Higher June 2025 Paper 1 Q23
23
Diagram NOT accurately drawn
\(OAB\) is a triangle.
\(P\) is the midpoint of \(OA\)
\(Q\) is a point on \(OB\)
\(ABR\) and \(PQR\) are straight lines.
\(\overrightarrow{OA} = 12\mathbf{a}\) \(\overrightarrow{OB} = 8\mathbf{b}\)
(a) Express \(\overrightarrow{AB}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\) (1)
\(AB : BR = 1 : 2\) \(\overrightarrow{OQ} = n\mathbf{b}\)
(b) Use a vector method to find the value of \(n\) (4)
| Scheme | Marks |
|---|---|
| \(-12\mathbf{a} + 8\mathbf{b}\) | B1 |
| (1) |
Notes
B1: oe
| Scheme | Marks |
|---|---|
\(\left(\overrightarrow{PR} =\right)6\mathbf{a} + 3\left(\text{``}{-12\mathbf{a} + 8\mathbf{b}}\text{''}\right)\left(= -30\mathbf{a} + 24\mathbf{b}\right)\) oe or \(\left(\overrightarrow{RP} =\right) -3\left(\text{``}{-12\mathbf{a} + 8\mathbf{b}}\text{''}\right) - 6\mathbf{a}\left(= 30\mathbf{a} - 24\mathbf{b}\right)\) | M1 |
\(\left(\overrightarrow{PQ} =\right) -6\mathbf{a} + n\mathbf{b}\) oe eg \(\left(\overrightarrow{PQ} =\right) -6\mathbf{a} + m \times 8\mathbf{b}\) or \(\left(\overrightarrow{OQ} =\right)\) 6a + \(k\)(“–30a + 24b”) or \(\left(\overrightarrow{AQ} =\right) -12\mathbf{a} + n\mathbf{b}\) | M1 |
\(\overrightarrow{PQ} = \lambda \overrightarrow{PR}\) eg \(-6\mathbf{a} + n\mathbf{b} = \lambda\left(\text{``}{-30\mathbf{a} + 24\mathbf{b}}\text{''}\right)\) or \(-6 = -30\lambda\) oe or \(\lambda = \dfrac{1}{5}\) oe OR \(\mu \overrightarrow{PQ} = \overrightarrow{PR}\) eg \(\mu\left(-6\mathbf{a} + n\mathbf{b}\right) = \text{``}{-30\mathbf{a} + 24\mathbf{b}}\text{''}\) or \(-6\mu = -30\) oe or \(\mu = 5\) OR \(\overrightarrow{OQ} = \overrightarrow{OP} + k\overrightarrow{PR}\) eg \(n\)b = 6a + \(k\)(“–30a + 24b”) or 0 = 6 – 30\(k\) or \(k = \dfrac{1}{5}\) oe OR \(\overrightarrow{AQ} = \overrightarrow{AR} + x\overrightarrow{RP}\) eg \(-12\mathbf{a} + n\mathbf{b} = 3\left(\text{``}{-12\mathbf{a} + 8\mathbf{b}}\text{''}\right) + x\left(\text{``}{30\mathbf{a} - 24\mathbf{b}}\text{''}\right)\) or \(-36 + 30x = -12\) or \(x = \dfrac{4}{5}\) oe | M1 |
| Working required Answer: 4.8 | A1 |
| (4) | |
| (5 marks) |
Notes
M1: for method to find \(\overrightarrow{PR}\) or \(\overrightarrow{RP}\), ft their \(\overrightarrow{AB}\)
M1: for method to find \(\overrightarrow{PQ}\) or \(\overrightarrow{OQ}\) or \(\overrightarrow{AQ}\)
M1: for setting up an equation to find the value of the unknown coefficient(s)
A1: oe eg \(\dfrac{24}{5}\), dep on M1