A system \(ax+by=c\) and \(dx+ey=f\) has infinitely many solutions when both equations describe the same line — i.e., \(\dfrac{a}{d}=\dfrac{b}{e}=\dfrac{c}{f}\). This means one equation is a scalar multiple of the other.
✏️ Worked Example
\(2x+4y=8\) and \(x+2y=k\): multiply the second by 2 → \(2x+4y=2k\). For infinite solutions, \(2k=8\), so \(k=4\).
For what value of \(k\) does the system \(3x - 6y = 12\) and \(kx - 2y = 4\) have infinitely many solutions?
Problem 2
QUADRATIC EQUATIONSHARD
📚 Key Concept
The discriminant \(\Delta = b^2 - 4ac\) determines the nature of roots: \(\Delta > 0\) → two distinct real roots; \(\Delta = 0\) → one repeated root; \(\Delta < 0\) → no real roots (complex). For problems involving the sum and product of roots, use Vieta's Formulas: \(r_1+r_2 = -\dfrac{b}{a}\) and \(r_1 \cdot r_2 = \dfrac{c}{a}\).
✏️ Worked Example
For \(2x^2 - 5x + 3 = 0\): sum of roots \(= \tfrac{5}{2}\), product \(= \tfrac{3}{2}\). Discriminant \(= 25 - 24 = 1 > 0\), so two distinct real roots.
The quadratic equation \(x^2 - (k+2)x + (2k+1) = 0\) has two equal real roots. What is the value of \(k\)?
Problem 3
LINEAR INEQUALITIESHARD
📚 Key Concept
When solving compound inequalities and absolute value inequalities: \(|ax+b| \leq c\) becomes \(-c \leq ax+b \leq c\); \(|ax+b| \geq c\) splits into \(ax+b \leq -c\) OR \(ax+b \geq c\). Remember to flip the inequality sign when multiplying/dividing by a negative number.
The solution set of \(|3x - 2| > 7\) is expressed as \(x < a\) or \(x > b\). What is the value of \(a + b\)?
Problem 4
FUNCTIONS & TRANSFORMATIONSHARD
📚 Key Concept
For \(f(x) = a(x-h)^2 + k\): the vertex is \((h, k)\), the axis of symmetry is \(x = h\), and \(|a|\) controls width. The graph is reflected over the x-axis when \(a < 0\). The range is \([k, \infty)\) if \(a>0\), or \((-\infty, k]\) if \(a<0\).
✏️ Worked Example
\(f(x) = -2(x+3)^2 + 5\): vertex \((-3, 5)\), opens downward, max value is 5, range: \((-\infty, 5]\).
The function \(f(x) = -3(x-2)^2 + 12\). A new function \(g(x)\) is obtained by shifting \(f(x)\) left 3 and down 4. What is the maximum value of \(g(x)\)?
Problem 5
RATIONAL EXPRESSIONSHARD
📚 Key Concept
When solving rational equations, multiply through by the LCD to clear denominators. Always check for extraneous solutions — values that make any denominator zero are excluded from the domain.
If \(27^{x-1} = 9^{x+2}\), what is the value of \(x\)?
Problem 7
WORD PROBLEMS — MIXTUREHARD
📚 Key Concept
For mixture problems, use: Amount × Concentration = Total substance. Set up a table with columns for amount, concentration, and total, then write an equation using the final mixture row.
✏️ Worked Example
Mix 30L of 20% acid with \(x\)L of 50% acid to get 40% acid solution:
\(0.20(30) + 0.50x = 0.40(30+x)\) → \(6 + 0.5x = 12 + 0.4x\) → \(0.1x = 6\) → \(x = 60\)L.
A chemist has a 15% salt solution and a 45% salt solution. How many liters of the 45% solution must be added to 20 liters of the 15% solution to create a 30% salt solution?
Problem 8
ARITHMETIC SEQUENCESHARD
📚 Key Concept
Arithmetic sequence formulas: \(a_n = a_1 + (n-1)d\) where \(d\) is the common difference. Sum of first \(n\) terms: \(S_n = \dfrac{n}{2}(a_1 + a_n) = \dfrac{n}{2}(2a_1 + (n-1)d)\).
In an arithmetic sequence, the 5th term is 23 and the 11th term is 47. What is the sum of the first 20 terms?
Problem 9
FACTORING & POLYNOMIALSHARD
📚 Key Concept
Special factoring patterns: difference of cubes \(a^3 - b^3 = (a-b)(a^2+ab+b^2)\); sum of cubes \(a^3+b^3=(a+b)(a^2-ab+b^2)\). For the Rational Root Theorem: possible rational roots of \(a_nx^n+\cdots+a_0\) are \(\pm\dfrac{p}{q}\) where \(p \mid a_0\) and \(q \mid a_n\).
