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Q1. Right Circular Cylinder - Volume & Surface Area Q2. Right Triangle - Pythagoras & Trigonometry Q3. Sector - Arc Length & Area Q4. Cosine Rule in Triangle Q5. Exact Trigonometric Values Q6. Period & Amplitude of Sine/Cosine Q7. Vector Operations Q8. Pyramid - Slant Height, Surface Area, Volume Q9. Triangle - Cosine Rule & Area Q10. Voronoi Diagram - Perpendicular Bisectors & Vertex Q11. Lines in 3D - Angle & Skew Lines Q12. Graph Theory - Degree, Eulerian, Hamiltonian, MST (HL) Q13. Cone - Height, Angle, Volume, Similarity Q14. Angle of Elevation - Two Positions Q15. Unit Circle & Trigonometric Equations Q16. Vector Cross Product & Parallelogram Area Q17. 3D Coordinates - Vectors, Angle, Triangle Area Q18. Composite 3D Shape - Cylinder & Cone Q19. Voronoi Diagram - Edge, Nearest Site, Vertex, New Site Q20. Trigonometric Functions - Intersections & Area Q21. Lines in 3D - Cross Product, Distance, Plane (HL) Q22. Triangular Prism - Space Diagonal, Angle, Surface Area Q23. Graph Theory - MST, Dijkstra, Eulerian, Hamiltonian (HL) Q24. Quadrilateral - Cosine Rule & Area Q25. Geometric Transformations - Rotation & Reflection Matrices (HL)Practice questions covering cylinders, cones, pyramids, triangles (sine rule, cosine rule, area), sectors, vectors (dot product, cross product, lines, planes, angles), Voronoi diagrams, graph theory (Eulerian, Hamiltonian, MST, Dijkstra), 3D coordinates, trigonometric equations, geometric transformations, and more.
A right circular cylinder has radius 4 cm and height 9 cm. Find:
(a) Volume
(b) Total surface area.
Answer: (a) V=144π≈452 cm³
(b) SA=104π≈327 cm²
In triangle ABC, angle A=90°, AB=5cm, AC=12cm. Find BC, angle B, and angle C.
Answer: BC=13 cm; B≈67.4°; C≈22.6°
A sector has radius 8 cm and angle θ = 1.2 radians. Find:
(a) arc length
(b) area of sector.
Answer: (a) 9.6 cm
(b) 38.4 cm²
In triangle PQR, PQ=7, QR=10, angle Q=48°. Find PR using the cosine rule.
Answer: PR ≈ 7.42
Find the exact values of sin(60°), cos(30°), and tan(45°) using the unit circle.
Answer: sin(60°)=√3/2; cos(30°)=√3/2; tan(45°)=1
State the period and amplitude of:
(a) y=3sin(4x)
(b) y=−2cos(x/3).
Answer: (a) Period=π/2, amp=3
(b) Period=6π, amp=2
Vector a=[3,−1,2] and b=[−2,4,1]. Find:
(a) a+b
(b) |a|, (c) a·b.
Answer: (a) [1,3,3]
(b) √14≈3.74 (c) −8
A pyramid has a square base of side 6m and vertical height 8m. Find:
(a) slant height
(b) total surface area
(c) volume.
Answer: (a) ≈8.54 m
(b) ≈138.5 m²
(c) 96 m³
In triangle ABC: a=9, b=12, c=7. Find:
(a) angle A
(b) area of triangle.
Answer: (a) A≈48.2°
(b) Area≈31.4 sq units
A Voronoi diagram has sites at A(0,0), B(4,0), C(2,4).
(a) Find the perpendicular bisector of AB.
(b) Find the perpendicular bisector of AC.
(c) Find the circumcentre (Voronoi vertex).
(d) Explain what the Voronoi cell of A represents.
Answer: (a) x=2
(b) y=−x/2+5/2
(c) (2, 1.5)
(d) Points nearer to A than any other site
Vector r = [2,−1,3] + λ[1,2,−1] is a line. Another line s = [0,3,1] + μ[2,1,1].
(a) Find the angle between the direction vectors.
(b) Show the lines do not intersect (are skew).
Answer: (a) θ=60°
(b) System is inconsistent → lines are skew
Graph theory: A graph G has vertices {A,B,C,D,E} and edges {AB,AC,BC,CD,DE,AE}.
(a) State the degree of each vertex.
(b) Is G Eulerian?
(c) Find a Hamiltonian path if one exists.
(d) Find the minimum spanning tree.
Answer: (a) A:3,B:2,C:3,D:2,E:2
(b) Not Eulerian (two odd-degree vertices)
(c) B→A→C→D→E
(d) Any spanning tree
A cone has base radius 5cm and slant height 13cm.
(a) Find the height.
(b) Find the half-angle at the apex.
(c) Find the volume.
(d) A smaller similar cone has volume 1/8 of the original. Find its dimensions.
Answer: (a) 12 cm
(b) ≈22.6°
(c) 100π≈314 cm³
(d) r=2.5cm, h=6cm
From a boat, the angle of elevation to the top of a cliff is 28°. After moving 50m closer, the angle is 38°. Find the height of the cliff to the nearest metre.
Answer: h ≈ 116 m
The unit circle: P is a point at angle 5π/6.
(a) Write the exact coordinates of P.
(b) Hence write exact values of sin(5π/6), cos(5π/6), tan(5π/6).
