DP Mathematics SL Questionbank

• Syllabus

4.1

Sub sections and their related questions

Vectors as displacements in the plane and in three dimensions.

• 13N.2.sl.TZ0.9c: The point \({\text{Q}}(7, 5, 3)\) lies on \({L_1}\). The point \({\text{R}}\) is the reflection...
• 15M.1.sl.TZ1.8a: (i)     Show that...
• 15N.1.sl.TZ0.9c: A point \(D\) lies on line \({L_2}\) so that...
• 17N.1.sl.TZ0.9a.i: Show that...
• 17N.1.sl.TZ0.9a.ii: Find a vector equation for \(L\).
• 17N.1.sl.TZ0.9b: Find the value of \(p\).
• 17N.1.sl.TZ0.9c: The point D has coordinates \(({q^2},{\text{ }}0,{\text{ }}q)\). Given that...
• 18M.1.sl.TZ1.6: Six equilateral triangles, each with side length 3 cm, are arranged to form a hexagon.This is...
• 18M.1.sl.TZ1.9a: Show...
• 18M.1.sl.TZ1.9b.i: Find a vector equation for L.
• 18M.1.sl.TZ1.9b.ii: Point C (k , 12 , −k) is on L. Show that k = 14.
• 18M.1.sl.TZ1.9c.i: Find \(\mathop {{\text{OB}}}\limits^ \to  \, \bullet \mathop {{\text{AB}}}\limits^ \to  \).
• 18M.1.sl.TZ1.9c.ii: Write down the value of angle OBA.
• 18M.1.sl.TZ1.9d: Point D is also on L and has coordinates (8, 4, −9). Find the area of triangle OCD.
• 18M.2.sl.TZ2.8a.i: Find \(\mathop {{\text{PQ}}}\limits^ \to  \).
• 18M.2.sl.TZ2.8a.ii: Find \(\left| {\mathop {{\text{PQ}}}\limits^ \to  } \right|\).
• 18M.2.sl.TZ2.8b: Find the angle between PQ and PR.
• 18M.2.sl.TZ2.8c: Find the area of triangle PQR.
• 18M.2.sl.TZ2.8d: Hence or otherwise find the shortest distance from R to the line through P and Q.

Components of a vector; column representation; \(v = \left( {\begin{array}{*{20}{c}} {{v_1}} \\ {{v_2}} \\ {{v_3}} \end{array}} \right) = {v_1}i + {v_2}j + {v_3}k\) .

• 15M.1.sl.TZ1.8a: (i)     Show that...
• 15N.1.sl.TZ0.9c: A point \(D\) lies on line \({L_2}\) so that...
• 18M.1.sl.TZ1.6: Six equilateral triangles, each with side length 3 cm, are arranged to form a hexagon.This is...
• 18M.1.sl.TZ1.9a: Show...
• 18M.1.sl.TZ1.9b.i: Find a vector equation for L.
• 18M.1.sl.TZ1.9b.ii: Point C (k , 12 , −k) is on L. Show that k = 14.
• 18M.1.sl.TZ1.9c.i: Find \(\mathop {{\text{OB}}}\limits^ \to  \, \bullet \mathop {{\text{AB}}}\limits^ \to  \).
• 18M.1.sl.TZ1.9c.ii: Write down the value of angle OBA.
• 18M.1.sl.TZ1.9d: Point D is also on L and has coordinates (8, 4, −9). Find the area of triangle OCD.
• 18M.2.sl.TZ2.8a.i: Find \(\mathop {{\text{PQ}}}\limits^ \to  \).
• 18M.2.sl.TZ2.8a.ii: Find \(\left| {\mathop {{\text{PQ}}}\limits^ \to  } \right|\).
• 18M.2.sl.TZ2.8b: Find the angle between PQ and PR.
• 18M.2.sl.TZ2.8c: Find the area of triangle PQR.
• 18M.2.sl.TZ2.8d: Hence or otherwise find the shortest distance from R to the line through P and Q.

Algebraic and geometric approaches to the sum and difference of two vectors; the zero vector, the vector \( - v\).

