On this page:
Q1. Rounding to Significant Figures Q2. Standard Form (Scientific Notation) Q3. Arithmetic Sequence - Term & Sum Q4. Geometric Sequence - Term & Sum Q5. Depreciation Q6. Percentage Error Q7. Sigma Notation Q8. Arithmetic Sequence - nth term formula Q9. Geometric Sequence - r and S∞ Q10. Compound Interest (Quarterly) Q11. Geometric Sequence with Logarithms Q12. Loan Repayment (Finance Solver) Q13. Annuity - Deposits at Start Q14. Matrix Equation - Inverse Method Q15. Sigma Notation Evaluation Q16. Depreciation (Different Rates) Q17. Arithmetic Sequence - Positive Term, Min Sum Q18. Complex Numbers - Argand Diagram Q19. Infinite Geometric Series Q20. Matrix Eigenvalues & Eigenvectors (HL) Q21. Complex Numbers - De Moivre (HL) Q22. System of Linear Equations - Matrix Q23. Arithmetic & Geometric Sequence Combined Q24. Pension Scheme - Annuity & Perpetuity Q25. Markov Chains & Transition Matrices (HL)Practice questions covering rounding significant figures, standard form, arithmetic and geometric sequences and series, compound interest, loans, annuities, matrices, determinants, inverses, complex numbers (modulus, argument, polar form, De Moivre's theorem), Markov chains, and eigenvalues.
Round 0.003 472 to 2 significant figures.
Answer: 0.0035
Write 0.000 056 7 in the form a × 10ᵏ, where 1 ≤ a < 10 and k ∈ ℤ.
Answer: 5.67 × 10⁻⁵
An arithmetic sequence has first term u₁ = 5 and common difference d = 4. Find: (a) the 12th term, (b) the sum of the first 12 terms.
(a) u₁₂ = ?
(b) S₁₂ = ?
Answer: (a) u₁₂ = 49 (b) S₁₂ = 324
A geometric sequence has u₁ = 3 and common ratio r = 2. Write down u₅ and find the sum S₅.
Answer: u₅ = 93
A car bought for $24 000 depreciates at 15% per year. Find its value after 3 years, giving your answer to the nearest dollar.
Answer: 24000 × (0.85)³ = $14 739
The approximate value of √2 is 1.41. The exact value is 1.41421… Calculate the percentage error in the approximation.
Answer: ε ≈ 0.298%
Evaluate Σ (k=1 to 6) of (2k + 3).
Answer: = 60
The nth term of an arithmetic sequence is given by uₙ = 3n − 7.
(a) Find u₁ and d.
(b) Find the value of n for which uₙ = 62.
(c) Find S₂₅.
Answer: (a) u₁ = −4, d = 3 (b) n = 23 (c) S₂₅ = 800
A geometric sequence has u₁ = 64 and u₄ = 8.
(a) Find the common ratio r.
(b) Find the sum to infinity S∞.
(c) Find the smallest n for which uₙ < 0.001.
Answer: (a) r = 0.5 (b) S∞ = 128 (c) n = 18
Francesca invests €8 500 at a nominal annual interest rate of 5.4%, compounded quarterly.
(a) Find the value of her investment after 6 years.
(b) Find the number of full years needed for the investment to exceed €15 000.
Answer: (a) €11 591.92 (b) 12 years
The first three terms of a geometric sequence are ln(2x+7), ln(2x+7)/3, ln(2x+7)/9.
(a) Show that the common ratio is 1/3.
(b) State the condition for the sum to infinity to exist.
(c) Find S∞ in terms of x.
Answer: (a) r = [ln(2x+7)/3] ÷ ln(2x+7) = 1/3 (b) Since |1/3| < 1, sum to infinity always exists (c) S∞ = (3/2)ln(2x+7)
Ahmed takes a loan of $15 000 at an annual interest rate of 6%, compounded monthly, to be repaid in equal monthly instalments over 4 years.
(a) Find the monthly repayment using the Finance Solver.
(b) Find the total amount repaid.
(c) Find the total interest paid.
Answer: (a) $352.28/month (b) $16 909.44 (c) $1 909.44
Lucia deposits $250 at the start of each month into a savings account earning 3.6% per annum compounded monthly. She makes deposits for 8 years.
(a) Find the total value of her savings at the end of 8 years.
(b) Find the total amount deposited.
Answer: (a) FV ≈ $28 428.69 (b) $250 × 96 = $24 000
Consider the matrix equation AX = B where A = [[3, −1], [2, 4]] and B = [[7], [14]].
(a) Find det(A).
(b) Find A⁻¹.
(c) Hence solve for X = [[x], [y]].
Answer: (a) det(A) = 14 (b) A⁻¹ = (1/14)[[4,1],[−2,3]] (c) x = 3, y = 2
Evaluate: Σ (r=3 to 10) of (5r − 2). Show all working using arithmetic series formulas.
