IB Mathematics · Analysis & Approaches

CORE 20

Higher Level — All Units · Exam-Style Questions

20 Questions HL Standard All 5 Units 45 min
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● CONCEPT REFERENCE & KEY FORMULAS
U1Number & Algebra
\(S_n = \frac{n}{2}(2a_1+(n-1)d)\) — AP Sum
\(S_\infty = \frac{a_1}{1-r}, \; |r|<1\) — GP Infinite Sum
\((a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k\) — Binomial
arithmetic geometric binomial theorem logarithm laws complex numbers
U2Functions
\(f(x)=ax^2+bx+c\): vertex \(x=-\frac{b}{2a}\)
Rational: VA where denom\(=0\); HA from degree compare
domain & range transformations inverse functions asymptotes logarithm / exponential
U3Geometry & Trigonometry
\(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\) — Sine Rule
\(c^2=a^2+b^2-2ab\cos C\) — Cosine Rule
\(s=r\theta,\; A=\tfrac{1}{2}r^2\theta\) — Arc & Sector
unit circle trig identities 3D vectors dot product
U4Statistics & Probability
\(P(A|B)=\frac{P(A\cap B)}{P(B)}\) — Conditional Prob.
\(X\sim B(n,p):\; P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}\)
\(X\sim N(\mu,\sigma^2):\; z=\frac{x-\mu}{\sigma}\)
combinations binomial dist. normal dist. hypothesis testing
U5Calculus
Chain rule: \(\frac{d}{dx}[f(g(x))]=f'(g(x))\cdot g'(x)\)
IBP: \(\int u\,dv = uv - \int v\,du\)
FTC: \(\int_a^b f(x)\,dx = F(b)-F(a)\)
differentiation rules integration definite integrals optimization
● EXAM QUESTIONS — 20 PROBLEMS
1
Unit 1 · Arithmetic Sequences
An arithmetic sequence has first term \(a_1=3\) and common difference \(d=4\). What is the sum of the first 20 terms, \(S_{20}\)?
  • A \(760\)
  • B \(800\)
  • C \(820\)
  • D \(840\)
✦ Full Solution
Use \(S_n = \dfrac{n}{2}\bigl(2a_1 + (n-1)d\bigr)\).
\(S_{20} = \dfrac{20}{2}\bigl(2(3) + 19(4)\bigr) = 10(6 + 76) = 10 \times 82 = \boxed{820}\)
2
Unit 1 · Infinite Geometric Series
A geometric series has first term \(a_1 = 6\) and common ratio \(r = \tfrac{1}{3}\). What is the sum to infinity \(S_\infty\)?
  • A \(7\)
  • B \(9\)
  • C \(12\)
  • D \(18\)
✦ Full Solution
\(|r|=\tfrac{1}{3}<1\), so the sum converges.
\(S_\infty = \dfrac{a_1}{1-r} = \dfrac{6}{1-\tfrac{1}{3}} = \dfrac{6}{\tfrac{2}{3}} = 6 \times \tfrac{3}{2} = \boxed{9}\)
3
Unit 1 · Binomial Theorem
Find the coefficient of \(x^2\) in the expansion of \((x+2)^5\).
  • A \(40\)
  • B \(80\)
  • C \(60\)
  • D \(160\)
✦ Full Solution
The general term is \(\dbinom{5}{k}x^{5-k}\cdot 2^k\). For \(x^2\): \(5-k=2 \Rightarrow k=3\).
Coefficient \(= \dbinom{5}{3}\cdot 2^3 = 10 \times 8 = \boxed{80}\)
4
Unit 1 · Logarithms
Solve for \(x\): \(\log_2(8x) = 5\).
  • A \(x = 2\)
  • B \(x = 4\)
  • C \(x = 6\)
  • D \(x = 8\)
✦ Full Solution
\(\log_2(8x)=5 \Rightarrow 8x = 2^5 = 32 \Rightarrow x = \dfrac{32}{8} = \boxed{4}\)
5
Unit 2 · Quadratic Functions
Find the vertex of the parabola \(f(x) = 2x^2 - 8x + 3\).
  • A \((2,\,-5)\)
  • B \((2,\,-3)\)
  • C \((4,\,-5)\)
  • D \((-2,\,27)\)
✦ Full Solution
Axis of symmetry: \(x = -\dfrac{b}{2a} = -\dfrac{-8}{4} = 2\).