• 08N.2.sl.TZ0.8b: Find the coordinates of point C.
• 08M.1.sl.TZ2.8a(i) and (ii): Find (i)     \(\overrightarrow {{\rm{AB}}} \) ; (ii)    \(\overrightarrow {{\rm{AD}}} \) giving...
• 08M.1.sl.TZ2.8c: The point C is such that...
• 09M.1.sl.TZ1.9a: Find (i)     \(\overrightarrow {{\rm{PQ}}} \) ; (ii)    \(\overrightarrow {{\rm{PR}}} \) .
• 09M.1.sl.TZ2.10d: The point C is at (2, 1, − 4). Let D be the point such that ABCD is a parallelogram. Find...
• SPNone.1.sl.TZ0.1a: \(\overrightarrow {{\rm{AE}}} \)
• SPNone.1.sl.TZ0.1b: \(\overrightarrow {{\rm{EC}}} \)
• 14M.2.sl.TZ1.4a: Let \(w = u - v\). Represent \(w\) on the diagram above.
• 13M.1.sl.TZ1.1a: Find   (i)     \(2\boldsymbol{a} + \boldsymbol{b}\) ;   (ii)    ...
• 13M.1.sl.TZ1.1b: Find \(\boldsymbol{c}\) .
• 18M.1.sl.TZ1.6: Six equilateral triangles, each with side length 3 cm, are arranged to form a hexagon.This is...
• 18M.1.sl.TZ1.9a: Show...
• 18M.1.sl.TZ1.9b.i: Find a vector equation for L.
• 18M.1.sl.TZ1.9b.ii: Point C (k , 12 , −k) is on L. Show that k = 14.
• 18M.1.sl.TZ1.9c.i: Find \(\mathop {{\text{OB}}}\limits^ \to  \, \bullet \mathop {{\text{AB}}}\limits^ \to  \).
• 18M.1.sl.TZ1.9c.ii: Write down the value of angle OBA.
• 18M.1.sl.TZ1.9d: Point D is also on L and has coordinates (8, 4, −9). Find the area of triangle OCD.
• 18M.2.sl.TZ2.8a.i: Find \(\mathop {{\text{PQ}}}\limits^ \to  \).
• 18M.2.sl.TZ2.8a.ii: Find \(\left| {\mathop {{\text{PQ}}}\limits^ \to  } \right|\).
• 18M.2.sl.TZ2.8b: Find the angle between PQ and PR.
• 18M.2.sl.TZ2.8c: Find the area of triangle PQR.
• 18M.2.sl.TZ2.8d: Hence or otherwise find the shortest distance from R to the line through P and Q.

Algebraic and geometric approaches to multiplication by a scalar, \(kv\) ; parallel vectors.

• 08M.2.sl.TZ1.7: Let  \({\boldsymbol{v}} = 3{\boldsymbol{i}} + 4{\boldsymbol{j}} + {\boldsymbol{k}}\) and...
• SPNone.1.sl.TZ0.1a: \(\overrightarrow {{\rm{AE}}} \)
• SPNone.1.sl.TZ0.1b: \(\overrightarrow {{\rm{EC}}} \)
• 13M.1.sl.TZ1.1a: Find   (i)     \(2\boldsymbol{a} + \boldsymbol{b}\) ;   (ii)    ...
• 13M.1.sl.TZ1.1b: Find \(\boldsymbol{c}\) .
• 18M.1.sl.TZ1.6: Six equilateral triangles, each with side length 3 cm, are arranged to form a hexagon.This is...
• 18M.1.sl.TZ1.9a: Show...
• 18M.1.sl.TZ1.9b.i: Find a vector equation for L.
• 18M.1.sl.TZ1.9b.ii: Point C (k , 12 , −k) is on L. Show that k = 14.
• 18M.1.sl.TZ1.9c.i: Find \(\mathop {{\text{OB}}}\limits^ \to  \, \bullet \mathop {{\text{AB}}}\limits^ \to  \).
• 18M.1.sl.TZ1.9c.ii: Write down the value of angle OBA.
• 18M.1.sl.TZ1.9d: Point D is also on L and has coordinates (8, 4, −9). Find the area of triangle OCD.
• 18M.2.sl.TZ2.8a.i: Find \(\mathop {{\text{PQ}}}\limits^ \to  \).
• 18M.2.sl.TZ2.8a.ii: Find \(\left| {\mathop {{\text{PQ}}}\limits^ \to  } \right|\).
• 18M.2.sl.TZ2.8b: Find the angle between PQ and PR.
• 18M.2.sl.TZ2.8c: Find the area of triangle PQR.
• 18M.2.sl.TZ2.8d: Hence or otherwise find the shortest distance from R to the line through P and Q.

