Questions P1 (1401 questions)

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CAIE P1 2019 June Q8
8 marks Standard +0.3
A curve is such that \(\frac{\text{d}y}{\text{d}x} = 3x^2 + ax + b\). The curve has stationary points at \((-1, 2)\) and \((3, k)\). Find the values of the constants \(a\), \(b\) and \(k\). [8]
CAIE P1 2019 June Q9
10 marks Moderate -0.3
\includegraphics{figure_9} The function f : \(x \mapsto p \sin^2 2x + q\) is defined for \(0 \leqslant x \leqslant \pi\), where \(p\) and \(q\) are positive constants. The diagram shows the graph of \(y = \text{f}(x)\).
  1. In terms of \(p\) and \(q\), state the range of f. [2]
  2. State the number of solutions of the following equations.
    1. \(\text{f}(x) = p + q\) [1]
    2. \(\text{f}(x) = q\) [1]
    3. \(\text{f}(x) = \frac{1}{2}p + q\) [1]
  3. For the case where \(p = 3\) and \(q = 2\), solve the equation \(\text{f}(x) = 4\), showing all necessary working. [5]
CAIE P1 2019 June Q10
13 marks Standard +0.3
\includegraphics{figure_10} The diagram shows part of the curve with equation \(y = (3x + 4)^{\frac{1}{3}}\) and the tangent to the curve at the point \(A\). The \(x\)-coordinate of \(A\) is 4.
  1. Find the equation of the tangent to the curve at \(A\). [5]
  2. Find, showing all necessary working, the area of the shaded region. [5]
  3. A point is moving along the curve. At the point \(P\) the \(y\)-coordinate is increasing at half the rate at which the \(x\)-coordinate is increasing. Find the \(x\)-coordinate of \(P\). [3]
CAIE P1 2019 March Q1
3 marks Moderate -0.8
The coefficient of \(x^3\) in the expansion of \((1 - px)^5\) is \(-2160\). Find the value of the constant \(p\). [3]
CAIE P1 2019 March Q2
5 marks Moderate -0.3
A curve with equation \(y = f(x)\) passes through the points \((0, 2)\) and \((3, -1)\). It is given that \(f'(x) = kx^2 - 2x\), where \(k\) is a constant. Find the value of \(k\). [5]
CAIE P1 2019 March Q3
6 marks Standard +0.3
\includegraphics{figure_3} In the diagram, \(CXD\) is a semicircle of radius \(7\) cm with centre \(A\) and diameter \(CD\). The straight line \(YAX\) is perpendicular to \(CD\), and the arc \(CYD\) is part of a circle with centre \(B\) and radius \(8\) cm. Find the total area of the region enclosed by the two arcs. [6]
CAIE P1 2019 March Q4
7 marks Moderate -0.3
A curve has equation \(y = (2x - 1)^{-1} + 2x\).
  1. Find \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\). [3]
  2. Find the \(x\)-coordinates of the stationary points and, showing all necessary working, determine the nature of each stationary point. [4]
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]