Higher January 2022 Paper 1R Q24
24 \(OXYZ\) is a trapezium.
Diagram NOT accurately drawn
\(\overrightarrow{OX} = \mathbf{a}\)
\(\overrightarrow{XY} = \mathbf{b}\)
\(\overrightarrow{OZ} = 3\mathbf{b}\)
\(M\) is the midpoint of \(OX\)
\(N\) is the point such that \(MNZ\) and \(ONY\) are straight lines.
Given that \(ON : OY = \lambda : 1\)
use a vector method to find the value of \(\lambda\)
(5)
| Scheme | Marks |
|---|---|
| \((\overrightarrow{ON} =)\ \lambda(\mathbf{a} + \mathbf{b})(= \lambda\mathbf{a} + \lambda\mathbf{b})\) or \((\overrightarrow{NY} =)\ (1 - \lambda)(\mathbf{a} + \mathbf{b})(= (1 - \lambda)\mathbf{a} + (1 - \lambda)\mathbf{b})\) | M1 |
| \((\overrightarrow{MN} = \overrightarrow{MO} + \overrightarrow{ON} =) -0.5\mathbf{a} + \lambda\mathbf{a} + \lambda\mathbf{b}(= (\lambda - 0.5)\mathbf{a} + \lambda\mathbf{b})\) or \((\overrightarrow{MZ} = \overrightarrow{MO} + \overrightarrow{OZ} =) -0.5\mathbf{a} + 3\mathbf{b}\) or \((\overrightarrow{MN} = \overrightarrow{MX} + \overrightarrow{XY} + \overrightarrow{YN} =)\ 0.5\mathbf{a} + \mathbf{b} + (\lambda - 1)(\mathbf{a} + \mathbf{b})(= (\lambda - 0.5)\mathbf{a} + \lambda\mathbf{b})\) | M1 |
| \((\overrightarrow{MN} = \mu\overrightarrow{MZ} =)\ \mu(-0.5\mathbf{a} + 3\mathbf{b})(= -0.5\mu\mathbf{a} + 3\mu\mathbf{b})\) or \((\overrightarrow{ON} = \overrightarrow{OM} + \overrightarrow{MN} =)\ 0.5\mathbf{a} + \mu(-0.5\mathbf{a} + 3\mathbf{b})(= (0.5 - 0.5\mu)\mathbf{a} + 3\mu\mathbf{b})\) or \((\overrightarrow{NY} = \overrightarrow{NM} + \overrightarrow{MX} + \overrightarrow{XY} =) -\mu(-0.5\mathbf{a} + 3\mathbf{b}) + 0.5\mathbf{a} + \mathbf{b}(= (0.5 + 0.5\mu)\mathbf{a} + (1 - 3\mu)\mathbf{b})\) | M1 |
| \(-0.5\mu = -0.5 + \lambda\) oe \(3\mu = \lambda\) oe or \(1 - \lambda = 0.5\mu + 0.5\) oe \(1 - \lambda = 1 - 3\mu\) oe | M1 |
Working required Answer: \(\dfrac{3}{7}\) | A1 |
| (5) | |
| (5 marks) |
Notes
M1: for finding a vector for \(\overrightarrow{ON}\) or \(\overrightarrow{NY}\) or \(\overrightarrow{NO}\) or \(\overrightarrow{YN}\) in terms a and b and using \(\lambda\) oe (can be embedded)
M1: for finding a vector for \(\overrightarrow{MN}\) or \(\overrightarrow{NM}\) or \(\overrightarrow{MZ}\) or \(\overrightarrow{ZM}\)
M1: for finding a vector for \(\overrightarrow{MN}\) or \(\overrightarrow{ON}\) or \(\overrightarrow{NY}\) or \(\overrightarrow{NM}\) or \(\overrightarrow{NO}\) or \(\overrightarrow{YN}\) using another variable e.g. \(\mu\) oe
M1: for setting up two simultaneous equations using the components of a and b for \(\overrightarrow{MN}\) or \(\overrightarrow{ON}\) or \(\overrightarrow{NY}\) oe
A1: (allow \(\dfrac{3}{7}\) = 0.42(8571…) to 2 sf truncated or rounded)
| Scheme | Marks |
|---|---|
| \((\overrightarrow{ON} =)\ \lambda(\mathbf{a} + \mathbf{b})(= \lambda\mathbf{a} + \lambda\mathbf{b})\) or \((\overrightarrow{NY} =)\ (1 - \lambda)(\mathbf{a} + \mathbf{b})(= (1 - \lambda)\mathbf{a} + (1 - \lambda)\mathbf{b})\) | M1 |
| \((\overrightarrow{MN} = \overrightarrow{MO} + \overrightarrow{ON} =) -0.5\mathbf{a} + \lambda\mathbf{a} + \lambda\mathbf{b}(= (\lambda - 0.5)\mathbf{a} + \lambda\mathbf{b})\) or \((\overrightarrow{MN} = \overrightarrow{MX} + \overrightarrow{XY} + \overrightarrow{YN} =)\ 0.5\mathbf{a} + \mathbf{b} + (\lambda - 1)(\mathbf{a} + \mathbf{b})(= (\lambda - 0.5)\mathbf{a} + \lambda\mathbf{b})\) | M1 |
| \((\overrightarrow{NZ} = \overrightarrow{NO} + \overrightarrow{OZ} =) -\lambda(\mathbf{a} + \mathbf{b}) + 3\mathbf{b}(= -\lambda\mathbf{a} + (3 - \lambda)\mathbf{b})\) or \((\overrightarrow{NZ} = \overrightarrow{NY} + \overrightarrow{YZ} =)\ (1 - \lambda)(\mathbf{a} + \mathbf{b}) - \mathbf{b} - \mathbf{a} + 3\mathbf{b}(= -\lambda\mathbf{a} + (3 - \lambda)\mathbf{b})\) | M1 |
| \(\dfrac{\lambda - 0.5}{-\lambda} = \dfrac{\lambda}{3 - \lambda}\) oe | M1 |
Working required Answer: \(\dfrac{3}{7}\) | A1 |
Notes
M1: for finding a vector for \(\overrightarrow{ON}\) or \(\overrightarrow{NY}\) or \(\overrightarrow{NO}\) or \(\overrightarrow{YN}\) in terms a and b and using \(\lambda\) oe
M1: for finding a vector for \(\overrightarrow{MN}\) or \(\overrightarrow{NM}\) in terms a and b and using \(\lambda\) oe
M1: for finding a vector for \(\overrightarrow{NZ}\) or \(\overrightarrow{ZN}\) in terms a and b and using \(\lambda\) oe
M1: for setting up an equation using the components of \(\overrightarrow{MN}\) and \(\overrightarrow{NZ}\) oe
A1: (allow \(\dfrac{3}{7}\) = 0.42(8571…) to 2 sf truncated or rounded)