Edexcel C1 (Core Mathematics 1)

Given that \((2 + \sqrt{7})(4 - \sqrt{7}) = a + b\sqrt{7}\), where \(a\) and \(b\) are integers,

1. find the value of \(a\) and the value of \(b\). [2]

Given that \(\frac{2 + \sqrt{7}}{4 + \sqrt{7}} = c + d\sqrt{7}\) where \(c\) and \(d\) are rational numbers,

1. find the value of \(c\) and the value of \(d\). [3]

1. Solve the inequality \(3x - 8 > x + 13\). [2]
2. Solve the inequality \(x^2 - 5x - 14 > 0\). [3]

1. Prove, by completing the square, that the roots of the equation \(x^2 + 2kx + c = 0\), where \(k\) and \(c\) are constants, are \(-k \pm \sqrt{(k^2 - c)}\). [4]

The equation \(x^2 + 2kx + 81 = 0\) has equal roots.

1. Find the possible values of \(k\). [2]

Solve the simultaneous equations \(x - 3y + 1 = 0\), \(x^2 - 3xy + y^2 = 11\). [7]

$$\frac{dy}{dx} = 5 + \frac{1}{x^2}.$$

1. Use integration to find \(y\) in terms of \(x\). [3]
2. Given that \(y = 7\) when \(x = 1\), find the value of \(y\) at \(x = 2\). [4]

The points \(A\) and \(B\) have coordinates \((4, 6)\) and \((12, 2)\) respectively. The straight line \(l_1\) passes through \(A\) and \(B\).

1. Find an equation for \(l_1\) in the form \(ax + by = c\), where \(a\), \(b\) and \(c\) are integers. [4]

The straight line \(l_2\) passes through the origin and has gradient \(-4\).

1. Write down an equation for \(l_2\). [1]

The lines \(l_1\) and \(l_2\) intercept at the point \(C\).

1. Find the exact coordinates of the mid-point of \(AC\). [5]

A curve \(C\) has equation \(y = x^3 - 5x^2 + 5x + 2\).

1. Find \(\frac{dy}{dx}\) in terms of \(x\). [2]

The points \(P\) and \(Q\) lie on \(C\). The gradient of \(C\) at both \(P\) and \(Q\) is 2. The \(x\)-coordinate of \(P\) is 3.

1. Find the \(x\)-coordinate of \(Q\). [2]
2. Find an equation for the tangent to \(C\) at \(P\), giving your answer in the form \(y = mx + c\), where \(m\) and \(c\) are constants. [3]

This tangent intersects the coordinate axes at the points \(R\) and \(S\).

1. Find the length of \(RS\), giving your answer as a surd. [4]

\includegraphics{figure_1} The points \(A(3, 0)\) and \(B(0, 4)\) are two vertices of the rectangle \(ABCD\), as shown in Fig. 1.

1. Write down the gradient of \(AB\) and hence the gradient of \(BC\). [3]

The point \(C\) has coordinates \((8, k)\), where \(k\) is a positive constant.

1. Find the length of \(BC\) in terms of \(k\). [2]

Given that the length of \(BC\) is 10 and using your answer to part (b),

1. find the value of \(k\), [4]
2. find the coordinates of \(D\). [2]