I. Multiple Choices based on the lecture notes 1. Which of the following statements is NOT correct ? (a) Ancient Egyptian recorded their mathematical work on papyrus. (b) Babylonian knew how to find areas for triangles and trapezoids (c) Egyptians had two different number systems. (d) Babylonian invented zero. 2. Which one of the following Schools did Zeno belong to in the classical period ? (a) Sophist School. (b) Platonic School. (c) Ionian School (d) Eleatic School 1 History of Mathematics 3. Which of the following statements was discovered or claimed by Thales ? (a) A triangle inscribed in a semicircle has a right angle (b) The planets and starts moved according to mathematical rules which corresponded to musical notes and thus produced a symphony. (c) A flying arrow is motionless. (d) Study of geometry is for drawing the soul towards truth. 4. Which of the following triples is NOT Pythagorean numbers? (a) (5, 12, 13) (b) (21, 20, 29) (c) (65, 72, 97) (d) (45, 26, 53) 5. Which of the following CANNOT be true: (a) Plato was a student of Socrates. (b) Euclid was a student of Apollonius. (c) Pythagoras was a student of Pherekydes. (d) Aristotle was a student of Plato. 6. For the following statements • (1) π is an irrational number. • (2) √3 2 is an irrational number. • (3) e is an irrational number. which of them are correct? (a) (1) and (2). (b) (2) and (3). (c) (3) and (1). (d) all of them. 7. In Palto’s Academy, which of the following objects was NOT included in the mathematical part of the education: 2 (a) Algebra (b) Geometry (c) Astronomy (d) Arithmetic 8. Which mathematical results were NOT included into Euclid’s Elements? (a) Apollonius’ conic section (b) Eudoxus’ proportions theory (c) Pythagoras theorem (d) Five postulates 9. What is the heart of Aristotle’s indirect method: (a) The law of contradiction (b) The law of exclude middle (c) The law of identity (d) Both 1 and 2 (e) Both 1 and 3 (f) Both 2 and 3 10. Euclid’s Elements was used as a textbook on geometry throughout the Western world for about (a) 1500 years (b) 2000 years (c) 1700 years (d) 1300 years 3 11. The theory of conic sections is useful because of (a) Copernicus’s fundamental work on the solar system. (b) Ptolemy’s model for the solar system. (c) Galileo’s observation on the solar system. (d) Kepler’s discovery on the solar system. 12. The word “ Museum” in the Alexandrian Greek Period means (a) a building which houses and cares for a collection of objects for public viewing through exhibits. (b) a private collections of wealthy individuals, families or institutions of art and rare or curious natural objects and artifacts. (c) a building for storing documents and scholars’ work and for general public use. (d) a center for scholars, including poets, philologists, astronomers and mathematicians. 13. What did Diophantus study or contribute ? (a) solve the equation x n + y n = z n for n ≥ 3. (b) curves of equations. (c) solutions of all cubic equations. (d) algebraic equation with integer coefficients. 14. Before Greece was destroyed, in Roma by a “mathematician”, it means (a) an astrologer (b) a mathematical teacher (c) a scientist (d) a geometer 15. In medieval China, Zhu Zhongzhi’s most important work was determined π to 7 digits: 3.1415926, by using Liu Hui’s algorithm applied to a 12288-gon. This value had remained the most accurate approximation of π available for the next 900 years. He also proposed use a fraction for close approximation value of π. Which of the fraction is proposed by him? 4 (a) 335 113 (b) 345 113 (c) 355 113 (d) 365 113 16. Which one of the following contributions is NOT due to Indian mathematicians? (a) Find a formula for the area of a cyclic quadrilateral. (b) Invention of zero. (c) Solve two equations with two unknowns. (d) Using negative numbers. 17. In Medieval Arab, which of the following was(were) translated by Arabic mathematicians from Greek mathematical manuscripts: (a) Euclidean’s Elements (b) Ptolemy’s Mathematical Syntaxis (c) the works of Apollonius (d) both (a) and (b) (e) both (a) and (c) (f) both (b) and (c) (g) All of (a)(b)(c) 18. In Medieval Europe, to read mathematical books, one needed to know (a) German (b) Dutch (c) Latin (d) French 5 19. Until the late 15th century, Hindu-Arabic numerals seem to have predominated among mathematicians, while merchants in Italy preferred to use the Roman numerals. Why did the merchants prefer not to use Hindu-Arabic numerals? (a) They did not know Latin (b) They did not trust it (c) They did not know mathematics (d) They had no chance to learn 20. The fact that at first only mathematicians supported Copernicus’ heliocentric theory because (a) people regarded Copernicus’s work merely as a mathematical hypothesis (b) the mathematical details of Copernicus’s work