IB Math Grade 10 — Study Notebook

Q1

Question 01 · Quadratic Formula

Solve the equation using the quadratic formula.

⚡ Key Memory

DISCRIMINANT = \(b^2 - 4ac\)
▸ \(>0\) → 2 real roots ▸ \(=0\) → 1 repeated root ▸ \(<0\) → no real roots
Formula: \(x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\)

📝 Example

Solve \(x^2 + 5x + 6 = 0\)
Here \(a=1,\ b=5,\ c=6\). Discriminant: \(25-24=1 > 0\) → 2 roots
\(x = \dfrac{-5 \pm 1}{2}\) → \(x = -2\) or \(x = -3\) ✓

✦ Now You Try ✦

Solve: \(\ 2x^2 - 7x + 3 = 0\)

💡 Explanation

\(a=2,\ b=-7,\ c=3\). \(\Delta = 49-24 = 25\)
\(x = \dfrac{7 \pm 5}{4}\) → \(x = \dfrac{12}{4}=3\) or \(x = \dfrac{2}{4}=\tfrac{1}{2}\)
⚠ Trap: Don't forget ±! Many students only take +.

Q2

Question 02 · Factoring

Which factored form is correct for \(x^2 - x - 12\)?

⚡ Key Memory

FACTORING TRICK: Find two numbers that
✔ Multiply to \(c\) ✔ Add to \(b\)
→ "MULTIPLY to the bottom, ADD to the middle"

📝 Example

\(x^2 + 2x - 15\): need × = −15, + = 2 → 5 and −3
Answer: \((x+5)(x-3)\)

💡 Explanation

Need: × = −12, + = −1 → +3 and −4
\((x+3)(x-4)\). Check: \(3 \times (-4)=-12\) ✓, \(3+(-4)=-1\) ✓
⚠ Trap: A and C look the same but C has correct signs!

Q3

Question 03 · Completing the Square

Complete the square for \(x^2 + 6x + 2\).

⚡ Key Memory

COMPLETING THE SQUARE
Step: \(x^2 + bx = \left(x + \dfrac{b}{2}\right)^2 - \left(\dfrac{b}{2}\right)^2\)
HALF the coefficient of x, then SQUARE it!
→ "HALF → SQUARE → SUBTRACT"

💡 Explanation

Half of 6 = 3. Square = 9.
\(x^2+6x+2 = (x+3)^2 - 9 + 2 = (x+3)^2 - 7\)
⚠ Trap: Many students forget to subtract the 9 and just add it!

Q4

Question 04 · Systems of Equations

Find the solution to the system: \(3x + 2y = 12\) and \(x - y = 1\).

⚡ Key Memory

SUBSTITUTION: Isolate one variable, plug in.
ELIMINATION: Multiply to match coefficients, add/subtract.
Always check your answer in BOTH equations!

💡 Explanation

From eq.2: \(x = y+1\). Substitute: \(3(y+1)+2y=12\) → \(5y=9\) … wait let me redo:
\(3(y+1)+2y=12\) → \(3y+3+2y=12\) → \(5y=9\)? No: \(y=\frac{9}{5}\)? Let's try elimination:
Multiply eq.2 by 2: \(2x-2y=2\). Add to eq.1: \(5x=14\)… Hmm.
Actually sub: \(x=y+1\), so \(3(y+1)+2y=12\) → \(5y=9\) → this doesn't give integer. So check A: \(3(2)+2(1)=8≠12\). Let me recheck eq: from \(x-y=1\), \(x=y+1\). \(3(y+1)+2y=12\) → \(5y+3=12\) → \(5y=9\) → \(y=9/5\). Hmm. But for teaching: The answer is \(x=\frac{14}{5}, y=\frac{9}{5}\).
⚠ Always verify by substituting back into BOTH equations!

Q5

Question 05 · Vertex of a Parabola

Find the vertex of \(f(x) = 2x^2 - 8x + 5\).

⚡ Key Memory

VERTEX formula: \(x = -\dfrac{b}{2a}\), then find \(y\)
Or use: vertex form \(f(x) = a(x-h)^2+k\) → vertex is \((h,k)\)
→ "NEGATIVE b over 2a" = x of vertex

💡 Explanation

\(x = -\dfrac{-8}{2 \cdot 2} = \dfrac{8}{4} = 2\)
\(f(2) = 2(4) - 8(2) + 5 = 8 - 16 + 5 = -3\)
Vertex = \((2, -3)\) ✓
⚠ Trap: Don't forget to substitute x back to find y!

Q6

Question 06 · Domain & Range

What is the domain of \(f(x) = \sqrt{3x - 6}\)?

⚡ Key Memory

DOMAIN RULES:
▸ Square root: inside ≥ 0
▸ Fraction: denominator ≠ 0
▸ Log: inside > 0
→ "What x values make this LEGAL?"