Algebraic and geometric approaches to magnitude of a vector, \(\left| v \right|\) .

• 12N.1.sl.TZ0.9a: Show that...
• 12N.1.sl.TZ0.9b: Let C and D be points such that ABCD is a rectangle. Given that...
• 12N.1.sl.TZ0.9c: Let C and D be points such that ABCD is a rectangle. Find the coordinates of point C.
• 12N.1.sl.TZ0.9d: Let C and D be points such that ABCD is a rectangle. Find the area of rectangle ABCD.
• 10M.1.sl.TZ1.10a: Write down a vector equation for \({L_2}\) in the form...
• 10M.1.sl.TZ1.10b: A third line \({L_3}\) is perpendicular to \({L_1}\) and is represented by...
• 10M.1.sl.TZ1.10c: The lines \({L_1}\) and \({L_3}\) intersect at the point A. Find the coordinates of A.
• 10M.1.sl.TZ1.10d(i) and (ii): The lines \({L_2}\)and \({L_3}\)intersect at point C where...
• 10M.2.sl.TZ2.9a(i) and (ii): (i)     Write down the coordinates of A. (ii)    Find the speed of the airplane in...
• 10M.2.sl.TZ2.9b(i) and (ii): After seven seconds the airplane passes through a point B. (i)     Find the coordinates of...
• 10M.2.sl.TZ2.9c: Airplane 2 passes through a point C. Its position q seconds after it passes through C is given by...
• 11M.1.sl.TZ1.2a: Write down a vector equation for L in the form...
• 11M.1.sl.TZ1.2b(i) and (ii): Find (i)     \(\overrightarrow {{\rm{OP}}} \) ; (ii)    \(|\overrightarrow {{\rm{OP}}} |\) .
• 11M.1.sl.TZ1.9a(i) and (ii): (i)     Write down \(\overrightarrow {{\rm{BA}}} \) . (ii)    Find...
• 11M.1.sl.TZ1.9b(i) and (ii): (i)     Find \(\cos {\rm{A}}\widehat {\rm{B}}{\rm{C}}\) . (ii)    Hence, find...
• 09N.1.sl.TZ0.2b: Let \({\boldsymbol{v}} = \left( {\begin{array}{*{20}{c}}  1 \\   q \\   5 \end{array}} \right)\)...
• 13M.1.sl.TZ1.1a: Find   (i)     \(2\boldsymbol{a} + \boldsymbol{b}\) ;   (ii)    ...
• 13M.1.sl.TZ1.1b: Find \(\boldsymbol{c}\) .
• 15M.1.sl.TZ1.8a: (i)     Show that...
• 15M.1.sl.TZ1.8e: Hence or otherwise, find the area of triangle \(OAB\).
• 15M.2.sl.TZ2.2a: Find (i)     \(u \bullet v\); (ii)     \(\left| {{u}} \right|\); (iii)    ...
• 15N.1.sl.TZ0.9c: A point \(D\) lies on line \({L_2}\) so that...
• 18M.1.sl.TZ1.6: Six equilateral triangles, each with side length 3 cm, are arranged to form a hexagon.This is...
• 18M.1.sl.TZ1.9a: Show...
• 18M.1.sl.TZ1.9b.i: Find a vector equation for L.
• 18M.1.sl.TZ1.9b.ii: Point C (k , 12 , −k) is on L. Show that k = 14.
• 18M.1.sl.TZ1.9c.i: Find \(\mathop {{\text{OB}}}\limits^ \to  \, \bullet \mathop {{\text{AB}}}\limits^ \to  \).
• 18M.1.sl.TZ1.9c.ii: Write down the value of angle OBA.
• 18M.1.sl.TZ1.9d: Point D is also on L and has coordinates (8, 4, −9). Find the area of triangle OCD.
• 18M.2.sl.TZ2.8a.i: Find \(\mathop {{\text{PQ}}}\limits^ \to  \).
• 18M.2.sl.TZ2.8a.ii: Find \(\left| {\mathop {{\text{PQ}}}\limits^ \to  } \right|\).
• 18M.2.sl.TZ2.8b: Find the angle between PQ and PR.
• 18M.2.sl.TZ2.8c: Find the area of triangle PQR.
• 18M.2.sl.TZ2.8d: Hence or otherwise find the shortest distance from R to the line through P and Q.

