Comprehensive Mathematics Review and Problem Set Analysis

Surds and Radical Expressions

• Simplifying Basic Surds:

  • To simplify $\sqrt{72} + \sqrt{50} - \sqrt{8}$

    • $\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}$

    • $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$

    • $\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$

    • Final calculation: $6\sqrt{2} + 5\sqrt{2} - 2\sqrt{2} = 9\sqrt{2}$

  • To simplify $\sqrt{48} - 2\sqrt{12} + \sqrt{75}$

    • $\sqrt{48} = 4\sqrt{3}$

    • $2\sqrt{12} = 2(2\sqrt{3}) = 4\sqrt{3}$

    • $\sqrt{75} = 5\sqrt{3}$

    • Result: $4\sqrt{3} - 4\sqrt{3} + 5\sqrt{3} = 5\sqrt{3}$

  • To simplify $\sqrt{200} - \sqrt{72}$

    • $\sqrt{200} = 10\sqrt{2}$

    • $\sqrt{72} = 6\sqrt{2}$

    • Result: $4\sqrt{2}$

• Expansion and Rationalization:

  • Difference of Squares: $(\sqrt{11} + \sqrt{3})(\sqrt{11} - \sqrt{3})$ simplifies to $(\sqrt{11})^2 - (\sqrt{3})^2 = 11 - 3 = 8$.

  • Rationalizing the Denominator:

    • Expression: $\frac{8}{\sqrt{3} + 1}$

    • Method: Multiply numerator and denominator by the conjugate $(\sqrt{3} - 1)$.

    • Calculation: $\frac{8(\sqrt{3}-1)}{(\sqrt{3})^2 - 1^2} = \frac{8(\sqrt{3}-1)}{3-1} = \frac{8(\sqrt{3}-1)}{2} = 4(\sqrt{3}-1)$

  • Simple Rationalization: $\frac{6}{\sqrt{2}}$ becomes $\frac{6\sqrt{2}}{2} = 3\sqrt{2}$.

• Operations with Radicals:

  • Division: $\frac{\sqrt{45} + \sqrt{20}}{\sqrt{5}}$ is calculated as $\frac{3\sqrt{5} + 2\sqrt{5}}{\sqrt{5}} = \frac{5\sqrt{5}}{\sqrt{5}} = 5$.

  • Division of Quotients: $\frac{\sqrt{98}}{\sqrt{2}} = \sqrt{\frac{98}{2}} = \sqrt{49} = 7$.

  • Multiplication: $\sqrt{18} \times \sqrt{8} = \sqrt{144} = 12$.

  • Complex Expressions: $(2\sqrt{5})^2 - \sqrt{80} = 4(5) - 4\sqrt{5} = 20 - 4\sqrt{5}$.

  • Mixed Fractions: $\frac{\sqrt{108}}{3} + \sqrt{12} = \frac{6\sqrt{3}}{3} + 2\sqrt{3} = 2\sqrt{3} + 2\sqrt{3} = 4\sqrt{3}$.

Algebra

• Factorization Techniques:

  • Difference of Perfect Squares: $81x^2 - 49y^2 = (9x - 7y)(9x + 7y)$.

  • Quadratic Factorization (Monic): $x^2 - 4x - 45 = (x - 9)(x + 5)$.

  • Quadratic Factorization (Non-monic): $10x^2 - 13x - 3 = (5x + 1)(2x - 3)$.

  • Grouping Terms: $ab + 6a - 5b - 30 = a(b + 6) - 5(b + 6) = (a - 5)(b + 6)$.

• Expanding and Solving Equations:

  • Perfect Squares: $(3x - 4)^2 = 9x^2 - 24x + 16$.

  • Solving Linear Equalities: $\frac{5x + 8}{6} = 7 \rightarrow 5x + 8 = 42 \rightarrow 5x = 34 \rightarrow x = \frac{34}{5}$.

  • Radical Equations:

    • $\sqrt{2x + 1} = x - 1 \rightarrow 2x + 1 = (x - 1)^2 \rightarrow 2x + 1 = x^2 - 2x + 1 \rightarrow x^2 - 4x = 0$. Solutions are $x=0$ or $x=4$. (Note: Check for extraneous solutions; $x=4$ is valid, $x=0$ results in $1 = -1$, which is invalid).

