Edexcel C1 (Core Mathematics 1)

1. Express \(\frac{18}{\sqrt{3}}\) in the form \(k\sqrt{3}\). [2]
2. Express \((1 - \sqrt{3})(4 - 2\sqrt{3})\) in the form \(a + b\sqrt{3}\) where \(a\) and \(b\) are integers. [2]

Solve the equation $$3x - \frac{5}{x} = 2.$$ [4]

The straight line \(l\) has the equation \(x - 5y = 7\). The straight line \(m\) is perpendicular to \(l\) and passes through the point \((-4, 1)\). Find an equation for \(m\) in the form \(y = mx + c\). [5]

A sequence of terms is defined by $$u_n = 3^n - 2, \quad n \geq 1.$$

1. Write down the first four terms of the sequence. [2]

The same sequence can also be defined by the recurrence relation $$u_{n+1} = au_n + b, \quad n \geq 1, \quad u_1 = 1,$$ where \(a\) and \(b\) are constants.

1. Find the values of \(a\) and \(b\). [4]

\includegraphics{figure_1} Figure 1 shows the curve with equation \(y = 8x - x^{\frac{3}{2}}\), \(x \geq 0\). The curve meets the \(x\)-axis at the origin, \(O\), and at the point \(A\).

1. Find the \(x\)-coordinate of \(A\). [3]
2. Find the gradient of the tangent to the curve at \(A\). [4]

$$\text{f}(x) = 2x^2 - 4x + 1.$$

1. Find the values of the constants \(a\), \(b\) and \(c\) such that $$\text{f}(x) = a(x + b)^2 + c.$$ [4]
2. State the equation of the line of symmetry of the curve \(y = \text{f}(x)\). [1]
3. Solve the equation \(\text{f}(x) = 3\), giving your answers in exact form. [3]

$$\text{f}(x) = \frac{(x-4)^2}{2x^{\frac{1}{2}}}, \quad x > 0.$$

1. Find the values of the constants \(A\), \(B\) and \(C\) such that $$\text{f}(x) = Ax^{\frac{3}{2}} + Bx^{\frac{1}{2}} + Cx^{-\frac{1}{2}}.$$ [3]
2. Show that $$\text{f}'(x) = \frac{(3x+4)(x-4)}{4x^{\frac{3}{2}}}.$$ [6]

1. Describe fully the single transformation that maps the graph of \(y = \text{f}(x)\) onto the graph of \(y = \text{f}(x - 1)\). [2]
2. Showing the coordinates of any points of intersection with the coordinate axes and the equations of any asymptotes, sketch the graph of \(y = \frac{1}{x-1}\). [3]
3. Find the \(x\)-coordinates of any points where the graph of \(y = \frac{1}{x-1}\) intersects the graph of \(y = 2 + \frac{1}{x}\). Give your answers in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are rational. [5]

A store begins to stock a new range of DVD players and achieves sales of £1500 of these products during the first month. In a model it is assumed that sales will decrease by £\(x\) in each subsequent month, so that sales of £\((1500 - x)\) and £\((1500 - 2x)\) will be achieved in the second and third months respectively. Given that sales total £8100 during the first six months, use the model to

1. find the value of \(x\), [4]
2. find the expected value of sales in the eighth month, [2]
3. show that the expected total of sales in pounds during the first \(n\) months is given by \(kn(51 - n)\), where \(k\) is an integer to be found. [3]
4. Explain why this model cannot be valid over a long period of time. [1]

The curve \(C\) with equation \(y = \text{f}(x)\) is such that $$\frac{\text{d}y}{\text{d}x} = 3x^2 + 4x + k,$$ where \(k\) is a constant. Given that \(C\) passes through the points \((0, -2)\) and \((2, 18)\),

1. show that \(k = 2\) and find an equation for \(C\), [7]
2. show that the line with equation \(y = x - 2\) is a tangent to \(C\) and find the coordinates of the point of contact. [5]