Algebraic and geometric approaches to unit vectors; base vectors; \(i\), \(j\) and \(k\).

• 08N.1.sl.TZ0.2a(i) and (ii): (i)     Find the velocity vector, \(\overrightarrow {{\rm{AB}}} \) . (ii)    Find the speed of...
• 08N.2.sl.TZ0.8a(i), (ii) and (iii): (i)     Show that...
• 08M.2.sl.TZ1.9a(i) and (ii): (i)     Show that...
• 10M.1.sl.TZ2.2a: Find \(\overrightarrow {{\rm{BC}}} \) .
• 10M.1.sl.TZ2.2b: Find a unit vector in the direction of \(\overrightarrow {{\rm{AB}}} \) .
• 10M.1.sl.TZ2.2c: Show that \(\overrightarrow {{\rm{AB}}} \) is perpendicular to \(\overrightarrow {{\rm{AC}}} \) .
• 09N.2.sl.TZ0.10a: Show...
• 09M.1.sl.TZ1.9a: Find (i)     \(\overrightarrow {{\rm{PQ}}} \) ; (ii)    \(\overrightarrow {{\rm{PR}}} \) .
• 09M.1.sl.TZ2.10c: Let B be the point of intersection of lines \({L_1}\) and \({L_2}\) . (i)     Show that...
• 14M.1.sl.TZ1.8a: Show that \(\overrightarrow {{\text{AB}}}  =...
• 13N.1.sl.TZ0.1a: \(\overrightarrow {{\text{QP}}} \);
• 13N.1.sl.TZ0.1b: \(\overrightarrow {{\text{OT}}} \).
• 13N.2.sl.TZ0.9c: The point \({\text{Q}}(7, 5, 3)\) lies on \({L_1}\). The point \({\text{R}}\) is the reflection...
• 17N.2.sl.TZ0.3a: Find \(\left| {\overrightarrow {{\text{AB}}} } \right|\).
• 17N.2.sl.TZ0.3b: Let ...
• 18M.1.sl.TZ1.6: Six equilateral triangles, each with side length 3 cm, are arranged to form a hexagon.This is...
• 18M.1.sl.TZ1.9a: Show...
• 18M.1.sl.TZ1.9b.i: Find a vector equation for L.
• 18M.1.sl.TZ1.9b.ii: Point C (k , 12 , −k) is on L. Show that k = 14.
• 18M.1.sl.TZ1.9c.i: Find \(\mathop {{\text{OB}}}\limits^ \to  \, \bullet \mathop {{\text{AB}}}\limits^ \to  \).
• 18M.1.sl.TZ1.9c.ii: Write down the value of angle OBA.
• 18M.1.sl.TZ1.9d: Point D is also on L and has coordinates (8, 4, −9). Find the area of triangle OCD.
• 18M.2.sl.TZ2.8a.i: Find \(\mathop {{\text{PQ}}}\limits^ \to  \).
• 18M.2.sl.TZ2.8a.ii: Find \(\left| {\mathop {{\text{PQ}}}\limits^ \to  } \right|\).
• 18M.2.sl.TZ2.8b: Find the angle between PQ and PR.
• 18M.2.sl.TZ2.8c: Find the area of triangle PQR.
• 18M.2.sl.TZ2.8d: Hence or otherwise find the shortest distance from R to the line through P and Q.

