Chapter A: The Enchanted Foundation Gate
SL + HL · Shared Core
Every traveler — SL or HL — must pass through the same golden gate. Here lie the shared foundations of Algebra, Functions, Trigonometry, Statistics and Calculus that both kingdoms are built upon.
Q1. A1
Sequences
★☆☆☆☆
An arithmetic sequence has \(u_1=5\) and \(d=3\). Find \(u_{10}\).
A 30
B 32
C 35
D 27
💡 Hint
📖 Concept
Use \(u_n = u_1 + (n-1)d\).
Arithmetic sequence formula: \(u_n=u_1+(n-1)d\).
Explanation: \(u_{10}=5+9(3)=32\).
Q2. A2
Sequences
★☆☆☆☆
A geometric sequence has \(u_1=2\) and \(r=3\). Find \(u_4\).
A 18
B 54
C 24
D 36
💡 Hint
📖 Concept
Use \(u_n = u_1 r^{n-1}\).
Geometric sequence formula: \(u_n=u_1 r^{n-1}\).
Explanation: \(u_4=2(3^3)=2(27)=54\).
Q3. A3
Exponents
★☆☆☆☆
Simplify \(2^3 \times 2^4\).
A 64
B 128
C 32
D 256
💡 Hint
📖 Concept
Add the exponents when multiplying same base.
Exponent law: \(a^m\cdot a^n=a^{m+n}\).
Explanation: \(2^3\times2^4=2^7=128\).
Q4. A4
Logarithms
★☆☆☆☆
Evaluate \(\log_2 8\).
A 2
B 4
C 3
D 8
💡 Hint
📖 Concept
Ask: 2 to what power gives 8?
\(\log_a x=y \iff a^y=x\).
Explanation: \(2^3=8\), so \(\log_2 8=3\).
Q5. A5
Sequences
★★☆☆☆
Find the sum of the first 5 terms of an arithmetic sequence with \(u_1=4, d=2\).
A 30
B 35
C 40
D 45
💡 Hint
📖 Concept
Use \(S_n=\tfrac{n}{2}(2u_1+(n-1)d)\).
Arithmetic series sum formula.
Explanation: \(S_5=\tfrac{5}{2}(8+8)=40\).
Q6. A6
Binomial
★★☆☆☆
Find the coefficient of \(x^2\) in the expansion of \((1+x)^5\).
A 5
B 10
C 15
D 20
💡 Hint
📖 Concept
Use \(\binom{5}{2}\).
Binomial coefficient \(\binom{n}{r}\) gives the coefficient of \(x^r\) in \((1+x)^n\).
Explanation: \(\binom{5}{2}=10\).
Q7. A7
Functions
★☆☆☆☆
If \(f(x)=2x+3\), find \(f(4)\).
A 7
B 10
C 11
D 14
💡 Hint
📖 Concept
Substitute \(x=4\) directly.
Function evaluation: replace \(x\) with the input value.
Explanation: \(f(4)=2(4)+3=11\).
Q8. A8
Functions
★☆☆☆☆
State the value excluded from the domain of \(f(x)=\dfrac{1}{x-2}\).
A 0
B 2
C -2
D 1
💡 Hint
📖 Concept
The denominator cannot equal zero.
Rational functions exclude values that make the denominator zero.
Explanation: \(x-2=0 \Rightarrow x=2\) is excluded.
Q9. A9
Functions
★★☆☆☆
If \(f(x)=x^2\) and \(g(x)=x+1\), find \(f(g(2))\).
A 5
B 6
C 9
D 4
💡 Hint
📖 Concept
First find \(g(2)\), then apply \(f\).
Composite functions: evaluate the inner function first.
Explanation: \(g(2)=3\), so \(f(3)=9\).
Q10. A10
Functions
★★☆☆☆
If \(f(x)=2x-4\), find \(f^{-1}(6)\).
A 4
B 5
C 3
D 6
💡 Hint
📖 Concept
Solve \(2x-4=6\) for \(x\).
\(f^{-1}(a)=b\) means \(f(b)=a\).
Explanation: \(2x-4=6 \Rightarrow x=5\).
Q11. A11
Functions
★☆☆☆☆
Find the vertex of \(y=(x-3)^2+1\).
A (3,1)
B (-3,1)
C (3,-1)
D (1,3)
💡 Hint
📖 Concept
The vertex form \(y=(x-h)^2+k\) has vertex \((h,k)\).
Vertex form of a parabola.
Explanation: Vertex is \((3,1)\).
Q12. A12
Functions
★★☆☆☆
Find the sum of the roots of \(x^2-5x+6=0\).
A 5
B 6
C 1
D -5
💡 Hint
📖 Concept
Factorise or use sum of roots \(=-b/a\).
For \(ax^2+bx+c=0\), sum of roots \(=-b/a\).
Explanation: Roots are 2 and 3; sum \(=5\).
Q13. A13
Trigonometry
★☆☆☆☆
Evaluate \(\sin(30^\circ)\).
A \(\tfrac{1}{2}\)
B \(\tfrac{\sqrt3}{2}\)
C 1
D 0
💡 Hint
📖 Concept
Recall the special angle values.
Special angle: \(\sin30^\circ=\tfrac12\).
Explanation: \(\sin30^\circ=\tfrac12\).
Q14. A14
Trigonometry
★☆☆☆☆
Evaluate \(\cos(60^\circ)\).
A \(\tfrac{\sqrt3}{2}\)
B \(\tfrac{1}{2}\)
C 1
D 0
💡 Hint
📖 Concept
Recall the special angle values.
Special angle: \(\cos60^\circ=\tfrac12\).
Explanation: \(\cos60^\circ=\tfrac12\).
