Questions — CAIE P1 (1228 questions)

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CAIE P1 2017 June Q7
7 marks Standard +0.3
\includegraphics{figure_7} The diagram shows two circles with centres \(A\) and \(B\) having radii 8 cm and 10 cm respectively. The two circles intersect at \(C\) and \(D\) where \(CAD\) is a straight line and \(AB\) is perpendicular to \(CD\).
  1. Find angle \(ABC\) in radians. [1]
  2. Find the area of the shaded region. [6]
CAIE P1 2017 June Q8
8 marks Moderate -0.3
\(A(-1, 1)\) and \(P(a, b)\) are two points, where \(a\) and \(b\) are constants. The gradient of \(AP\) is 2.
  1. Find an expression for \(b\) in terms of \(a\). [2]
  2. \(B(10, -1)\) is a third point such that \(AP = AB\). Calculate the coordinates of the possible positions of \(P\). [6]
CAIE P1 2017 June Q9
9 marks Moderate -0.8
  1. Express \(9x^2 - 6x + 6\) in the form \((ax + b)^2 + c\), where \(a\), \(b\) and \(c\) are constants. [3]
The function f is defined by \(\text{f}(x) = 9x^2 - 6x + 6\) for \(x \geqslant p\), where \(p\) is a constant.
  1. State the smallest value of \(p\) for which f is a one-one function. [1]
  2. For this value of \(p\), obtain an expression for \(\text{f}^{-1}(x)\), and state the domain of \(\text{f}^{-1}\). [4]
  3. State the set of values of \(q\) for which the equation \(\text{f}(x) = q\) has no solution. [1]
CAIE P1 2017 June Q10
11 marks Standard +0.3
  1. \includegraphics{figure_1} Fig. 1 shows part of the curve \(y = x^2 - 1\) and the line \(y = h\), where \(h\) is a constant.
    1. The shaded region is rotated through \(360°\) about the \(y\)-axis. Show that the volume of revolution, \(V\), is given by \(V = \pi\left(\frac{1}{2}h^2 + h\right)\). [3]
    2. Find, showing all necessary working, the area of the shaded region when \(h = 3\). [4]
  2. \includegraphics{figure_2} Fig. 2 shows a cross-section of a bowl containing water. When the height of the water level is \(h\) cm, the volume, \(V\) cm\(^3\), of water is given by \(V = \pi\left(\frac{1}{4}h^2 + h\right)\). Water is poured into the bowl at a constant rate of 2 cm\(^3\) s\(^{-1}\). Find the rate, in cm s\(^{-1}\), at which the height of the water level is increasing when the height of the water level is 3 cm. [4]
CAIE P1 2017 June Q11
11 marks Standard +0.3
The function f is defined for \(x \geqslant 0\). It is given that f has a minimum value when \(x = 2\) and that \(\text{f}''(x) = (4x + 1)^{-\frac{1}{2}}\).
  1. Find \(\text{f}'(x)\). [3]
It is now given that \(\text{f}''(0)\), \(\text{f}'(0)\) and \(\text{f}(0)\) are the first three terms respectively of an arithmetic progression.
  1. Find the value of \(\text{f}(0)\). [3]
  2. Find \(\text{f}(x)\), and hence find the minimum value of f. [5]
CAIE P1 2019 June Q1
5 marks Moderate -0.8
The function f is defined by \(\text{f}(x) = x^2 - 4x + 8\) for \(x \in \mathbb{R}\).
  1. Express \(x^2 - 4x + 8\) in the form \((x - a)^2 + b\). [2]
  2. Hence find the set of values of \(x\) for which \(\text{f}(x) < 9\), giving your answer in exact form. [3]
CAIE P1 2019 June Q2
5 marks Moderate -0.8
  1. In the binomial expansion of \(\left(2x - \frac{1}{2x}\right)^5\), the first three terms are \(32x^5 - 40x^3 + 20x\). Find the remaining three terms of the expansion. [3]
  2. Hence find the coefficient of \(x\) in the expansion of \((1 + 4x^2)\left(2x - \frac{1}{2x}\right)^5\). [2]
CAIE P1 2019 June Q3
6 marks Standard +0.3
\includegraphics{figure_3} The diagram shows triangle \(ABC\) which is right-angled at \(A\). Angle \(ABC = \frac{1}{4}\pi\) radians and \(AC = 8\) cm. The points \(D\) and \(E\) lie on \(BC\) and \(BA\) respectively. The sector \(ADE\) is part of a circle with centre \(A\) and is such that \(BDC\) is the tangent to the arc \(DE\) at \(D\).