\(f(2) = 2(4) - 8(2) + 3 = 8 - 16 + 3 = -5\).
Vertex \(= \boxed{(2,\,-5)}\)
6
Unit 2 · Rational Functions & Asymptotes
For \(f(x) = \dfrac{2x+1}{x-3}\), what are the vertical and horizontal asymptotes?
  • A VA: \(x=3\), HA: \(y=2\)
  • B VA: \(x=-3\), HA: \(y=2\)
  • C VA: \(x=3\), HA: \(y=1\)
  • D VA: \(x=1\), HA: \(y=3\)
✦ Full Solution
VA: set denominator \(= 0\): \(x - 3 = 0 \Rightarrow x = 3\).
HA: same degree (both degree 1), so \(y = \dfrac{\text{leading coeff of num}}{\text{leading coeff of denom}} = \dfrac{2}{1} = 2\).
Answer: \(\boxed{x=3,\; y=2}\)
7
Unit 3 · Trigonometric Equations
Solve \(2\sin x = \sqrt{3}\) for \(x \in [0, 2\pi]\). How many solutions are there, and what are they?
  • A One solution: \(x = \dfrac{\pi}{3}\)
  • B Two solutions: \(x = \dfrac{\pi}{3}\) and \(x = \dfrac{2\pi}{3}\)
  • C Two solutions: \(x = \dfrac{\pi}{6}\) and \(x = \dfrac{5\pi}{6}\)
  • D Two solutions: \(x = \dfrac{\pi}{3}\) and \(x = \pi\)
✦ Full Solution
\(\sin x = \dfrac{\sqrt{3}}{2}\).
Reference angle: \(\arcsin\!\left(\dfrac{\sqrt{3}}{2}\right) = \dfrac{\pi}{3}\).
\(\sin x > 0\) in Q1 and Q2: \(x = \dfrac{\pi}{3}\) and \(x = \pi - \dfrac{\pi}{3} = \dfrac{2\pi}{3}\).
Solutions: \(\boxed{x = \dfrac{\pi}{3},\; \dfrac{2\pi}{3}}\)
8
Unit 3 · Sine Rule
In triangle \(ABC\), side \(a = 8\), angle \(A = 30°\), and angle \(B = 45°\). Find side \(b\).
  • A \(b = 4\sqrt{2}\)
  • B \(b = 8\sqrt{2}\)
  • C \(b = 4\sqrt{3}\)
  • D \(b = 8\sqrt{3}\)
✦ Full Solution
Sine Rule: \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B}\)
\(\dfrac{8}{\sin 30°} = \dfrac{b}{\sin 45°}\)
\(\dfrac{8}{0.5} = \dfrac{b}{\frac{\sqrt{2}}{2}} \Rightarrow 16 = \dfrac{b}{\frac{\sqrt{2}}{2}} \Rightarrow b = 16 \cdot \dfrac{\sqrt{2}}{2} = \boxed{8\sqrt{2}}\)
9
Unit 3 · Arc Length
A sector has radius \(r = 5\,\text{cm}\) and central angle \(\theta = 1.2\) radians. What is the arc length?
  • A \(5\,\text{cm}\)
  • B \(6\,\text{cm}\)
  • C \(7\,\text{cm}\)
  • D \(8\,\text{cm}\)
✦ Full Solution
\(s = r\theta = 5 \times 1.2 = \boxed{6\,\text{cm}}\)
10
Unit 3 · Vectors & Dot Product
Given \(\mathbf{a} = (3, 4)\) and \(\mathbf{b} = (1, -2)\), what is the angle between them (to 1 decimal place)?
  • A \(90.0°\)
  • B \(116.6°\)
  • C \(63.4°\)
  • D \(180.0°\)
✦ Full Solution
\(\mathbf{a}\cdot\mathbf{b} = 3(1)+4(-2) = 3-8 = -5\)
\(|\mathbf{a}|=\sqrt{9+16}=5,\quad |\mathbf{b}|=\sqrt{1+4}=\sqrt{5}\)
\(\cos\theta = \dfrac{-5}{5\sqrt{5}} = \dfrac{-1}{\sqrt{5}}\)
\(\theta = \arccos\!\left(\dfrac{-1}{\sqrt{5}}\right) \approx \boxed{116.6°}\)
11
Unit 4 · Combinatorics
How many ways can a committee of 3 people be chosen from a group of 8?
  • A \(24\)
  • B \(336\)
  • C \(56\)
  • D \(512\)
✦ Full Solution
Order does not matter → use combinations.
