CAIE P1 (Pure Mathematics 1) 2024 June

1. Express \(3y^2 - 12y - 15\) in the form \(3(y + a)^2 + b\), where \(a\) and \(b\) are constants. [2]
2. Hence find the exact solutions of the equation \(3x^4 - 12x^2 - 15 = 0\). [3]

\includegraphics{figure_2} The diagram shows two curves. One curve has equation \(y = \sin x\) and the other curve has equation \(y = \text{f}(x)\).

1. In order to transform the curve \(y = \sin x\) to the curve \(y = \text{f}(x)\), the curve \(y = \sin x\) is first reflected in the \(x\)-axis. Describe fully a sequence of two further transformations which are required. [4]
2. Find f\((x)\) in terms of \(\sin x\). [2]

The coefficient of \(x^3\) in the expansion of \((3 + ax)^6\) is 160.

1. Find the value of the constant \(a\). [2]
2. Hence find the coefficient of \(x^5\) in the expansion of \((3 + ax)^6(1 - 2x)\). [3]

The equation of a curve is \(y = \text{f}(x)\), where f\((x) = (2x - 1)\sqrt{3x - 2} - 2\). The following points lie on the curve. Non-exact values have been given correct to 5 decimal places. \(A(2, 4)\), \(B(2.0001, k)\), \(C(2.001, 4.00625)\), \(D(2.01, 4.06261)\), \(E(2.1, 4.63566)\), \(F(3, 11.22876)\)

1. Find the value of \(k\). Give your answer correct to 5 decimal places. [1]

The table shows the gradients of the chords \(AB\), \(AC\), \(AD\) and \(AF\).

1. Find the gradient of the chord \(AE\). Give your answer correct to 4 decimal places. [1]
2. Deduce the value of f\('(2)\) using the values in the table. [1]

1. Prove the identity \(\frac{\sin^2 x - \cos x - 1}{1 + \cos x} \equiv -\cos x\). [3]
2. Hence solve the equation \(\frac{\sin^2 x - \cos x - 1}{2 + 2\cos x} = \frac{1}{4}\) for \(0° \leq x \leq 360°\). [3]

\includegraphics{figure_6} The function f is defined by f\((x) = \frac{2}{x^2} + 4\) for \(x < 0\). The diagram shows the graph of \(y = \text{f}(x)\).

1. On this diagram, sketch the graph of \(y = \text{f}^{-1}(x)\). Show any relevant mirror line. [2]
2. Find an expression for f\(^{-1}(x)\). [3]
3. Solve the equation f\((x) = 4.5\). [1]
4. Explain why the equation f\(^{-1}(x) = \text{f}(x)\) has no solution. [1]

\includegraphics{figure_7} In the diagram, \(AOD\) and \(BC\) are two parallel straight lines. Arc \(AB\) is part of a circle with centre \(O\) and radius \(15\text{cm}\). Angle \(BOA = \theta\) radians. Arc \(CD\) is part of a circle with centre \(O\) and radius \(10\text{cm}\). Angle \(COD = \frac{1}{3}\pi\) radians.

1. Show that \(\theta = 0.7297\), correct to 4 decimal places. [1]
2. Find the perimeter and the area of the shape \(ABCD\). Give your answers correct to 3 significant figures. [7]

1. The first three terms of an arithmetic progression are \(25\), \(4p - 1\) and \(13 - p\), where \(p\) is a constant. Find the value of the tenth term of the progression. [4]
2. The first three terms of a geometric progression are \(25\), \(4q - 1\) and \(13 - q\), where \(q\) is a positive constant. Find the sum to infinity of the progression. [4]

\includegraphics{figure_9} The diagram shows part of the curve with equation \(y = \frac{1}{(5x - 4)^3}\) and the lines \(x = 2.4\) and \(y = 1\). The curve intersects the line \(y = 1\) at the point \((1, 1)\). Find the exact volume of the solid generated when the shaded region is rotated through \(360°\) about the \(x\)-axis. [6]

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]

\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]