Prep 2 - Mathematics
Unit 2: Linear Functions, Slope & Real Numbers Operations
Unit 2 Capstone Review: Radicals, Conjugates, Solid Geometry, Linear Equations, and Slopes
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Congratulations on completing Unit 2! In this comprehensive capstone review, we integrate all the essential concepts and computational methods mastered across the unit. We review: the algebraic laws of square and cube roots, simplification by extracting perfect powers, rationalizing binomial denominators using conjugate numbers, geometric applications on cubes, spheres, and cylinders, graphing linear equations ax + by = c using ordered pairs, determining intercepts with the coordinate axes, computing and interpreting slopes, verifying collinearity of points, and solving real-life rate-of-change word problems. This thorough synthesis ensures complete mathematical mastery for the standardized digital exams.
Radicals & Conjugates
- √(ab) = √a × √b | ³√(ab) = ³√a × ³√b.
- Conjugates: (√a + √b)(√a - √b) = a - b.
- Cube: V = s³, Area = 6s².
- Sphere: V = 4/3 π r³, Area = 4 π r².
- Cylinder: V = π r² h, Lat. Area = 2 π r h.
Linear Relations
ax + by = crepresents a straight line.- x-intercept: set y = 0 → (c/a, 0).
- y-intercept: set x = 0 → (0, c/b).
- x-axis: y = 0 | y-axis: x = 0.
- Passes through origin if c = 0.
Slope Matrix
m = (y&sub2; - y&sub1;) / (x&sub2; - x&sub1;).- Parallel lines: m&sub1; = m&sub2;.
- Perpendicular lines: m&sub1; × m&sub2; = -1.
- Horizontal: m = 0 | Vertical: undefined.
- Distance-time slope = velocity.
Comprehensive Diagnostic Problems:
Q1: Find the value of x that makes the line passing through (x, 3) and (2, 7) have a slope of 2.
Solution:
m = (7 - 3) / (2 - x) = 4 / (2 - x) = 2 → 2(2 - x) = 4 → 2 - x = 2 → x = 0.
Solution:
m = (7 - 3) / (2 - x) = 4 / (2 - x) = 2 → 2(2 - x) = 4 → 2 - x = 2 → x = 0.
Q2: If the volume of a sphere is equal to the volume of a right circular cylinder of radius 3 cm and height 4 cm, find the radius of the sphere.
Solution:
1. Volume of cylinder = π r² h = π (3)² (4) = 36π cm³.
2. Volume of sphere = 4/3 π R³ = 36π → R³ = 36 × (3/4) = 27.
3. Radius R = ³√27 = 3 cm.
Solution:
1. Volume of cylinder = π r² h = π (3)² (4) = 36π cm³.
2. Volume of sphere = 4/3 π R³ = 36π → R³ = 36 × (3/4) = 27.
3. Radius R = ³√27 = 3 cm.
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synthesis
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تكامل المفاهيم
📖 Def: combining separate algebraic and geometric ideas into a unified system
diagnostic
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تشخيصي وتقييمي
📖 Def: identifying areas of strength and topics needing further practice
gradient
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معامل الانحدار والميل
📖 Def: the measure of steepness and rate of change of a line
perpendicular
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متعامد وبزاوية قائمة
📖 Def: intersecting at a 90° right angle
parallel
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متوازٍ ولا يتقاطع
📖 Def: extending in the same direction, equidistant at all points
rationalization
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إنطاق المقام وتبسيطه
📖 Def: the algebraic procedure of converting an irrational denominator into a rational one
geometric solid
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المجسم الفضائي
📖 Def: a three-dimensional figure with volume and surface area
ordered pair
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الزوج المرتب الإحداثي
📖 Def: a point (x, y) on the coordinate plane
rate of change
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معدل التغير المنتظم
📖 Def: the ratio between changes in two related variables
linear function
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الدالة الخطية
📖 Def: a function whose graph forms a non-vertical straight line
intercept
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نقطة التقاطع المحورية
📖 Def: the point where a graph intersects a coordinate axis
collinearity
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الوقوع على استقامة واحدة
📖 Def: the condition where three or more points lie on the same line
Click on an option to check your answer and view the detailed explanation:
1
What is the simplest radical form of √50 - √18?
2
If the slope of the line passing through (2, 5) and (4, k) is 3, what is the value of k?
3
What is the total surface area of a cube whose edge length is 4 cm?
4
What is the slope of the line perpendicular to a line with slope -1/3?
5
Find the point of intersection of the line 5x - 2y = 10 with the x-axis:
6
What is the product of the conjugate pair (√11 - √2)(√11 + √2)?
7
What is the volume of a sphere whose radius is 3 cm in terms of π?
8
The equation of the line passing through (0, 0) with slope 4 is:
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