\(\dbinom{8}{3} = \dfrac{8!}{3!\,5!} = \dfrac{8\times7\times6}{3\times2\times1} = \dfrac{336}{6} = \boxed{56}\)
12
Unit 4 · Binomial Distribution
Let \(X \sim B(10,\, 0.3)\). What is \(P(X = 3)\), correct to 4 decimal places?
  • A \(0.2668\)
  • B \(0.1211\)
  • C \(0.3000\)
  • D \(0.0090\)
✦ Full Solution
\(P(X=3) = \dbinom{10}{3}(0.3)^3(0.7)^7\)
\(= 120 \times 0.027 \times 0.0823543\)
\(= 120 \times 0.002223566 \approx \boxed{0.2668}\)
13
Unit 4 · Normal Distribution
If \(Z\) is the standard normal variable, find \(P(Z < 1.5)\) (to 4 decimal places).
  • A \(0.8413\)
  • B \(0.9332\)
  • C \(0.9772\)
  • D \(0.9938\)
✦ Full Solution
From the standard normal table:
\(P(Z < 1.0) = 0.8413\)
\(P(Z < 1.5) = \boxed{0.9332}\)
\(P(Z < 2.0) = 0.9772\)
\(P(Z < 2.5) = 0.9938\)
14
Unit 4 · Conditional Probability
Events \(A\) and \(B\) satisfy \(P(A \cap B) = 0.15\) and \(P(B) = 0.4\). Find \(P(A \mid B)\).
  • A \(0.25\)
  • B \(0.375\)
  • C \(0.55\)
  • D \(0.60\)
✦ Full Solution
\(P(A \mid B) = \dfrac{P(A \cap B)}{P(B)} = \dfrac{0.15}{0.4} = \boxed{0.375}\)
15
Unit 5 · Differentiation (Power Rule)
Find \(f'(x)\) for \(f(x) = 3x^4 - 2x^3 + 5x\).
  • A \(12x^3 - 6x^2 + 5\)
  • B \(12x^3 - 6x^2\)
  • C \(3x^3 - 2x^2 + 5\)
  • D \(12x^4 - 6x^3 + 5x\)
✦ Full Solution
Apply the power rule term-by-term:
\(\dfrac{d}{dx}[3x^4] = 12x^3,\quad \dfrac{d}{dx}[-2x^3] = -6x^2,\quad \dfrac{d}{dx}[5x] = 5\)
\(f'(x) = \boxed{12x^3 - 6x^2 + 5}\)
16
Unit 5 · Chain Rule
Differentiate \(f(x) = \sin(3x^2)\).
  • A \(\cos(3x^2)\)
  • B \(3x\cos(3x^2)\)
  • C \(6x\cos(3x^2)\)
  • D \(-6x\cos(3x^2)\)
✦ Full Solution
Chain rule: outer function \(\sin(u)\), inner \(u = 3x^2\).
\(\dfrac{d}{dx}[\sin(3x^2)] = \cos(3x^2) \cdot \dfrac{d}{dx}[3x^2] = \cos(3x^2) \cdot 6x = \boxed{6x\cos(3x^2)}\)
17
Unit 5 · Indefinite Integration
Find \(\displaystyle\int (2x^3 - 3x)\,dx\).
  • A \(\dfrac{x^4}{2} - \dfrac{3x^2}{2} + C\)
  • B \(6x^2 - 3 + C\)
  • C \(2x^4 - 3x^2 + C\)
  • D \(\dfrac{x^4}{2} + 3x^2 + C\)
✦ Full Solution
\(\int 2x^3\,dx = \dfrac{2x^4}{4} = \dfrac{x^4}{2},\quad \int (-3x)\,dx = -\dfrac{3x^2}{2}\)
Result: \(\boxed{\dfrac{x^4}{2} - \dfrac{3x^2}{2} + C}\)
Check: \(\dfrac{d}{dx}\!\left[\dfrac{x^4}{2}-\dfrac{3x^2}{2}\right] = 2x^3 - 3x\) ✓
18
Unit 5 · Definite Integral
Evaluate \(\displaystyle\int_0^2 (x^2 + 1)\,dx\).