(c) Solve sin(2θ)=cos(θ) for 0≤θ<2π.
Answer: (a) (−√3/2, 1/2)
(b) sin=1/2, cos=−√3/2, tan=−√3/3
(c) θ=π/6, π/2, 5π/6, 3π/2
Two vectors: u=[2,3,−1] and v=[1,−2,4].
(a) Find u×v (vector product).
(b) Verify u×v is perpendicular to both u and v.
(c) Find the area of the parallelogram formed by u and v.
Answer: (a) [10,−9,−7]
(b) Verified
(c) √230≈15.2 sq units
In a 3D room, point A is at (1,0,0), B at (4,3,0), and C at (2,1,5) (all in metres).
(a) Find AB and AC
(b) Find the angle BAC
(c) Find the area of triangle ABC.
Answer: (a) AB=[3,3,0], AC=[1,1,5]
(b) ≈73.2°
(c) ≈10.6 m²
The diagram shows a compound 3D shape: a cylinder of radius r=2cm, height=3cm, topped by a cone of same base radius and height=2cm.
(a) Find the total volume
(b) Find the total surface area (excluding where they join)
(c) Find the slant height of the cone
(d) If the entire shape is scaled by factor k, find the value of k such that the total volume equals 5× the original.
Answer: (a) 44π/3≈46.1cm³
(b) π(16+4√2)≈64.5cm²
(c) 2√2≈2.83cm
(d) k=∛5≈1.71
The Voronoi diagram shows 5 sites: A(1,1), B(4,2), C(2,5), D(5,5), E(3,3.5).
(a) Find the equation of the Voronoi edge between sites A and B
(b) Determine which site is nearest to the point P(3.5, 4)
(c) A toxic waste dump must be placed at the point equidistant from A, B, and E (Voronoi vertex). Find its coordinates.
(d) A new site F(6,1) is added. Describe how the Voronoi diagram changes.
Answer: (a) y=−3x+9
(b) Site E
(c) Circumcentre via GDC
(d) New cell for F appears
The graph shows f(x)=4cos(x/2+π/6) (blue) and g(x)=2sin(x)−1 (red) with intersection points marked ★.
(a) Find the amplitude, period, and phase shift of f
(b) Find all x ∈ [0,4π] where f(x)=g(x) to 3 s.f.
(c) Find the area enclosed between f and g between the first two intersections
(d) Show that g can be written in the form R·sin(x+φ)−1 and find R and φ.
Answer: (a) Amp=4, Period=4π, Phase=π/3 to the right
(b) ≈1.05,3.50,7.33,9.78
(c) Via GDC (d) R=2, φ=0
Vector product applications: Line L₁: r=[1,2,3]+λ[2,1,−1] and Line L₂: r=[3,1,0]+μ[1,3,2].
(a) Find d₁×d₂ (normal to both)
(b) Find the shortest distance between the lines
(c) Plane π contains L₁ and is parallel to L₂. Find the equation of π
(d) Find the angle between π and the xy-plane.
Answer: (a) [1,−1,1]
(b) Lines intersect (dist=0)
(c) Plane via point+normal
(d) Compute via GDC
A triangular prism has vertices: A(0,0,0), B(4,0,0), C(2,3,0), D(0,0,5), E(4,0,5), F(2,3,5).
(a) Find the length of the space diagonal AE⃗
(b) Find angle DAF (between face AD and diagonal AF)
(c) Find the surface area of the prism
(d) A bug travels from A along the surface to F by the shortest path. Find this distance.
Answer: (a) √41≈6.40
(b) ≈35.7°
(c) 72 sq units
(d) 10 units (by unfolding net)
Graph Theory: Weighted graph with vertices A,B,C,D,E and edges: AB=3, AC=5, BC=2, BD=6, BE=4, CD=7, DE=3, CE=8.
(a) Find the minimum spanning tree using Kruskal's algorithm. State total weight
(b) Starting from A, apply Dijkstra's algorithm to find shortest path to E
(c) Does an Eulerian circuit exist? Justify
(d) Find a Hamiltonian cycle if possible.
Answer: (a) MST: BC+AB+DE+BE=12
(b) Shortest A→E=7 (via B)
(c) No (C,E odd degree)
(d) ABCDE weight=23
Sine and cosine rule combined: In a quadrilateral ABCD (vertices in order): AB=6cm, BC=8cm, CD=5cm, DA=7cm, diagonal AC=9cm.
(a) Find angle ABC using cosine rule in triangle ABC
(b) Find the area of triangle ABC
(c) Find angle ADC using cosine rule in triangle ACD
(d) Find the total area of quadrilateral ABCD
(e) Find angle BAD if BD=10.2cm using cosine rule in ABD.
Answer: (a) ≈78.6°
(b) ≈23.5cm²
(c) ≈95.7°
(d) ≈40.9cm²
(e) ≈103.1°
Matrix M=[[0,−1],[1,0]] represents a rotation.
(a) Show M represents a 90° anticlockwise rotation
(b) Apply M to the triangle with vertices (1,0),(3,0),(2,2) and state the new vertices
(c) Find M⁴ and interpret geometrically
(d) Matrix N=[[0,1],[1,0]] represents a reflection. Find MN and NM and show they represent different transformations.
Answer: (b) (0,1),(0,3),(−2,2)
(c) M⁴=I (identity/360°)
(d) MN≠NM (y-axis vs x-axis reflection)