Edexcel P1 (Pure Mathematics 1) 2018 Specimen

Given that \(y = 4x^3 - \frac{5}{x^2}\), \(x \neq 0\), find in their simplest form

1. \(\frac{dy}{dx}\). [3]
2. \(\int y \, dx\) [3]

1. Given that \(3^{-1.5} = a\sqrt{3}\) find the exact value of \(a\) [2]
2. Simplify fully \(\frac{(2x^{\frac{1}{2}})^3}{4x^2}\) [3]

Solve the simultaneous equations $$y + 4x + 1 = 0$$ $$y^2 + 5x^2 + 2y = 0$$ [6]

The straight line with equation \(y = 4x + c\), where \(c\) is a constant, is a tangent to the curve with equation \(y = 2x^2 + 8x + 3\) Calculate the value of \(c\) [5]

1. On the same axes, sketch the graphs of \(y = x + 2\) and \(y = x^2 - x - 6\) showing the coordinates of all points at which each graph crosses the coordinate axes. [4]
2. On your sketch, show, by shading, the region \(R\) defined by the inequalities $$y < x + 2 \text{ and } y > x^2 - x - 6$$ [1]
3. Hence, or otherwise, find the set of values of \(x\) for which \(x^2 - 2x - 8 < 0\) [3]

\includegraphics{figure_1} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \text{f}(x)\) The curve \(C\) passes through the origin and through \((6, 0)\) The curve \(C\) has a minimum at the point \((3, -1)\) On separate diagrams, sketch the curve with equation

1. \(y = \text{f}(2x)\) [3]
2. \(y = \text{f}(x + p)\), where \(p\) is a constant and \(0 < p < 3\) [4]

On each diagram show the coordinates of any points where the curve intersects the \(x\)-axis and of any minimum or maximum points.

A curve with equation \(y = \text{f}(x)\) passes through the point \((4, 25)\) Given that $$\text{f}'(x) = \frac{3}{8}x^2 - 10x^{-\frac{1}{2}} + 1, \quad x > 0$$ find \(\text{f}(x)\), simplifying each term. [5]

\includegraphics{figure_2} The line \(l_1\) shown in Figure 2 has equation \(2x + 3y = 26\) The line \(l_2\) passes through the origin \(O\) and is perpendicular to \(l_1\)

1. Find an equation for the line \(l_2\) [4]

The line \(l_1\) intersects the line \(l_1\) at the point \(C\). Line \(l_1\) crosses the \(y\)-axis at the point \(B\) as shown in Figure 2.

1. Find the area of triangle \(OBC\). Give your answer in the form \(\frac{a}{b}\), where \(a\) and \(b\) are integers to be found. [6]

\includegraphics{figure_3} A sketch of part of the curve \(C\) with equation $$y = 20 - 4x - \frac{18}{x}, \quad x > 0$$ is shown in Figure 3. Point \(A\) lies on \(C\) and has \(x\) coordinate equal to 2

1. Show that the equation of the normal to \(C\) at \(A\) is \(y = -2x + 7\). [6]

The normal to \(C\) at \(A\) meets \(C\) again at the point \(B\), as shown in Figure 3.

1. Use algebra to find the coordinates of \(B\). [5]

\includegraphics{figure_4} The triangle \(XYZ\) in Figure 4 has \(XY = 6\) cm, \(YZ = 9\) cm, \(ZX = 4\) cm and angle \(ZXY = a\). The point \(W\) lies on the line \(XY\). The circular arc \(ZW\), in Figure 4, is a major arc of the circle with centre \(X\) and radius 4 cm.

1. Show that, to 3 significant figures, \(a = 2.22\) radians. [2]
2. Find the area, in cm\(^2\), of the major sector \(XZWX\). [3]

The region, shown shaded in Figure 4, is to be used as a design for a logo. Calculate

1. the area of the logo [3]
2. the perimeter of the logo. [4]