Mathematics AA: Standard Level
Geometry and trigonometry — Topic 3
- 1.
Convert 150° to radians, giving your answer as a multiple of π.
[2 marks] · no calculator - 2.
Marking analysis: A learner attempts the following task: “Convert 150° to radians, giving your answer as a multiple of π.” Their response addresses only this point: “Uses the conversion 180° = π radians.” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks] · no calculator - 3.
Use exact values to evaluate sin(π/3) + cos(π/6), giving your answer in simplest surd form.
[3 marks] · no calculator - 4.
Marking analysis: A learner attempts the following task: “Use exact values to evaluate sin(π/3) + cos(π/6), giving your answer in simplest surd form.” Their response addresses only this point: “States sin(π/3) = √3/2.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] · no calculator - 5.
Given that sin θ = 3/5 and θ is obtuse, find the exact value of cos θ.
[4 marks] · no calculator - 6.
Marking analysis: A learner attempts the following task: “Given that sin θ = 3/5 and θ is obtuse, find the exact value of cos θ.” Their response addresses only this point: “Uses the Pythagorean identity sin²θ + cos²θ = 1.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] · no calculator - 7.
Solve the equation 2 sin x − 1 = 0 for 0 ≤ x ≤ 2π, giving all solutions exactly.
[4 marks] · no calculator - 8.
Marking analysis: A learner attempts the following task: “Solve the equation 2 sin x − 1 = 0 for 0 ≤ x ≤ 2π, giving all solutions exactly.” Their response addresses only this point: “Rearranges to sin x = 1/2.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] · no calculator - 9.
In triangle ABC, angle A = 40°, side a = 12 cm and side b = 15 cm. Find angle B, giving your answer to one decimal place.
[4 marks] - 10.
Marking analysis: A learner attempts the following task: “In triangle ABC, angle A = 40°, side a = 12 cm and side b = 15 cm. Find angle B, giving your answer to one decimal place.” Their response addresses only this point: “Uses the sine rule sin A/a = sin B/b.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 11.
In triangle PQR, PQ = 7 cm, QR = 9 cm and angle PQR = 65°. Find the length of PR.
[4 marks] - 12.
Marking analysis: A learner attempts the following task: “In triangle PQR, PQ = 7 cm, QR = 9 cm and angle PQR = 65°. Find the length of PR.” Their response addresses only this point: “Uses the cosine rule PR² = PQ² + QR² − 2(PQ)(QR)cos(PQR).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 13.
Triangle XYZ has XY = 8 cm, XZ = 10 cm and angle YXZ = 55°. Find the area of the triangle.
[3 marks] - 14.
Marking analysis: A learner attempts the following task: “Triangle XYZ has XY = 8 cm, XZ = 10 cm and angle YXZ = 55°. Find the area of the triangle.” Their response addresses only this point: “Uses the area formula Area = ½ab sin C with the given sides and included angle.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 15.
The function f(x) = 3 sin(2x) + 1 models a periodic quantity. State the amplitude, period and vertical shift of f.
[3 marks] · no calculator - 16.
Marking analysis: A learner attempts the following task: “The function f(x) = 3 sin(2x) + 1 models a periodic quantity. State the amplitude, period and vertical shift of f.” Their response addresses only this point: “States the amplitude as 3.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] · no calculator - 17.
Prove the identity (1 − cos²x)/sin x = sin x for sin x ≠ 0.
[3 marks] · no calculator - 18.
Marking analysis: A learner attempts the following task: “Prove the identity (1 − cos²x)/sin x = sin x for sin x ≠ 0.” Their response addresses only this point: “Uses the Pythagorean identity to rewrite the numerator: 1 − cos²x = sin²x.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] · no calculator - 19.
A vertical cliff is observed from two points A and B on level ground, with B further from the cliff than A. AB = 50 m, the angle of elevation of the top of the cliff from A is 42° and from B is 27°. Find the height of the cliff.
[6 marks] - 20.
Marking analysis: A learner attempts the following task: “A vertical cliff is observed from two points A and B on level ground, with B further from the cliff than A. AB = 50 m, the angle of elevation of the top of the cliff from A is 42° and from B is 27°. Find the height of the cliff.” Their response addresses only this point: “Sets up a triangle with the cliff top, A and B, identifying the angle at the cliff top using angle properties.” Evaluate the response against the complete 6-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[6 marks] - 21.
Points A(1, 2, 0) and B(4, −2, 3) are given. Find the vector AB and hence determine the distance |AB|.
[4 marks] - 22.
