The diagram shows a set of rectangular axes Ox, Oy and Oz, and four points A, B, C and D with position vectors **OA** = 3**i**, **OB** = 3**i** + 4**j**, **OC** = **i** + 3**j** and **OD** = 2**i** + 3**j** + 5**k**.

Mark scheme

Marking points from the official mark scheme (May/June 2019). The mark or mark type is shown in brackets.

1. [B1] (i) Obtain a vector parallel to the plane, e.g. CB = 2i + j
2. [M1] (i) Use scalar product to obtain an equation in a, b, c, e.g. 2a + b = 0, a + 5c = 0, a + b − 5c = 0
3. [A1] (i) Obtain two correct equations in a, b, c
4. [M1] (i) Solve to obtain a : b : c,
5. [A1] (i) Obtain a:b: c = 5 : - 10 : – 1, or equivalent
6. [A1] (i) Obtain equation 5x-10y - z = -25, or equivalent
7. [B1] (i) Obtain a vector parallel to the plane, e.g. CD = i + 5k
8. [M1] (i) Obtain a second such vector and calculate their vector product, e.g. (2i+j)×(i+5k)
9. [A1] (i) Obtain two correct components
10. [A1] (i) Obtain correct answer, e.g. 5i – 10j – k
11. [M1] (i) Substitute to find d
12. [A1] (i) Obtain equation 5x-10y - z = -25, or equivalent
13. [B1] (ii) State or imply a normal vector for the plane OABC is k
14. [M1] (ii) Carry out correct process for evaluating a scalar product of two relevant vectors, e.g. (5i – 10j – k).(k)
15. [M1] (ii) Using the correct process for calculating the moduli, divide the scalar product by the product of the moduli and evaluate the inverse cosine of the result
16. [A1] (ii) Obtain answer 84.9° or 1.48 radians