Edexcel C1 (Core Mathematics 1)

Find the value of \(y\) such that $$4^{y + 3} = 8.$$ [3]

Find $$\int \left( 3x^2 + \frac{1}{2x^2} \right) dx.$$ [4]

\includegraphics{figure_1} Figure 1 shows the rectangles \(ABCD\) and \(EFGH\) which are similar. Given that \(AB = (3 - \sqrt{5})\) cm, \(AD = \sqrt{5}\) cm and \(EF = (1 + \sqrt{5})\) cm, find the length \(EH\) in cm, giving your answer in the form \(a + b\sqrt{5}\) where \(a\) and \(b\) are integers. [6]

1. Sketch on the same diagram the curves \(y = x^2 - 4x\) and \(y = -\frac{1}{x}\). [4]
2. State, with a reason, the number of real solutions to the equation $$x^2 - 4x + \frac{1}{x} = 0.$$ [2]

1. By completing the square, find in terms of the constant \(k\) the roots of the equation $$x^2 + 2kx + 4 = 0.$$ [4]
2. Hence find the exact roots of the equation $$x^2 + 6x + 4 = 0.$$ [2]

1. Evaluate $$\sum_{r=1}^{50} (80 - 3r).$$ [3]
2. Show that $$\sum_{r=1}^{n} \frac{r + 3}{2} = k n(n + 7),$$ where \(k\) is a rational constant to be found. [4]

Solve the simultaneous equations \begin{align} x - 3y + 7 &= 0
x^2 + 2xy - y^2 &= 7 \end{align} [7]

Given that $$\frac{dy}{dx} = \frac{x^3 - 4}{x^2}, \quad x \neq 0,$$

1. find \(\frac{d^2y}{dx^2}\). [3]

Given also that \(y = 0\) when \(x = -1\),

1. find the value of \(y\) when \(x = 2\). [6]

A curve has the equation \(y = (\sqrt{x} - 3)^2\), \(x \geq 0\).

1. Show that \(\frac{dy}{dx} = 1 - \frac{3}{\sqrt{x}}\). [4]

The point \(P\) on the curve has \(x\)-coordinate 4.

1. Find an equation for the normal to the curve at \(P\) in the form \(y = mx + c\). [5]
2. Show that the normal to the curve at \(P\) does not intersect the curve again. [4]

The straight line \(l\) has gradient 3 and passes through the point \(A(-6, 4)\).

1. Find an equation for \(l\) in the form \(y = mx + c\). [2]

The straight line \(m\) has the equation \(x - 7y + 14 = 0\). Given that \(m\) crosses the \(y\)-axis at the point \(B\) and intersects \(l\) at the point \(C\),

1. find the coordinates of \(B\) and \(C\), [4]
2. show that \(\angle BAC = 90°\), [4]
3. find the area of triangle \(ABC\). [4]