CAT 2008

Area of polygon

2008-02-12T16:47:00.000+05:30

Polygon Tutorial Polygon Definition:
A polygon is a plane figure that is bounded by a closed path or circuit, composed of a finite sequence of straight line segments.

Polygon Formula:
Using length of a side :
Area of Polygon = ((side)² * N) / (4Tan(π / N))
Perimeter of Polygon = N * (side)
Using radius (circumradius) :
Area of Polygon = ½ * R² * Sin(2π / N)
Using apothem (inradius) :
Area of Polygon = A² * N * Tan(π / N)
where A = R * Cos(π / N)
Using apothem and length of a side :
Area of Polygon = (A * P) / 2
where A = side / (2 * Tan(π / N))
where,
N = Number of sides, A = Apothem, R = Radius, P = Perimeter

Polygon Image/Diagram

Polygon Example:
Case 1: Find the area and perimeter of a polygon with the length 2 and the number of sides is 4.

Step 1: Find the area.
Area = ((side)² * N) / (4Tan(π / N))
= ((2)² * 4) / (4 * Tan(3.14 / 4))
= (4 * 4) / 4 * Tan(0.785)
= 16 / 4 * 0.999
= 16 / 3.996
Area = 4.

Step 2: Find the perimeter.
Perimeter = (N * (side) = 4 * 2 = 8

Case 2: Find the area of a polygon with the given radius 2 and the number of sides is 5.

Step 1: Find the area.
Area = ½ * R² * Sin(2π / N)
= (0.5) * 2² * Sin(2 * 3.14 / 5)
= 0.5 * 4 * Sin(6.28 / 5)
= 2 * Sin(1.26)
= 2 * 0.95
Area = 1.9.

Case 3:Find the area of a polygon with the given radius 2 and the number of sides is 5 using Apothem.

Step 1: Find the apothem.
Apothem = R * Cos(π / N)
= 2 * Cos(3.14 / 5)
= 2 * Cos(0.63)
= 2 * 0.81
Apothem = 1.62.

Step 2: Find the area.
Area = A² * N * Tan(π / N)
= 1.62² * 5 * Tan(3.14 / 5)
= 2.62 * 5 * Tan(0.63)
= 13.1 * 0.73
Area = 9.5.

Case 4: Find the area of a polygon with the length 2 and the number of sides is 4 using Apothem.

Step 1: Find the apothem.
Apothem = side / (2 * Tan(π / N))
= 2 / (2 * Tan(π / 4))
= 2 / (2 * Tan(0.785))
= 2 / (2 * 0.999)
= 2 / 1.998
Apothem = 1.

Step 2: Find the perimeter.
Perimeter = (N * (side) = 4 * 2 = 8

Step 3: Find the area.
Area = (A * P) / 2
= (1 * 8) / 2
= 8 / 2
Area = 4.

The above example will clearly illustrates how to calculate the Area, Perimeter of a Polygon manually.

Some GK for XAT

2007-12-21T12:29:00.000+05:30

Inventions

Adrenaline:

(isolation of) John Jacob Abel, U.S., 1897.

Aerosol can:

Erik Rotheim, Norway, 1926.

Air brake:

George Westinghouse, U.S., 1868.

Air conditioning:

Willis Carrier, U.S., 1911.

Airship:

(non-rigid) Henri Giffard, France, 1852; (rigid) Ferdinand von Zeppelin, Germany, 1900.

Aluminum manufacture:

(by electrolytic action) Charles M. Hall, U.S., 1866.

Anatomy, human:

(De fabrica corporis humani, an illustrated systematic study of the human body) Andreas Vesalius, Belgium, 1543; (comparative: parts of an organism are correlated to the functioning whole) Georges Cuvier, France, 1799–1805.

Anesthetic:

(first use of anesthetic—ether—on humans) Crawford W. Long, U.S., 1842.

Antibiotics:

(first demonstration of antibiotic effect) Louis Pasteur, Jules-François Joubert, France, 1887; (discovery of penicillin, first modern antibiotic) Alexander Fleming, Scotland, 1928; (penicillin's infection-fighting properties) Howard Florey, Ernst Chain, England, 1940.

Antiseptic:

(surgery) Joseph Lister, England, 1867.

Antitoxin, diphtheria:

Emil von Behring, Germany, 1890.

Appliances, electric:

(fan) Schuyler Wheeler, U.S., 1882; (flatiron) Henry W. Seely, U.S., 1882; (stove) Hadaway, U.S., 1896; (washing machine) Alva Fisher, U.S., 1906.

Aqualung:

Jacques-Yves Cousteau, Emile Gagnan, France, 1943.

Aspirin:

Dr. Felix Hoffman, Germany, 1899.

Astronomical calculator:

The Antikythera device, Greece, first century B.C.. Found off island of Antikythera in 1900.

Atom:

(nuclear model of) Ernest Rutherford, England, 1911.

Atomic structure:

(formulated nuclear model of atom, Rutherford model) Ernest Rutherford, England, 1911; (proposed current concept of atomic structure, the Bohr model) Niels Bohr, Denmark, 1913.

Atomic theory:

(ancient) Leucippus, Democritus, Greece, c. 500 B.C.; Lucretius, Rome c.100 B.C.; (modern) John Dalton, England, 1808.

Automobile:

(first with internal combustion engine, 250 rpm) Karl Benz, Germany, 1885; (first with practical high-speed internal combustion engine, 900 rpm) Gottlieb Daimler, Germany, 1885; (first true automobile, not carriage with motor) René Panhard, Emile Lavassor, France, 1891; (carburetor, spray) Charles E. Duryea, U.S., 1892.

Autopilot:

(for aircraft) Elmer A. Sperry, U.S., c.1910, first successful test, 1912, in a Curtiss flying boat.

Avogadro's law:

(equal volumes of all gases at the same temperature and pressure contain equal number of molecules) Amedeo Avogadro, Italy, 1811.

Bacteria:

Anton van Leeuwenhoek, The Netherlands, 1683.

Balloon, hot-air:

Joseph and Jacques Montgolfier, France, 1783.

Barbed wire:

(most popular) Joseph E. Glidden, U.S., 1873.

Bar codes (computer-scanned binary signal code):

(retail trade use) Monarch Marking, U.S. 1970; (industrial use) Plessey Telecommunications, England, 1970.

Barometer:

Evangelista Torricelli, Italy, 1643.

Bicycle:

Karl D. von Sauerbronn, Germany, 1816; (first modern model) James Starley, England, 1884.

Big Bang theory:

(the universe originated with a huge explosion) George LeMaitre, Belgium, 1927; (modified LeMaitre theory labeled “Big Bang”) George A. Gamow, U.S., 1948; (cosmic microwave background radiation discovered, confirms theory) Arno A. Penzias and Robert W. Wilson, U.S., 1965.

Blood, circulation of:

William Harvey, England, 1628.

Boyle's law:

(relation between pressure and volume in gases) Robert Boyle, Ireland, 1662.

Braille:

Louis Braille, France, 1829.

Bridges:

(suspension, iron chains) James Finley, Pa., 1800; (wire suspension) Marc Seguin, Lyons, 1825; (truss) Ithiel Town, U.S., 1820.

Bullet:

(conical) Claude Minié, France, 1849.

Calculating machine:

(logarithms: made multiplying easier and thus calculators practical) John Napier, Scotland, 1614; (slide rule) William Oughtred, England, 1632; (digital calculator) Blaise Pascal, 1642; (multiplication machine) Gottfried Leibniz, Germany, 1671; (important 19th-century contributors to modern machine) Frank S. Baldwin, Jay R. Monroe, Dorr E. Felt, W. T. Ohdner, William Burroughs, all U.S.; (“analytical engine” design, included concepts of programming, taping) Charles Babbage, England, 1835.

Calculus:

Isaac Newton, England, 1669; (differential calculus) Gottfried Leibniz, Germany, 1684.

Camera:

(hand-held) George Eastman, U.S., 1888; (Polaroid Land) Edwin Land, U.S., 1948.

