Edexcel C1 (Core Mathematics 1)

Express \(\sqrt{50} + 3\sqrt{8}\) in the form \(k\sqrt{2}\). [3]

Differentiate with respect to \(x\) $$3x^2 - \sqrt{x} + \frac{1}{2x}.$$ [4]

A sequence is defined by the recurrence relation $$u_{n+1} = u_n - 2, \quad n > 0, \quad u_1 = 50.$$

1. Write down the first four terms of the sequence. [1]
2. Evaluate $$\sum_{r=1}^{20} u_r.$$ [3]

1. Find the value of the constant \(k\) such that the equation $$x^2 - 6x + k = 0$$ has equal roots. [2]
2. Solve the inequality $$2x^2 - 9x + 4 < 0.$$ [4]

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

Given that $$\frac{dy}{dx} = 3\sqrt{x} - x^2,$$ and that \(y = \frac{2}{3}\) when \(x = 1\), find the value of \(y\) when \(x = 4\). [7]

The first three terms of an arithmetic series are \((12 - p)\), \(2p\) and \((4p - 5)\) respectively, where \(p\) is a constant.

1. Find the value of \(p\). [2]
2. Show that the sixth term of the series is 50. [3]
3. Find the sum of the first 15 terms of the series. [2]
4. Find how many terms of the series have a value of less than 400. [3]

$$f(x) = 2x^2 + 3x - 2.$$

1. Solve the equation \(f(x) = 0\). [2]
2. Sketch the curve with equation \(y = f(x)\), showing the coordinates of any points of intersection with the coordinate axes. [2]
3. Find the coordinates of the points where the curve with equation \(y = f(\frac{1}{2}x)\) crosses the coordinate axes. [3]

When the graph of \(y = f(x)\) is translated by 1 unit in the positive \(x\)-direction it maps onto the graph with equation \(y = ax^2 + bx + c\), where \(a\), \(b\) and \(c\) are constants.

1. Find the values of \(a\), \(b\) and \(c\). [3]

\includegraphics{figure_1} Figure 1 shows the curve \(C\) with the equation \(y = x^3 + 3x^2 - 4x\) and the straight line \(l\). The curve \(C\) crosses the \(x\)-axis at the origin, \(O\), and at the points \(A\) and \(B\).

1. Find the coordinates of \(A\) and \(B\). [3]

The line \(l\) is the tangent to \(C\) at \(O\).

1. Find an equation for \(l\). [4]
2. Find the coordinates of the point where \(l\) intersects \(C\) again. [4]

The straight line \(l_1\) has equation \(2x + y - 14 = 0\) and crosses the \(x\)-axis at the point \(A\).

1. Find the coordinates of \(A\). [2]

The straight line \(l_2\) is parallel to \(l_1\) and passes through the point \(B(-6, 6)\).

1. Find an equation for \(l_2\) in the form \(y = mx + c\). [3]

The line \(l_2\) crosses the \(x\)-axis at the point \(C\).

1. Find the coordinates of \(C\). [1]

The point \(D\) lies on \(l_1\) and is such that \(CD\) is perpendicular to \(l_1\).

1. Show that \(D\) has coordinates \((5, 4)\). [5]
2. Find the area of triangle \(ACD\). [2]