Marking analysis: A learner attempts the following task: “Points A(1, 2, 0) and B(4, −2, 3) are given. Find the vector AB and hence determine the distance |AB|.” Their response addresses only this point: “Uses AB = OB − OA.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 23.
Vectors a = 2i − j + 2k and b = i + 2j − 2k are given. Calculate a · b and hence find the angle between a and b.
[6 marks] - 24.
Marking analysis: A learner attempts the following task: “Vectors a = 2i − j + 2k and b = i + 2j − 2k are given. Calculate a · b and hence find the angle between a and b.” Their response addresses only this point: “Uses a · b = (2)(1) + (−1)(2) + (2)(−2).” Evaluate the response against the complete 6-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[6 marks] - 25.
A line passes through the point (1, 0, −2) and is parallel to the vector (2, 1, 3). Write a vector equation for the line and state the coordinates of the point on the line when the parameter equals 2.
[3 marks] · no calculator - 26.
Marking analysis: A learner attempts the following task: “A line passes through the point (1, 0, −2) and is parallel to the vector (2, 1, 3). Write a vector equation for the line and state the coordinates of the point on the line when the parameter equals 2.” Their response addresses only this point: “Writes r = (1, 0, −2) + t(2, 1, 3).” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] · no calculator - 27.
Find a unit vector in the same direction as v = (3, −4, 12).
[3 marks] - 28.
Marking analysis: A learner attempts the following task: “Find a unit vector in the same direction as v = (3, −4, 12).” Their response addresses only this point: “Calculates |v| = √(3² + (−4)² + 12²).” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 29.
Points A(2, 1, −1), B(4, 0, 2) and C(3, 3, 0) form a triangle. Find the vectors AB and AC, and hence calculate the area of triangle ABC using the cross product.
[6 marks] - 30.
Marking analysis: A learner attempts the following task: “Points A(2, 1, −1), B(4, 0, 2) and C(3, 3, 0) form a triangle. Find the vectors AB and AC, and hence calculate the area of triangle ABC using the cross product.” Their response addresses only this point: “Obtains AB = (2, −1, 3).” Evaluate the response against the complete 6-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[6 marks] - 31.
Two lines have equations r = (1, 2, 0) + s(1, −1, 2) and r = (3, 0, 4) + t(2, 1, −1). Show that the lines intersect and find the point of intersection.
[6 marks] - 32.
Marking analysis: A learner attempts the following task: “Two lines have equations r = (1, 2, 0) + s(1, −1, 2) and r = (3, 0, 4) + t(2, 1, −1). Show that the lines intersect and find the point of intersection.” Their response addresses only this point: “Equates corresponding components: 1 + s = 3 + 2t, 2 − s = t, 2s = 4 − t.” Evaluate the response against the complete 6-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[6 marks] - 33.
Find the angle that the vector d = (1, 1, 0) makes with the positive x-axis.
[4 marks] - 34.
Marking analysis: A learner attempts the following task: “Find the angle that the vector d = (1, 1, 0) makes with the positive x-axis.” Their response addresses only this point: “Identifies the x-axis direction vector as i = (1, 0, 0).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 35.
The vector equation of a line is r = (2, −1, 3) + t(1, 2, −2). Determine whether the point (5, 5, −3) lies on this line.
[4 marks] · no calculator - 36.
Marking analysis: A learner attempts the following task: “The vector equation of a line is r = (2, −1, 3) + t(1, 2, −2). Determine whether the point (5, 5, −3) lies on this line.” Their response addresses only this point: “Sets up three equations: 2 + t = 5, −1 + 2t = 5, 3 − 2t = −3.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] · no calculator - 37.
Vectors p and q are such that |p| = 5, |q| = 3 and the angle between them is 60°. Calculate |p + q|.
[5 marks] - 38.
Marking analysis: A learner attempts the following task: “Vectors p and q are such that |p| = 5, |q| = 3 and the angle between them is 60°. Calculate |p + q|.” Their response addresses only this point: “Uses |p + q|² = |p|² + |q|² + 2|p||q|cos θ.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[5 marks] - 39.
A plane contains the point (1, 0, 2) and has normal vector n = (2, −1, 3). Write the Cartesian equation of the plane.
[3 marks] · no calculator - 40.
Marking analysis: A learner attempts the following task: “A plane contains the point (1, 0, 2) and has normal vector n = (2, −1, 3). Write the Cartesian equation of the plane.” Their response addresses only this point: “Uses the form n · (r − r₀) = 0 with the given point and normal.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] · no calculator