“Canals” of Mars:

Giovanni Schiaparelli, Italy, 1877.

Carpet sweeper:

Melville R. Bissell, U.S., 1876.

Car radio:

William Lear, Elmer Wavering, U.S., 1929, manufactured by Galvin Manufacturing Co., “Motorola.”

Cells:

(word used to describe microscopic examination of cork) Robert Hooke, England, 1665; (theory: cells are common structural and functional unit of all living organisms) Theodor Schwann, Matthias Schleiden, 1838–1839.

Cement, Portland:

Joseph Aspdin, England, 1824.

Chewing gum:

(spruce-based) John Curtis, U.S., 1848; (chicle-based) Thomas Adams, U.S., 1870.

Cholera bacterium:

Robert Koch, Germany, 1883.

Circuit, integrated:

(theoretical) G.W.A. Dummer, England, 1952; (phase-shift oscillator) Jack S. Kilby, Texas Instruments, U.S., 1959.

Classification of plants:

(first modern, based on comparative study of forms) Andrea Cesalpino, Italy, 1583; (classification of plants and animals by genera and species) Carolus Linnaeus, Sweden, 1737–1753.

Clock, pendulum:

Christian Huygens, The Netherlands, 1656.

Coca-Cola:

John Pemberton, U.S., 1886.

Combustion:

(nature of) Antoine Lavoisier, France, 1777.

Compact disk:

RCA, U.S., 1972.

Computers:

(first design of analytical engine) Charles Babbage, 1830s; (ENIAC, Electronic Numerical Integrator and Calculator, first all-electronic, completed) John Presper Eckert, Jr., John Mauchly, U.S., 1945; (dedicated at University of Pennsylvania) 1946; (UNIVAC, Universal Automatic Computer, handled both numeric and alphabetic data) 1951; (personal computer) Steve Wozniak, U.S., 1976.

Concrete:

(reinforced) Joseph Monier, France, 1877.

Condensed milk:

Gail Borden, U.S., 1853.

Conditioned reflex:

Ivan Pavlov, Russia, c.1910.

Conservation of electric charge:

(the total electric charge of the universe or any closed system is constant) Benjamin Franklin, U.S., 1751–1754.

Contagion theory:

(infectious diseases caused by living agent transmitted from person to person) Girolamo Fracastoro, Italy, 1546.

Continental drift theory:

(geographer who pieced together continents into a single landmass on maps) Antonio Snider-Pellegrini, France, 1858; (first proposed in lecture) Frank Taylor, U.S. 1912; (first comprehensive detailed theory) Alfred Wegener, Germany, 1912.

Contraceptive, oral:

Gregory Pincus, Min Chuch Chang, John Rock, Carl Djerassi, U.S., 1951.

Converter, Bessemer:

William Kelly, U.S., 1851.

Cosmetics:

Egypt, c. 4000 B.C.

Cosmic string theory:

(first postulated) Thomas Kibble, UK, 1976.

Cotton gin:

Eli Whitney, U.S., 1793.

Crossbow:

China, c. 300 B.C.

Cyclotron:

Ernest O. Lawrence, U.S., 1931.

Defibrillator:

Dr. William Bennett Kouwenhoven, U.S., 1932; (implantable) M. Stephen Heilman, MD, Dr. Alois Langer, Morton Mower, MD, Michel Mirowski, MD, 1980.

Deuterium:

(heavy hydrogen) Harold Urey, U.S., 1931.

Disease:

(chemicals in treatment of) crusaded by Philippus Paracelsus, 1527–1541; (germ theory) Louis Pasteur, France, 1862–1877.

DNA:

(deoxyribonucleic acid) Friedrich Meischer, Germany, 1869; (determination of double-helical structure) F. H. Crick, England and James D. Watson, U.S., 1953.

Dye:

(aniline, start of synthetic dye industry) William H. Perkin, England, 1856.

Dynamite:

Alfred Nobel, Sweden, 1867.

Electric cooking utensil:

(first) patented by St. George Lane-Fox, England, 1874.

Electric generator (dynamo):

(laboratory model) Michael Faraday, England, 1832; Joseph Henry, U.S., c.1832; (hand-driven model) Hippolyte Pixii, France, 1833; (alternating-current generator) Nikola Tesla, U.S., 1892.

Electric lamp:

(arc lamp) Sir Humphrey Davy, England, 1801; (fluorescent lamp) A.E. Becquerel, France, 1867; (incandescent lamp) Sir Joseph Swann, England, Thomas A. Edison, U.S., contemporaneously, 1870s; (carbon arc street lamp) Charles F. Brush, U.S., 1879; (first widely marketed incandescent lamp) Thomas A. Edison, U.S., 1879; (mercury vapor lamp) Peter Cooper Hewitt, U.S., 1903; (neon lamp) Georges Claude, France, 1911; (tungsten filament) Irving Langmuir, U.S., 1915.

Electrocardiography:

Demonstrated by Augustus Waller, Switzerland, 1887; (first practical device for recording activity of heart) Willem Einthoven, 1903, Netherlands.

Electromagnet:

William Sturgeon, England, 1823.

Electron:

Sir Joseph J. Thompson, England, 1897.

Electronic mail:

Ray Tomlinson, U.S., 1972.

Elevator, passenger:

(safety device permitting use by passengers) Elisha G. Otis, U.S., 1852; (elevator utilizing safety device) 1857.

E = mc²:

(equivalence of mass and energy) Albert Einstein, Switzerland, 1907.

Engine, internal combustion:

No single inventor. Fundamental theory established by Sadi Carnot, France, 1824; (two-stroke) Etienne Lenoir, France, 1860; (ideal operating cycle for four-stroke) Alphonse Beau de Roche, France, 1862; (operating four-stroke) Nikolaus Otto, Germany, 1876; (diesel) Rudolf Diesel, Germany, 1892; (rotary) Felix Wankel, Germany, 1956.

Evolution:

(organic) Jean-Baptiste Lamarck, France, 1809; (by natural selection) Charles Darwin, England, 1859.

Exclusion principle:

(no two electrons in an atom can occupy the same energy level) Wolfgang Pauli, Germany, 1925.

Expanding universe theory:

(first proposed) George LeMaitre, Belgium, 1927; (discovered first direct evidence that the universe is expanding) Edwin P. Hubble, U.S., 1929; (Hubble constant: a measure of the rate at which the universe is expanding) Edwin P. Hubble, U.S., 1929.

Falling bodies, law of:

Galileo Galilei, Italy, 1590.

Fermentation:

(microorganisms as cause of) Louis Pasteur, France, c.1860.

Fiber optics:

Narinder Kapany, England, 1955.

Fibers, man-made:

(nitrocellulose fibers treated to change flammable nitrocellulose to harmless cellulose, precursor of rayon) Sir Joseph Swann, England, 1883; (rayon) Count Hilaire de Chardonnet, France, 1889; (Celanese) Henry and Camille Dreyfuss, U.S., England, 1921; (research on polyesters and polyamides, basis for modern man-made fibers) U.S., England, Germany, 1930s; (nylon) Wallace H. Carothers, U.S., 1935.

Frozen food:

Clarence Birdseye, U.S., 1924.

Gene transfer:

(recombinant DNA organism) Herbert Boyer, Stanley Cohen, U.S., 1973; (human) Steven Rosenberg, R. Michael Blaese, W. French Anderson, U.S., 1989.

Geometry, elements of:

Euclid, Alexandria, Egypt, c. 300 B.C.; (analytic) René Descartes, France; and Pierre de Fermat, Switzerland, 1637.

Gravitation, law of:

Sir Isaac Newton, England, c.1665 (published 1687).

Gunpowder:

China, c.700.

Gyrocompass:

Elmer A. Sperry, U.S., 1905.

Gyroscope:

Jean Léon Foucault, France, 1852.

Halley's Comet:

Edmund Halley, England, 1705.

Heart implanted in human, permanent artificial:

Dr. Robert Jarvik, U.S., 1982.

