Higher November 2022 Paper 1 Q22
22 (4, 8) is a point on a circle, centre \(O\).
The tangent at (4, 8) intersects the \(x\)-axis at \(P\).
Not drawn accurately
Work out the \(x\)-coordinate of \(P\). [5 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(\dfrac{8 - 0}{4 - 0}\) or 2 | M1 | oe gradient from origin to point |
| \(-\dfrac{1}{2}\) or \(y = -\dfrac{1}{2}x \ldots\) | M1 | oe gradient of tangent negative inverse of their gradient |
| \(8 =\) their \(-\dfrac{1}{2} \times 4 + c\) or \(c = 10\) | M1dep | oe equation in \(c\) (any letter) dep on previous mark |
| \(0 =\) their \(-\dfrac{1}{2}x +\) their 10 | M1 | oe equation in \(x\) ft their equation of the form \(y = mx + c\) where \(m\) and \(c\) are numbers \(\neq 0\) |
| 20 | A1 | condone (20, 0) |
| Alternative method 2 | ||
| \(\dfrac{8 - 0}{4 - 0}\) or 2 | M1 | oe gradient from origin to point |
| \(-\dfrac{1}{2}\) or \(y = -\dfrac{1}{2}x \ldots\) | M1 | oe gradient of tangent negative inverse of their gradient |
| \(\dfrac{8 - 0}{4 - x} =\) their \(-\dfrac{1}{2}\) | M1dep | oe equation in \(x\) dep on previous mark |
| their \(2 \times (8 - 0) =\) their \(-1 \times (4 - x)\) or \(16 = -4 + x\) | M1dep | oe linear equation in \(x\) |
| 20 | A1 | condone (20, 0) |
| Alternative method 3 | ||
| \(\dfrac{8 - 0}{4 - 0}\) or 2 | M1 | oe gradient from origin to point |
| \(-\dfrac{1}{2}\) or \(y = -\dfrac{1}{2}x \ldots\) | M1 | oe gradient of tangent negative inverse of their gradient |
| \(y - 8 =\) their \(-\dfrac{1}{2} \times (x - 4)\) | M1dep | oe equation eg \(x + 2y = 20\) dep on previous mark |
| \(0 - 8 =\) their \(-\dfrac{1}{2} \times (x - 4)\) | M1 | oe linear equation in \(x\) ft their equation in \(y\) and \(x\) |
| 20 | A1 | condone (20, 0) |
| Alternative method 4 | ||
| \(4^2 + 8^2\) and \((x - 4)^2 + 8^2\) | M1 | |
| \(x^2 = 4^2 + 8^2 + (x - 4)^2 + 8^2\) | M1dep | oe equation in \(x\) |
| \(x^2 = 16 + 64 + x^2 - 8x + 16 + 64\) | M1dep | oe equation in \(x\) with brackets expanded and squares evaluated |
| \(8x = 16 + 64 + 16 + 64\) or \(8x = 160\) | M1dep | oe linear equation in \(x\) |
| 20 | A1 | condone (20, 0) |