Edexcel C1 (Core Mathematics 1)

Find in exact form the real solutions of the equation $$x^4 = 5x^2 + 14.$$ [3]

Express $$\frac{2}{3\sqrt{5} + 7}$$ in the form \(a + b\sqrt{5}\) where \(a\) and \(b\) are rational. [3]

1. Solve the equation $$x^{\frac{3}{2}} = 27.$$ [2]
2. Express \((2\frac{1}{4})^{-\frac{3}{2}}\) as an exact fraction in its simplest form. [2]

\includegraphics{figure_1} Figure 1 shows the curve with equation \(y = x^3 + ax^2 + bx + c\), where \(a\), \(b\) and \(c\) are constants. The curve crosses the \(x\)-axis at the point \((-1, 0)\) and touches the \(x\)-axis at the point \((3, 0)\). Show that \(a = -5\) and find the values of \(b\) and \(c\). [5]

Given that $$y = \frac{x^4 - 3}{2x^2},$$

1. find \(\frac{dy}{dx}\), [4]
2. show that \(\frac{d^2y}{dx^2} = \frac{x^4 - 9}{x^4}\). [2]

1. Sketch on the same diagram the curve with equation \(y = (x - 2)^2\) and the straight line with equation \(y = 2x - 1\). Label on your sketch the coordinates of any points where each graph meets the coordinate axes. [5]
2. Find the set of values of \(x\) for which $$(x - 2)^2 > 2x - 1.$$ [3]

A curve has the equation \(y = \frac{x}{2} + 3 - \frac{1}{x}\), \(x \neq 0\). The point \(A\) on the curve has \(x\)-coordinate 2.

1. Find the gradient of the curve at \(A\). [4]
2. Show that the tangent to the curve at \(A\) has equation $$3x - 4y + 8 = 0.$$ [3]

The tangent to the curve at the point \(B\) is parallel to the tangent at \(A\).

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

The straight line \(l_1\) has gradient \(\frac{3}{4}\) and passes through the point \(A(5, 3)\).

1. Find an equation for \(l_1\) in the form \(y = mx + c\). [2]

The straight line \(l_2\) has the equation \(3x - 4y + 3 = 0\) and intersects \(l_1\) at the point \(B\).

1. Find the coordinates of \(B\). [3]
2. Find the coordinates of the mid-point of \(AB\). [2]
3. Show that the straight line parallel to \(l_2\) which passes through the mid-point of \(AB\) also passes through the origin. [4]

The third term of an arithmetic series is \(5\frac{1}{2}\). The sum of the first four terms of the series is \(22\frac{3}{4}\).

1. Show that the first term of the series is \(6\frac{1}{4}\) and find the common difference. [7]
2. Find the number of positive terms in the series. [3]
3. Hence, find the greatest value of the sum of the first \(n\) terms of the series. [2]

The curve \(C\) has the equation \(y = f(x)\). Given that $$\frac{dy}{dx} = 8x - \frac{2}{x^3}, \quad x \neq 0,$$ and that the point \(P(1, 1)\) lies on \(C\),

1. find an equation for the tangent to \(C\) at \(P\) in the form \(y = mx + c\), [3]
2. find an equation for \(C\), [5]
3. find the \(x\)-coordinates of the points where \(C\) meets the \(x\)-axis, giving your answers in the form \(k\sqrt{2}\). [5]