Edexcel C1 (Core Mathematics 1)

Evaluate $$\sum_{r=1}^{20} (3r + 4).$$ [3]

1. Express \(x^2 + 6x + 7\) in the form \((x + a)^2 + b\). [3]
2. State the coordinates of the minimum point of the curve \(y = x^2 + 6x + 7\). [1]

The straight line \(l_1\) has the equation \(3x - y = 0\). The straight line \(l_2\) has the equation \(x + 2y - 4 = 0\).

1. Sketch \(l_1\) and \(l_2\) on the same diagram, showing the coordinates of any points where each line meets the coordinate axes. [3]
2. Find, as exact fractions, the coordinates of the point where \(l_1\) and \(l_2\) intersect. [3]

Find the pairs of values \((x, y)\) which satisfy the simultaneous equations $$3x^2 + y^2 = 21$$ $$5x + y = 7$$ [7]

1. Sketch on the same diagram the graphs of \(y = (x - 1)^2(x - 5)\) and \(y = 8 - 2x\). Label on your diagram the coordinates of any points where each graph meets the coordinate axes. [5]
2. Explain how your diagram shows that there is only one solution, \(\alpha\), to the equation $$(x - 1)^2(x - 5) = 8 - 2x.$$ [1]
3. State the integer, \(n\), such that $$n < \alpha < n + 1.$$ [1]

The curve with equation \(y = x^2 + 2x\) passes through the origin, \(O\).

1. Find an equation for the normal to the curve at \(O\). [5]
2. Find the coordinates of the point where the normal to the curve at \(O\) intersects the curve again. [3]

Given that $$y = \sqrt{x} - \frac{4}{\sqrt{x}},$$

1. find \(\frac{dy}{dx}\). [3]
2. find \(\frac{d^2y}{dx^2}\). [2]
3. show that $$4x\frac{d^2y}{dx^2} + 4x\frac{dy}{dx} - y = 0.$$ [3]

1. Prove that the sum of the first \(n\) positive integers is given by $$\frac{1}{2}n(n + 1).$$ [4]
2. Hence, find the sum of
   1. the integers from 100 to 200 inclusive,
   2. the integers between 300 to 600 inclusive which are divisible by 3.
   [5]

1. Express each of the following in the form \(p + q\sqrt{2}\) where \(p\) and \(q\) are rational.
   1. \((4 - 3\sqrt{2})^2\)
   2. \(\frac{1}{2 + \sqrt{2}}\) [5]
2. 1. Solve the equation $$y^2 + 8 = 9y.$$
   2. Hence solve the equation $$x^3 + 8 = 9x^{\frac{1}{2}}.$$ [5]

\includegraphics{figure_1} Figure 1 shows the curve with equation \(y = \text{f}(x)\). The curve meets the \(x\)-axis at the origin and at the point \(A\). Given that $$\text{f}'(x) = 3x^{\frac{1}{2}} - 4x^{-\frac{1}{2}},$$

1. find f\((x)\). [5]
2. Find the coordinates of \(A\). [2]

The point \(B\) on the curve has \(x\)-coordinate 2.

1. Find an equation for the tangent to the curve at \(B\) in the form \(y = mx + c\). [6]