Math Master — Algebra 2 & Geometry

Q 01 Quadratic ★☆☆

The vertex form of a parabola is $f(x) = 2(x - 3)^2 + 5$.
What are the coordinates of the vertex?

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⚡ Quick Memory

f(x) = a(x−h)² + k → vertex is always (h, k). Watch the sign flip: (x − 3) means h = +3, not −3!

📖 Explanation

In vertex form $f(x) = a(x - h)^2 + k$, compare with $2(x - 3)^2 + 5$.
Here $h = 3$ and $k = 5$, so the vertex is $(3, 5)$.
Key trap: The formula has $(x - h)$, so when you see $(x - 3)$, $h = +3$, not $-3$.

Q 02 Quadratic ★☆☆

Solve: $x^2 - 5x + 6 = 0$
Which values of $x$ satisfy the equation?

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⚡ Quick Memory

Find two numbers that multiply to c and add to b. For $x^2 - 5x + 6$: need × = 6, + = −5 → −2 and −3.

📖 Explanation

Factor: $x^2 - 5x + 6 = (x - 2)(x - 3) = 0$.
Set each factor to zero: $x - 2 = 0 \Rightarrow x = 2$; \ $x - 3 = 0 \Rightarrow x = 3$.
Verify: $2^2 - 5(2) + 6 = 4 - 10 + 6 = 0$ ✓ and $3^2 - 5(3) + 6 = 9 - 15 + 6 = 0$ ✓

Q 03 Discriminant ★★☆

For $2x^2 - 4x + 5 = 0$, evaluate the discriminant.
How many real solutions does this equation have?

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⚡ Quick Memory

Discriminant = b² − 4ac. If > 0: 2 real roots | = 0: 1 real root | < 0: NO real roots (imaginary).

📖 Explanation

$a=2,\ b=-4,\ c=5$
$\Delta = b^2 - 4ac = (-4)^2 - 4(2)(5) = 16 - 40 = -24$
Since $\Delta = -24 < 0$, there are no real solutions — the roots are complex (imaginary).

Q 04 Polynomials ★☆☆

A polynomial $P(x)$ has the property that $P(3) = 0$.
Which of the following must be a factor of $P(x)$?

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⚡ Quick Memory

Factor Theorem: If P(c) = 0, then (x − c) is a factor. Zero of polynomial → factor form uses minus sign.

📖 Explanation

By the Factor Theorem: if $P(c) = 0$, then $(x - c)$ is a factor.
Since $P(3) = 0$, the factor is $(x - 3)$.
Note: $(x+3)$ would be a factor only if $P(-3)=0$, which we don't know.

Q 05 Logarithms ★☆☆

Simplify: $\log_2 8 + \log_2 4$
What is the value of this expression?

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⚡ Quick Memory

Log Product Rule: log(A) + log(B) = log(A × B). Same base only! Think: add logs = multiply numbers.

📖 Explanation

Method 1 — Product Rule: $\log_2 8 + \log_2 4 = \log_2(8 \times 4) = \log_2 32 = 5$ (since $2^5 = 32$).
Method 2 — Direct: $\log_2 8 = 3$ (since $2^3=8$) and $\log_2 4 = 2$ (since $2^2=4$).
So $3 + 2 = \mathbf{5}$.

Q 06 Logarithms ★★☆

Solve for $x$: $\log_3(x - 1) = 2$
What is the value of $x$?

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⚡ Quick Memory

Log ↔ Exponent flip: log_b(x) = y ↔ b^y = x. Convert log to exponential form, then solve. Always check: argument must be > 0.

📖 Explanation

Convert to exponential form: $\log_3(x-1) = 2 \Rightarrow 3^2 = x - 1$.
$9 = x - 1 \Rightarrow x = 10$.
Check: $\log_3(10 - 1) = \log_3 9 = 2$ ✓ (since $3^2 = 9$).

Q 07 Sequences ★☆☆

An arithmetic sequence has first term $a_1 = 4$ and common difference $d = 3$.
What is the 10th term, $a_{10}$?

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⚡ Quick Memory

Arithmetic nth term: a_n = a₁ + (n−1)d. Think: first term + (how many jumps) × (jump size). n−1 jumps, not n jumps!

📖 Explanation

Formula: $a_n = a_1 + (n-1)d$
$a_{10} = 4 + (10-1)(3) = 4 + 9 \times 3 = 4 + 27 = \mathbf{31}$
Common mistake: Using $n \cdot d$ instead of $(n-1) \cdot d$ gives 34, which is wrong.