Q15. A15
Trigonometry
★☆☆☆☆
Convert \(90^\circ\) to radians.
A \(\pi\)
B \(\tfrac{\pi}{2}\)
C \(\tfrac{\pi}{4}\)
D \(2\pi\)
💡 Hint
📖 Concept
Use \(180^\circ=\pi\) radians.
Radian conversion: \(\theta_{rad}=\theta_{deg}\times\tfrac{\pi}{180}\).
Explanation: \(90 \times \tfrac{\pi}{180}=\tfrac{\pi}{2}\).
Q16. A16
Trigonometry
★★☆☆☆
In a right triangle, the side opposite an angle is 3 and the hypotenuse is 5. Find \(\sin\theta\).
A \(\tfrac{4}{5}\)
B \(\tfrac{3}{4}\)
C \(\tfrac{3}{5}\)
D \(\tfrac{5}{3}\)
💡 Hint
📖 Concept
\(\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}\).
SOH-CAH-TOA.
Explanation: \(\sin\theta=\tfrac{3}{5}\).
Q17. A17
Trigonometry
★★☆☆☆
A circle has radius 4. Find the arc length for \(\theta=\tfrac{\pi}{2}\) rad.
A \(4\pi\)
B \(2\pi\)
C \(\pi\)
D \(8\pi\)
💡 Hint
📖 Concept
Arc length \(=r\theta\).
Formula: arc length \(=r\theta\) (\(\theta\) in radians).
Explanation: \(4\times\tfrac{\pi}{2}=2\pi\).
Q18. A18
Trigonometry
★★☆☆☆
A circle has radius 6. Find the sector area for \(\theta=\tfrac{\pi}{3}\) rad.
A \(6\pi\)
B \(3\pi\)
C \(12\pi\)
D \(9\pi\)
💡 Hint
📖 Concept
Sector area \(=\tfrac12 r^2\theta\).
Formula: sector area \(=\tfrac12 r^2\theta\).
Explanation: \(\tfrac12(36)(\tfrac{\pi}{3})=6\pi\).
Q19. A19
Statistics
★☆☆☆☆
Find the mean of \(\{2,4,6,8,10\}\).
A 5
B 6
C 7
D 8
💡 Hint
📖 Concept
Add all values and divide by how many there are.
Mean \(=\dfrac{\sum x}{n}\).
Explanation: \((2+4+6+8+10)/5=6\).
Q20. A20
Statistics
★☆☆☆☆
Find the mode of \(\{3,3,5,7,9\}\).
A 9
B 7
C 5
D 3
💡 Hint
📖 Concept
The mode is the most frequent value.
Mode = most frequently occurring value.
Explanation: 3 appears twice; mode \(=3\).
Q21. A21
Statistics
★☆☆☆☆
Find the median of \(\{1,3,5,7,9\}\).
A 3
B 5
C 7
D 1
💡 Hint
📖 Concept
Order the data and pick the middle value.
Median = middle value of ordered data.
Explanation: Middle value is 5.
Q22. A22
Statistics
★★☆☆☆
Find the population variance of \(\{1,2,3,4,5\}\).
A 1
B 1.5
C 2
D 2.5
💡 Hint
📖 Concept
Variance \(=\dfrac{\sum (x-\bar x)^2}{n}\), with \(\bar x=3\).
Population variance formula.
Explanation: \((4+1+0+1+4)/5=2\).
Q23. A23
Probability
★☆☆☆☆
A fair die is rolled. Find \(P(\text{even number})\).
A \(\tfrac13\)
B \(\tfrac12\)
C \(\tfrac23\)
D \(\tfrac16\)
💡 Hint
📖 Concept
3 of the 6 outcomes are even.
Probability \(=\dfrac{\text{favourable outcomes}}{\text{total outcomes}}\).
Explanation: \(3/6=\tfrac12\).
Q24. A24
Probability
★★☆☆☆
Events A and B are independent with \(P(A)=0.5\), \(P(B)=0.4\). Find \(P(A \cap B)\).
A 0.9
B 0.2
C 0.1
D 0.45
💡 Hint
📖 Concept
For independent events, multiply the probabilities.
Independence: \(P(A\cap B)=P(A)P(B)\).
Explanation: \(0.5\times0.4=0.2\).
Q25. A25
Calculus
★☆☆☆☆
If \(f(x)=x^3\), find \(f'(2)\).
A 6
B 8
C 12
D 16
💡 Hint
📖 Concept
Differentiate then substitute \(x=2\).
Power rule: \(\dfrac{d}{dx}x^n=nx^{n-1}\).
Explanation: \(f'(x)=3x^2\), \(f'(2)=12\).
Q26. A26
Calculus
★★☆☆☆
Find \(f'(1)\) if \(f(x)=5x^2+3x\).
A 10
B 13
C 8
D 5
💡 Hint
📖 Concept
Differentiate term by term, then substitute \(x=1\).
Power rule applied term-by-term.
Explanation: \(f'(x)=10x+3\), \(f'(1)=13\).
Q27. A27
Calculus
★★☆☆☆
Evaluate \(\displaystyle\int_0^3 2x\,dx\).
A 6
B 9
C 12
D 3
💡 Hint
📖 Concept
Antiderivative of \(2x\) is \(x^2\).
Fundamental theorem of calculus.
Explanation: \([x^2]_0^3=9-0=9\).
Q28. A28
Calculus
★☆☆☆☆
Find the gradient of the tangent to \(y=x^2\) at \(x=1\).
A 1
B 2
C 3
D 4
💡 Hint
📖 Concept
Differentiate and substitute \(x=1\).
Gradient of tangent \(=f'(x)\) at that point.
Explanation: \(y'=2x\), at \(x=1\): gradient \(=2\).
