Higher November 2017 Paper 2 Q26
26
ABCDE is a pentagon.
Not drawn accurately
Show that BCDE is a parallelogram. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 Shows that CB (or BC) is equal and parallel to DE (or ED) | ||
| (\(\overrightarrow{CB}\) =) \(-(\mathbf{b} - 2\mathbf{a}) - 2\mathbf{b} - \mathbf{a}\) or (\(\overrightarrow{BC}\) =) \(\mathbf{b} - 2\mathbf{a} + 2\mathbf{b} + \mathbf{a}\) | M1 | oe method |
| (\(\overrightarrow{CB}\) =) \(\mathbf{a} - 3\mathbf{b}\) or (\(\overrightarrow{BC}\) =) \(3\mathbf{b} - \mathbf{a}\) | A1 | Must see correct method for \(\overrightarrow{CB}\) or \(\overrightarrow{BC}\) |
| CB is equal and parallel to DE | A1 | Must see a correct vector for first A1 and have a statement oe eg CB is equal and parallel to ED |
| Alternative method 2 Shows that BE (or EB) is equal and parallel to CD (or DC) | ||
| (\(\overrightarrow{BE}\) =) \(\mathbf{a} + 2\mathbf{b}\) or (\(\overrightarrow{CD}\) =) \(-(\mathbf{b} - 2\mathbf{a}) - (\mathbf{a} - 3\mathbf{b})\) or (\(\overrightarrow{EB}\) =) \(-\mathbf{a} - 2\mathbf{b}\) or (\(\overrightarrow{DC}\) =) \((\mathbf{a} - 3\mathbf{b}) + (\mathbf{b} - 2\mathbf{a})\) | M1 | oe method |
| (\(\overrightarrow{BE}\) =) \(\mathbf{a} + 2\mathbf{b}\) and (\(\overrightarrow{CD}\) =) \(\mathbf{a} + 2\mathbf{b}\) or (\(\overrightarrow{EB}\) =) \(-\mathbf{a} - 2\mathbf{b}\) and (\(\overrightarrow{DC}\) =) \(-\mathbf{a} - 2\mathbf{b}\) | A1 | Must see correct method for \(\overrightarrow{CD}\) or \(\overrightarrow{DC}\) oe eg (\(\overrightarrow{BE}\) =) \(\mathbf{a} + 2\mathbf{b}\) and (\(\overrightarrow{DC}\) =) \(-\mathbf{a} - 2\mathbf{b}\) |
| BE is equal and parallel to CD | A1 | Must see two correct vectors for first A1 and have a statement oe eg BE is equal and parallel to DC |
| Alternative method 3 Shows that two pairs of opposite sides are parallel | ||
| (\(\overrightarrow{CB}\) =) \(-(\mathbf{b} - 2\mathbf{a}) - 2\mathbf{b} - \mathbf{a}\) or (\(\overrightarrow{BC}\) =) \(\mathbf{b} - 2\mathbf{a} + 2\mathbf{b} + \mathbf{a}\) or (\(\overrightarrow{BE}\) =) \(\mathbf{a} + 2\mathbf{b}\) or (\(\overrightarrow{CD}\) =) \(-(\mathbf{b} - 2\mathbf{a}) - (\mathbf{a} - 3\mathbf{b})\) or (\(\overrightarrow{EB}\) =) \(-\mathbf{a} - 2\mathbf{b}\) or (\(\overrightarrow{DC}\) =) \((\mathbf{a} - 3\mathbf{b}) + (\mathbf{b} - 2\mathbf{a})\) | M1 | oe method |
| (\(\overrightarrow{CB}\) =) \(\mathbf{a} - 3\mathbf{b}\) or (\(\overrightarrow{BC}\) =) \(3\mathbf{b} - \mathbf{a}\) or (\(\overrightarrow{BE}\) =) \(\mathbf{a} + 2\mathbf{b}\) and (\(\overrightarrow{CD}\) =) \(\mathbf{a} + 2\mathbf{b}\) or (\(\overrightarrow{EB}\) =) \(-\mathbf{a} - 2\mathbf{b}\) and (\(\overrightarrow{DC}\) =) \(-\mathbf{a} - 2\mathbf{b}\) | A1 | Must see correct method for \(\overrightarrow{CB}\) or \(\overrightarrow{BC}\) or \(\overrightarrow{CD}\) or \(\overrightarrow{DC}\) oe eg (\(\overrightarrow{BE}\) =) \(\mathbf{a} + 2\mathbf{b}\) and (\(\overrightarrow{DC}\) =) \(-\mathbf{a} - 2\mathbf{b}\) |
