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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]
CAIE P1 2024 November Q3
5 marks Standard +0.3
\includegraphics{figure_3} The diagram shows a sector of a circle, centre \(O\), where \(OB = OC = 15\) cm. The size of angle \(BOC\) is \(\frac{2}{5}\pi\) radians. Points \(A\) and \(D\) on the lines \(OB\) and \(OC\) respectively are joined by an arc \(AD\) of a circle with centre \(O\). The shaded region is bounded by the arcs \(AD\) and \(BC\) and by the straight lines \(AB\) and \(DC\). It is given that the area of the shaded region is \(\frac{90}{7}\pi\) cm\(^2\). Find the perimeter of the shaded region. Give your answer in terms of \(\pi\). [5]
CAIE P1 2024 November Q4
5 marks Standard +0.3
Show that the curve with equation \(x^2 - 3xy - 40 = 0\) and the line with equation \(3x + y + k = 0\) meet for all values of the constant \(k\). [5]
CAIE P1 2024 November Q5
7 marks Moderate -0.8
The equation of a curve is such that \(\frac{dy}{dx} = 4x - 3\sqrt{x} + 1\).
  1. Find the \(x\)-coordinate of the point on the curve at which the gradient is \(\frac{11}{2}\). [3]
  2. Given that the curve passes through the point \((4, 11)\), find the equation of the curve. [4]
CAIE P1 2024 November Q6
7 marks Moderate -0.3
Circles \(C_1\) and \(C_2\) have equations $$x^2 + y^2 + 6x - 10y + 18 = 0 \text{ and } (x-9)^2 + (y+4)^2 - 64 = 0$$ respectively.
  1. Find the distance between the centres of the circles. [4] \(P\) and \(Q\) are points on \(C_1\) and \(C_2\) respectively. The distance between \(P\) and \(Q\) is denoted by \(d\).
  2. Find the greatest and least possible values of \(d\). [3]
CAIE P1 2024 November Q7
8 marks Moderate -0.3
\includegraphics{figure_7} The diagram shows part of the curve with equation \(y = \frac{12}{\sqrt{2x+1}}\). The point \(A\) on the curve has coordinates \(\left(\frac{7}{2}, 6\right)\).
  1. Find the equation of the tangent to the curve at \(A\). Give your answer in the form \(y = mx + c\). [4]
  2. Find the area of the region bounded by the curve and the lines \(x = 0\), \(x = \frac{7}{2}\) and \(y = 0\). [4]
CAIE P1 2024 November Q8
8 marks Moderate -0.3
  1. It is given that \(\beta\) is an angle between \(90°\) and \(180°\) such that \(\sin \beta = a\). Express \(\tan^2 \beta - 3 \sin \beta \cos \beta\) in terms of \(a\). [3]
  2. Solve the equation \(\sin^2 \theta + 2 \cos^2 \theta = 4 \sin \theta + 3\) for \(0° < \theta < 360°\). [5]
CAIE P1 2024 November Q9
8 marks Standard +0.3
The equation of a curve is \(y = 4 + 5x + 6x^2 - 3x^3\).
  1. Find the set of values of \(x\) for which \(y\) decreases as \(x\) increases. [4]
  2. It is given that \(y = 9x + k\) is a tangent to the curve. Find the value of the constant \(k\). [4]
CAIE P1 2024 November Q10
8 marks Standard +0.3
An arithmetic progression has first term 5 and common difference \(d\), where \(d > 0\). The second, fifth and eleventh terms of the arithmetic progression, in that order, are the first three terms of a geometric progression.
  1. Find the value of \(d\). [3]
  2. The sum of the first 77 terms of the arithmetic progression is denoted by \(S_{77}\). The sum of the first 10 terms of the geometric progression is denoted by \(G_{10}\). Find the value of \(S_{77} - G_{10}\). [5]
CAIE P1 2024 November Q11
11 marks Moderate -0.3
The function f is defined by f\((x) = 3 + 6x - 2x^2\) for \(x \in \mathbb{R}\).
  1. Express f\((x)\) in the form \(a - b(x - c)^2\), where \(a\), \(b\) and \(c\) are constants, and state the range of f. [3]
  2. The graph of \(y = \)f\((x)\) is transformed to the graph of \(y = \)h\((x)\) by a reflection in one of the axes followed by a translation. It is given that the graph of \(y = \)h\((x)\) has a minimum point at the origin. Give details of the reflection and translation involved. [2] The function g is defined by g\((x) = 3 + 6x - 2x^2\) for \(x \leqslant 0\).
  3. Sketch the graph of \(y = \)g\((x)\) and explain why g is a one-one function. You are not required to find the coordinates of any intersections with the axes. [2]
  4. Sketch the graph of \(y = \)g\(^{-1}(x)\) on your diagram in (c), and find an expression for g\(^{-1}(x)\). You should label the two graphs in your diagram appropriately and show any relevant mirror line. [4]
CAIE P1 2024 November Q1
5 marks Moderate -0.8
\includegraphics{figure_1} The diagram shows the curve with equation \(y = a\sin(bx) + c\) for \(0 \leqslant x \leqslant 2\pi\), where \(a\), \(b\) and \(c\) are positive constants.
  1. State the values of \(a\), \(b\) and \(c\). [3]
  2. For these values of \(a\), \(b\) and \(c\), determine the number of solutions in the interval \(0 \leqslant x \leqslant 2\pi\) for each of the following equations:
    1. \(a\sin(bx) + c = 7 - x\) [1]
    2. \(a\sin(bx) + c = 2\pi(x - 1)\). [1]
CAIE P1 2024 November Q2
5 marks Moderate -0.8
The first term of an arithmetic progression is \(-20\) and the common difference is \(5\).
  1. Find the sum of the first 20 terms of the progression. [2]
It is given that the sum of the first \(2k\) terms is 10 times the sum of the first \(k\) terms.
  1. Find the value of \(k\). [3]
CAIE P1 2024 November Q3
5 marks Moderate -0.8
The equation of a curve is \(y = 2x^2 - 3\). Two points \(A\) and \(B\) with \(x\)-coordinates 2 and \((2 + h)\) respectively lie on the curve.
  1. Find and simplify an expression for the gradient of the chord \(AB\) in terms of \(h\). [3]
  2. Explain how the gradient of the curve at the point \(A\) can be deduced from the answer to part (a), and state the value of this gradient. [2]