was too difficult for most people. (c) general public did not care about it. (d) both (a) and (b) (e) both (a) and (c) (f) both (b) and (c) (g) all of (a)(b)(c) 21. Besides the cubic equations, the discovery the solution of quartic equations was credited by (a) Lodovico Ferrari (b) Niccolo Tartaglia (c) Gerolamo Cardano (d) Scipione del Ferro 22. Based on the work by John Napier, slide rules had been used by all engineers and college engineering students during the period of (a) from 1600s to 1920s (b) from 1600s to 1950s (c) from 1600s to 1900s (d) from 1600s to 1960s 6 23. In the improvement of the theory of equations, Vi`ete’s main achievement were (a) letters near the beginning of the alphabet, such as a, b, c, represent known quantities (b) letters near the end, such as x, y, z, represent unknown quantities (c) represent numbers by letters (d) use × to denote multiplication 24. Ren´e Descartes’ main contribution to analytic geometry is that (a) he discovered the fact that each curve in plane or in the space can be represented as the zero set of some equation(s) (b) he discovered the techniques to manipulate equations to obtain information about curves. (c) he discovered a way to compute normal line. (d) both (a) and (b) (e) both (a) and (c) (f) both (b) and (c) (g) all of (a)(b)(c). 25. Which of the following is NOT true? (a) Fermat is generally regarded as one of the greatest mathematicians of all times. (b) Fermat’s work entitles him to be regarded as one of the founders of calculus. (c) Fermat was in Paris to meet some mathematician. (d) Fermat and Descartes had a dispute. 26. The origin of projective geometry was from (a) philosophers (b) artists (c) mathematicians (d) geometers 7 27. Isaac Newton was a student of (a) John Wallis (b) Isaac Barrow (c) Evangelista Torricelli (d) Bonaventura Cavalieri 28. During the two years period at his home in Woolsthorpe in 1665, Isaac Newton DID NOT develop (a) his theory on variation of calculus (b) his theory on calculus (c) his theory on optics (d) his theory on law of gravitation 29. Which of the following was proved by Jacob Bernoulli? (a) 1 + 1 2 + 1 3 + ... + 1 n + ... diverges. (b) 1 + 1 2 2 + 1 3 2 + ... + 1 n2 + .... converges to a number less than 2. (c) 1 + 1 2 2 + 1 3 2 + ... + 1 n2 + .... = π 2 6 . (d) 1 − 1 2 + 1 3 + ... + (−1)n+1 1 n + ... converges. 30. Which of the following discoveries was NOT made by Euler? (a) 1 + 1 2 2 + 1 3 2 + 1 4 2 + ... = π 2 6 (b) cos θ + i sin θ = e iθ . (c) V + F = E + 2 for any polyhedron. (d) There are five regular polyhedrons. 31. When he was 19 years old, Lagrange wrote letters to Euler in which he described (a) Lagrange multiplier (b) Lagrange mean value theorem (c) Lagrange error estimate (d) Eulier-Lagrange equation 8 32. Which contribution did Gauss make in the field of differential geometry? (a) Gaussian function (b) Gaussian curvature (c) Gaussian elimination (d) Gaussian distribution 33. While J´anos Bolyai discovered the non-Euclidean geometry in 1832 and Nicolai Lobachevsky discovered the non-Euclidean geometry in 1829, when did Gauss discover it ? (a) 1792 (b) 1823 (c) 1825 (d) 1817 34. The imaginary number i = √ −1 was introduced motivated by (a) some geometric problem (b) imagination (c) some polynomial equation of degree 3 (d) some arithmetic problem 35. Which modern definition was NOT given by Augustin Cauchy ? (a) limit (b) uniform convergence (c) derivative (d) continuous 36. The geometries proposed by Riemann (a) contained Euclidean geometry. (b) contained Euclidean geometry and the Gauss-Bolyai-Lobachesky non-Euclidean gemetry. (c) more geometries including Euclidean geometry and the Gauss-Bolyai-Lobachesky non-Euclidean gemetry. 9 37. Einstein believed that geometry can be regarded as the most ancient branch of physics because (a) Gravity’s action is instaneous. (b) Grossman is a mathematician. (c) Mathematical tool for his general relativity theory is geometry. (d) Some mathematicians studied both ohysics and mathematics. 38. Which of the following problems Niels Abel did not make contribution: (a) Polynomial equations with radical solutions (b) Riemannian geometry (c) elliptic integrals (d) Abelian groups 39. What did Evariste Galois try to prove before he died? ´ (a) A radical solution to all quintic equations was impossible. (b) A necessary and sufficient for a polynomial equation can be solved by radicals. (c) elliptic integrals (d) Uniform convergence theorem 40. Which field does Poincar´e conjecture belong to ? (a) Differential equations (b) Topology (c) Differential geometry (d) Complex analysis 10 2. Mathematical problems 1. Prove that √ 3 is an irrational number. Proof: 2. For the Fibonacci series: F1 = 1, F2 = 1 and Fn = Fn−1 + Fn−2 for n ≥ 3. Prove F 2 1 + F 2 2 + ... + F 2 n = FnFn+1. Proof: