OCR MEI C1 (Core Mathematics 1)

Expand \((2x + 5)(x - 1)(x + 3)\), simplifying your answer. [3]

Find the discriminant of \(3x^2 + 5x + 2\). Hence state the number of distinct real roots of the equation \(3x^2 + 5x + 2 = 0\). [3]

Make \(x\) the subject of the formula \(y = \frac{1 - 2x}{x + 3}\). [4]

Factorise \(n^3 + 3n^2 + 2n\). Hence prove that, when \(n\) is a positive integer, \(n^3 + 3n^2 + 2n\) is always divisible by 6. [3]

Express \(5x^2 + 20x + 6\) in the form \(a(x + b)^2 + c\). [4]

Rearrange the formula \(c = \sqrt{\frac{a + b}{2}}\) to make \(a\) the subject. [3]

Make \(a\) the subject of the formula \(s = ut + \frac{1}{2}at^2\). [3]

Prove that, when \(n\) is an integer, \(n^3 - n\) is always even. [3]

Solve the equation \(\frac{3x + 1}{2x} = 4\). [3]

Find the range of values of \(k\) for which the equation \(2x^2 + kx + 18 = 0\) does not have real roots. [4]

Rearrange \(y + 5 = x(y + 2)\) to make \(y\) the subject of the formula. [4]