    • $\sqrt{x} + \sqrt{9x} = 16 \rightarrow \sqrt{x} + 3\sqrt{x} = 16 \rightarrow 4\sqrt{x} = 16 \rightarrow \sqrt{x} = 4 \rightarrow x = 16$.

    • $\sqrt{x - 3} = x - 5 \rightarrow x - 3 = (x - 5)^2 \rightarrow x - 3 = x^2 - 10x + 25 \rightarrow x^2 - 11x + 28 = 0 \rightarrow (x - 7)(x - 4) = 0$. Valid solution is $x=7$.

• Simultaneous Equations and Intersections:

  • System: $4x + y = 29$ and $2x - 3y = -1$.

    • To find $y$, eliminate $x$ by multiplying the second equation by 2: $4x - 6y = -2$.

    • Subtract from first: $(4x + y) - (4x - 6y) = 29 - (-2) \rightarrow 7y = 31 \rightarrow y = \frac{31}{7}$.

  • Intersection of a Parabola and Line: $y = x^2$ and $y = 2x + 8$.

    • Set equations equal: $x^2 = 2x + 8 \rightarrow x^2 - 2x - 8 = 0 \rightarrow (x - 4)(x + 2) = 0$.

    • Coordinates: $(4, 16)$ and $(-2, 4)$.

• Inequalities:

  • Solve: $14 - 3x < -22$:

    • $-3x < -36$

    • Dividing by a negative number flips the inequality sign: $x > 12$.

• Indices (Exponents):

  • Simplify: $\frac{5^8 \times 5^{-3}}{5^2} = \frac{5^5}{5^2} = 5^3$.

  • Product rule: $4a^{-2}b^5 \times 3a^6b^{-1} = 12a^4b^4$.

  • Fractional exponents: $125^{\frac{2}{3}} = (\sqrt[3]{125})^2 = 5^2 = 25$.

Coordinate Geometry

• Gradient (Slope):

  • For points $(x_1, y_1)$ and $(x_2, y_2)$, $m = \frac{y_2 - y_1}{x_2 - x_1}$.

  • Example: Passing through $(-4, 9)$ and $(8, -6)$, $m = \frac{-6 - 9}{8 - (-4)} = \frac{-15}{12} = -\frac{5}{4}$.

• Midpoint:

  • Formula: $(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$.

  • Example: Midpoint of $(-12, 7)$ and $(4, -9)$ is $(\frac{-12 + 4}{2}, \frac{7 - 9}{2}) = (-4, -1)$.

• Distance Formula:

  • Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

  • Example: Distance between $(2, -5)$ and $(10, 10)$ is $\sqrt{(10 - 2)^2 + (10 - (-5))^2} = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17$.

• Graphing Lines and Inequalities:

  • The equation of a line is $y = mx + c$.

  • For a line passing through $(0, 4)$ and $(4, -2)$, the gradient is $\frac{-2 - 4}{4 - 0} = -\frac{3}{2}$. The y-intercept is $4$, so the equation is $y = -\frac{3}{2}x + 4$.

  • Shaded Regions: A region bounded by a line and the axes. For a line with intercepts $(6, 0)$ and $(0, 3)$, the equation is $y = -\frac{1}{2}x + 3$. If the region is below the line and in the first quadrant, the inequalities are $y \le -\frac{1}{2}x + 3, x \ge 0, y \ge 0$.

Geometry and Trigonometry

• Right-Angled Triangles:

  • Pythagorean Triple: In a triangle with legs $15$ and $20$, the hypotenuse is $\sqrt{15^2 + 20^2} = \sqrt{225 + 400} = 25$.

  • Trigonometric Ratios:

    • $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$ (e.g., $\frac{40}{41}$ for a triangle with sides $9, 40, 41$).

    • $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$.

    • Find $x$ in a triangle where $\theta = 60^\circ$ and the hypotenuse is $18$: if $x$ is the adjacent side, $x = 18\cos(60^\circ)$.