  1. Find the length of \(AD\). [3]
  2. Find the area of the shaded region. [3]
CAIE P1 2019 June Q4
5 marks Moderate -0.3
The function f is defined by \(\text{f}(x) = \frac{48}{x - 1}\) for \(3 \leqslant x \leqslant 7\). The function g is defined by \(\text{g}(x) = 2x - 4\) for \(a \leqslant x \leqslant b\), where \(a\) and \(b\) are constants.
  1. Find the greatest value of \(a\) and the least value of \(b\) which will permit the formation of the composite function gf. [2] It is now given that the conditions for the formation of gf are satisfied.
  2. Find an expression for \(\text{gf}(x)\). [1]
  3. Find an expression for \((\text{gf})^{-1}(x)\). [2]
CAIE P1 2019 June Q5
7 marks Moderate -0.8
Two heavyweight boxers decide that they would be more successful if they competed in a lower weight class. For each boxer this would require a total weight loss of 13 kg. At the end of week 1 they have each recorded a weight loss of 1 kg and they both find that in each of the following weeks their weight loss is slightly less than the week before. Boxer A's weight loss in week 2 is 0.98 kg. It is given that his weekly weight loss follows an arithmetic progression.
  1. Write down an expression for his total weight loss after \(x\) weeks. [1]
  2. He reaches his 13 kg target during week \(n\). Use your answer to part (i) to find the value of \(n\). [2]
Boxer B's weight loss in week 2 is 0.92 kg and it is given that his weekly weight loss follows a geometric progression.
  1. Calculate his total weight loss after 20 weeks and show that he can never reach his target. [4]
CAIE P1 2019 June Q6
7 marks Moderate -0.8
\includegraphics{figure_6} The diagram shows a solid figure \(ABCDEF\) in which the horizontal base \(ABC\) is a triangle right-angled at \(A\). The lengths of \(AB\) and \(AC\) are 8 units and 4 units respectively and \(M\) is the mid-point of \(AB\). The point \(D\) is 7 units vertically above \(A\). Triangle \(DEF\) lies in a horizontal plane with \(DE\), \(DF\) and \(FE\) parallel to \(AB\), \(AC\) and \(CB\) respectively and \(N\) is the mid-point of \(FE\). The lengths of \(DE\) and \(DF\) are 4 units and 2 units respectively. Unit vectors \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\) are parallel to \(\overrightarrow{AB}\), \(\overrightarrow{AC}\) and \(\overrightarrow{AD}\) respectively.
  1. Find \(\overrightarrow{MF}\) in terms of \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\). [1]
  2. Find \(\overrightarrow{FN}\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\). [1]
  3. Find \(\overrightarrow{MN}\) in terms of \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\). [1]
  4. Use a scalar product to find angle \(FMN\). [4]
CAIE P1 2019 June Q7
9 marks Moderate -0.8
The coordinates of two points \(A\) and \(B\) are \((1, 3)\) and \((9, -1)\) respectively and \(D\) is the mid-point of \(AB\). A point \(C\) has coordinates \((x, y)\), where \(x\) and \(y\) are variables.
  1. State the coordinates of \(D\). [1]
  2. It is given that \(CD^2 = 20\). Write down an equation relating \(x\) and \(y\). [1]
  3. It is given that \(AC\) and \(BC\) are equal in length. Find an equation relating \(x\) and \(y\) and show that it can be simplified to \(y = 2x - 9\). [3]
  4. Using the results from parts (ii) and (iii), and showing all necessary working, find the possible coordinates of \(C\). [4]
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]