Q29. A29
Calculus
★★☆☆☆
Find the stationary point of \(y=x^2-4x+5\).
A (2,1)
B (1,2)
C (2,5)
D (4,5)
💡 Hint
📖 Concept
Set \(y'=0\) and solve for \(x\), then find \(y\).
Stationary points occur where \(f'(x)=0\).
Explanation: \(y'=2x-4=0 \Rightarrow x=2\), \(y=1\).
Q30. A30
Calculus
★☆☆☆☆
Evaluate \(\displaystyle\int_1^4 3\,dx\).
A 3
B 9
C 12
D 15
💡 Hint
📖 Concept
Integral of a constant \(c\) over \([a,b]\) is \(c(b-a)\).
Integrating a constant gives \(c\times\text{width}\).
Explanation: \(3\times(4-1)=9\).
Chapter B: The Village of Word Problems
SL-style Applications
Beyond the gate, a lively village where numbers meet real life: interest, growth, triangles, and motion. These are SL-style applications of the shared core.
Q31. B1
Algebra Applications
★★☆☆☆
\$1000 is invested at 5% annual compound interest. Find the value after 2 years.
A \$1050.00
B \$1100.00
C \$1102.50
D \$1150.00
💡 Hint
📖 Concept
Use \(A=P(1+r)^n\).
Compound interest formula: \(A=P(1+r)^n\).
Explanation: \(1000(1.05)^2=1102.50\).
Q32. B2
Algebra Applications
★★☆☆☆
A population of 100 grows at 10% per year. Find the population after 3 years (nearest whole).
A 121
B 130
C 133
D 140
💡 Hint
📖 Concept
Use \(P_n=P_0(1+r)^n\).
Geometric growth model.
Explanation: \(100(1.1)^3=133.1\approx133\).
Q33. B3
Algebra Applications
★★☆☆☆
A runner covers 2 km on day 1, increasing by 0.5 km each day. Find the total distance after 10 days.
A 40 km
B 42.5 km
C 45 km
D 47.5 km
💡 Hint
📖 Concept
Use the arithmetic series sum formula.
Arithmetic series: \(S_n=\tfrac n2(2u_1+(n-1)d)\).
Explanation: \(S_{10}=5(4+4.5)=42.5\) km.
Q34. B4
Algebra Applications
★★☆☆☆
Find the sum to infinity of a geometric series with \(u_1=8\), \(r=\tfrac12\).
A 12
B 14
C 16
D 20
💡 Hint
📖 Concept
Use \(S_\infty=\dfrac{u_1}{1-r}\).
Infinite geometric series formula (valid for \(|r|<1\)).
Explanation: \(8/(1-0.5)=16\).
Q35. B5
Function Modelling
★★☆☆☆
The cost of production is \(C(x)=50+2x\) dollars. Find the cost of producing 20 units.
A \$70
B \$90
C \$100
D \$110
💡 Hint
📖 Concept
Substitute \(x=20\).
Linear cost model: fixed cost + variable cost per unit.
Explanation: \(C(20)=50+40=90\).
Q36. B6
Function Modelling
★★☆☆☆
A drug concentration is modelled by \(C(t)=\dfrac{100}{t+1}\) mg/L. Find the concentration at \(t=4\) hours.
A 10 mg/L
B 20 mg/L
C 25 mg/L
D 40 mg/L
💡 Hint
📖 Concept
Substitute \(t=4\) into the model.
Rational function modelling decay over time.
Explanation: \(C(4)=100/5=20\).
Q37. B7
Function Modelling
★★★☆☆
Revenue is modelled by \(R(x)=-2x^2+40x\). Find the maximum revenue.
A \$150
B \$180
C \$200
D \$220
💡 Hint
📖 Concept
Find the vertex: \(x=-b/2a\), then evaluate \(R(x)\).
Maximum of a downward parabola occurs at the vertex.
Explanation: \(x=10\), \(R(10)=200\).
Q38. B8
Function Modelling
★★☆☆☆
Find the horizontal asymptote of \(f(x)=\dfrac{3}{x-2}+1\).
A y = 0
B y = 1
C y = 2
D y = 3
💡 Hint
📖 Concept
As \(x\to\infty\), the fraction term \(\to0\).
Horizontal asymptotes describe end behaviour.
Explanation: As \(x\to\infty\), \(f(x)\to1\); asymptote \(y=1\).
Q39. B9
Trig Applications
★★★☆☆
Triangle sides \(a=7\), \(b=9\), included angle \(C=60^\circ\). Find \(c\) (nearest 0.01).
A 7.94
B 8.19
C 8.54
D 9.00
💡 Hint
📖 Concept
Use the cosine rule \(c^2=a^2+b^2-2ab\cos C\).
Cosine rule for a non-right triangle.
Explanation: \(c^2=49+81-63=67\), \(c\approx8.19\).
Q40. B10
Trig Applications
★★★☆☆
In a triangle, \(A=30^\circ\), \(a=5\), \(B=45^\circ\). Find side \(b\) (exact form).
A \(5\sqrt2\)
B \(5\sqrt3\)
C \(\tfrac{5}{\sqrt2}\)
D 10
💡 Hint
📖 Concept
Use the sine rule \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}\).
Sine rule for non-right triangles.
Explanation: \(b=\dfrac{5\sin45^\circ}{\sin30^\circ}=5\sqrt2\).
Q41. B11
Vectors
★★☆☆☆
A ship sails 10 km east, then 10 km north. Find the distance from the start (exact form).
A \(10\)
B \(10\sqrt2\)
C \(20\)
D \(10\sqrt3\)
💡 Hint
📖 Concept
Use Pythagoras on the two perpendicular legs.
