CAIE P1 (Pure Mathematics 1) 2023 November

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]

Find the exact solution of the equation $$\frac{1}{6}\pi + \tan^{-1}(4x) = -\cos^{-1}(\frac{1}{3}\sqrt{3}).$$ [2]

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]

\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]

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]

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]

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]

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]
2. 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]

\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]

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]

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]