CAIE P1 (Pure Mathematics 1) 2024 November

\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]

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]

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]

Find the term independent of \(x\) in the expansion of each of the following:

1. \(\left(x + \frac{3}{x^2}\right)^6\) [2]
2. \((4x^3 - 5)\left(x + \frac{3}{x^2}\right)^6\) [4]

The function f is defined by \(\mathrm{f}(x) = \frac{2x + 1}{2x - 1}\) for \(x < \frac{1}{2}\).

1. 1. State the value of f\((-1)\). [1]
   2. \includegraphics{figure_5} The diagram shows the graph of \(y = \mathrm{f}(x)\). Sketch the graph of \(y = \mathrm{f}^{-1}(x)\) on this diagram. Show any relevant mirror line. [2]
   3. Find an expression for \(\mathrm{f}^{-1}(x)\) and state the domain of the function \(\mathrm{f}^{-1}\). [4]

The function g is defined by \(\mathrm{g}(x) = 3x + 2\) for \(x \in \mathbb{R}\).

1. Solve the equation \(\mathrm{f}(x) = \mathrm{gf}\left(\frac{1}{4}\right)\). [3]

\includegraphics{figure_6} The diagram shows a metal plate \(OABCDEF\) consisting of sectors of two circles, each with centre \(O\). The radii of sectors \(AOB\) and \(EOF\) are \(r\) cm and the radius of sector \(COD\) is \(2r\) cm. Angle \(AOB =\) angle \(EOF = \theta\) radians and angle \(COD = 2\theta\) radians. It is given that the perimeter of the plate is 14 cm and the area of the plate is 10 cm\(^2\). Given that \(r \geqslant \frac{3}{2}\) and \(\theta < \frac{3}{4}\), find the values of \(r\) and \(\theta\). [6]

1. By expressing \(-2x^2 + 8x + 11\) in the form \(-a(x - h)^2 + c\), where \(a\), \(b\) and \(c\) are positive integers, find the coordinates of the vertex of the graph with equation \(y = -2x^2 + 8x + 11\). [3]
2. \includegraphics{figure_7} The diagram shows part of the curve with equation \(y = -2x^2 + 8x + 11\) and the line with equation \(y = 8x + 9\). Find the area of the shaded region. [5]

The equation of a circle is \(x^2 + y^2 + px + 2y + q = 0\), where \(p\) and \(q\) are constants.

1. Express the equation in the form \((x - a)^2 + (y - b)^2 = r^2\), where \(a\) is to be given in terms of \(p\) and \(r^2\) is to be given in terms of \(p\) and \(q\). [2]

The line with equation \(x + 2y = 10\) is the tangent to the circle at the point \(A(4, 3)\).

1. 1. Find the equation of the normal to the circle at the point \(A\). [3]
   2. Find the values of \(p\) and \(q\). [5]

The equation of a curve is \(y = \frac{1}{3}k^2x^2 - 2kx + 2\) and the equation of a line is \(y = kx + p\), where \(k\) and \(p\) are constants with \(0 < k < 1\).

1. It is given that one of the points of intersection of the curve and the line has coordinates \(\left(\frac{6}{5}, \frac{3}{5}\right)\). Find the values of \(k\) and \(p\), and find the coordinates of the other point of intersection. [7]
2. It is given instead that the line and the curve do not intersect. Find the set of possible values of \(p\). [3]

A function f with domain \(x > 0\) is such that \(\mathrm{f}'(x) = 8(2x - 3)^{\frac{1}{3}} - 10x^{\frac{3}{5}}\). It is given that the curve with equation \(y = \mathrm{f}(x)\) passes through the point \((1, 0)\).

1. Find the equation of the normal to the curve at the point \((1, 0)\). [3]
2. Find f\((x)\). [4]

It is given that the equation \(\mathrm{f}'(x) = 0\) can be expressed in the form $$125x^2 - 128x + 192 = 0.$$

1. Determine, making your reasoning clear, whether f is an increasing function, a decreasing function or neither. [3]