Heart, temporary artificial:

Willem Kolff, Netherlands, U.S., 1957.

Helicopter:

(double rotor) Heinrich Focke, Germany, 1936; (single rotor) Igor Sikorsky, U.S., 1939.

Helium first observed on sun:

Sir Joseph Lockyer, England, 1868.

Heredity, laws of:

Gregor Mendel, Austria, 1865.

Holograph:

Dennis Gabor, England, 1947.

Home videotape systems (VCR):

(Betamax) Sony, Japan, 1975; (VHS) Matsushita, Japan, 1975.

Ice age theory:

Louis Agassiz, Swiss-American, 1840.

Induction, electric:

Joseph Henry, U.S., 1828.

Insulin:

(first isolated) Sir Frederick G. Banting and Charles H. Best, Canada, 1921; (discovery first published) Banting and Best, 1922; (Nobel Prize awarded for purification for use in humans) John Macleod and Banting, 1923; (first synthesized), China, 1966.

Intelligence testing:

Alfred Binet, Theodore Simon, France, 1905.

Interferon:

Alick Isaacs, England, Jean Lindemann, Switzerland, 1957.

Isotopes:

(concept of) Frederick Soddy, England, 1912; (stable isotopes) J. J. Thompson, England, 1913; (existence demonstrated by mass spectrography) Francis W. Aston, England, 1919.

Jet propulsion:

(engine) Sir Frank Whittle, England, Hans von Ohain, Germany, 1936; (aircraft) Heinkel He 178, 1939.

Kinetic theory of gases:

(molecules of a gas are in a state of rapid motion) Daniel Bernoulli, Switzerland, 1738.

Laser:

(theoretical work on) Charles H. Townes, Arthur L. Schawlow, U.S., N. Basov, A. Prokhorov, U.S.S.R., 1958; (first working model) T. H. Maiman, U.S., 1960.

Lawn mower:

Edwin Budding, John Ferrabee, England, 1830–1831.

LCD (liquid crystal display):

Hoffmann-La Roche, Switzerland, 1970.

Lens, bifocal:

Benjamin Franklin, U.S., c.1760.

Leyden jar:

(prototype electrical condenser) Canon E. G. von Kleist of Kamin, Pomerania, 1745; independently evolved by Cunaeus and P. van Musschenbroek, University of Leyden, Holland, 1746, from where name originated.

Light, nature of:

(wave theory) Christian Huygens, The Netherlands, 1678; (electromagnetic theory) James Clerk Maxwell, England, 1873.

Light, speed of:

(theory that light has finite velocity) Olaus Roemer, Denmark, 1675.

Lightning rod:

Benjamin Franklin, U.S., 1752.

Lock, cylinder:

Linus Yale, U.S., 1851.

Locomotive:

(steam powered) Richard Trevithick, England, 1804; (first practical, due to multiple-fire-tube boiler) George Stephenson, England, 1829; (largest steam-powered) Union Pacific's “Big Boy,” U.S., 1941.

Loom:

(horizontal, two-beamed) Egypt, c. 4400 B.C.; (Jacquard drawloom, pattern controlled by punch cards) Jacques de Vaucanson, France, 1745, Joseph-Marie Jacquard, 1801; (flying shuttle) John Kay, England, 1733; (power-driven loom) Edmund Cartwright, England, 1785.

Machine gun:

(hand-cranked multibarrel) Richard J. Gatling, U.S., 1862; (practical single barrel, belt-fed) Hiram S. Maxim, Anglo-American, 1884.

Magnet, Earth is:

William Gilbert, England, 1600.

Match:

(phosphorus) François Derosne, France, 1816; (friction) Charles Sauria, France, 1831; (safety) J. E. Lundstrom, Sweden, 1855.

Measles vaccine:

John F. Enders, Thomas Peebles, U.S., 1953.

Metric system:

revolutionary government of France, 1790–1801.

Microphone:

Charles Wheatstone, England, 1827.

Microscope:

(compound) Zacharias Janssen, The Netherlands, 1590; (electron) Vladimir Zworykin et al., U.S., Canada, Germany, 1932–1939.

Microwave oven:

Percy Spencer, U.S., 1947.

Motion, laws of:

Isaac Newton, England, 1687.

Motion pictures:

Thomas A. Edison, U.S., 1893.

Motion pictures, sound:

Product of various inventions. First picture with synchronized musical score: Don Juan, 1926; with spoken dialogue: The Jazz Singer, 1927; both Warner Bros.

Motor, electric:

Michael Faraday, England, 1822; (alternating-current) Nikola Tesla, U.S., 1892.

Motorcycle:

(motor tricycle) Edward Butler, England, 1884; (gasoline-engine motorcycle) Gottlieb Daimler, Germany, 1885.

Moving assembly line:

Henry Ford, U.S., 1913.

Neptune:

(discovery of) Johann Galle, Germany, 1846.

Neptunium:

(first transuranic element, synthesis of) Edward M. McMillan, Philip H. Abelson, U.S., 1940.

Neutron:

James Chadwick, England, 1932.

Neutron-induced radiation:

Enrico Fermi et al., Italy, 1934.

Nitroglycerin:

Ascanio Sobrero, Italy, 1846.

Nuclear fission:

Otto Hahn, Fritz Strassmann, Germany, 1938.

Nuclear reactor:

Enrico Fermi, Italy, et al., 1942.

Ohm's law:

(relationship between strength of electric current, electromotive force, and circuit resistance) Georg S. Ohm, Germany, 1827.

Oil well:

Edwin L. Drake, U.S., 1859.

Oxygen:

(isolation of) Joseph Priestley, England, 1774; Karl Scheele, Sweden, 1773.

Ozone:

Christian Schönbein, Germany, 1839.

Pacemaker:

(internal) Clarence W. Lillehie, Earl Bakk, U.S., 1957.

Paper:

China, c.100 A.D.

Parachute:

Louis S. Lenormand, France, 1783.

Pen:

(fountain)Lewis E. Waterman, U.S., 1884; (ball-point, for marking on rough surfaces) John H. Loud, U.S., 1888; (ball-point, for handwriting) Lazlo Biro, Argentina, 1944.

Periodic law:

(that properties of elements are functions of their atomic weights) Dmitri Mendeleev, Russia, 1869.

Periodic table:

(arrangement of chemical elements based on periodic law) Dmitri Mendeleev, Russia, 1869.

Phonograph:

Thomas A. Edison, U.S., 1877.

Photography:

(first paper negative, first photograph, on metal) Joseph Nicéphore Niepce, France, 1816–1827; (discovery of fixative powers of hyposulfite of soda) Sir John Herschel, England, 1819; (first direct positive image on silver plate, the daguerreotype) Louis Daguerre, based on work with Niepce, France, 1839; (first paper negative from which a number of positive prints could be made) William Talbot, England, 1841. Work of these four men, taken together, forms basis for all modern photography. (First color images) Alexandre Becquerel, Claude Niepce de Saint-Victor, France, 1848–1860; (commercial color film with three emulsion layers, Kodachrome) U.S., 1935.

Photovoltaic effect:

(light falling on certain materials can produce electricity) Edmund Becquerel, France, 1839.

Piano:

(Hammerklavier) Bartolommeo Cristofori, Italy, 1709; (pianoforte with sustaining and damper pedals) John Broadwood, England, 1873.

Planetary motion, laws of:

Johannes Kepler, Germany, 1609, 1619.

Plant respiration and photosynthesis:

Jan Ingenhousz, Holland, 1779.

Plastics:

(first material, nitrocellulose softened by vegetable oil, camphor, precursor to Celluloid) Alexander Parkes, England, 1855; (Celluloid, involving recognition of vital effect of camphor) John W. Hyatt, U.S., 1869; (Bakelite, first completely synthetic plastic) Leo H. Baekeland, U.S., 1910; (theoretical background of macromolecules and process of polymerization on which modern plastics industry rests) Hermann Staudinger, Germany, 1922; (polypropylene and low-pressure method for producing high-density polyethylene) Robert Banks, Paul Hogan, U.S., 1958.