| (\(\overrightarrow{CB}\) =) \(\mathbf{a} - 3\mathbf{b}\) and (\(\overrightarrow{BE}\) =) \(\mathbf{a} + 2\mathbf{b}\) and (\(\overrightarrow{CD}\) =) \(\mathbf{a} + 2\mathbf{b}\) and CB is parallel to DE and BE is parallel to CD | A1 | Must see three correct vectors and have two statements oe eg (\(\overrightarrow{BC}\) =) \(3\mathbf{b} - \mathbf{a}\) and (\(\overrightarrow{BE}\) =) \(\mathbf{a} + 2\mathbf{b}\) and (\(\overrightarrow{DC}\) =) \(-\mathbf{a} - 2\mathbf{b}\) and BC is parallel to DE and BE is parallel to DC |
| Alternative method 4 Shows that two pairs of opposite sides are equal | ||
| (\(\overrightarrow{CB}\) =) \(-(\mathbf{b} - 2\mathbf{a}) - 2\mathbf{b} - \mathbf{a}\) or (\(\overrightarrow{BC}\) =) \(\mathbf{b} - 2\mathbf{a} + 2\mathbf{b} + \mathbf{a}\) or (\(\overrightarrow{BE}\) =) \(\mathbf{a} + 2\mathbf{b}\) or (\(\overrightarrow{CD}\) =) \(-(\mathbf{b} - 2\mathbf{a}) - (\mathbf{a} - 3\mathbf{b})\) or (\(\overrightarrow{EB}\) =) \(-\mathbf{a} - 2\mathbf{b}\) or (\(\overrightarrow{DC}\) =) \((\mathbf{a} - 3\mathbf{b}) + (\mathbf{b} - 2\mathbf{a})\) | M1 | oe |
| (\(\overrightarrow{CB}\) =) \(\mathbf{a} - 3\mathbf{b}\) or (\(\overrightarrow{BC}\) =) \(3\mathbf{b} - \mathbf{a}\) or (\(\overrightarrow{BE}\) =) \(\mathbf{a} + 2\mathbf{b}\) and (\(\overrightarrow{CD}\) =) \(\mathbf{a} + 2\mathbf{b}\) or (\(\overrightarrow{EB}\) =) \(-\mathbf{a} - 2\mathbf{b}\) and (\(\overrightarrow{DC}\) =) \(-\mathbf{a} - 2\mathbf{b}\) | A1 | Must see correct method for \(\overrightarrow{CB}\) or \(\overrightarrow{BC}\) or \(\overrightarrow{CD}\) or \(\overrightarrow{DC}\) oe eg (\(\overrightarrow{BE}\) =) \(\mathbf{a} + 2\mathbf{b}\) and (\(\overrightarrow{DC}\) =) \(-\mathbf{a} - 2\mathbf{b}\) |
| (\(\overrightarrow{CB}\) =) \(\mathbf{a} - 3\mathbf{b}\) and (\(\overrightarrow{BE}\) =) \(\mathbf{a} + 2\mathbf{b}\) and (\(\overrightarrow{CD}\) =) \(\mathbf{a} + 2\mathbf{b}\) and CB is equal to DE and BE is equal to CD | A1 | Must see three correct vectors and have two statements oe eg (\(\overrightarrow{BC}\) =) \(3\mathbf{b} - \mathbf{a}\) and (\(\overrightarrow{BE}\) =) \(\mathbf{a} + 2\mathbf{b}\) and (\(\overrightarrow{DC}\) =) \(-\mathbf{a} - 2\mathbf{b}\) and BC is equal to DE and BE is equal to DC |
Additional guidance
| Choose the method that gives most marks | |
| Ignore incorrect vectors if not contradictory | |
| For parallel allow in the same direction or in the opposite direction | |
| For equal to allow \(=\) or the same as | |
| Condone incorrect notation if unambiguous eg CB \(= -(b - 2a) - 2b - a\) | M1 |