Algebraic and geometric approaches to position vectors \(\overrightarrow {OA} = a\) .

• 12N.1.sl.TZ0.9a: Show that...
• 12N.1.sl.TZ0.9b: Let C and D be points such that ABCD is a rectangle. Given that...
• 12N.1.sl.TZ0.9c: Let C and D be points such that ABCD is a rectangle. Find the coordinates of point C.
• 12N.1.sl.TZ0.9d: Let C and D be points such that ABCD is a rectangle. Find the area of rectangle ABCD.
• 12M.2.sl.TZ2.8a(i), (ii) and (iii): (i)     Find \(\overrightarrow {{\rm{OB}}} \) . (ii)    Find \(\overrightarrow {{\rm{OF}}} \)...
• 12M.2.sl.TZ2.8b(i) and (ii): Write down a vector equation for (i)     the line OF; (ii)    the line AG.
• 12M.2.sl.TZ2.8c: Find the obtuse angle between the lines OF and AG.
• 11M.1.sl.TZ1.2a: Write down a vector equation for L in the form...
• 11M.1.sl.TZ1.2b(i) and (ii): Find (i)     \(\overrightarrow {{\rm{OP}}} \) ; (ii)    \(|\overrightarrow {{\rm{OP}}} |\) .
• 15M.1.sl.TZ1.8e: Hence or otherwise, find the area of triangle \(OAB\).
• 18M.1.sl.TZ1.6: Six equilateral triangles, each with side length 3 cm, are arranged to form a hexagon.This is...
• 18M.1.sl.TZ1.9a: Show...
• 18M.1.sl.TZ1.9b.i: Find a vector equation for L.
• 18M.1.sl.TZ1.9b.ii: Point C (k , 12 , −k) is on L. Show that k = 14.
• 18M.1.sl.TZ1.9c.i: Find \(\mathop {{\text{OB}}}\limits^ \to  \, \bullet \mathop {{\text{AB}}}\limits^ \to  \).
• 18M.1.sl.TZ1.9c.ii: Write down the value of angle OBA.
• 18M.1.sl.TZ1.9d: Point D is also on L and has coordinates (8, 4, −9). Find the area of triangle OCD.
• 18M.2.sl.TZ2.8a.i: Find \(\mathop {{\text{PQ}}}\limits^ \to  \).
• 18M.2.sl.TZ2.8a.ii: Find \(\left| {\mathop {{\text{PQ}}}\limits^ \to  } \right|\).
• 18M.2.sl.TZ2.8b: Find the angle between PQ and PR.
• 18M.2.sl.TZ2.8c: Find the area of triangle PQR.
• 18M.2.sl.TZ2.8d: Hence or otherwise find the shortest distance from R to the line through P and Q.

Algebraic and geometric approaches to \(\overrightarrow {AB} = \overrightarrow {OB} - \overrightarrow {OA} = b - a\) .