Q 08 Sequences ★★☆

A geometric sequence has $a_1 = 5$ and common ratio $r = 2$.
Find the sum of the first 6 terms, $S_6$.

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⚡ Quick Memory

Geometric sum: S_n = a₁(rⁿ − 1) / (r − 1) when r ≠ 1. Numerator: r to the power n minus 1. Don't mix up r and n!

📖 Explanation

$S_6 = \dfrac{a_1(r^6 - 1)}{r - 1} = \dfrac{5(2^6 - 1)}{2 - 1} = \dfrac{5(64-1)}{1} = 5 \times 63 = \mathbf{315}$
The terms are: 5, 10, 20, 40, 80, 160 → sum = 315 ✓

Q 09 Systems ★☆☆

Solve the system of equations:
$y = 2x + 1$ and $y = -x + 7$
What is the solution $(x, y)$?

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⚡ Quick Memory

Substitution shortcut: Both equal y → set them equal to each other. Solve for x first, then plug back in to get y. Always check both equations!

📖 Explanation

Set equal: $2x + 1 = -x + 7 \Rightarrow 3x = 6 \Rightarrow x = 2$.
Substitute: $y = 2(2) + 1 = 5$. So the solution is $(2, 5)$.
Check eq 2: $y = -2 + 7 = 5$ ✓

Q 10 Rational Expressions ★★☆

A store sells two types of coffee. Brand A costs \$4 per bag and Brand B costs \$6 per bag. A customer buys a total of 10 bags and spends \$48. How many bags of Brand A did they buy?

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⚡ Quick Memory

Set up TWO equations: quantity eq + cost eq. Let A = bags of Brand A, B = bags of Brand B. Two unknowns → need two equations!

📖 Explanation

Let $a$ = bags of A, $b$ = bags of B.
System: $a + b = 10$ and $4a + 6b = 48$.
From eq 1: $b = 10 - a$. Substitute: $4a + 6(10-a) = 48 \Rightarrow 4a + 60 - 6a = 48 \Rightarrow -2a = -12 \Rightarrow a = 6$.
So Brand A: 6 bags. Check: $4(6) + 6(4) = 24 + 24 = 48$ ✓

Q 11 Triangle ★☆☆

A right triangle has legs of length 6 and 8.
What is the length of the hypotenuse?

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⚡ Quick Memory

Pythagorean Theorem: a² + b² = c². c is always the hypotenuse (longest side, opposite right angle). Memorize triples: 3-4-5, 5-12-13, 6-8-10!

📖 Explanation

$c^2 = 6^2 + 8^2 = 36 + 64 = 100$, so $c = \sqrt{100} = 10$.
This is the classic 3-4-5 triple scaled by 2: $6 = 3 \times 2$, $8 = 4 \times 2$, $10 = 5 \times 2$.

Q 12 Angles ★☆☆

Two parallel lines are cut by a transversal.
One of the co-interior angles (same-side interior) measures 65°.
What is the measure of the other co-interior angle?

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⚡ Quick Memory

Parallel line angle pairs: Alternate interior = EQUAL | Co-interior (same-side) = 180° | Corresponding = EQUAL. "Co-interior" → Co = cooperate to make 180°!

📖 Explanation

Co-interior (same-side interior) angles are supplementary — they add up to 180°.
$180° - 65° = \mathbf{115°}$.
Note: If the question had asked about alternate interior angles, the answer would be 65° (equal). Don't mix these up!

Q 13 Circle ★☆☆

A circle has a radius of 7 cm.
What is the area of the circle? (Leave answer in terms of $\pi$)

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⚡ Quick Memory

Circle formulas: Area = πr² | Circumference = 2πr. Memory trick: "Pie Are Square" → Area = π·r·r = πr². r not d!

📖 Explanation

$A = \pi r^2 = \pi (7)^2 = 49\pi \text{ cm}^2$
Common mistakes:
• Using diameter (14) instead of radius (7) → gives $196\pi$
• Forgetting to square → gives $7\pi$

Q 14 Triangle Area ★☆☆

A triangle has a base of 12 cm and a height of 9 cm.
Two students disagree on the area. Student A says 108 cm², Student B says 54 cm². Who is correct?

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⚡ Quick Memory

Triangle area = ½ × base × height. The ½ is always there — a triangle is half a rectangle. Height must be perpendicular to base!

📖 Explanation

$A = \tfrac{1}{2} \times b \times h = \tfrac{1}{2} \times 12 \times 9 = \tfrac{1}{2} \times 108 = \mathbf{54 \text{ cm}^2}$
Student A forgot the $\tfrac{1}{2}$ — a very common mistake! The triangle is exactly half the area of a rectangle with the same base and height.

