June 2019 Paper 1 Q13
13 Prove that \(\quad (3x + 5)^2 - 5x(x + 10) \geqslant 0 \quad\) for all values of \(x\). [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(9x^2 + 15x + 15x + 25 - 5x^2 - 50x\) or \(9x^2 + 30x + 25 - 5x^2 - 50x\) or \(9x^2 + 15x + 15x + 25\) and \(-5x^2 - 50x\) or \(5x^2 + 50x\) | M1 | allow only one error in sign, omission or coefficient but not in more than one of these could be written as 2 separate expansions or in a grid |
| \(4x^2 - 20x + 25\) | A1 | |
| \(4x^2 - 20x + 25\) and \((2x - 5)^2\) or \((2x - 5)(2x - 5)\) or \(4(x - 2.5)^2\) or \(x = 2.5\) or \(b^2 - 4ac = 0\) from quadratic formula | M1dep | factorises or completes the square or uses the quadratic formula correctly. Answer required for M1 dep |
| \((2x - 5)^2\) or \(4(x - 2.5)^2\) (are squared terms) and so are always \(\geqslant 0\) | A1 | oe there must be a stated conclusion eg equal roots and positive quadratic so must be greater than or equal to zero |
| Alternative method 2 | ||
| \(9x^2 + 15x + 15x + 25 - 5x^2 - 50x\) or \(9x^2 + 30x + 25 - 5x^2 - 50x\) or \(9x^2 + 15x + 15x + 25\) and \(-5x^2 - 50x\) or \(5x^2 + 50x\) | M1 | allow only one error in sign, omission or coefficient but not in more than one of these could be written as 2 separate expansions or in a grid |
| \(4x^2 - 20x + 25\) | A1 | |
| \(4x^2 - 20x + 25\) and \(\dfrac{\mathrm{d}}{\mathrm{d}x} = 8x - 20\) and is zero when \(x = 2.5\) | M1dep | uses calculus to find stationary point |
| Tests for minimum by using eg \(x = 2\) and \(x = 3\) or by using 2nd derivative or concludes argument by saying this is a positive quadratic curve with minimum point \((2.5,\ 0)\), hence always \(\geqslant 0\) | A1 | oe there must be a stated conclusion |