A+ Level Question
Solutions to Mathematical Problems
Page 1: Solutions to Second Degree Equations
Overview of methods for solving second degree equations.
Page 2: Rectangle Problem
Perimeter: 24 cm
Area: 35 cm²
a) Length + breadth = 12 cm
b) If length = x cm, breadth = 12-x cm
c) Form second degree equation: x(12-x) = 35
Page 3: Finding Lengths
Given equations to solve for dimensions of rectangle:
x² - 12x + 35 = 0
Solutions yield dimensions of the rectangle.
Page 4: Intersecting Chords
Chords AB and CD intersect at P:
AB = 13 cm, CD = 15 cm, PD = 3 cm
a) Length of PC = 12 cm
b) PB = x, PA = 13-x
Page 5: Length Calculations
Uses equations from intersecting chords:
Solve for PB using specified relationships and values.
Page 6: Arithmetic Sequence
Sequence: 6, 10, 14...
a) Common difference = 4
b) Number of terms to sum to 510:
Use formula for the sum of an arithmetic series.
Page 7: Trigonometry Solutions
Types of trigonometric problems addressed.
Page 8: Right Triangle Problems
Given angle measures to find unknown angles and relationships:
tan 44°, tan 46° relationship to prove.
Page 9: Triangle ABC
Given dimensions <A=50°, <B=65°:
a) Determine <C.
b) Determine special properties (isosceles).
c) Find perpendicular distance from C to AB.
Page 10: Area Calculations
Formulas to calculate area and dimensions of triangle ABC.
Page 11: Circumcircle of a Triangle
Find diameters and sides using given angles and lengths.
Page 12: Height and Distance from Tower
Given right triangle details to find height and distance:
a) Height of tower calculated using angles of elevation and depression.
Page 13: Similar Discussions in Malayalam
Problems related to height and angles of vision to a tower.
Page 14: Coordinates Solutions
Introduction to coordinate geometry.
Page 15: Square and Coordinates
Square ABCD coordinates determined by given angles and distances.
Page 16: Semicircle Problems
Using diameters and perpendiculars to find lengths and coordinates.
Page 17: Equilateral Triangle
Prove characteristics of triangle OAB with certain coordinates.
Page 18: Circle and Points
Radius calculation and checks for points on a circle.
Page 19: Circle Properties
Establishing properties of points related to the circle.
Page 20: Circle Geometry
Determine properties and equations related to circles.
Page 21: Center and Coordinates
Questions related to coordinates concerning circles.
Page 22: Further Point Relations
Information on points related to circles and their coordinates.
Page 23: Tangent Lines
Concepts of tangent lines and intersections with circles.
Page 24: Angle Calculations
Given tangents and points to find various angles in circle geometry.
Page 25: More Tangent Problems
Use tangent properties to calculate relevant angles.
Page 26: Tangent Lengths
Calculate lengths based on tangent properties from a point.
Page 27: Tangent and Circle Relations
Find lengths and radius based on tangents and circle parameters.
Page 28: Solid Geometry
Introduction to volume problems related to solids.
Page 29: Square Pyramid Calculations
Dimensions and cost calculations for a toy in the pyramid shape.
Page 30: Surface Area
Calculating surface area and total costs based on pyramidal geometry.
Page 31: Length and Volume of Pyramid
Methods to determine dimensions and volumes based on given area.
Page 32: Surface Area Calculations
Calculations related to the surface area of various solids.
Page 33: Pyramid Dimensions
Problem-solving for dimensions and surface area of a square pyramid.
Page 34: Sector to Cone Conversion
Calculate dimensions and area when forming a cone from a sector.
Page 35: Cone Dimensions
Find slant height and surface area for cones with given dimensions.
Page 36: Repeated Cone Calculations
Similar calculations for identifying the dimensions of cones.
Page 37: Composite Solid
Properties and volumes for solids comprising a hemisphere, cylinder, and cone.
Page 38: Volume Calculations
Extend calculations of dimensions to volume for composite solids.
Page 39: Geometry and Algebra Solutions
Mixed problem definitions involving coordinates and geometry.
Page 40: Parallelogram Coordinates
Identifying vertices and intersection points for a parallelogram.
Page 41: Division of Line Segments
Finding coordinates of points dividing line segments in given ratios.
Page 42: Calculating Ratios
Methods for calculating ratios within geometric contexts.