° < \theta < 180°\). [5]
  2. \includegraphics{figure_7b} The diagram shows part of the graph of \(y = a + \tan bx\), where \(x\) is measured in radians and \(a\) and \(b\) are constants. The curve intersects the \(x\)-axis at \(\left(-\frac{1}{6}\pi, 0\right)\) and the \(y\)-axis at \((0, \sqrt{3})\). Find the values of \(a\) and \(b\). [3]
CAIE P1 2019 March Q8
10 marks Moderate -0.8
  1. Express \(x^2 - 4x + 7\) in the form \((x + a)^2 + b\). [2]
The function \(f\) is defined by \(f(x) = x^2 - 4x + 7\) for \(x < k\), where \(k\) is a constant.
  1. State the largest value of \(k\) for which \(f\) is a decreasing function. [1]
The value of \(k\) is now given to be \(1\).
  1. Find an expression for \(f^{-1}(x)\) and state the domain of \(f^{-1}\). [3]
  2. The function \(g\) is defined by \(g(x) = \frac{2}{x-1}\) for \(x > 1\). Find an expression for \(gf(x)\) and state the range of \(gf\). [4]
CAIE P1 2019 March Q9
10 marks Standard +0.3
\includegraphics{figure_9} The diagram shows part of the curve with equation \(y = \sqrt{x^3 + x^2}\). The shaded region is bounded by the curve, the \(x\)-axis and the line \(x = 3\).
  1. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360°\) about the \(x\)-axis. [4]
  2. \(P\) is the point on the curve with \(x\)-coordinate \(3\). Find the \(y\)-coordinate of the point where the normal to the curve at \(P\) crosses the \(y\)-axis. [6]
CAIE P1 2019 March Q10
12 marks Standard +0.3
\includegraphics{figure_10} The diagram shows the curve with equation \(y = 4x^{\frac{1}{3}}\).
  1. The straight line with equation \(y = x + 3\) intersects the curve at points \(A\) and \(B\). Find the length of \(AB\). [6]
  2. The tangent to the curve at a point \(T\) is parallel to \(AB\). Find the coordinates of \(T\). [3]
  3. Find the coordinates of the point of intersection of the normal to the curve at \(T\) with the line \(AB\). [3]
CAIE P1 2011 November Q1
3 marks Moderate -0.5
The coefficient of \(x^2\) in the expansion of \(\left(k + \frac{1}{x}\right)^5\) is 30. Find the value of the constant \(k\). [3]
CAIE P1 2011 November Q2
4 marks Easy -1.2
The first and second terms of a progression are 4 and 8 respectively. Find the sum of the first 10 terms given that the progression is
  1. an arithmetic progression, [2]
  2. a geometric progression. [2]
CAIE P1 2011 November Q3
5 marks Moderate -0.8
\includegraphics{figure_3} The diagram shows the curve \(y = 2x^5 + 3x^3\) and the line \(y = 2x\) intersecting at points \(A\), \(O\) and \(B\).
  1. Show that the \(x\)-coordinates of \(A\) and \(B\) satisfy the equation \(2x^4 + 3x^2 - 2 = 0\). [2]
  2. Solve the equation \(2x^4 + 3x^2 - 2 = 0\) and hence find the coordinates of \(A\) and \(B\), giving your answers in an exact form. [3]
CAIE P1 2011 November Q4
6 marks Moderate -0.3
\includegraphics{figure_4} In the diagram, \(ABCD\) is a parallelogram with \(AB = BD = DC = 10\) cm and angle \(ABD = 0.8\) radians. \(APD\) and \(BQC\) are arcs of circles with centres \(B\) and \(D\) respectively.
  1. Find the area of the parallelogram \(ABCD\). [2]
  2. Find the area of the complete figure \(ABQCDP\). [2]
  3. Find the perimeter of the complete figure \(ABQCDP\). [2]
CAIE P1 2011 November Q5
7 marks Moderate -0.3
  1. Given that $$3\sin^2 x - 8\cos x - 7 = 0,$$ show that, for real values of \(x\), $$\cos x = -\frac{2}{3}.$$ [3]
  2. Hence solve the equation $$3\sin^2(\theta + 70°) - 8\cos(\theta + 70°) - 7 = 0$$ for \(0° \leqslant \theta \leqslant 180°\). [4]
CAIE P1 2011 November Q6
8 marks Moderate -0.8
Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are \(\mathbf{3i} + 4\mathbf{j} - \mathbf{k}\) and \(5\mathbf{i} - 2\mathbf{j} - 3\mathbf{k}\) respectively.
  1. Use a scalar product to find angle \(BOA\). [4]
The point \(C\) is the mid-point of \(AB\). The point \(D\) is such that \(\overrightarrow{OD} = 2\overrightarrow{OB}\).
  1. Find \(\overrightarrow{DC}\). [4]
CAIE P1 2011 November Q7
9 marks Moderate -0.3
  1. A straight line passes through the point \((2, 0)\) and has gradient \(m\). Write down the equation of the line. [1]
  2. Find the two values of \(m\) for which the line is a tangent to the curve \(y = x^2 - 4x + 5\). For each value of \(m\), find the coordinates of the point where the line touches the curve. [6]
  3. Express \(x^2 - 4x + 5\) in the form \((x + a)^2 + b\) and hence, or otherwise, write down the coordinates of the minimum point on the curve. [2]
CAIE P1 2011 November Q8
10 marks Moderate -0.3
A curve \(y = \mathrm{f}(x)\) has a stationary point at \(P(3, -10)\). It is given that \(\mathrm{f}'(x) = 2x^2 + kx - 12\), where \(k\) is a constant.