  • 45-45-90 Triangle: If the hypotenuse is $12$, each shorter side is $\frac{12}{\sqrt{2}} = 6\sqrt{2}$.

• Angles and Parallel Lines:

  • External Angles and Triangles: In a configuration with angles $74^\circ$ and $131^\circ$, the missing angle $a$ can be found using the property that the external angle equals the sum of interior opposite angles or by linear pairs: $131 - 74 = 57^\circ$.

  • Parallel Lines: Co-interior angles sum to $180^\circ$. Given an angle of $68^\circ$, its supplementary angle is $180 - 68 = 112^\circ$.

• Circles, Sectors, and Arcs:

  • Sector Area: $Area = \frac{\text{angle}}{360} \times \pi r^2$. For radius $9\,cm$ and angle $120^\circ$, $Area = \frac{120}{360} \times \pi(9)^2 = \frac{1}{3} \times 81\pi = 27\pi\,cm^2$.

  • Arc Length and Circumference: $\text{Arc Length} = \frac{\text{angle}}{360} \times \text{Circumference}$. Given arc length $24\,cm$ and angle $80^\circ$, $24 = \frac{80}{360} \times C \rightarrow 24 = \frac{2}{9} \times C \rightarrow C = \frac{24 \times 9}{2} = 108\,cm$.

• Mensuration (Area and Perimeter):

  • Area of a Rectangle: $(5x + 4)(3x - 2) = 15x^2 - 10x + 12x - 8 = 15x^2 + 2x - 8$.

  • Rectangle Problem: Perimeter of rectangle is $112\,cm$. Length is $4\,cm$ less than three times width.

    • Let width be $w$, length be $3w - 4$.

    • Perimeter $2(w + 3w - 4) = 112 \rightarrow 8w - 8 = 112 \rightarrow 8w = 120 \rightarrow w = 15\,cm$.

  • Area to Perimeter: Rectangle with length $(x + 6)$, width $(x - 2)$, and area $88\,cm^2$.

    • $(x + 6)(x - 2) = 88 \rightarrow x^2 + 4x - 12 = 88 \rightarrow x^2 + 4x - 100 = 0$. (Equation results in $x = 8.19$ approx; solving for perimeter $2(x + 6 + x - 2) = 4x + 8$).

Statistics and Probability

• Probability:

  • Rolling a fair six-sided die twice to get a sum of exactly 7:

    • Possible pairs: $(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)$.

    • Total outcomes: $6 \times 6 = 36$.

    • Probability = $\frac{6}{36} = \frac{1}{6}$.

• Mean:

  • The mean of $x, x + 6, x + 14,$ and $x + 20$ is $31$.

  • Equation: $\frac{x + x + 6 + x + 14 + x + 20}{4} = 31$

  • $4x + 40 = 124 \rightarrow 4x = 84 \rightarrow x = 21$.

• Venn Diagram Problem (Sets):

  • Total students: $120$. Basketball ($B$): $64$, Soccer ($S$): $53$, Both: $28$.

  • Union formula: $P(B \cup S) = P(B) + P(S) - P(B \cap S) = 64 + 53 - 28 = 89$.

  • Students playing neither: $120 - 89 = 31$.

General Problem Solving

• Train Dynamics:

  • A train takes $16\,s$ to pass a pole (its own length) and $46\,s$ to pass a $540\,m$ platform (length of train + platform).

  • Let length be $L$ and speed be $v$.

  • $v = \frac{L}{16}$ and $v = \frac{L + 540}{46}$.

  • $\frac{L}{16} = \frac{L + 540}{46} \rightarrow 46L = 16L + 8640 \rightarrow 30L = 8640 \rightarrow L = 288\,m$.

• Value for Money:

  • Compare cost per practice test across plans:

    • Plan A: $30 \div 5 = \$6/\text{test}$

    • Plan B: $48 \div 8 = \$6/\text{test}$

    • Plan C: $54 \div 12 = \$4.5/\text{test}$

    • Plan D: $70 \div 14 = \$5/\text{test}$

  • Best value: Plan C.