Answer: 244
A piece of equipment bought for $50 000 loses 20% of its value in the first year, then 15% per year after that. Find the value after (a) 1 year, (b) 5 years from new. Give answers to 3 s.f.
Answer: (a) $40 000 (b) $20 900 (3 s.f.)
The first term of an arithmetic sequence is −86 and common difference is 4.
(a) Find the first positive term and state which term it is.
(b) Find the minimum value of Sₙ.
Answer: (a) First positive term = 2, it is the 23rd term (b) Min Sₙ = −968
The Argand diagram shows three complex numbers z₁ = 2+3i, z₂ = −1+2i, and z₃ = 3−2i.
(a) Calculate |z₁| and arg(z₁), giving arg in radians to 3 d.p.
(b) Write z₁ in polar form r(cosθ + i sinθ) and in Euler form reⁱᶿ.
(c) Find z₁·z₂ and express in rectangular form.
(d) Find z₁/z₃ in rectangular form a+bi.
Answer: (a) |z₁|=√13≈3.606, arg=0.983 rad (b) √13·eⁱ⁰·⁹⁸³ (c) −8+i (d) 0+i = i
The graph shows the terms and partial sums of the geometric sequence uₙ = 48·(½)ⁿ⁻¹.
(a) Use the formula to find S∞ and verify with the graph.
(b) Find the smallest n such that |Sₙ − S∞| < 0.5.
(c) The sum of the first n terms exceeds 95. Find the smallest such n.
(d) [HL] Find all values of x for which the series Σxⁿ·48·(½)ⁿ⁻¹ converges, and state S∞ in terms of x.
Answer: (a) 96 (b) n=8 (c) n=7 (d) −2
Matrix A = [[2,1],[0,3]] transforms the unit square as shown.
(a) Find the eigenvalues of A.
(b) For each eigenvalue, find the corresponding eigenvector.
(c) Write A = PDP⁻¹ where D is the diagonal matrix of eigenvalues.
(d) [HL] Hence find A⁵.
Answer: (a) λ=2 and λ=3 (b) v₁=[1,0]ᵀ, v₂=[1,1]ᵀ (c) P=[[1,1],[0,1]] (d) A⁵=[[32,211],[0,243]]
Complex number z = 1 + i.
(a) Write z in polar form and Euler form.
(b) Use De Moivre's theorem to find z⁸.
(c) Find all fourth roots of z, giving answers in Euler form.
(d) Show that the sum of all fourth roots of any complex number is always 0.
Answer: (a) √2·eⁱᵖⁱ/⁴ (b) z⁸ = 16 (c) 2^(1/8)·eⁱ⁽ᵖⁱ/¹⁶⁺ᵏᵖⁱ/²⁾, k=0,1,2,3 (d) Shown
Solve the system of equations using matrices: 2x + y − z = 4, x − 2y + 3z = −6, 3x + 4y − 2z = 11.
(a) Write in matrix form AX = B.
(b) Find det(A).
(c) Find A⁻¹ using GDC.
(d) Hence find x, y, z.
Answer: (a) Written (b) det(A) = −15 (c) Computed via GDC (d) x=1, y=2, z=1
The first term of a sequence is u₁ = a and uₙ₊₁ = uₙ + d (arithmetic, d>0). Another sequence has v₁ = a and vₙ₊₁ = vₙ·r (geometric, r>1). Given: u₄ = v₃, u₈ = v₅, and a=2.
(a) Write two equations in d and r.
(b) Show that r² − 3r + 2 = 0.
(c) Since r>1, find r and d.
(d) Find which term is the first where vₙ > uₙ by 100 or more.
Answer: (a) 2+3d=2r², 2+7d=2r⁴ (b) Shown (c) r=2, d=6 (d) n=8
A pension scheme: Marco pays $400 at the start of each month into an account earning 4.8% per annum compounded monthly for 30 years.
(a) Find the total value of his pension fund after 30 years.
(b) He then withdraws $2 800 per month from this fund (same rate applies). Find how many months the fund will last.
(c) Find the total amount withdrawn and compare to total deposited.
(d) [HL] If he wanted to withdraw permanently (fund never depletes), find the maximum monthly withdrawal.
Answer: (a) ≈$318 382 (b) ≈170 months (≈14.2 years) (c) Withdrawn $476K vs deposited $144K (d) $1273.53/month
A town has two internet providers: Aster (A) and Bolt (B). Each month, 20% of Aster customers switch to Bolt, and 15% of Bolt customers switch to Aster.
(a) Write the transition matrix T.
(b) If initially 6000 use Aster and 4000 use Bolt, find the state after 3 months.
(c) Find the steady-state (long-run) distribution.
(d) Find the eigenvalues of T and verify the steady state.
Answer: (a) [[0.80,0.15],[0.20,0.85]] (b) via GDC (c) ≈4286 Aster, 5714 Bolt (d) λ=1, λ=0.65