Questions — CAIE P1 (1228 questions)

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CAIE P1 2024 June Q10
8 marks Standard +0.8
The equation of a circle is \((x - 3)^2 + y^2 = 18\). The line with equation \(y = mx + c\) passes through the point \((0, -9)\) and is a tangent to the circle. Find the two possible values of \(m\) and, for each value of \(m\), find the coordinates of the point at which the tangent touches the circle. [8]
CAIE P1 2024 June Q11
13 marks Standard +0.8
\includegraphics{figure_11} A function is defined by f\((x) = \frac{4}{x^3} - \frac{3}{x} + 2\) for \(x \neq 0\). The graph of \(y = \text{f}(x)\) is shown in the diagram.
  1. Find the set of values of \(x\) for which f\((x)\) is decreasing. [5]
  2. A triangle is bounded by the \(y\)-axis, the normal to the curve at the point where \(x = 1\) and the tangent to the curve at the point where \(x = -1\). Find the area of the triangle. Give your answer correct to 3 significant figures. [8]
CAIE P1 2024 June Q1
3 marks Moderate -0.5
The coefficient of \(x^2\) in the expansion of \((1-4x)^6\) is 12 times the coefficient of \(x^2\) in the expansion of \((2+ax)^5\). Find the value of the positive constant \(a\). [3]
CAIE P1 2024 June Q2
5 marks Moderate -0.8
The curve \(y = x^2\) is transformed to the curve \(y = 4(x-3)^2 - 8\). Describe fully a sequence of transformations that have been combined, making clear the order in which the transformations have been applied. [5]
CAIE P1 2024 June Q3
6 marks Moderate -0.5
  1. Show that the equation \(\frac{7\tan\theta}{\cos\theta} + 12 = 0\) can be expressed as $$12\sin^2\theta - 7\sin\theta - 12 = 0.$$ [3]
  2. Hence solve the equation \(\frac{7\tan\theta}{\cos\theta} + 12 = 0\) for \(0° < \theta \leqslant 360°\). [3]
CAIE P1 2024 June Q4
7 marks Standard +0.3
The function f is defined as follows: $$f(x) = \sqrt{x-1} \text{ for } x > 1.$$ \begin{enumerate}[label=(\alph*)] \item Find an expression for \(f^{-1}(x)\). [1] \end enumerate} \includegraphics{figure_4} The diagram shows the graph of \(y = g(x)\) where \(g(x) = \frac{1}{x^2+2}\) for \(x \in \mathbb{R}\). \begin{enumerate}[label=(\alph*)] \setcounter{enumi}{1} \item State the range of g and explain whether \(g^{-1}\) exists. [2] \end enumerate} The function h is defined by \(h(x) = \frac{1}{x^2+2}\) for \(x \geqslant 0\). \begin{enumerate}[label=(\alph*)] \setcounter{enumi}{2} \item Solve the equation \(hf(x) = f\left(\frac{25}{16}\right)\). Give your answer in the form \(a + b\sqrt{c}\), where \(a\), \(b\) and \(c\) are integers. [4] \end enumerate}
CAIE P1 2024 June Q5
9 marks Standard +0.3
The first and second terms of an arithmetic progression are \(\tan\theta\) and \(\sin\theta\) respectively, where \(0 < \theta < \frac{1}{2}\pi\). \begin{enumerate}[label=(\alph*)] \item Given that \(\theta = \frac{1}{4}\pi\), find the exact sum of the first 40 terms of the progression. [4] \end enumerate} The first and second terms of a geometric progression are \(\tan\theta\) and \(\sin\theta\) respectively, where \(0 < \theta < \frac{1}{2}\pi\).