Displacement magnitude \(=\sqrt{x^2+y^2}\).
Explanation: \(\sqrt{10^2+10^2}=10\sqrt2\).
Q42. B12
Vectors
★☆☆☆☆
Find the magnitude of \(\mathbf{a}=(1,2,2)\).
A 2
B 3
C 4
D 5
💡 Hint
📖 Concept
Use \(|\mathbf a|=\sqrt{x^2+y^2+z^2}\).
Magnitude of a 3D vector.
Explanation: \(\sqrt{1+4+4}=3\).
Q43. B13
Statistics Applications
★★★☆☆
\(X\sim N(100,15^2)\). Find \(P(X<115)\) (nearest 0.001).
A 0.500
B 0.683
C 0.841
D 0.977
💡 Hint
📖 Concept
Standardise: \(z=\dfrac{115-100}{15}=1\).
Normal distribution, \(z\)-score approach.
Explanation: \(z=1\Rightarrow P(X<115)\approx0.841\).
Q44. B14
Statistics Applications
★★★☆☆
\(X\sim B(10,0.5)\). Find \(P(X=5)\) (nearest 0.001).
A 0.176
B 0.246
C 0.205
D 0.302
💡 Hint
📖 Concept
Use \(P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}\).
Binomial distribution formula.
Explanation: \(\binom{10}{5}(0.5)^{10}=0.246\).
Q45. B15
Statistics Applications
★★☆☆☆
A game pays \$10 if a die shows 6, and costs \$1 otherwise. Find the expected value (nearest 0.01).
A \$0.83
B \$1.00
C \$1.50
D \$0.50
💡 Hint
📖 Concept
\(E(X)=\sum x\,P(x)\).
Expected value of a discrete random variable.
Explanation: \(\tfrac16(10)+\tfrac56(-1)=\tfrac{5}{6}\approx0.83\).
Q46. B16
Statistics Applications
★☆☆☆☆
A correlation coefficient is \(r=0.9\). Describe the relationship.
A Weak negative
B Strong positive
C No correlation
D Strong negative
💡 Hint
📖 Concept
Values close to \(+1\) mean a strong positive linear relationship.
Interpreting the correlation coefficient \(r\), \(-1\le r\le1\).
Explanation: \(r=0.9\) indicates a strong positive correlation.
Q47. B17
Kinematics
★★☆☆☆
Displacement is \(s(t)=t^3-3t^2\). Find the velocity at \(t=2\).
A -6
B 0
C 6
D 12
💡 Hint
📖 Concept
Velocity is \(s'(t)\); substitute \(t=2\).
Velocity = derivative of displacement.
Explanation: \(v(t)=3t^2-6t\), \(v(2)=12-12=0\).
Q48. B18
Optimisation
★★★☆☆
A rectangle has perimeter 20. Find the maximum possible area.
A 20
B 24
C 25
D 30
💡 Hint
📖 Concept
The maximum area for fixed perimeter occurs for a square.
Optimisation with a perimeter constraint.
Explanation: Side \(=5\), area \(=25\).
Q49. B19
Kinematics
★★☆☆☆
Velocity is \(v(t)=6t-6\). Find the acceleration at \(t=1\).
A -6
B 0
C 6
D 1
💡 Hint
📖 Concept
Acceleration is \(v'(t)\).
Acceleration = derivative of velocity.
Explanation: \(a(t)=v'(t)=6\) (constant), so \(a(1)=6\).
Q50. B20
Related Rates
★★★☆☆
A circle's radius grows at \(\tfrac{dr}{dt}=2\) when \(r=5\). Find \(\tfrac{dA}{dt}\).
A \(10\pi\)
B \(20\pi\)
C \(25\pi\)
D \(40\pi\)
💡 Hint
📖 Concept
Use \(A=\pi r^2\), so \(\tfrac{dA}{dt}=2\pi r\tfrac{dr}{dt}\).
Related rates via the chain rule.
Explanation: \(2\pi(5)(2)=20\pi\).
Chapter C: The Bridge of Twin Kingdoms
SL + HL · Advanced Shared Core
A high stone bridge connecting both kingdoms — the questions here are still shared material, but the path grows steeper and the puzzles trickier.
Q51. C1
Logarithms
★★★☆☆
Solve \(\log_2 x = 5\).
A 10
B 16
C 25
D 32
💡 Hint
📖 Concept
Rewrite in exponential form.
\(\log_a x=y \iff x=a^y\).
Explanation: \(x=2^5=32\).
Q52. C2
Logarithms
★★★★☆
Solve \(\log x + \log(x-3) = 1\) (base 10, \(x>3\)).
A 2
B 4
C 5
D 10
💡 Hint
📖 Concept
Combine logs: \(\log[x(x-3)]=1\).
Log law: \(\log a+\log b=\log(ab)\).
Explanation: \(x(x-3)=10 \Rightarrow x^2-3x-10=0 \Rightarrow x=5\) (reject \(x=-2\)).
Q53. C3
Series
★★★☆☆
An infinite geometric series has \(u_1=4\) and \(S_\infty=8\). Find \(r\).
A 0.25
B 0.5
C 0.75
D 2
💡 Hint
📖 Concept
Use \(S_\infty=\dfrac{u_1}{1-r}\) and solve for \(r\).
Infinite geometric series formula.
Explanation: \(8=4/(1-r)\Rightarrow 1-r=0.5\Rightarrow r=0.5\).
Q54. C4
Binomial
★★★☆☆
Find the coefficient of \(x^3\) in the expansion of \((1+2x)^5\).
A 40
B 60
C 80
D 100
💡 Hint
📖 Concept
Use \(\binom{5}{3}(2)^3\).