  1. Show that \(k = -2\) and hence find the \(x\)-coordinate of the other stationary point, \(Q\). [4]
  2. Find \(\mathrm{f}''(x)\) and determine the nature of each of the stationary points \(P\) and \(Q\). [2]
  3. Find \(\mathrm{f}(x)\). [4]
CAIE P1 2011 November Q9
11 marks Standard +0.3
Functions \(\mathrm{f}\) and \(\mathrm{g}\) are defined by \begin{align} \mathrm{f} : x \mapsto 2x + 3 \quad &\text{for } x \leqslant 0,
\mathrm{g} : x \mapsto x^2 - 6x \quad &\text{for } x \leqslant 3. \end{align}
  1. Express \(\mathrm{f}^{-1}(x)\) in terms of \(x\) and solve the equation \(\mathrm{f}(x) = \mathrm{f}^{-1}(x)\). [3]
  2. On the same diagram sketch the graphs of \(y = \mathrm{f}(x)\) and \(y = \mathrm{f}^{-1}(x)\), showing the coordinates of their point of intersection and the relationship between the graphs. [3]
  3. Find the set of values of \(x\) which satisfy \(\mathrm{gf}(x) \leqslant 16\). [5]
CAIE P1 2011 November Q10
12 marks Standard +0.3
\includegraphics{figure_10} The diagram shows the line \(y = x + 1\) and the curve \(y = \sqrt{(x + 1)}\), meeting at \((-1, 0)\) and \((0, 1)\).
  1. Find the area of the shaded region. [5]
  2. Find the volume obtained when the shaded region is rotated through \(360°\) about the \(y\)-axis. [7]
CAIE P1 2014 November Q1
4 marks Standard +0.3
\includegraphics{figure_1} The diagram shows part of the curve \(y = x^2 + 1\). Find the volume obtained when the shaded region is rotated through \(360°\) about the \(y\)-axis. [4]
CAIE P1 2014 November Q2
6 marks Standard +0.3
\includegraphics{figure_2} The diagram shows a triangle \(AOB\) in which \(OA\) is 12 cm, \(OB\) is 5 cm and angle \(AOB\) is a right angle. Point \(P\) lies on \(AB\) and \(OP\) is an arc of a circle with centre \(A\). Point \(Q\) lies on \(AB\) and \(OQ\) is an arc of a circle with centre \(B\).
  1. Show that angle \(BAO\) is 0.3948 radians, correct to 4 decimal places. [1]
  2. Calculate the area of the shaded region. [5]
CAIE P1 2014 November Q3
5 marks Moderate -0.8
  1. Find the first 3 terms, in ascending powers of \(x\), in the expansion of \((1 + x)^5\). [2]
The coefficient of \(x^2\) in the expansion of \((1 + (px + x^2))^5\) is 95.
  1. Use the answer to part (i) to find the value of the positive constant \(p\). [3]
CAIE P1 2014 November Q4
6 marks Standard +0.3
A curve has equation \(y = \frac{12}{5 - 2x}\).
  1. Find \(\frac{dy}{dx}\). [2]
A point moves along this curve. As the point passes through \(A\), the \(x\)-coordinate is increasing at a rate of 0.15 units per second and the \(y\)-coordinate is increasing at a rate of 0.4 units per second.
  1. Find the possible \(x\)-coordinates of \(A\). [4]
CAIE P1 2014 November Q5
6 marks Moderate -0.3
  1. Show that the equation \(1 + \sin x \tan x = 5 \cos x\) can be expressed as $$6 \cos^2 x - \cos x - 1 = 0.$$ [3]
  2. Hence solve the equation \(1 + \sin x \tan x = 5 \cos x\) for \(0° \leqslant x \leqslant 180°\). [3]
CAIE P1 2014 November Q6
6 marks Moderate -0.3
The equation of a curve is \(y = x^3 + ax^2 + bx\), where \(a\) and \(b\) are constants.
  1. In the case where the curve has no stationary point, show that \(a^2 < 3b\). [3]
  2. In the case where \(a = -6\) and \(b = 9\), find the set of values of \(x\) for which \(y\) is a decreasing function of \(x\). [3]
CAIE P1 2014 November Q7
7 marks Moderate -0.8
\includegraphics{figure_7} The diagram shows a pyramid \(OABCX\). The horizontal square base \(OABC\) has side 8 units and the centre of the base is \(D\). The top of the pyramid, \(X\), is vertically above \(D\) and \(XD = 10\) units. The mid-point of \(OX\) is \(M\). The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are parallel to \(\overrightarrow{OA}\) and \(\overrightarrow{OC}\) respectively and the unit vector \(\mathbf{k}\) is vertically upwards.
  1. Express the vectors \(\overrightarrow{AM}\) and \(\overrightarrow{AC}\) in terms of \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\). [3]
  2. Use a scalar product to find angle \(MAC\). [4]
CAIE P1 2014 November Q8
8 marks Moderate -0.3
  1. The sum, \(S_n\), of the first \(n\) terms of an arithmetic progression is given by \(S_n = 32n - n^2\). Find the first term and the common difference. [3]
  2. A geometric progression in which all the terms are positive has sum to infinity 20. The sum of the first two terms is 12.8. Find the first term of the progression. [5]
CAIE P1 2014 November Q9
8 marks Moderate -0.3
\includegraphics{figure_9} The diagram shows a trapezium \(ABCD\) in which \(AB\) is parallel to \(DC\) and angle \(BAD\) is \(90°\). The coordinates of \(A\), \(B\) and \(C\) are \((2, 6)\), \((5, -3)\) and \((8, 3)\) respectively.
  1. Find the equation of \(AD\). [3]
  2. Find, by calculation, the coordinates of \(D\). [3]
The point \(E\) is such that \(ABCE\) is a parallelogram.
  1. Find the length of \(BE\). [2]