💡 Explanation

Need: \(3x - 6 \geq 0\) → \(3x \geq 6\) → \(x \geq 2\)
Domain: \([2, \infty)\)
⚠ Trap: Use ≥ (not >) because √0 = 0 is valid!

Q7

Question 07 · Inverse Functions

Find \(f^{-1}(x)\) if \(f(x) = 3x - 5\).

⚡ Key Memory

INVERSE steps:
1. Replace \(f(x)\) with \(y\)
2. Swap x and y
3. Solve for \(y\) → that's \(f^{-1}(x)\)
→ "SWAP then SOLVE"

💡 Explanation

\(y=3x-5\) → swap: \(x=3y-5\) → solve: \(3y=x+5\) → \(y=\dfrac{x+5}{3}\)
⚠ Trap: D is the reciprocal, NOT the inverse! Completely different!

Q8

Question 08 · Sine Rule

In triangle ABC, \(a=8,\ b=5,\ A=60°\). Find angle B.

⚡ Key Memory

SINE RULE: \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}\)
Use when: AAS, ASA, SSA (non-right triangles)
→ "side OVER sin of its OWN angle"

📝 Example

\(\dfrac{8}{\sin 60°} = \dfrac{5}{\sin B}\) → \(\sin B = \dfrac{5 \sin 60°}{8}\)

💡 Explanation

\(\sin B = \dfrac{5 \times \sin 60°}{8} = \dfrac{5 \times 0.866}{8} \approx \dfrac{4.33}{8} \approx 0.541\)
\(B = \sin^{-1}(0.541) \approx 32.8°\)
⚠ Trap: Remember the ambiguous case — SSA can give 2 possible angles!

Q9

Question 09 · Cosine Rule

Find side \(c\) if \(a=7,\ b=5,\ C=60°\).

⚡ Key Memory

COSINE RULE: \(c^2 = a^2 + b^2 - 2ab\cos C\)
Use when: SAS or SSS
→ "Pythagorean theorem + correction term"
When \(C=90°\): \(\cos 90° = 0\) → becomes Pythagoras! 🤯

💡 Explanation

\(c^2 = 49 + 25 - 2(7)(5)(0.5) = 74 - 35 = 39\)
\(c = \sqrt{39} \approx 6.24\)
⚠ Trap: \(\cos 60° = 0.5\), NOT 0.866 (that's sin 60°!)

Q10

Question 10 · Unit Circle

What is the exact value of \(\sin 135°\)?

⚡ Key Memory

CAST RULE (which trig is positive):
Q1: All Q2: Sin Q3: Tan Q4: Cos
→ "All Students Take Calculus"
Reference angle of 135° = 180° − 135° = 45°

💡 Explanation

135° is in Q2 → sin is positive (by CAST).
Reference angle = 45°, so \(\sin 135° = \sin 45° = \dfrac{\sqrt{2}}{2}\)
⚠ Trap: Don't confuse sin(135°) with cos(135°) which IS negative!

Q11

Question 11 · Probability

A bag has 4 red and 6 blue balls. Two drawn WITHOUT replacement. P(both red) = ?

⚡ Key Memory

WITHOUT REPLACEMENT: denominator decreases!
\(P(A \text{ and } B) = P(A) \times P(B \mid A)\)
→ "After each pick, pool SHRINKS"

💡 Explanation

\(P = \dfrac{4}{10} \times \dfrac{3}{9} = \dfrac{12}{90} = \dfrac{2}{15}\)
⚠ Trap: B is WITH replacement (\(\frac{4}{10} \times \frac{4}{10}\)) — wrong here!

Q12

Question 12 · Standard Deviation

Dataset: {2, 4, 4, 4, 5, 5, 7, 9}. What is the mean?

⚡ Key Memory

MEAN = \(\dfrac{\text{Sum of all values}}{\text{Number of values}}\)
MEDIAN = middle value (sort first!)
MODE = most frequent
→ "Mean: Add ALL, divide by COUNT"

💡 Explanation

Sum = 2+4+4+4+5+5+7+9 = 40. Count = 8.
Mean = 40 ÷ 8 = 5
Median = (4+5)/2 = 4.5, Mode = 4
⚠ Trap: Mean ≠ Median ≠ Mode. Know WHICH one is asked!

Q13

Question 13 · Arithmetic Sequence

The 5th term of an arithmetic sequence is 17 and the common difference is 3. Find the 1st term.

⚡ Key Memory

ARITHMETIC: \(u_n = u_1 + (n-1)d\)
\(u_n\) = nth term, \(u_1\) = first term, \(d\) = common difference
→ "First term + (n−1) JUMPS of d"

💡 Explanation

\(17 = u_1 + (5-1)(3) = u_1 + 12\)
\(u_1 = 17 - 12 = 5\)
⚠ Trap: It's (n−1), NOT n! Many students write \(u_1+5d\) instead of \(u_1+4d\).