✏️ Worked Example
Factor \(x^3 - 8\): \(= (x-2)(x^2+2x+4)\). The second factor \(x^2+2x+4\) has discriminant \(4-16 = -12 < 0\), so no real roots.
The polynomial \(p(x) = 2x^3 - 3x^2 - 11x + 6\). Given that \(x = 3\) is a root, which of the following shows the complete factorization?
Problem 10
DIRECT & INVERSE VARIATIONHARD
📚 Key Concept
Combined variation: if \(y\) varies directly as \(x\) and inversely as \(z\), then \(y = \dfrac{kx}{z}\). Find constant \(k\) from a known data point, then use it to find the unknown. Joint variation: \(y = kxz\).
✏️ Worked Example
\(y\) varies directly as \(x\) and inversely as \(z^2\). When \(x=4, z=2, y=6\): \(k = \dfrac{yz^2}{x} = \dfrac{6 \cdot 4}{4} = 6\). So \(y = \dfrac{6x}{z^2}\).
The variable \(w\) varies directly as \(x^2\) and inversely as \(\sqrt{y}\). When \(x = 3\) and \(y = 4\), \(w = 18\). Find \(w\) when \(x = 6\) and \(y = 16\).
Geometry Q 11 – 20
Problem 11
SIMILAR TRIANGLESHARD
📚 Key Concept
For similar triangles (\(\triangle ABC \sim \triangle DEF\)): corresponding sides are proportional and corresponding angles are equal. The ratio of areas equals the square of the ratio of corresponding sides: \(\dfrac{[\triangle ABC]}{[\triangle DEF]} = \left(\dfrac{AB}{DE}\right)^2\). The ratio of perimeters equals the similarity ratio.
✏️ Worked Example
Two similar triangles have sides in ratio \(3:5\). If the smaller triangle has area \(27\,\text{cm}^2\), the larger has area \(27 \times \left(\tfrac{5}{3}\right)^2 = 27 \times \tfrac{25}{9} = 75\,\text{cm}^2\).
Triangle \(ABC\) is similar to triangle \(DEF\). The perimeter of \(\triangle ABC\) is 36 and the perimeter of \(\triangle DEF\) is 60. If the area of \(\triangle ABC\) is \(54\,\text{cm}^2\), what is the area of \(\triangle DEF\)?
Problem 12
CIRCLE THEOREMSHARD
📚 Key Concept
Key circle angle theorems: (1) Inscribed angle = half the intercepted arc; (2) Central angle = intercepted arc; (3) Angle formed by two chords inside a circle = \(\dfrac{1}{2}(\text{arc}_1 + \text{arc}_2)\); (4) Tangent–chord angle = half the intercepted arc.
✏️ Worked Example
Two chords intersect inside a circle. One intercepts arcs of 80° and 120°. The angle at intersection: \(\dfrac{80°+120°}{2} = 100°\).
In a circle, two chords \(AB\) and \(CD\) intersect at point \(P\) inside the circle. Arc \(AC = 72°\) and arc \(BD = 108°\). What is the measure of \(\angle APD\)?
Problem 13
COORDINATE GEOMETRYHARD
📚 Key Concept
The equation of a circle with center \((h,k)\) and radius \(r\): \((x-h)^2 + (y-k)^2 = r^2\). To find the equation from a general form \(x^2+y^2+Dx+Ey+F=0\), complete the square for both \(x\) and \(y\). The radius is \(r = \sqrt{h^2+k^2-F}\).
✏️ Worked Example
\(x^2+y^2-4x+6y-3=0\): \((x-2)^2-4+(y+3)^2-9-3=0\) → \((x-2)^2+(y+3)^2=16\). Center \((2,-3)\), radius \(4\).
A circle passes through points \(A(1, 0)\), \(B(5, 0)\), and \(C(3, 4)\). What is the radius of this circle?
Problem 14
PYTHAGOREAN THEOREM & 3DHARD
📚 Key Concept
The space diagonal of a rectangular box (cuboid) with dimensions \(l \times w \times h\) is \(d = \sqrt{l^2+w^2+h^2}\). This is a two-step Pythagorean application: first find the face diagonal, then apply Pythagoras again with the height.
✏️ Worked Example
A box of \(3 \times 4 \times 5\): face diagonal \(= \sqrt{9+16} = 5\). Space diagonal \(= \sqrt{25+25} = 5\sqrt{2}\).
A rectangular box has dimensions \(6\,\text{cm} \times 8\,\text{cm} \times 10\,\text{cm}\). A spider crawls along the space diagonal of the box. What is the length of this diagonal, rounded to the nearest tenth?