General term of binomial expansion: \(\binom{n}{r}a^{n-r}b^r\).
Explanation: \(\binom{5}{3}(2^3)=10\times8=80\).
Q55. C5
Exponentials
★★★☆☆
Solve \(3^{x+1}=27\).
A 1
B 2
C 3
D 4
💡 Hint
📖 Concept
Write 27 as a power of 3.
Equating exponents when bases match.
Explanation: \(3^{x+1}=3^3 \Rightarrow x=2\).
Q56. C6
Transformations
★★★☆☆
If \(f(x)=x^2\), find the vertex of \(y=f(x-2)+3\).
A (2,3)
B (-2,3)
C (2,-3)
D (3,2)
💡 Hint
📖 Concept
The graph shifts right 2 and up 3.
Transformation: \(y=f(x-h)+k\) shifts right \(h\), up \(k\).
Explanation: New vertex: \((2,3)\).
Q57. C7
Composite/Inverse
★★★★☆
\(f(x)=2x+1\), \(g(x)=x-3\). Find \((f\circ g)^{-1}(7)\).
A 5
B 6
C 7
D 8
💡 Hint
📖 Concept
First find \(f(g(x))\), then its inverse, then substitute.
Inverse of a composite function.
Explanation: \(f(g(x))=2x-5\); inverse is \(\tfrac{x+5}{2}\); at \(x=7\): \(6\).
Q58. C8
Functions
★★★☆☆
Which function is self-inverse (i.e. \(f^{-1}(x)=f(x)\))?
A \(f(x)=x^2\)
B \(f(x)=2x\)
C \(f(x)=\dfrac{1}{x}\)
D \(f(x)=x+1\)
💡 Hint
📖 Concept
Test: does applying \(f\) twice return \(x\)?
A self-inverse function satisfies \(f(f(x))=x\).
Explanation: \(f(f(x))=1/(1/x)=x\), so \(f(x)=1/x\) is self-inverse.
Q59. C9
Transformations
★★★☆☆
If \(f(3)=4\), find \(y\) at \(x=3\) for \(y=2f(x)-1\).
A 6
B 7
C 8
D 9
💡 Hint
📖 Concept
Substitute \(f(3)=4\) into the transformation.
Vertical stretch and shift: \(y=2f(x)-1\).
Explanation: \(y=2(4)-1=7\).
Q60. C10
Equations
★★★☆☆
Find the sum of all solutions to \(|2x-3|=7\).
A 1
B 2
C 3
D 4
💡 Hint
📖 Concept
Split into two linear equations.
Absolute value equations give two cases.
Explanation: \(x=5\) or \(x=-2\); sum \(=3\).
Q61. C11
Trig Identities
★★★★☆
If \(\sin\theta=\tfrac35\), \(\cos\theta=\tfrac45\) (\(\theta\) acute), find \(\sin(2\theta)\).
A \(\tfrac{7}{25}\)
B \(\tfrac{12}{25}\)
C \(\tfrac{24}{25}\)
D \(\tfrac{7}{5}\)
💡 Hint
📖 Concept
Use \(\sin2\theta=2\sin\theta\cos\theta\).
Double angle formula for sine.
Explanation: \(2(\tfrac35)(\tfrac45)=\tfrac{24}{25}\).
Q62. C12
Trig Identities
★★★★☆
If \(\sin\theta=\tfrac12\), find \(\cos(2\theta)\).
A \(\tfrac12\)
B \(-\tfrac12\)
C \(\tfrac14\)
D 1
💡 Hint
📖 Concept
Use \(\cos2\theta=1-2\sin^2\theta\).
Double angle formula for cosine.
Explanation: \(1-2(\tfrac14)=\tfrac12\).
Q63. C13
Trig Equations
★★★★☆
Find the sum of all solutions to \(2\sin x - 1 = 0\) for \(0\le x\le 2\pi\).
A \(\tfrac{\pi}{3}\)
B \(\tfrac{2\pi}{3}\)
C \(\pi\)
D \(\tfrac{5\pi}{3}\)
💡 Hint
📖 Concept
Solve \(\sin x=\tfrac12\) on the given interval.
Two solutions in \([0,2\pi]\) with the same sine value.
Explanation: \(x=\tfrac{\pi}{6}\) or \(\tfrac{5\pi}{6}\); sum \(=\pi\).
Q64. C14
Trig Identities
★★★★☆
Find the exact value of \(\tan(75^\circ)\).
A \(2+\sqrt3\)
B \(2-\sqrt3\)
C \(1+\sqrt3\)
D \(\sqrt3\)
💡 Hint
📖 Concept
Use \(\tan(45^\circ+30^\circ)\) and the compound angle formula.
Compound angle formula: \(\tan(A+B)=\dfrac{\tan A+\tan B}{1-\tan A\tan B}\).
Explanation: \(\tan75^\circ=2+\sqrt3\).
Q65. C15
Trig Equations
★★★★☆
Find the sum of all solutions to \(2\cos^2 x - 1 = 0\) for \(0\le x\le\pi\).
A \(\tfrac{\pi}{2}\)
B \(\pi\)
C \(\tfrac{3\pi}{2}\)
D \(2\pi\)
💡 Hint
📖 Concept
Note \(2\cos^2x-1=\cos2x\).
Double angle identity simplifies the equation.
Explanation: \(\cos2x=0 \Rightarrow x=\tfrac{\pi}{4},\tfrac{3\pi}{4}\); sum \(=\pi\).
Q66. C16
Conditional Probability
★★★☆☆
\(P(A)=0.3\), \(P(B)=0.5\), \(P(A\cap B)=0.2\). Find \(P(A\mid B)\).