Plate tectonics:

Alfred Wegener, Germany, 1912–1915.

Plow, forked:

Mesopotamia, before 3000 B.C.

Plutonium, synthesis of:

Glenn T. Seaborg, Edwin M. McMillan, Arthur C. Wahl, Joseph W. Kennedy, U.S., 1941.

Polio, vaccine:

(experimentally safe dead-virus vaccine)Jonas E. Salk, U.S., 1952; (effective large-scale field trials) 1954; (officially approved) 1955; (safe oral live-virus vaccine developed) Albert B. Sabin, U.S., 1954; (available in the U.S.) 1960.

Positron:

Carl D. Anderson, U.S., 1932.

Pressure cooker:

(early version) Denis Papin, France, 1679.

Printing:

(block) Japan, c.700; (movable type) Korea, c.1400, Johann Gutenberg, Germany, c.1450; (lithography, offset) Aloys Senefelder, Germany, 1796; (rotary press) Richard Hoe, U.S., 1844; (linotype) Ottmar Mergenthaler, U.S., 1884.

Probability theory:

René Descartes, France, and Pierre de Fermat, Switzerland, 1654.

Proton:

Ernest Rutherford, England, 1919.

Prozac:

(antidepressant fluoxetine) Bryan B. Malloy, Scotland, and Klaus K. Schmiegel, U.S., 1972; (released for use in U.S.) Eli Lilly & Company, 1987.

Psychoanalysis:

Sigmund Freud, Austria, c.1904.

Pulsars:

Antony Hewish and Jocelyn Bell Burnel, England, 1967.

Quantum theory:

(general) Max Planck, Germany, 1900; (sub-atomic) Niels Bohr, Denmark, 1913; (quantum mechanics) Werner Heisenberg, Erwin Schrödinger, Germany, 1925.

Quarks:

Jerome Friedman, Henry Kendall, Richard Taylor, U.S., 1967.

Quasars:

Marten Schmidt, U.S., 1963.

Rabies immunization:

Louis Pasteur, France, 1885.

Radar:

(limited to one-mile range) Christian Hulsmeyer, Germany, 1904; (pulse modulation, used for measuring height of ionosphere) Gregory Breit, Merle Tuve, U.S., 1925; (first practical radar—radio detection and ranging) Sir Robert Watson-Watt, England, 1934–1935.

Radio:

(electromagnetism, theory of) James Clerk Maxwell, England, 1873; (spark coil, generator of electromagnetic waves) Heinrich Hertz, Germany, 1886; (first practical system of wireless telegraphy) Guglielmo Marconi, Italy, 1895; (first long-distance telegraphic radio signal sent across the Atlantic) Marconi, 1901; (vacuum electron tube, basis for radio telephony) Sir John Fleming, England, 1904; (triode amplifying tube) Lee de Forest, U.S., 1906; (regenerative circuit, allowing long-distance sound reception) Edwin H. Armstrong, U.S., 1912; (frequency modulation—FM) Edwin H. Armstrong, U.S., 1933.

Radioactivity:

(X-rays) Wilhelm K. Roentgen, Germany, 1895; (radioactivity of uranium) Henri Becquerel, France, 1896; (radioactive elements, radium and polonium in uranium ore) Marie Sklodowska-Curie, Pierre Curie, France, 1898; (classification of alpha and beta particle radiation) Pierre Curie, France, 1900; (gamma radiation) Paul-Ulrich Villard, France, 1900.

Radiocarbon dating, carbon-14 method:

(discovered) Willard F. Libby, U.S., 1947; (first demonstrated) U.S., 1950.

Radio signals, extraterrestrial:

first known radio noise signals were received by U.S. engineer, Karl Jansky, originating from the Galactic Center, 1931.

Radio waves:

(cosmic sources, led to radio astronomy) Karl Jansky, U.S., 1932.

Razor:

(safety, successfully marketed) King Gillette, U.S., 1901; (electric) Jacob Schick, U.S., 1928, 1931.

Reaper:

Cyrus McCormick, U.S., 1834.

Refrigerator:

Alexander Twining, U.S., James Harrison, Australia, 1850; (first with a compressor device) the Domelse, Chicago, U.S., 1913.

Refrigerator ship:

(first) the Frigorifique, cooling unit designed by Charles Teller, France, 1877.

Relativity:

(special and general theories of) Albert Einstein, Switzerland, Germany, U.S., 1905–1953.

Revolver:

Samuel Colt, U.S., 1835.

Richter scale:

Charles F. Richter, U.S., 1935.

Rifle:

(muzzle-loaded) Italy, Germany, c.1475; (breech-loaded) England, France, Germany, U.S., c.1866; (bolt-action) Paul von Mauser, Germany, 1889; (automatic) John Browning, U.S., 1918.

Rocket:

(liquid-fueled) Robert Goddard, U.S., 1926.

Roller bearing:

(wooden for cartwheel) Germany or France, c.100 B.C.

Rotation of Earth:

Jean Bernard Foucault, France, 1851.

Royal Observatory, Greenwich:

established in 1675 by Charles II of England; John Flamsteed first Astronomer Royal.

Rubber:

(vulcanization process) Charles Goodyear, U.S., 1839.

Saccharin:

Constantine Fuhlberg, Ira Remsen, U.S., 1879.

Safety pin:

Walter Hunt, U.S., 1849.

Saturn, ring around:

Christian Huygens, The Netherlands, 1659.

“Scotch” tape:

Richard Drew, U.S., 1929.

Screw propeller:

Sir Francis P. Smith, England, 1836; John Ericsson, England, worked independently of and simultaneously with Smith, 1837.

Seat belt:

(three point) Nils Bohlin, Sweden, 1962.

Seismograph:

(first accurate) John Milne, England, 1880.

Sewing machine:

Elias Howe, U.S., 1846; (continuous stitch) Isaac Singer, U.S., 1851.

Solar energy:

First realistic application of solar energy using parabolic solar reflector to drive caloric engine on steam boiler, John Ericsson, U.S., 1860s.

Solar system, universe:

(Sun-centered universe) Nicolaus Copernicus, Warsaw, 1543; (establishment of planetary orbits as elliptical) Johannes Kepler, Germany, 1609; (infinity of universe) Giordano Bruno, Italian monk, 1584.

Spectrum:

(heterogeneity of light) Sir Isaac Newton, England, 1665–1666.

Spectrum analysis:

Gustav Kirchhoff, Robert Bunsen, Germany, 1859.

Spermatozoa:

Anton van Leeuwenhoek, The Netherlands, 1683.

Spinning:

(spinning wheel) India, introduced to Europe in Middle Ages; (Saxony wheel, continuous spinning of wool or cotton yarn) England, c.1500–1600; (spinning jenny) James Hargreaves, England, 1764; (spinning frame) Sir Richard Arkwright, England, 1769; (spinning mule, completed mechanization of spinning, permitting production of yarn to keep up with demands of modern looms) Samuel Crompton, England, 1779.

Star catalog:

(first modern) Tycho Brahe, Denmark, 1572.

Steam engine:

(first commercial version based on principles of French physicist Denis Papin) Thomas Savery, England, 1639; (atmospheric steam engine) Thomas Newcomen, England, 1705; (steam engine for pumping water from collieries) Savery, Newcomen, 1725; (modern condensing, double acting) James Watt, England, 1782; (high-pressure) Oliver Evans, U.S., 1804.

Steamship:

Claude de Jouffroy d'Abbans, France, 1783; James Rumsey, U.S., 1787; John Fitch, U.S., 1790; (high-pressure) Oliver Evans, U.S., 1804. All preceded Robert Fulton, U.S., 1807, credited with launching first commercially successful steamship.

Stethoscope:

René Laënnec, France, 1819.

Sulfa drugs:

(parent compound, para-aminobenzenesulfanomide) Paul Gelmo, Austria, 1908; (antibacterial activity) Gerhard Domagk, Germany, 1935.

Superconductivity:

(theory) John Bardeen, Leon Cooper, John Scheiffer, U.S., 1957.