• 10N.1.sl.TZ0.8a(i), (ii) and (iii): (i)     Show that...
• 10N.1.sl.TZ0.8b(i) and (ii): The line (AC) has equation \({\boldsymbol{r}} = {\boldsymbol{u}} + s{\boldsymbol{v}}\) . (i)    ...
• 10N.1.sl.TZ0.8c: The lines (AC) and (BD) intersect at the point \({\text{P}}(3{\text{, }}k)\) . Show that...
• 10N.1.sl.TZ0.8d: The lines (AC) and (BD) intersect at the point \({\text{P}}(3{\text{, }}k)\) . Hence find the...
• 10M.1.sl.TZ1.10a: Write down a vector equation for \({L_2}\) in the form...
• 10M.1.sl.TZ1.10b: A third line \({L_3}\) is perpendicular to \({L_1}\) and is represented by...
• 10M.1.sl.TZ1.10c: The lines \({L_1}\) and \({L_3}\) intersect at the point A. Find the coordinates of A.
• 10M.1.sl.TZ1.10d(i) and (ii): The lines \({L_2}\)and \({L_3}\)intersect at point C where...
• 10M.1.sl.TZ2.2a: Find \(\overrightarrow {{\rm{BC}}} \) .
• 10M.1.sl.TZ2.2b: Find a unit vector in the direction of \(\overrightarrow {{\rm{AB}}} \) .
• 10M.1.sl.TZ2.2c: Show that \(\overrightarrow {{\rm{AB}}} \) is perpendicular to \(\overrightarrow {{\rm{AC}}} \) .
• 10M.2.sl.TZ2.9a(i) and (ii): (i)     Write down the coordinates of A. (ii)    Find the speed of the airplane in...
• 10M.2.sl.TZ2.9b(i) and (ii): After seven seconds the airplane passes through a point B. (i)     Find the coordinates of...
• 10M.2.sl.TZ2.9c: Airplane 2 passes through a point C. Its position q seconds after it passes through C is given by...
• 11N.1.sl.TZ0.8a(i) and (ii): (i)     Find \(\overrightarrow {{\rm{PQ}}} \) . (ii)    Hence write down a vector equation for...
• 11N.1.sl.TZ0.8b(i) and (ii): (i)     Find the value of p . (ii)    Given that \({L_2}\) passes through...
• 11N.1.sl.TZ0.8c: The lines \({L_1}\) and \({L_2}\) intersect at the point A. Find the x-coordinate of A.
• 11M.1.sl.TZ1.9a(i) and (ii): (i)     Write down \(\overrightarrow {{\rm{BA}}} \) . (ii)    Find...
• 11M.1.sl.TZ1.9b(i) and (ii): (i)     Find \(\cos {\rm{A}}\widehat {\rm{B}}{\rm{C}}\) . (ii)    Hence, find...
• 11M.1.sl.TZ1.9c(i) and (ii): The point D is such that...
• 11M.1.sl.TZ2.3a: Find \(\overrightarrow {{\rm{BC}}} \) .
• 11M.1.sl.TZ2.3b: Show...
• 11M.1.sl.TZ2.3c: Show that vectors \(\overrightarrow {{\rm{BD}}} \) and \(\overrightarrow {{\rm{AC}}} \) are...
• 11M.2.sl.TZ2.8a: Find \(\overrightarrow {{\rm{AB}}} \) .
• 11M.2.sl.TZ2.8b: Find an equation for \({L_1}\) in the form...
• 11M.2.sl.TZ2.8c: Find the angle between \({L_1}\) and \({L_2}\) .
• 11M.2.sl.TZ2.8d: The lines \({L_1}\) and \({L_2}\) intersect at point C. Find the coordinates of C.
• 13M.1.sl.TZ1.8a.i: Find \(\overrightarrow {{\rm{AB}}} \) .
• 13M.2.sl.TZ2.8a: Find   (i)     \(\overrightarrow {{\rm{AB}}} \) ;   (ii)     \(\overrightarrow {{\rm{AC}}} \) .
• 18M.1.sl.TZ1.6: Six equilateral triangles, each with side length 3 cm, are arranged to form a hexagon.This is...
• 18M.1.sl.TZ1.9a: Show...
• 18M.1.sl.TZ1.9b.i: Find a vector equation for L.
• 18M.1.sl.TZ1.9b.ii: Point C (k , 12 , −k) is on L. Show that k = 14.
• 18M.1.sl.TZ1.9c.i: Find \(\mathop {{\text{OB}}}\limits^ \to  \, \bullet \mathop {{\text{AB}}}\limits^ \to  \).
• 18M.1.sl.TZ1.9c.ii: Write down the value of angle OBA.
• 18M.1.sl.TZ1.9d: Point D is also on L and has coordinates (8, 4, −9). Find the area of triangle OCD.
• 18M.2.sl.TZ2.8a.i: Find \(\mathop {{\text{PQ}}}\limits^ \to  \).
• 18M.2.sl.TZ2.8a.ii: Find \(\left| {\mathop {{\text{PQ}}}\limits^ \to  } \right|\).
• 18M.2.sl.TZ2.8b: Find the angle between PQ and PR.
• 18M.2.sl.TZ2.8c: Find the area of triangle PQR.
• 18M.2.sl.TZ2.8d: Hence or otherwise find the shortest distance from R to the line through P and Q.