Calculate the coordinates of the points where the line y=x+2 cuts the curve x² + y² = 10.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2014). The mark or mark type is shown in brackets.

1. [B1] (i) (x + 2)² + x² = 10
2. [M1] (i) x²+2x-3=0 → (x+3)(x-1)=0. 3 term quadratic with attempt to solve
3. [A1] (i) Points (1, 3), (-3, -1) isw. Both x or a pair
4. [A1] (i) Both y or second pair
5. [B1] (ii) m²x² + 10mx + 25 + x² = 10
6. [B1] (ii) (m² + 1)x² + 10mx + 15 = 0
7. [M1] (ii) b² - 4ac = (0) → 100m² -60(m² +1)=0. Attempt to use discriminant on three term quadratic. Allow unsimplified
8. [A1] (ii) m = ±√(3/2) oe isw. cao ± is required
9. [B1] (ii) Alternative solution: dy/dx = -x/√(10-x²) or dy/dx = -x/y. allow unsimplified
10. [M1] (ii) Alternative solution: y² = x² + 5y after inserted in y = mx + 5. Attempt to solve with x² + y² = 10. Eliminate x or y
11. [A1] (ii) Alternative solution: y = 2, x = ±√6. both
12. [A1] (ii) Alternative solution: m = ±(√6)/2 oe