Symbolic logic:

George Boule, 1854; (modern) Bertrand Russell, Alfred North Whitehead, England, 1910–1913.

Tank, military:

Sir Ernest Swinton, England, 1914.

Tape recorder:

(magnetic steel tape) Valdemar Poulsen, Denmark, 1899.

Teflon:

DuPont, U.S., 1943.

Telegraph:

Samuel F. B. Morse, U.S., 1837.

Telephone:

Alexander Graham Bell, U.S., 1876.

Telescope:

Hans Lippershey, The Netherlands, 1608; (astronomical) Galileo Galilei, Italy, 1609; (reflecting) Isaac Newton, England, 1668.

Television:

(Iconoscope–T.V. camera table) Vladimir Zworykin, U.S., 1923, and also kinescope (cathode ray tube) 1928; (mechanical disk-scanning method) successfully demonstrated by J.L. Baird, Scotland, C.F. Jenkins, U.S., 1926; (first all-electric television image) Philo T. Farnsworth, U.S., 1927; (color, mechanical disk) Baird, 1928; (color, compatible with black and white) George Valensi, France, 1938; (color, sequential rotating filter) Peter Goldmark, U.S., first introduced, 1951; (color, compatible with black and white) commercially introduced in U.S., National Television Systems Committee, 1953.

Thermodynamics:

(first law: energy cannot be created or destroyed, only converted from one form to another) Julius von Mayer, Germany, 1842; James Joule, England, 1843; (second law: heat cannot of itself pass from a colder to a warmer body) Rudolph Clausius, Germany, 1850; (third law: the entropy of ordered solids reaches zero at the absolute zero of temperature) Walter Nernst, Germany, 1918.

Thermometer:

(open-column) Galileo Galilei, c.1593; (clinical) Santorio Santorio, Padua, c.1615; (mercury, also Fahrenheit scale) Gabriel D. Fahrenheit, Germany, 1714; (centigrade scale) Anders Celsius, Sweden, 1742; (absolute-temperature, or Kelvin, scale) William Thompson, Lord Kelvin, England, 1848.

Tire, pneumatic:

Robert W. Thompson, England, 1845; (bicycle tire) John B. Dunlop, Northern Ireland, 1888.

Toilet, flush:

Product of Minoan civilization, Crete, c. 2000 B.C. Alleged invention by “Thomas Crapper” is untrue.

Tractor:

Benjamin Holt, U.S., 1900.

Transformer, electric:

William Stanley, U.S., 1885.

Transistor:

John Bardeen, Walter H. Brattain, William B. Shockley, U.S., 1947.

Tuberculosis bacterium:

Robert Koch, Germany, 1882.

Typewriter:

Christopher Sholes, Carlos Glidden, U.S., 1867.

Uncertainty principle:

(that position and velocity of an object cannot both be measured exactly, at the same time) Werner Heisenberg, Germany, 1927.

Uranus:

(first planet discovered in recorded history) William Herschel, England, 1781.

Vaccination:

Edward Jenner, England, 1796.

Vacuum cleaner:

(manually operated) Ives W. McGaffey, U.S., 1869; (electric) Hubert C. Booth, England, 1901; (upright) J. Murray Spangler, U.S., 1907.

Van Allen (radiation) Belt:

(around Earth) James Van Allen, U.S., 1958.

Video disk:

Philips Co., The Netherlands, 1972.

Vitamins:

(hypothesis of disease deficiency) Sir F. G. Hopkins, Casimir Funk, England, 1912; (vitamin A) Elmer V. McCollum, M. Davis, U.S., 1912–1914; (vitamin B) McCollum, U.S., 1915–1916; (thiamin, B₁) Casimir Funk, England, 1912; (riboflavin, B₂) D. T. Smith, E. G. Hendrick, U.S., 1926; (niacin) Conrad Elvehjem, U.S., 1937; (B₆) Paul Gyorgy, U.S., 1934; (vitamin C) C. A. Hoist, T. Froelich, Norway, 1912; (vitamin D) McCollum, U.S., 1922; (folic acid) Lucy Wills, England, 1933.

Voltaic pile:

(forerunner of modern battery, first source of continuous electric current) Alessandro Volta, Italy, 1800.

Wallpaper:

Europe, 16th and 17th century.

Wassermann test:

(for syphilis) August von Wassermann, Germany, 1906.

Wheel:

(cart, solid wood) Mesopotamia, c.3800–3600 B.C.

Windmill:

Persia, c.600.

World Wide Web:

(developed while working at CERN) Tim Berners-Lee, England, 1989; (development of Mosaic browser makes WWW available for general use) Marc Andreeson, U.S., 1993.

Xerography:

Chester Carlson, U.S., 1938.

Yellow Fever:

(transmission of)Walter Reed, U.S., 1900.

Zero:

India, c. 600; (absolute zero temperature, cessation of all molecular energy) William Thompson, Lord Kelvin, England, 1848.

Zipper:

W. L. Judson, U.S., 1891.

RC section Revisited

2007-12-21T12:27:00.000+05:30

------------------------------------------------------------------------------------------------------------
Q1:
During her presidency of the short-lived Woman’s State Temperance Society (1852-1853), Elizabeth Cady Stanton, as she was a staunch advocate of liberalized divorce laws, scandalized many of her most ardent supporters in her suggestion that drunkenness should be made sufficient cause for divorce.

A. as she was a staunch advocate of liberalized divorce laws, scandalized many of her most ardent supporters in her suggestion that drunkenness should be
B. as she was a staunch advocate for liberalized divorce laws, scandalized many of her most ardent supporters by her suggestion of drunkenness being
C. in being a staunch advocate for liberalized divorce laws, had scandalized many of her most ardent supporters with the suggestion of drunkenness being
D. a staunch advocate of liberalized divorce laws, scandalized many of her most ardent supporters by suggesting that drunkenness be
E. a staunch advocate of liberalized divorce laws, she scandalized many of her most ardent supporters in suggesting that drunkenness should be

------------------------------------------------------------------------------------------------------------
Q2:
By merging its two publishing divisions, the company will increase their share of the country’s $21 billion book market from 6 percent to 10 percent, a market ranging from obscure textbooks to mass-market paperbacks.

A. their share of the country’s $21 billion book market from 6 percent to 10 percent, a market ranging
B. from 6 percent to 10 percent its share of the $21 billion book market in the country, which ranges
C. to 10 percent from 6 percent in their share of the $21 billion book market in the country, a market ranging
D. in its share, from 6 percent to 10 percent, of the $21 billion book market in the country, which ranges
E. to 10 percent from 6 percent its share of the country’s $21 billion book market, which ranges

------------------------------------------------------------------------------------------------------------
Q3:
A product that represents a clear technological advance over competing products can generally command a high price. Because technological advances tend to be quickly surpassed and companies want to make large profits while they still can, many companies charge the greatest price the market will bear when they have such a product. But large profits on the mew product will give competitors a strong incentive to quickly match the mew product’s capabilities. Consequently, the strategy to maximize overall profit from a new product is to charge less than the greatest possible price.

In the argument above, the two portions in boldface play which of the following roles?