A 0.2
B 0.4
C 0.6
D 0.67
💡 Hint
📖 Concept
Use \(P(A\mid B)=\dfrac{P(A\cap B)}{P(B)}\).
Conditional probability formula.
Explanation: \(0.2/0.5=0.4\).
Q67. C17
Probability
★★★★☆
A box has 5 red and 3 blue balls. Two are drawn without replacement. Find \(P(\text{both red})\).
A \(\tfrac{5}{14}\)
B \(\tfrac{5}{8}\)
C \(\tfrac{25}{64}\)
D \(\tfrac{4}{7}\)
💡 Hint
📖 Concept
Multiply the probabilities for each draw (without replacement).
Dependent events: sample space shrinks after each draw.
Explanation: \(\tfrac58\times\tfrac47=\tfrac{5}{14}\).
Q68. C18
Regression
★★☆☆☆
A regression line is \(y=2x+3\). Predict \(y\) when \(x=10\).
A 20
B 21
C 23
D 25
💡 Hint
📖 Concept
Substitute \(x=10\) into the equation.
Using a regression line for prediction.
Explanation: \(y=2(10)+3=23\).
Q69. C19
Statistics
★★★☆☆
A dataset has mean 70 and standard deviation 5. Find the \(z\)-score for \(x=80\).
A 1
B 1.5
C 2
D 2.5
💡 Hint
📖 Concept
Use \(z=\dfrac{x-\mu}{\sigma}\).
Standardisation (z-score) formula.
Explanation: \(z=(80-70)/5=2\).
Q70. C20
Statistics
★★★☆☆
Class A (20 students) averages 80. Class B (30 students) averages 90. Find the combined mean.
A 84
B 85
C 86
D 87
💡 Hint
📖 Concept
Use a weighted average.
Combined mean \(=\dfrac{n_1\bar x_1+n_2\bar x_2}{n_1+n_2}\).
Explanation: \((20\times80+30\times90)/50=86\).
Q71. C21
Chain Rule
★★★☆☆
Find \(f'(0)\) if \(f(x)=(2x+1)^3\).
A 3
B 6
C 9
D 12
💡 Hint
📖 Concept
Use the chain rule.
Chain rule: \(\dfrac{d}{dx}[g(x)]^n=n[g(x)]^{n-1}g'(x)\).
Explanation: \(f'(x)=3(2x+1)^2(2)\); \(f'(0)=6\).
Q72. C22
Product Rule
★★★★☆
Find \(f'(0)\) if \(f(x)=x^2\sin x\).
A -1
B 0
C 1
D 2
💡 Hint
📖 Concept
Use the product rule.
Product rule: \((uv)'=u'v+uv'\).
Explanation: \(f'(x)=2x\sin x+x^2\cos x\); \(f'(0)=0\).
Q73. C23
Quotient Rule
★★★★☆
Find \(f'(1)\) if \(f(x)=\dfrac{x}{x+1}\).
A \(\tfrac14\)
B \(\tfrac12\)
C \(\tfrac13\)
D 1
💡 Hint
📖 Concept
Use the quotient rule.
Quotient rule: \(\left(\dfrac uv\right)'=\dfrac{u'v-uv'}{v^2}\).
Explanation: \(f'(x)=\dfrac{1}{(x+1)^2}\); \(f'(1)=\tfrac14\).
Q74. C24
Definite Integrals
★★★☆☆
Evaluate \(\displaystyle\int_0^2 x^2\,dx\).
A \(\tfrac83\)
B 4
C \(\tfrac43\)
D 8
💡 Hint
📖 Concept
Find the antiderivative, then evaluate at the bounds.
Fundamental theorem of calculus.
Explanation: \([\tfrac{x^3}{3}]_0^2=\tfrac83\).
Q75. C25
Area Between Curves
★★★★☆
Find the area under \(y=4-x^2\) between \(x=-2\) and \(x=2\).
A \(\tfrac{16}{3}\)
B \(\tfrac{32}{3}\)
C 16
D 32
💡 Hint
📖 Concept
Integrate \(4-x^2\) between the bounds.
Definite integral gives the area under a curve.
Explanation: \(\int_{-2}^2(4-x^2)dx=\tfrac{32}{3}\).
Chapter D: The HL Sorcerer's Tower
HL Exclusive
Only HL adventurers climb this tower. Complex numbers, proof, further vectors and further calculus await at the summit.
Q76. D1
Complex Numbers
★★★☆☆
Simplify \((3+2i)+(1-4i)\).
A 4+2i
B 2-2i
C 4-2i
D 2+2i
💡 Hint
📖 Concept
Add real parts and imaginary parts separately.
Complex number addition.
Explanation: \((3+1)+(2-4)i=4-2i\).
Q77. D2
Complex Numbers
★★★☆☆
Simplify \((2+i)(3-i)\).
A 5+i
B 7+i
C 6-i
D 7-i
💡 Hint
📖 Concept
Expand using FOIL and \(i^2=-1\).
Multiplying complex numbers.
Explanation: \(6-2i+3i-i^2=6+i+1=7+i\).
Q78. D3
Complex Numbers
★★☆☆☆
Find \(|z|\) for \(z=3+4i\).
A 5
B 7
C 25
D 4
💡 Hint
📖 Concept
Use \(|z|=\sqrt{a^2+b^2}\).
Modulus of a complex number.
Explanation: \(\sqrt{9+16}=5\).
Q79. D4
Complex Numbers
★★★★☆
Write \(z=1+i\) in polar form \(r\,\text{cis}\theta\).
A \(\sqrt2\,\text{cis}\tfrac{\pi}{4}\)
B \(2\,\text{cis}\tfrac{\pi}{4}\)
C \(\sqrt2\,\text{cis}\tfrac{\pi}{2}\)
D \(1\,\text{cis}\tfrac{\pi}{4}\)
💡 Hint
📖 Concept
Find \(r=|z|\) and \(\theta=\arctan(b/a)\).