    1. Find the sum to infinity of the progression in terms of \(\theta\). [2]
    2. Given that \(\theta = \frac{1}{3}\pi\), find the sum of the first 10 terms of the progression. Give your answer correct to 3 significant figures. [3]
CAIE P1 2024 June Q6
9 marks Moderate -0.3
The curve with equation \(y = 2x - 8x^{\frac{1}{2}}\) has a minimum point at \(A\) and intersects the positive \(x\)-axis at \(B\). \begin{enumerate}[label=(\alph*)] \item Find the coordinates of \(A\) and \(B\). [4] \end enumerate}
\includegraphics{figure_6} The diagram shows the curve with equation \(y = 2x - 8x^{\frac{1}{2}}\) and the line \(AB\). It is given that the equation of \(AB\) is \(y = \frac{2x-32}{3}\). Find the area of the shaded region between the curve and the line. [5]
CAIE P1 2024 June Q7
8 marks Standard +0.3
The equation of a circle is \((x-6)^2 + (y+a)^2 = 18\). The line with equation \(y = 2a - x\) is a tangent to the circle.
  1. Find the two possible values of the constant \(a\). [5]
  2. For the greater value of \(a\), find the equation of the diameter which is perpendicular to the given tangent. [3]
CAIE P1 2024 June Q8
10 marks Moderate -0.8
\includegraphics{figure_8} The diagram shows a symmetrical plate \(ABCDEF\). The line \(ABCD\) is straight and the length of \(BC\) is 2cm. Each of the two sectors \(ABF\) and \(DCE\) is of radius \(r\)cm and each of the angles \(ABF\) and \(DCE\) is equal to \(\frac{1}{4}\pi\) radians.
  1. It is given that \(r = 0.4\)cm.
    1. Show that the length \(EF = 2.4\)cm. [2]
    2. Find the area of the plate. Give your answer correct to 3 significant figures. [4]
  2. It is given instead that the perimeter of the plate is 6cm. Find the value of \(r\). Give your answer correct to 3 significant figures. [4]
CAIE P1 2024 June Q9
8 marks Moderate -0.3
A function f is such that \(f'(x) = 6(2x-3)^2 - 6x\) for \(x \in \mathbb{R}\).
  1. Determine the set of values of \(x\) for which f\((x)\) is decreasing. [4]
  2. Given that f\((1) = -1\), find f\((x)\). [4]
CAIE P1 2024 June Q10
10 marks Standard +0.3
The equation of a curve is \(y = (5-2x)^{\frac{1}{2}} + 5\) for \(x < \frac{5}{2}\).
  1. A point \(P\) is moving along the curve in such a way that the \(y\)-coordinate of point \(P\) is decreasing at 5 units per second. Find the rate at which the \(x\)-coordinate of point \(P\) is increasing when \(y = 32\). [4]
  2. Point \(A\) on the curve has \(y\)-coordinate 32. Point \(B\) on the curve is such that the gradient of the curve at \(B\) is \(-3\). Find the equation of the perpendicular bisector of \(AB\). Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [6]
CAIE P1 2023 November Q1
4 marks Standard +0.3
The coefficient of \(x^3\) in the expansion of \((3 + 2ax)^5\) is six times the coefficient of \(x^2\) in the expansion of \((2 + ax)^6\). Find the value of the constant \(a\). [4]
CAIE P1 2023 November Q2
2 marks Standard +0.3
Find the exact solution of the equation $$\frac{1}{6}\pi + \tan^{-1}(4x) = -\cos^{-1}(\frac{1}{3}\sqrt{3}).$$ [2]
CAIE P1 2023 November Q3
6 marks Moderate -0.8
The equation of a curve is such that \(\frac{dy}{dx} = \frac{1}{2}x + \frac{72}{x^4}\). The curve passes through the point \(P(2, 8)\).
  1. Find the equation of the normal to the curve at \(P\). [2]
  2. Find the equation of the curve. [4]
CAIE P1 2023 November Q4
6 marks Moderate -0.5
\includegraphics{figure_4} The diagram shows the shape of a coin. The three arcs \(AB\), \(BC\) and \(CA\) are parts of circles with centres \(C\), \(A\) and \(B\) respectively. \(ABC\) is an equilateral triangle with sides of length 2 cm.
  1. Find the perimeter of the coin. [2]
  2. Find the area of the face \(ABC\) of the coin, giving the answer in terms of \(\pi\) and \(\sqrt{3}\). [4]
CAIE P1 2023 November Q5
6 marks Challenging +1.2
The first, second and third terms of a geometric progression are \(\sin\theta\), \(\cos\theta\) and \(2 - \sin\theta\) respectively, where \(\theta\) radians is an acute angle.