A. The first is an assumption that forms the basis for a course of action that the argument criticizes; the second presents the course of action endorsed by the argument.
B. The first is a consideration raised to explain the appeal of a certain strategy; the second is a consideration raised to call into question the wisdom of adopting that strategy.
C. The first is an assumption that has been used to justify a certain strategy; the second is a consideration that is used to cast doubt on that assumption.
D. The first is a consideration raised in support of a strategy the argument endorses; the second presents grounds in support of that consideration.
E. The first is a consideration raised to show that adopting a certain strategy is unlikely to achieve the intended effect; the second is presented to explain the appeal of that strategy.
------------------------------------------------------------------------------------------------------------
Q4 to Q6:
The fields of antebellum (pre-Civil
War) political history and women’s his-
tory use separate sources and focus
Line on separate issues. Political histori-
(5) ans, examining sources such as voting
records, newspapers, and politicians’
writings, focus on the emergence in the
1840’s of a new “American political
nation,” and since women were neither
(10) voters nor politicians, they receive little
discussion. Women’s historians, mean-
while, have shown little interest in the
subject of party politics, instead draw-
ing on personal papers, legal records
(15) such as wills, and records of female
associations to illuminate women’s
domestic lives, their moral reform
activities, and the emergence of the
woman’s rights movement.
(20) However, most historians have
underestimated the extent and signifi-
cance of women’s political allegiance
in the antebellum period. For example,
in the presidential election campaigns
(25) of the 1840’s, the Virginia Whig party
strove to win the allegiance of Virginia’s
women by inviting them to rallies and
speeches. According to Whig propa-
ganda, women who turned out at the
(30) party’s rallies gathered information
that enabled them to mold party-loyal
families, reminded men of moral values
that transcended party loyalty, and con-
ferred moral standing on the party.
(35) Virginia Democrats, in response,
began to make similar appeals to
women as well. By the mid-1850’s
the inclusion of women in the rituals of
party politics had become common-
(40) place, and the ideology that justified
such inclusion had been assimilated
by the Democrats.
--------------------------------------------------------------------------------
Q4:
The primary purpose of the passage as a whole is to

A. examine the tactics of antebellum political parties with regard to women
B. trace the effect of politics on the emergence of the woman’s rights movement
C. point out a deficiency in the study of a particular historical period
D. discuss the ideologies of opposing antebellum political parties
E. contrast the methodologies in two differing fields of historical inquiry

--------------------------------------------------------------------------------
Q5:
According to the second paragraph of the passage (lines 20-42), Whig propaganda included the assertion that

A. women should enjoy more political rights than they did
B. women were the most important influences on political attitudes within a family
C. women’s reform activities reminded men of important moral values
D. women’s demonstrations at rallies would influence men’s voting behavior
E. women’s presence at rallies would enhance the moral standing of the party

--------------------------------------------------------------------------------
Q6:
According to the passage, which of the following was true of Virginia Democrats in the mid-1850’s?

A. They feared that their party was losing its strong moral foundation.
B. They believed that the Whigs’ inclusion of women in party politics had led to the Whigs’ success in many elections.
C. They created an ideology that justified the inclusion of women in party politics.
D. They wanted to demonstrate that they were in support of the woman’s rights movement.
E. They imitated the Whigs’ efforts to include women in the rituals of party politics.

------------------------------------------------------------------------------------------------------------
Q7:
A recent review of pay scales indicates that CEO’s now earn an average of 419 times more pay than blue-collar workers, compared to a ratio of 42 times in 1980.

A. that CEO’s now earn an average of 419 times more pay than blue-collar workers, compared to a ratio of 42 times
B. that, on average, CEO’s now earn 419 times the pay of blue-collar workers, a ratio that compares to 42 times
C. that, on average, CEO’s now earn 419 times the pay of blue-collar workers, as compared to 42 times their pay, the ratio
D. CEO’s who now earn on average 419 times more pay than blue-collar workers, as compared to 42 times their pay, the ratio
E. CEO’s now earning an average of 419 times the pay of blue-collar workers, compared to the ratio of 42 times

------------------------------------------------------------------------------------------------------------
Q8:
The 32 species that make up the dolphin family are closely related to whales and in fact include the animal known as the killer whale, which can grow to be 30 feet long and is famous for its aggressive hunting pods.

A. include the animal known as the killer whale, which can grow to be 30 feet long and is
B. include the animal known as the killer whale, growing as big as 30 feet long and
C. include the animal known as the killer whale, growing up to 30 feet long and being
D. include the animal known as the killer whale, which can grow as big as 30 feet long and is
E. include the animal known as the killer whale, which can grow to be 30 feet long and it is

------------------------------------------------------------------------------------------------------------
Q9 to Q12:
Over the last 150 years, large
stretches of salmon habitat have
been eliminated by human activity:
Line mining, livestock grazing, timber
(5) harvesting, and agriculture as well
as recreational and urban devel-
opment. The numerical effect is
obvious: there are fewer salmon
in degraded regions than in pris-
(10) tine ones; however, habitat loss
also has the potential to reduce
genetic diversity. This is most
evident in cases where it results
in the extinction of entire salmon
(15) populations. Indeed, most
analysts believe that some kind
of environmental degradation
underlies the demise of many
extinct salmon populations.
(20) Although some rivers have
been recolonized, the unique
genes of the original populations
have been lost.
Large-scale disturbances in
(25) one locale also have the potential
to alter the genetic structure of
populations in neighboring areas,
even if those areas have pristine
habitats. Why? Although the
(30) homing instinct of salmon to their
natal stream is strong, a fraction
of the fish returning from the sea
(rarely more than 15 percent)
stray and spawn in nearby
(35) streams. Low levels of straying
are crucial, since the process
provides a source of novel
genes and a mechanism
by which a location can be
(40) repopulated should the fish
there disappear. Yet high rates
of straying can be problematic
because misdirected fish may
interbreed with the existing stock
(45) to such a degree that any local
adaptations that are present
become diluted. Straying
rates remain relatively low when
environmental conditions are
(50) stable, but can increase dramati-
cally when streams suffer severe
disturbance. The 1980 volcanic
eruption of Mount Saint Helens,
for example, sent mud and debris
(55) into several tributaries of the
Columbia River. For the next
couple of years, steelhead trout
(a species included among the
salmonids) returning from the
(60) sea to spawn were forced to
find alternative streams. As
a consequence, their rates of
straying, initially 16 percent,
rose to more than 40 percent
(65) overall.
Although no one has quantified
changes in the rate of straying
as a result of the disturbances
caused by humans, there is no
(70) reason to suspect that the effect
would be qualitatively different
than what was seen in the
aftermath of the Mount Saint
Helens eruption. Such a dra-
(75) matic increase in straying from
damaged areas to more pristine
streams results in substantial
gene flow, which can in turn lower
the overall fitness of subsequent
generations.

--------------------------------------------------------------------------------
Q9:
The primary purpose of the passage is to

A. argue against a conventional explanation for the extinction of certain salmon populations and suggest an alternative
B. correct a common misunderstanding about the behavior of salmon in response to environmental degradation caused by human activity
C. compare the effects of human activity on salmon populations with the effects of natural disturbances on salmon populations
D. differentiate the particular effects of various human activities on salmon habitats
E. describe how environmental degradation can cause changes in salmon populations that extend beyond a numerical reduction

--------------------------------------------------------------------------------
Q10:
It can be inferred from the passage that the occasional failure of some salmon to return to their natal streams in order to spawn provides a mechanism by which

A. pristine streams that are near polluted streams become polluted themselves
B. the particular adaptations of a polluted stream’s salmon population can be preserved without dilution
C. the number of salmon in pristine habitats decreases relative to the number in polluted streams
D. an environmentally degraded stream could be recolonized by new salmon populations should the stream recover
E. the extinction of the salmon populations that spawn in polluted streams is accelerated

--------------------------------------------------------------------------------
Q11:
According to the passage, human activity has had which of the following effects on salmon populations?

A. An increase in the size of salmon populations in some previously polluted rivers
B. A decline in the number of salmon in some rivers
C. A decrease in the number straying salmon in some rivers
D. A decrease in the gene flow between salmon populations that spawn in polluted streams and populations that spawn in pristine streams
E. A decline in the vulnerability of some salmon populations to the effects of naturally occurring habitat destruction

--------------------------------------------------------------------------------
Q12:
The author mentions the “aftermath of the Mount Saint Helens eruption” (lines 73-74) most likely in order to

A. provide an example of the process that allows the repopulation of rivers whose indigenous salmon population has become extinct
B. indicate the extent to which the disturbance of salmon habitat by human activity in one stream might affect the genetic structure of salmon populations elsewhere
C. provide a standard of comparison against which the impact of human activity on the gene flow among salmon populations should be measured
D. show how salmons’ homing instinct can be impaired as a result of severe environmental degradation of their natal streams
E. show why straying rates in salmon populations remain generally low except when spawning streams suffer severe environmental disturbance

------------------------------------------------------------------------------------------------------------
Q13:
In the United States, of the people who moved from one state to another when they retired, the percentage who retired to Florida has decreased by three percentage points over the past ten years. Since many local businesses in Florida cater to retirees, this decline is likely to have a noticeably negative economic effect on these businesses.

