CAIE P1 (Pure Mathematics 1) 2024 November

An arithmetic progression has fourth term 15 and eighth term 25. Find the 30th term of the progression. [3]

Find the exact solution of the equation $$\cos\frac{x}{6} + \tan 2x + \frac{\sqrt{3}}{2} = 0 \text{ for } -\frac{1}{4}\pi < x < \frac{1}{4}\pi.$$ [2]

1. Find the coefficients of \(x^3\) and \(x^4\) in the expansion of \((3 - ax)^5\), where \(a\) is a constant. Give your answers in terms of \(a\). [3]
2. Given that the coefficient of \(x^4\) in the expansion of \((ax + 7)(3 - ax)^5\) is 240, find the positive value of \(a\). [3]

Solve the equation \(4\sin^4\theta + 12\sin^2\theta - 7 = 0\) for \(0° \leqslant \theta \leqslant 360°\). [4]

\includegraphics{figure_5} In the diagram, the graph with equation \(y = \text{f}(x)\) is shown with solid lines and the graph with equation \(y = \text{g}(x)\) is shown with broken lines.

1. Describe fully a sequence of three transformations which transforms the graph of \(y = \text{f}(x)\) to the graph of \(y = \text{g}(x)\). [6]
2. Find an expression for g(x) in the form \(af(bx + c)\), where \(a\), \(b\) and \(c\) are integers. [2]

The first term of a convergent geometric progression is 10. The sum of the first 4 terms of the progression is \(p\) and the sum of the first 8 terms of the progression is \(q\). It is given that \(\frac{q}{p} = \frac{17}{16}\). Find the two possible values of the sum to infinity. [5]

\includegraphics{figure_7} The diagram shows a metal plate \(ABCDEF\) consisting of five parts. The parts \(BCD\) and \(DEF\) are semicircles. The part \(BAFO\) is a sector of a circle with centre \(O\) and radius 20 cm, and \(D\) lies on this circle. The parts \(OBD\) and \(ODF\) are triangles. Angles \(BOD\) and \(DOF\) are both \(\theta\) radians.

1. Given that \(\theta = 1.2\), find the area of the metal plate. Give your answer correct to 3 significant figures. [5]
2. Given instead that the area of each semicircle is \(50\pi \text{ cm}^2\), find the exact perimeter of the metal plate. [5]

1. Express \(3x^2 - 12x + 14\) in the form \(3(x + a)^2 + b\), where \(a\) and \(b\) are constants to be found. [2]

The function f(x) = \(3x^2 - 12x + 14\) is defined for \(x \geqslant k\), where \(k\) is a constant.

1. Find the least value of \(k\) for which the function \(\text{f}^{-1}\) exists. [1]

For the rest of this question, you should assume that \(k\) has the value found in part (b).

1. Find an expression for \(\text{f}^{-1}(x)\). [3]
2. Hence or otherwise solve the equation \(\text{f f}(x) = 29\). [3]

\includegraphics{figure_9} The diagram shows the curves with equations \(y = x^3 - 3x + 3\) and \(y = 2x^3 - 4x^2 + 3\).

1. Find the \(x\)-coordinates of the points of intersection of the curves. [3]
2. Find the area of the shaded region. [4]

Points \(A\) and \(B\) have coordinates \((4, 3)\) and \((8, -5)\) respectively. A circle with radius 10 passes through the points \(A\) and \(B\).

1. Show that the centre of the circle lies on the line \(y = \frac{1}{2}x - 4\). [4]
2. Find the two possible equations of the circle. [5]

The equation of a curve is \(y = kx^{\frac{1}{2}} - 4x^2 + 2\), where \(k\) is a constant.

1. Find \(\frac{\text{d}y}{\text{d}x}\) and \(\frac{\text{d}^2y}{\text{d}x^2}\) in terms of \(k\). [2]
2. It is given that \(k = 2\). Find the coordinates of the stationary point and determine its nature. [4]
3. Points \(A\) and \(B\) on the curve have \(x\)-coordinates 0.25 and 1 respectively. For a different value of \(k\), the tangents to the curve at the points \(A\) and \(B\) meet at a point with \(x\)-coordinate 0.6. Find this value of \(k\). [6]