Q 15 Similar Triangles ★★☆

Two similar triangles have corresponding sides in the ratio $3 : 5$.
If the area of the smaller triangle is 27 cm², what is the area of the larger triangle?

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⚡ Quick Memory

Similar figures: Area ratio = (side ratio)². If sides ratio = k, area ratio = k². Sides scale once, area scales twice!

📖 Explanation

Side ratio = $\tfrac{3}{5}$, so area ratio = $\left(\tfrac{3}{5}\right)^2 = \tfrac{9}{25}$.
$\dfrac{27}{A_{\text{large}}} = \dfrac{9}{25} \Rightarrow A_{\text{large}} = \dfrac{27 \times 25}{9} = 3 \times 25 = \mathbf{75 \text{ cm}^2}$

Q 16 Volume ★☆☆

A cylinder has a radius of 4 cm and a height of 10 cm.
What is its volume? (Leave in terms of $\pi$)

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⚡ Quick Memory

Cylinder Volume = πr²h = (circle area) × height. Think: stack circles up! Cone = ⅓πr²h. Cylinder = 3 × Cone (same r and h).

📖 Explanation

$V = \pi r^2 h = \pi (4)^2 (10) = \pi \times 16 \times 10 = \mathbf{160\pi \text{ cm}^3}$
Trap: Choice A uses $\pi \cdot r \cdot h$ (forgot to square r). Always square the radius!

Q 17 Coordinate Geometry ★☆☆

Find the midpoint of the segment connecting $A(2, -4)$ and $B(8, 6)$.

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⚡ Quick Memory

Midpoint = ((x₁+x₂)/2 , (y₁+y₂)/2). Think: average the x's, average the y's. Add both coordinates, then divide by 2 — each separately!

📖 Explanation

Midpoint $= \left(\dfrac{2+8}{2},\ \dfrac{-4+6}{2}\right) = \left(\dfrac{10}{2},\ \dfrac{2}{2}\right) = \mathbf{(5,\ 1)}$.
Note: $\dfrac{-4+6}{2} = \dfrac{2}{2} = 1$, not $\dfrac{4+6}{2}$ — keep the negative sign!

Q 18 Coordinate Geometry ★★☆

Line $\ell$ passes through $(1, 2)$ and $(4, 8)$.
What is the equation of a line parallel to $\ell$ passing through $(0, -1)$?

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⚡ Quick Memory

Parallel lines have the SAME slope. Perpendicular = negative reciprocal slope. Find slope first: m = (y₂−y₁)/(x₂−x₁), then use y = mx + b.

📖 Explanation

Step 1 — Slope of $\ell$: $m = \dfrac{8-2}{4-1} = \dfrac{6}{3} = 2$.
Step 2 — Parallel line has slope $m = 2$ and passes through $(0,-1)$.
Since the y-intercept is $(0,-1)$, directly: $y = 2x - 1$.
Trap: Choice C is the perpendicular line (slope = $-\tfrac{1}{2}$).

Q 19 Circle Theorems ★★☆

In a circle, a central angle measures $80°$.
What is the measure of the inscribed angle that intercepts the same arc?

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⚡ Quick Memory

Inscribed Angle Theorem: Inscribed angle = ½ × central angle (same arc). Central is always DOUBLE the inscribed. Think: center is closer to the arc = bigger angle!

📖 Explanation

The Inscribed Angle Theorem states that an inscribed angle is exactly half the central angle that intercepts the same arc.
Inscribed angle $= \dfrac{80°}{2} = \mathbf{40°}$.
Remember: this only works when both angles intercept the same arc.

Q 20 Surface Area ★★☆

A rectangular prism (box) has dimensions:
length $= 5$ cm, width $= 3$ cm, height $= 4$ cm.
What is its total surface area?

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⚡ Quick Memory

Rectangular prism SA = 2(lw + lh + wh). Think: 3 pairs of faces, multiply each pair by 2. The "2" outside is often forgotten — total 6 faces, not 3!

📖 Explanation

$SA = 2(lw + lh + wh) = 2(5 \times 3 + 5 \times 4 + 3 \times 4)$
$= 2(15 + 20 + 12) = 2(47) = \mathbf{94 \text{ cm}^2}$
Trap: Choice B (47) is only half the surface area — forgot the factor of 2!

Algebra 2 &Geometry

Algebra 2 &
Geometry