Which of the following, if true, most seriously weakens the argument?

A. Florida attracts more people who move from one state to another when they retire than does any other state.
B. The number of people who move out of Florida to accept employment in other states has increased over the past ten years.
C. There are far more local businesses in Florida that cater to tourists than there are local businesses that cater to retirees.
D. The total number of people who retired and moved to another state for their retirement has increased significantly over the past ten years.
E. The number of people who left Florida when they retired to live in another state was greater last year than it was ten years ago.

------------------------------------------------------------------------------------------------------------
Q14:
That the application of new technology can increase the productivity of existing coal mines is demonstrated by the case of Tribnia’s coal industry. Coal output per miner in Tribnia is double what it was five years ago, even though no new mines have opened.

Which of the following can be properly concluded from the statement about coal output per miner in the passage?

A. If the number of miners working in Tribnian coal mines has remained constant in the past five years, Tribnia’s total coal production has doubled in that period of time.
B. Any individual Tribnian coal mine that achieved an increase in overall output in the past five years has also experienced an increase in output per miner.
C. If any new coal mines had opened in Tribnia in the past five years, then the increase in output per miner would have been even greater than it actually was.
D. If any individual Tribnian coal mine has not increased its output per miner in the past five years, then that mine’s overall output has declined or remained constant.
E. In Tribnia the cost of producing a given quantity of coal has declined over the past five years.

------------------------------------------------------------------------------------------------------------
Q15:
In parts of South America, vitamin-A deficiency is a serious health problem, especially among children. In one region, agriculturists are attempting to improve nutrition by encouraging farmers to plant a new variety of sweet potato called SPK004 that is rich in beta-carotene, which the body converts into vitamin A. The plan has good chances of success, since sweet potato is a staple of the region’s diet and agriculture, and the varieties currently grown contain little beta-carotene.

Which of the following, if true, most strongly supports the prediction that the plan will succeed?

A. The growing conditions required by the varieties of sweet potato currently cultivated in the region are conditions in which SPK004 can flourish.
B. The flesh of SPK004 differs from that of the currently cultivated sweet potatoes in color and texture, so traditional foods would look somewhat different when prepared from SPK004.
C. There are no other varieties of sweet potato that are significantly richer in beta-carotene than SPK004 is.
D. The varieties of sweet potato currently cultivated in the region contain some important nutrients that are lacking in SPK004.
E. There are other vegetables currently grown in the region that contain more beta-carotene than the currently cultivated varieties of sweet potato do.

------------------------------------------------------------------------------------------------------------
Q16:
Soaring television costs accounted for more than half the spending in the presidential campaign of 1992, a greater proportion than it was in any previous election.

A. a greater proportion than it was
B. a greater proportion than
C. a greater proportion than they have been
D. which is greater than was so
E. which is greater than it has been

------------------------------------------------------------------------------------------------------------
Q17:
The spacing of the four holes on a fragment of a bone flute excavated at a Neanderthal campsite is just what is required to play the third through sixth notes of the diatonic scale—the seven-note musical scale used in much of Western music since the Renaissance. Musicologists therefore hypothesize that the diatonic musical scale was developed and used thousands of years before it was adopted by Western musicians.

Which of the following, if true, most strongly supports the hypothesis?

A. Bone flutes were probably the only musical instrument made by Neanderthals.
B. No musical instrument that is known to have used a diatomic scale is of an earlier date than the flute found at the Neanderthal campsite.
C. The flute was made from a cave-bear bone and the campsite at which the flute fragment was excavated was in a cave that also contained skeletal remains of cave bears.
D. Flutes are the simplest wind instrument that can be constructed to allow playing a diatonic scale.
E. The cave-bear leg bone used to make the Neanderthal flute would have been long enough to make a flute capable of playing a complete diatonic scale.

------------------------------------------------------------------------------------------------------------
Q18:
It is illegal to advertise prescription medications in Hedland except in professional medical journals or by mail directly to physicians. A proposed law would allow general advertising of prescription medications. Opponents object that, in general, laypersons lack the specialized knowledge to evaluate such advertisements and might ask their physicians for inappropriate medications. But since physicians have the final say as to whether to prescribe a medication for a patient, the objection provides no grounds for concern.

Which of the following would it be most useful to establish in order to evaluate the argument?

A. Whether nonprescription medications can interact with and block the action of any prescription medications that could be advertised to the general public
B. Whether most prescription medication advertisements directed at the general public would be advertisements for recently developed medications newly available by prescription
C. Whether prescription medication advertisements directed at the general public would appear on television and radio as well as in print
D. Whether physicians are more likely to pay attention to advertising directed to the general public than to advertising directed to physicians
E. Whether physicians are likely to succumb to pressure from patients to prescribe inappropriate medications

------------------------------------------------------------------------------------------------------------
Q19:
Recently physicians have determined that stomach ulcers are not caused by stress, alcohol, or rich foods, but a bacterium that dwells in the mucous lining of the stomach.

A. not caused by stress, alcohol, or rich foods, but
B. not caused by stress, alcohol, or rich foods, but are by
C. caused not by stress, alcohol, or rich foods, but by
D. caused not by stress, alcohol, and rich foods, but
E. caused not by stress, alcohol, and rich foods, but are by

------------------------------------------------------------------------------------------------------------
Q20:
Rivaling the pyramids of Egypt or even the ancient cities of the Maya as an achievement, the army of terra-cotta warriors created to protect Qin Shi Huang, China’s first emperor, in his afterlife is more than 2,000 years old and took 700,000 artisans more than 36 years to complete them.

A. took 700,000 artisans more than 36 years to complete them
B. took 700,000 artisans more than 36 years to complete it
C. took 700,000 artisans more than 36 years to complete
D. 700,000 artisans took more than 36 years to complete
E. to complete them too 700,000 artisans more than 36 years

------------------------------------------------------------------------------------------------------------
Q21:
That twenty-one ceramic dog figurines were discovered during the excavating of a 1,000-year-old Hohokam village in Tempe, Arizona, has nearly doubled the number of these artifacts known to exist.

A. That twenty-one ceramic dog figurines were discovered during the excavating
B. Twenty-one ceramic dog figurines discovered at the excavation
C. Discovering twenty-one ceramic dog figurines at the excavating
D. Ceramic dog figurines, twenty-one of which were discovered during excavating
E The discovery of twenty-one ceramic dog figurines during the excavation

------------------------------------------------------------------------------------------------------------
Q22:
City Official: At City Hospital, uninsured patients tend to have shorter stays and fewer procedures performed than do insured patients, even though insured patients, on average, have slightly less serious medical problems at the time of admission to the hospital than uninsured patients have. Critics of the hospital have concluded that the uninsured patients are mot receiving proper medical care. However, this conclusion is almost certainly false. Careful investigation has recently shown two things: insured patients have much longer stays in the hospital than necessary, and they tend to have more procedures performed than are medically necessary.

In the city official’s argument, the two boldface portions play which of the following roles?

A. The first states the conclusion of the city official’s argument; the second provides support for that conclusion.
B. The first is used to support the conclusion of the city official’s argument; the second states that conclusion.
C. The first was used to support the conclusion drawn by hospital critics; the second states the position that the city official’s argument opposes.
D. The first was used to support the conclusion drawn by hospital critics; the second provides support for the conclusion of the city official’s argument.
E. The first states the position that the city official’s argument opposes; the second states the conclusion of the city official’s argument.

------------------------------------------------------------------------------------------------------------
Q23:
Past assessments of the Brazilian rain forest have used satellite images to tally deforested areas, where farmers and ranchers have clear-cut and burned all the trees, but such work has not addressed either logging, which is the removal of only selected trees, as well as surface fires, burning down individual trees but do not denude the forest.

