Edexcel C1 (Core Mathematics 1)

Evaluate \(49^{\frac{1}{2}} + 8^{\frac{1}{3}}\). [3]

A sequence is defined by the recurrence relation $$u_{n+1} = \frac{u_n + 1}{3}, \quad n = 1, 2, 3, ...$$ Given that \(u_3 = 5\),

1. find the value of \(u_4\), [1]
2. find the value of \(u_1\). [3]

\(\text{f}(x) = 4x^2 + 12x + 9\).

1. Determine the number of real roots that exist for the equation \(\text{f}(x) = 0\). [2]
2. Solve the equation \(\text{f}(x) = 8\), giving your answers in the form \(a + b\sqrt{2}\) where \(a\) and \(b\) are rational. [4]

Find the set of values of \(x\) for which

1. \(6x - 11 > x + 4\), [2]
2. \(x^2 - 6x - 16 < 0\), [3]
3. both \(6x - 11 > x + 4\) and \(x^2 - 6x - 16 < 0\). [1]

\(\text{f}(x) = (2 - \sqrt{x})^2, \quad x > 0\).

1. Solve the equation \(\text{f}(x) = 0\). [2]
2. Find \(\text{f}(3)\), giving your answer in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are integers. [2]
3. Find $$\int \text{f}(x) \, dx.$$ [4]

The straight line \(l\) passes through the point \(P(-3, 6)\) and the point \(Q(1, -4)\).

1. Find an equation for \(l\) in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]

The straight line \(m\) has the equation \(2x + ky + 7 = 0\), where \(k\) is a constant. Given that \(l\) and \(m\) are perpendicular,

1. find the value of \(k\). [4]

Given that $$\text{f}'(x) = 5 + \frac{4}{x^2}, \quad x \neq 0,$$

1. find an expression for \(\text{f}(x)\). [3]

Given also that $$\text{f}(2) = 2\text{f}(1),$$

1. find \(\text{f}(4)\). [5]

\(\text{f}(x) = x^3 - 6x^2 + 5x + 12\).

1. Show that $$(x + 1)(x - 3)(x - 4) \equiv x^3 - 6x^2 + 5x + 12.$$ [3]
2. Sketch the curve \(y = \text{f}(x)\), showing the coordinates of any points of intersection with the coordinate axes. [3]
3. Showing the coordinates of any points of intersection with the coordinate axes, sketch on separate diagrams the curves
   1. \(y = \text{f}(x + 3)\),
   2. \(y = \text{f}(-x)\). [4]

The first two terms of an arithmetic series are \((t - 1)\) and \((t^2 - 5)\) respectively, where \(t\) is a positive constant.

1. Find and simplify expressions in terms of \(t\) for
   1. the common difference of the series,
   2. the third term of the series. [4]

Given also that the third term of the series is 19,

1. find the value of \(t\), [2]
2. show that the 10th term of the series is 75, [3]
3. find the sum of the first 40 terms of the series. [2]

\includegraphics{figure_1} Figure 1 shows the curve with equation \(y = 2 + 3x - x^2\) and the straight lines \(l\) and \(m\). The line \(l\) is the tangent to the curve at the point \(A\) where the curve crosses the \(y\)-axis.

1. Find an equation for \(l\). [5]

The line \(m\) is the normal to the curve at the point \(B\). Given that \(l\) and \(m\) are parallel,

1. find the coordinates of \(B\). [6]