Q14

Question 14 · Geometric Sequence

Find the 6th term of the geometric sequence: \(3, 6, 12, 24, \ldots\)

⚡ Key Memory

GEOMETRIC: \(u_n = u_1 \cdot r^{n-1}\)
Common ratio: \(r = \dfrac{u_2}{u_1}\)
→ "First term TIMES r to the power (n−1)"

💡 Explanation

\(r = 6/3 = 2\). \(u_6 = 3 \times 2^{6-1} = 3 \times 32 = 96\)
⚠ Trap: \(2^5 = 32\), NOT \(2^6 = 64\). Remember it's (n−1)!

Q15

Question 15 · Sum of Arithmetic Series

Find the sum of the first 10 terms: \(5 + 8 + 11 + \ldots\)

⚡ Key Memory

ARITHMETIC SUM: \(S_n = \dfrac{n}{2}(2u_1 + (n-1)d)\)
OR: \(S_n = \dfrac{n}{2}(u_1 + u_n)\) ← use this if you know the last term
→ "n over 2, times (first + last)"

💡 Explanation

\(u_1=5,\ d=3,\ n=10\)
\(S_{10} = \dfrac{10}{2}(2 \times 5 + 9 \times 3) = 5(10+27) = 5 \times 37 = 185\)
⚠ Trap: \((n-1)d\) uses 9, not 10!

Q16

Question 16 · Laws of Logarithms

Simplify: \(\log_3 81 - \log_3 9\)

⚡ Key Memory

LOG LAWS:
▸ \(\log(AB) = \log A + \log B\)
▸ \(\log\!\left(\dfrac{A}{B}\right) = \log A - \log B\)
▸ \(\log(A^n) = n\log A\)
→ "Log of DIVIDE = SUBTRACT the logs"

💡 Explanation

\(\log_3 81 - \log_3 9 = \log_3\!\left(\dfrac{81}{9}\right) = \log_3 9 = 2\)
(Because \(3^2 = 9\))
⚠ Trap: \(\log_3 81 = 4\) and \(\log_3 9 = 2\), so 4−2=2. Both methods work!

Q17

Question 17 · Exponential Equations

Solve: \(2^{3x} = 32\)

⚡ Key Memory

SAME BASE TRICK: If \(a^m = a^n\), then \(m = n\)
Steps: Rewrite both sides with the SAME base, then equate exponents.
→ "Make bases EQUAL, drop the base, set exponents equal"

💡 Explanation

\(32 = 2^5\) → \(2^{3x} = 2^5\) → \(3x = 5\) → \(x = \dfrac{5}{3}\)
⚠ Trap: \(32 = 2^5\), NOT \(2^6\)! Always double-check your powers of 2.

Q18

Question 18 · Differentiation

Differentiate: \(f(x) = 4x^3 - 5x^2 + 2x - 7\)

⚡ Key Memory

POWER RULE: \(\dfrac{d}{dx}(x^n) = nx^{n-1}\)
Steps: MULTIPLY by power, then REDUCE power by 1
Constant → disappears (derivative = 0)
→ "Bring the power DOWN, then MINUS ONE"

💡 Explanation

\(f'(x) = 4(3)x^2 - 5(2)x^1 + 2(1)x^0 - 0\)
\(= 12x^2 - 10x + 2\)
⚠ Trap: Constant (−7) disappears! A incorrectly keeps −7.

Q19

Question 19 · Gradient of Tangent

Find the gradient of \(f(x)=x^2+3x\) at \(x=2\).

⚡ Key Memory

GRADIENT at a point = value of \(f'(x)\) at that \(x\)
Step 1: Differentiate → get \(f'(x)\)
Step 2: Substitute the given x value
→ "DIFFERENTIATE first, PLUG IN second"

💡 Explanation

\(f'(x) = 2x + 3\)
At \(x=2\): \(f'(2) = 2(2)+3 = 7\)
⚠ Trap: Don't find f(2) = 4+6 = 10! That's the y-value, NOT the gradient.

Q20

Question 20 · Integration (Basic)

Evaluate: \(\displaystyle\int (6x^2 - 4x + 1)\,dx\)

⚡ Key Memory

INTEGRATION POWER RULE: \(\displaystyle\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C\)
Steps: ADD 1 to power, then DIVIDE by new power
ALWAYS add \(+C\) (constant of integration)!
→ "Plus one, divide, plus C — NEVER forget C!"

💡 Explanation

\(\int 6x^2\,dx = \dfrac{6x^3}{3} = 2x^3\)
\(\int -4x\,dx = \dfrac{-4x^2}{2} = -2x^2\)
\(\int 1\,dx = x\)
Answer: \(2x^3 - 2x^2 + x + C\)
⚠ Trap: B forgets +C. D is differentiating instead of integrating!

🎓 Final Score

0/20

📓 IB Math — Grade 10