Polar form: \(r\,\text{cis}\theta\), where \(r=\sqrt{a^2+b^2}\).
Explanation: \(r=\sqrt2\), \(\theta=\tfrac{\pi}{4}\).
Q80. D5
De Moivre
★★★★★
Using De Moivre's theorem, find \(\left(\text{cis}\tfrac{\pi}{6}\right)^3\).
A 1
B -1
C i
D -i
💡 Hint
📖 Concept
Multiply the angle by the power: \(\theta\times n\).
De Moivre: \((\text{cis}\theta)^n=\text{cis}(n\theta)\).
Explanation: \(\text{cis}\tfrac{\pi}{2}=i\).
Q81. D6
Complex Numbers
★★★☆☆
Solve \(z^2=-9\).
A \(z=\pm3\)
B \(z=\pm3i\)
C \(z=\pm9i\)
D \(z=\pm9\)
💡 Hint
📖 Concept
Take the square root of both sides, remembering \(i^2=-1\).
Solving quadratics with negative discriminant.
Explanation: \(z=\pm\sqrt{-9}=\pm3i\).
Q82. D7
Proof by Induction
★★★★☆
To prove \(\sum_{k=1}^n k = \tfrac{n(n+1)}{2}\) by induction, what is checked in the base case \(n=1\)?
A LHS \(=1\), RHS \(=1\)
B LHS \(=2\), RHS \(=1\)
C LHS \(=1\), RHS \(=2\)
D LHS \(=0\), RHS \(=0\)
💡 Hint
📖 Concept
Substitute \(n=1\) into both sides of the statement.
Induction base case verifies the statement for the smallest case.
Explanation: LHS \(=1\); RHS \(=1(2)/2=1\). They match.
Q83. D8
Proof by Contradiction
★★★★☆
In the classic proof that \(\sqrt2\) is irrational, we assume \(\sqrt2=\tfrac pq\) in lowest terms. What is derived next?
A \(p\) and \(q\) are both odd
B \(p^2=2q^2\), so \(p\) is even
C \(q=0\)
D \(p=q\)
💡 Hint
📖 Concept
Square both sides and isolate \(p^2\).
Contradiction proofs derive a statement that conflicts with the assumption.
Explanation: \(p^2=2q^2\) shows \(p\) must be even, eventually contradicting 'lowest terms'.
Q84. D9
Direct Proof
★★★☆☆
To prove the sum of two even numbers is even, we write them as \(2m\) and \(2n\). Which expression completes the proof?
A \(2m+2n=2(m+n)\)
B \(2m+2n=4mn\)
C \(2m+2n=m+n\)
D \(2m+2n=2mn\)
💡 Hint
📖 Concept
Factor out the common factor of 2.
A direct proof manipulates the general form algebraically.
Explanation: \(2m+2n=2(m+n)\), which is even since \(m+n\) is an integer.
Q85. D10
Proof by Induction
★★★★☆
Assuming \(\sum_{k=1}^{k}k=\tfrac{k(k+1)}{2}\) is true, which expression gives the sum up to \(k+1\)?
A \(\tfrac{k(k+1)}{2}+(k+1)\)
B \(\tfrac{k(k+1)}{2}+k\)
C \(\tfrac{(k+1)(k+1)}{2}\)
D \(\tfrac{k(k+1)}{2}\times(k+1)\)
💡 Hint
📖 Concept
Add the next term, \((k+1)\), to the inductive hypothesis.
Inductive step: add the \((k+1)\)th term to both sides.
Explanation: \(\tfrac{k(k+1)}{2}+(k+1)=\tfrac{(k+1)(k+2)}{2}\), completing the step.
Q86. D11
Vectors
★★★☆☆
Find \(\mathbf{a}\times\mathbf{b}\) for \(\mathbf{a}=(1,0,0)\), \(\mathbf{b}=(0,1,0)\).
A (0,0,1)
B (1,1,0)
C (0,1,1)
D (1,0,1)
💡 Hint
📖 Concept
Use the cross product formula or recognise \(\hat i\times\hat j=\hat k\).
Cross product of standard unit vectors.
Explanation: \(\mathbf{a}\times\mathbf{b}=(0,0,1)\).
Q87. D12
Vectors
★★★★☆
Find \(\mathbf{a}\times\mathbf{b}\) for \(\mathbf{a}=(2,1,-1)\), \(\mathbf{b}=(1,-1,2)\).
A (1,-5,-3)
B (1,5,-3)
C (-1,-5,3)
D (1,-5,3)
💡 Hint
📖 Concept
Use the determinant formula for the cross product.
Cross product: \(\mathbf a\times\mathbf b=(a_2b_3-a_3b_2,\,a_3b_1-a_1b_3,\,a_1b_2-a_2b_1)\).
Explanation: \((2,1,-1)\times(1,-1,2)=(1,-5,-3)\).
Q88. D13
Planes
★★★☆☆
Find the equation of the plane through \((1,2,3)\) with normal \((1,1,1)\).
A x+y+z=6
B x+y+z=3
C x+y+z=0
D x-y+z=6
💡 Hint
📖 Concept
Use \(n_1(x-x_0)+n_2(y-y_0)+n_3(z-z_0)=0\).
Plane equation from a point and a normal vector.
Explanation: \((x-1)+(y-2)+(z-3)=0 \Rightarrow x+y+z=6\).
Q89. D14
Lines & Planes
★★★★☆
A line is \(\mathbf r=(1,0,0)+t(0,0,1)\). Find where it meets the plane \(z=5\).