  • A \(\dfrac{10}{3}\)
  • B \(4\)
  • C \(\dfrac{14}{3}\)
  • D \(6\)
✦ Full Solution
\(\int_0^2(x^2+1)\,dx = \left[\dfrac{x^3}{3}+x\right]_0^2 = \left(\dfrac{8}{3}+2\right)-0 = \dfrac{8}{3}+\dfrac{6}{3} = \boxed{\dfrac{14}{3}}\)
19
Unit 1 · Complex Numbers
For the complex number \(z = 3 + 4i\), find \(|z|\) and \(\arg(z)\) (in degrees, to 2 decimal places).
  • A \(|z|=5,\; \arg(z) \approx 53.13°\)
  • B \(|z|=7,\; \arg(z) \approx 45.00°\)
  • C \(|z|=5,\; \arg(z) \approx 36.87°\)
  • D \(|z|=25,\; \arg(z) \approx 53.13°\)
✦ Full Solution
\(|z| = \sqrt{3^2+4^2} = \sqrt{9+16} = \sqrt{25} = 5\)
\(\arg(z) = \arctan\!\left(\dfrac{4}{3}\right) \approx \arctan(1.3333) \approx 53.13°\)
Answer: \(\boxed{|z|=5,\; \arg(z)\approx 53.13°}\)
20
Unit 5 · Integration by Parts
Evaluate \(\displaystyle\int x e^x\,dx\).
  • A \(x e^x + C\)
  • B \(e^x(x-1) + C\)
  • C \(\dfrac{x^2}{2}e^x + C\)
  • D \(e^x(x+1) + C\)
✦ Full Solution
IBP: \(\int u\,dv = uv - \int v\,du\). Let \(u=x\Rightarrow du=dx\), \(dv=e^x\,dx\Rightarrow v=e^x\).
\(\int xe^x\,dx = xe^x - \int e^x\,dx = xe^x - e^x + C = e^x(x-1) + C\)
Check: \(\dfrac{d}{dx}[e^x(x-1)] = e^x(x-1)+e^x = xe^x\) ✓
Answer: \(\boxed{e^x(x-1)+C}\)
Final Score
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Answer Key

1. C — 820
2. B — 9
3. B — 80
4. B — x=4
5. A — (2,−5)
6. A — x=3, y=2
7. B — π/3, 2π/3
8. B — 8√2
9. B — 6 cm
10. B — 116.6°
11. C — 56
12. A — 0.2668
13. B — 0.9332
14. B — 0.375
15. A — 12x³−6x²+5
16. C — 6x cos(3x²)
17. A — x⁴/2−3x²/2+C
18. C — 14/3
19. A — |z|=5, 53.13°
20. B — eˣ(x−1)+C

Detailed Solutions

Q1. S₂₀ = (20/2)(2·3 + 19·4) = 10(6+76) = 10×82 = 820
Q2. S∞ = 6/(1−1/3) = 6/(2/3) = 9
Q3. Term with x²: k=3, C(5,3)·2³ = 10·8 = 80
Q4. 8x = 2⁵ = 32, x = 4
Q5. x = −(−8)/(2·2) = 2; f(2) = 8−16+3 = −5; vertex (2,−5)
Q6. VA: denom = 0 → x = 3; HA: equal degrees → y = 2/1 = 2
Q7. sin x = √3/2; reference angle π/3; Q1 and Q2 → x = π/3, 2π/3
Q8. b/sin45° = 8/sin30° → b = 16·(√2/2) = 8√2
Q9. s = rθ = 5×1.2 = 6 cm
Q10. a·b = −5; |a|=5, |b|=√5; cosθ = −1/√5; θ ≈ 116.6°
Q11. C(8,3) = 8×7×6/(3×2×1) = 56
Q12. C(10,3)·(0.3)³·(0.7)⁷ = 120·0.027·0.082354 ≈ 0.2668
Q13. Standard normal table: P(Z<1.5) = 0.9332
Q14. P(A|B) = 0.15/0.4 = 0.375
Q15. f'(x) = 12x³ − 6x² + 5 (power rule)
Q16. Chain rule: cos(3x²)·6x = 6x cos(3x²)
Q17. ∫2x³dx = x⁴/2; ∫−3x dx = −3x²/2; answer: x⁴/2 − 3x²/2 + C
Q18. [x³/3 + x]₀² = (8/3+2) − 0 = 8/3 + 6/3 = 14/3
Q19. |z| = √(9+16) = 5; arg = arctan(4/3) ≈ 53.13°
Q20. IBP: u=x, dv=eˣdx → xeˣ − eˣ + C = eˣ(x−1) + C