OCR MEI C1 (Core Mathematics 1)

1. Solve the equation \(2x^2 + 3x = 0\). [2]
2. Find the set of values of \(k\) for which the equation \(2x^2 + 3x - k = 0\) has no real roots. [3]

Make \(x\) the subject of the equation \(y = \frac{x + 3}{x - 2}\). [4]

Solve the equation \(y^2 - 7y + 12 = 0\). Hence solve the equation \(x^4 - 7x^2 + 12 = 0\). [4]

1. Write \(\sqrt{48} + \sqrt{3}\) in the form \(a\sqrt{b}\), where \(a\) and \(b\) are integers and \(b\) is as small as possible. [2]
2. Simplify \(\frac{1}{5 + \sqrt{2}} + \frac{1}{5 - \sqrt{2}}\). [3]

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

Make \(a\) the subject of the equation $$2a + 5c = af + 7c.$$ [3]

Find the set of values of \(k\) for which the equation \(2x^2 + kx + 2 = 0\) has no real roots. [4]

One root of the equation \(x^3 + ax^2 + 7 = 0\) is \(x = -2\). Find the value of \(a\). [2]

\(n\) is a positive integer. Show that \(n^2 + n\) is always even. [2]

Make \(C\) the subject of the formula \(P = \frac{C}{C + 4}\). [4]

1. Find the range of values of \(k\) for which the equation \(x^2 + 5x + k = 0\) has one or more real roots. [3]
2. Solve the equation \(4x^2 + 20x + 25 = 0\). [2]