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.