A (1,0,5)
B (0,0,5)
C (1,5,0)
D (5,0,1)
💡 Hint
📖 Concept
Set the \(z\)-component of the line equal to 5, solve for \(t\).
Substitute the line's parametric equation into the plane equation.
Explanation: \(t=5\), giving point \((1,0,5)\).
Q90. D15
Distance
★★★★☆
Find the distance from \((0,0,0)\) to the plane \(x+y+z=6\).
A \(2\sqrt3\)
B \(3\sqrt2\)
C 6
D \(\sqrt6\)
💡 Hint
📖 Concept
Use \(d=\dfrac{|ax_0+by_0+cz_0-d|}{\sqrt{a^2+b^2+c^2}}\).
Distance from a point to a plane formula.
Explanation: \(\dfrac{|{-6}|}{\sqrt3}=2\sqrt3\).
Q91. D16
Implicit Differentiation
★★★★☆
For \(x^2+y^2=25\), find \(\dfrac{dy}{dx}\) at \((3,4)\).
A \(-\tfrac34\)
B \(\tfrac34\)
C \(-\tfrac43\)
D \(\tfrac43\)
💡 Hint
📖 Concept
Differentiate both sides with respect to \(x\), treating \(y\) as a function of \(x\).
Implicit differentiation.
Explanation: \(2x+2yy'=0 \Rightarrow y'=-x/y=-3/4\).
Q92. D17
Integration by Parts
★★★★★
Evaluate \(\displaystyle\int_0^1 xe^x\,dx\).
A 0
B 1
C e
D e-1
💡 Hint
📖 Concept
Use \(\int u\,dv=uv-\int v\,du\) with \(u=x\), \(dv=e^x dx\).
Integration by parts formula.
Explanation: \([xe^x-e^x]_0^1=0-(-1)=1\).
Q93. D18
Partial Fractions
★★★★☆
Decompose \(\dfrac{1}{(x-1)(x+1)}=\dfrac{A}{x-1}+\dfrac{B}{x+1}\). Find \(A\).
A \(\tfrac12\)
B \(-\tfrac12\)
C 1
D 2
💡 Hint
📖 Concept
Multiply through and substitute \(x=1\).
Partial fraction decomposition.
Explanation: \(A=\tfrac12\).
Q94. D19
Differential Equations
★★★★☆
Solve \(\dfrac{dy}{dx}=ky\), \(y(0)=5\), \(k=2\). Find \(y(1)\).
A \(5e\)
B \(5e^2\)
C \(10e\)
D \(2e^5\)
💡 Hint
📖 Concept
The general solution is \(y=Ce^{kx}\); use the initial condition to find \(C\).
Separable first-order differential equation.
Explanation: \(y=5e^{2x}\), so \(y(1)=5e^2\).
Q95. D20
Maclaurin Series
★★★★☆
Find the first two terms of the Maclaurin series for \(e^x\).
A 1 + x
B 1 + \(\tfrac{x^2}{2}\)
C x
D 1 - x
💡 Hint
📖 Concept
Use \(f(x)=f(0)+f'(0)x+\ldots\).
Maclaurin series: \(f(x)=\sum \dfrac{f^{(n)}(0)}{n!}x^n\).
Explanation: \(e^x=1+x+\dfrac{x^2}{2!}+\ldots\); first two terms: \(1+x\).
Q96. D21
Systems of Equations
★★★★☆
Solve: \(x+y+z=6\), \(2x-y+z=3\), \(x+2y-z=2\). Find \(x\).
A 1
B 2
C 3
D 0
💡 Hint
📖 Concept
Use elimination or substitution across the three equations.
Solving a 3x3 system of linear equations.
Explanation: The solution is \(x=1, y=2, z=3\).
Q97. D22
Further Binomial
★★★★☆
Find the coefficient of \(x^2\) in the expansion of \((1+x)^{-1}\) (as an infinite series).
A -1
B 0
C 1
D 2
💡 Hint
📖 Concept
Use the general binomial series \((1+x)^n=1+nx+\tfrac{n(n-1)}{2!}x^2+\ldots\) with \(n=-1\).
Extended binomial theorem for non-integer/negative \(n\).
Explanation: \((1+x)^{-1}=1-x+x^2-\ldots\); coefficient of \(x^2\) is 1.
Q98. D23
Further Binomial
★★★★★
Find the coefficient of \(x^2\) in the expansion of \((1+x)^{1/2}\).
A \(\tfrac18\)
B \(-\tfrac18\)
C \(\tfrac14\)
D \(-\tfrac14\)
💡 Hint
📖 Concept
Use \(n=\tfrac12\) in the general binomial series.
Extended binomial theorem with a fractional index.
Explanation: \((1+x)^{1/2}=1+\tfrac x2-\tfrac{x^2}{8}+\ldots\); coefficient is \(-\tfrac18\).
Q99. D24
Systems of Equations
★★★☆☆
Solve the system \(2x+3y=12\), \(x-y=1\). Find \(x+y\).
A 3
B 4
C 5
D 6
💡 Hint
📖 Concept
Solve for \(x\) and \(y\) individually, then add.
Solving simultaneous linear equations.
Explanation: \(x=3, y=2\); \(x+y=5\).
Q100. D25
Maclaurin Series
★★★★☆
Find the coefficient of \(x^2\) in the Maclaurin series for \(\cos x\).
A \(\tfrac12\)
B \(-\tfrac12\)
C 1
D -1
💡 Hint
📖 Concept
Use \(\cos x=1-\tfrac{x^2}{2!}+\tfrac{x^4}{4!}-\ldots\).
Standard Maclaurin series for cosine.
Explanation: Coefficient of \(x^2\) is \(-\tfrac12\).