OCR MEI C1 (Core Mathematics 1)

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

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

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

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

5 Express \(5 x ^ { 2 } + 20 x + 6\) in the form \(a ( x + b ) ^ { 2 } + c\).

6 Rearrange the formula \(c = \sqrt { \frac { a + b } { 2 } }\) to make \(a\) the subject.
\(7 \quad\) Make \(a\) the subject of the formula \(s = u t + \frac { 1 } { 2 } a t ^ { 2 }\).

8 Prove that, when \(n\) is an integer, \(n ^ { 3 } - n\) is always even.

9

1. Express \(x ^ { 2 } + 6 x + 5\) in the form \(( x + a ) ^ { 2 } + b\).
2. Write down the coordinates of the minimum point on the graph of \(y = x ^ { 2 } + 6 x + 5\).

10 Find the real roots of the equation \(x ^ { 4 } - 5 x ^ { 2 } - 36 = 0\) by considering it as a quadratic equation in \(x ^ { 2 }\).

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

12 Find the range of values of \(k\) for which the equation \(2 x ^ { 2 } + k x + 18 = 0\) does not have real roots.

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