  1. Find the value of \(\theta\). [3]
  2. Using this value of \(\theta\), find the sum of the first 10 terms of the progression. Give the answer in the form \(\frac{b}{\sqrt{c} - 1}\), where \(b\) and \(c\) are integers to be found. [3]
CAIE P1 2023 November Q6
8 marks Moderate -0.3
The equation of a curve is \(y = x^2 - 8x + 5\).
  1. Find the coordinates of the minimum point of the curve. [2]
The curve is stretched by a factor of 2 parallel to the \(y\)-axis and then translated by \(\begin{pmatrix} 4 \\ 1 \end{pmatrix}\).
  1. Find the coordinates of the minimum point of the transformed curve. [2]
  2. Find the equation of the transformed curve. Give the answer in the form \(y = ax^2 + bx + c\), where \(a\), \(b\) and \(c\) are integers to be found. [4]
CAIE P1 2023 November Q7
9 marks Standard +0.3
  1. Verify the identity \((2x - 1)(4x^2 + 2x - 1) \equiv 8x^3 - 4x + 1\). [1]
  2. Prove the identity \(\frac{\tan^2\theta + 1}{\tan^2\theta - 1} \equiv \frac{1}{1 - 2\cos^2\theta}\). [3]
  3. Using the results of (a) and (b), solve the equation $$\frac{\tan^2\theta + 1}{\tan^2\theta - 1} = 4\cos\theta,$$ for \(0° < \theta \leqslant 180°\). [5]
CAIE P1 2023 November Q8
8 marks Standard +0.3
Functions f and g are defined by $$f(x) = (x + a)^2 - a \text{ for } x \leqslant -a,$$ $$g(x) = 2x - 1 \text{ for } x \in \mathbb{R},$$ where \(a\) is a positive constant.
  1. Find an expression for \(f^{-1}(x)\). [3]
    1. State the domain of the function \(f^{-1}\). [1]
    2. State the range of the function \(f^{-1}\). [1]
  3. Given that \(a = \frac{7}{2}\), solve the equation \(gf(x) = 0\). [3]
CAIE P1 2023 November Q9
9 marks Standard +0.3
\includegraphics{figure_9} The diagram shows curves with equations \(y = 2x^{\frac{1}{2}} + 13x^{-\frac{1}{2}}\) and \(y = 3x^{-\frac{1}{4}} + 12\). The curves intersect at points \(A\) and \(B\).
  1. Find the coordinates of \(A\) and \(B\). [4]
  2. Hence find the area of the shaded region. [5]
CAIE P1 2023 November Q10
7 marks Standard +0.3
The equation of a curve is \(y = f(x)\), where \(f(x) = (4x - 3)^{\frac{3}{4}} - \frac{20}{3}x\).
  1. Find the \(x\)-coordinates of the stationary points of the curve and determine their nature. [6]
  2. State the set of values for which the function f is increasing. [1]
CAIE P1 2023 November Q11
10 marks Standard +0.3
The coordinates of points \(A\), \(B\) and \(C\) are \((6, 4)\), \((p, 7)\) and \((14, 18)\) respectively, where \(p\) is a constant. The line \(AB\) is perpendicular to the line \(BC\).
  1. Given that \(p < 10\), find the value of \(p\). [4]
A circle passes through the points \(A\), \(B\) and \(C\).
  1. Find the equation of the circle. [3]
  2. Find the equation of the tangent to the circle at \(C\), giving the answer in the form \(dx + ey + f = 0\), where \(d\), \(e\) and \(f\) are integers. [3]
CAIE P1 2024 November Q1
4 marks Moderate -0.3
In the expansion of \(\left(kx+\frac{2}{x}\right)^4\), where \(k\) is a positive constant, the term independent of \(x\) is equal to 150. Find the value of \(k\) and hence determine the coefficient of \(x^5\) in the expansion. [4]
CAIE P1 2024 November Q2
4 marks Moderate -0.5
The curve \(y = x^2 - \frac{a}{x}\) has a stationary point at \((-3, b)\). Find the values of the constants \(a\) and \(b\). [4]