Problem 15
AREA & PERIMETER — COMPOSITEHARD
📚 Key Concept
For composite figures, break the shape into simpler parts: rectangles, triangles, semicircles, etc. Area of a sector with radius \(r\) and central angle \(\theta°\): \(A = \dfrac{\theta}{360}\pi r^2\). Shaded region = Total area − Unshaded area.
✏️ Worked Example
Square of side 10 with inscribed circle (radius 5): shaded area (corners) \(= 100 - \pi(25) = 100 - 25\pi \approx 21.46\,\text{cm}^2\).
An equilateral triangle with side length \(12\,\text{cm}\) has a semicircle drawn outward on one side. What is the total area of the composite figure? (Use \(\sqrt{3} \approx 1.732\))
Problem 16
TRIANGLE CONGRUENCE & PROOFSHARD
📚 Key Concept
Triangle congruence shortcuts: SSS, SAS, ASA, AAS, and HL (right triangles only). Note that SSA is NOT a valid congruence shortcut. Once two triangles are proven congruent (CPCTC), all corresponding parts are congruent.
✏️ Worked Example
If \(\overline{AB} \cong \overline{DE}\), \(\angle B \cong \angle E\), \(\overline{BC} \cong \overline{EF}\), then \(\triangle ABC \cong \triangle DEF\) by SAS, so \(\angle A \cong \angle D\) by CPCTC.
In quadrilateral \(ABCD\), diagonal \(\overline{AC}\) bisects both \(\angle A\) and \(\angle C\). Which statement is sufficient to prove \(\triangle ABC \cong \triangle ADC\)?
Problem 17
SURFACE AREA & VOLUMEHARD
📚 Key Concept
For a cone: \(V = \dfrac{1}{3}\pi r^2 h\), lateral surface area \(= \pi r l\) (where \(l = \sqrt{r^2+h^2}\) is the slant height), total surface area \(= \pi r^2 + \pi r l\). For a sphere: \(V = \dfrac{4}{3}\pi r^3\), \(SA = 4\pi r^2\).
✏️ Worked Example
Cone with \(r=3, h=4\): \(l = \sqrt{9+16}=5\). Lateral SA \(= \pi(3)(5)=15\pi\). Total SA \(= 9\pi+15\pi=24\pi\,\text{cm}^2\).
A cone has a base radius of \(5\,\text{cm}\) and a height of \(12\,\text{cm}\). A hemisphere of radius \(3\,\text{cm}\) is placed on top of the cone (sitting inside, base aligned). What is the total exposed surface area? (Subtract the hemisphere base from the cone's base; use \(\pi \approx 3.14159\))
Problem 18
PARALLEL LINES & ANGLESMED-HARD
📚 Key Concept
When a transversal crosses parallel lines: alternate interior angles are equal; co-interior (same-side interior) angles are supplementary; corresponding angles are equal. For a triangle, the exterior angle theorem: an exterior angle equals the sum of the two non-adjacent interior angles.
✏️ Worked Example
Lines \(p \parallel q\) cut by transversal. If one angle is \((3x+20)°\) and its alternate interior angle is \((5x-10)°\): \(3x+20 = 5x-10\) → \(x=15\). Angle = \(65°\).
Lines \(m \parallel n\) are cut by a transversal. The co-interior (same-side interior) angles measure \((4x + 15)°\) and \((2x + 45)°\). What is the value of \(x\)?
Problem 19
TRIGONOMETRY IN TRIANGLESHARD
📚 Key Concept
In a right triangle: SOH-CAH-TOA. For non-right triangles: Law of Sines \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}\); Law of Cosines \(c^2 = a^2 + b^2 - 2ab\cos C\). Area of any triangle: \(A = \dfrac{1}{2}ab\sin C\).
✏️ Worked Example
Triangle with \(a=7, b=10, C=60°\): \(c^2 = 49+100-2(7)(10)(\tfrac{1}{2})=79\), so \(c = \sqrt{79} \approx 8.89\).
In triangle \(ABC\), \(AB = 8\,\text{cm}\), \(BC = 15\,\text{cm}\), and \(\angle ABC = 30°\). What is the area of triangle \(ABC\)?
Point \(P(3,4)\): reflect over y-axis → \((-3,4)\), then rotate 90° CCW → \((-4,-3)\).
Triangle \(PQR\) has vertices \(P(2,1)\), \(Q(5,1)\), and \(R(5,4)\). The triangle is first reflected over the line \(y = x\), then rotated 90° counterclockwise about the origin. What are the final coordinates of vertex \(R\)?