A. which is the removal of only selected trees, as well as surface fires, burning
B. which removes only selected trees, or surface fires that burn
C. which removes only selected trees, along with surface fires that burn
D. removing only selected trees, or surface fires, burning
E. removing only selected trees, as well as surface fires that burn

------------------------------------------------------------------------------------------------------------
Q24:
By sucking sap from the young twigs of the hemlock tree, tree growth is retarded by the woolly adelgid, causing needles to change color from deep green to grayish green and to drop prematurely.

A. tree growth is retarded by the woolly adelgid, causing needles to change color from deep green to grayish green and to drop
B. tree growth is retarded by the woolly adelgid, and this causes the color of needles to change from deep green to grayish green, and their dropping
C. the woolly adelgid retards tree growth, which causes needles to change color from deep green to grayish green, and dropping
D. the woolly adelgid retards tree growth, causing needles to change color from deep green to grayish green and to drop
E. the woolly adelgid retards tree growth, and this causes the color of needles to change from deep green to grayish green, and the their dropping

------------------------------------------------------------------------------------------------------------
Q25 to Q28:
Recently biologists have been
interested in a tide-associated
periodic behavior displayed by
Line the diatom Hantzschia virgata, a
(5) microscopic golden-brown alga that
inhabits that portion of a shoreline
washed by tides (the intertidal zone).
Diatoms of this species, sometimes
called “commuter” diatoms, remain
(10) burrowed in the sand during high
tide, and emerge on the sand sur-
face during the daytime low tide.
Just before the sand is inundated by
the rising tide, the diatoms burrow
(15) again. Some scientists hypothesize
that commuter diatoms know that it
is low tide because they sense an
environmental change, such as an
alteration in temperature or a change
(20) in pressure caused by tidal move-
ment. However, when diatoms are
observed under constant conditions
in a laboratory, they still display
periodic behavior, continuing to bur-
(25) row on schedule for several weeks.
This indicates that commuter diatoms,
rather than relying on environmental
cues to keep time, possess an inter-
nal pacemaker or biological clock
(30) that enables them to anticipate peri-
odic changes in the environment.
A commuter diatom has an unusu-
ally accurate biological clock, a
consequence of the unrelenting
(35) environmental pressures to which
it is subjected; any diatoms that do
not burrow before the tide arrives
are washed away.
This is not to suggest that the
(40) period of this biological clock is
immutably fixed. Biologists have
concluded that even though a
diatom does not rely on the envi-
ronment to keep time, environmental
(45) factors—including changes in the
tide’s hydrostatic pressure, salin-
ity, mechanical agitation, and
temperature—can alter the period
of its biological clock according to
(50) changes in the tidal cycle. In short,
the relation between an organism’s
biological clock and its environment
is similar to that between a wristwatch
and its owner: the owner cannot
(55) make the watch run faster or slower,
but can reset the hands. However,
this relation is complicated in intertidal
dwellers such as commuter diatoms
by the fact that these organisms are
(60) exposed to the solar-day cycle as
well as to the tidal cycle, and some-
times display both solar-day and
tidal periods in a single behavior.
Commuter diatoms, for example,
(65) emerge only during those low tides
that occur during the day.
--------------------------------------------------------------------------------
Q25:
The passage suggests which of the following about the accuracy of the commuter diatom’s biological clock?

A. The accuracy of the commuter diatom’s biological clock varies according to changes in the tidal cycle.
B. The unusual accuracy that characterizes the commuter diatom’s biological clock is rare among intertidal species.
C. The commuter diatom’s biological clock is likely to be more accurate than the biological clock of a species that is subject to less intense environmental pressures.
D. The commuter diatom’s biological clock tends to be more accurate than the biological clocks of most other species because of the consistency of the tidal cycle.
E. The accuracy of the commuter diatom’s biological clock tends to fluctuate when the diatom is observed under variable laboratory conditions.

--------------------------------------------------------------------------------
Q26:
The author of the passage compares the relationship between an organism’s biological clock and its environment to the relation between a wristwatch and its owner most probably in order to

A. point out a fundamental difference between the function of biological clocks in organisms and the use of mechanical clocks by humans
B. illustrate the way in which the period of an organism’s biological clock can be altered by environmental factors
C. suggest that there are important similarities between the biological clock in organisms such as the commuter diatom and the biological clock in humans
D. support an argument regarding the methods used by certain organisms to counteract the influence of the environment on their biological clocks
E. question the accuracy of the biological clock in organisms such as the commuter diatom

---------------------------------------------------------------------------------
Q27:
According to the passage, the periodic behavior displayed by commuter diatoms under constant laboratory conditions is characterized by which of the following?

A. Greater unpredictability than the corresponding behavior under natural conditions
B. A consistent periodic schedule in the short term
C. No difference over the long term from the corresponding behavior under natural conditions
D. Initial variability caused by the constant conditions of the laboratory
E. Greater sensitivity to environmental factors than is the case under natural conditions

--------------------------------------------------------------------------------
Q28:
The primary purpose of the passage is to

A. dispute the influence of environmental factors on the tide-associated behavioral rhythms displayed by the diatom Hantzschia virgata
B. describe how certain tide-associated behavioral rhythms displayed by the diatom Hantzschia virgata have changed over time
C. compare tide-associated behavioral rhythms to solar-day behavioral rhythms in the diatom Hantzschia virgata
D. examine how certain biological and environmental influences affect the tide-associated behavioral rhythms displayed by the diatom Hantzschia virgata
E. identify certain environmental factors that limit the effectiveness of the biological clock in the diatom Hantzschia virgata

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Q29:
Faulty voting equipment, confusing ballots, voter error, and problems at polling places have been cited by a new study of the 2000 United States presidential election, which estimated that they did not count 4 million to 6 million of the 100 million votes cast.

A. Faulty voting equipment, confusing ballots, voter error, and problems at polling places have been cited by a new study of the 2000 United States presidential election, which estimated that they did not count 4 million to 6 million of the 100 million votes cast.
B. Citing faulty voting equipment, confusing ballots, voter error, and problems at polling places, a new study of the 2000 United States presidential election has estimated that 4 million to 6 million of the 100 million votes cast were not counted.
C. Citing faulty voting equipment, confusing ballots, voter error, and problems at polling places, 4 million to 6 million of the 100 million votes cast were not counted in the 2000 United States presidential election, a new study estimates.
D. A new study has cited faulty voting equipment, confusing ballots, voter error, and problems at polling places in estimating that 4 million to 6 million of the 100 million votes that were cast had not been counted in the 2000 United States presidential election.
E. A new study of the 2000 United States presidential election, citing faulty voting equipment, confusing ballots, voter error, and problems at polling places, has estimated 4 million to 6 million votes had not been counted of the 100 million votes cast.

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Q30:
The ancient Anasazi harvested such native desert vegetation as the purple-flowered bee plant, what they now commonly call wild spinach in northern Arizona and other parts of the southwestern United States.

A. what they now commonly call
B. a plant that they now commonly call
C. now commonly called
D. and is mow commonly called
E. which it is now commonly called

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Q31:
Which of the following most logically completes the argument below?

Davison River farmers are currently deciding between planting winter wheat this fall or spring wheat next spring. Winter wheat and spring wheat are usually about equally profitable. Because of new government restrictions on the use of Davison River water for irrigation, per acre yields for winter wheat, though not for spring wheat, would be much lower than average. Therefore, planting spring wheat will be more profitable than planting winter wheat, since‗‗‗‗‗‗.

A. the smaller-than-average size of a winter wheat harvest this year would not be compensated for by higher winter wheat prices
B. new crops of spring wheat must be planted earlier than the time at which standing crops of winter wheat are ready to be harvested
C. the spring wheat that farmers in the Davison River region plant is well adapted to the soil of the region
D. spring wheat has uses that are different from those of winter wheat
E. planting spring wheat is more profitable than planting certain other crops, such as rye

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Q32:
To develop more accurate population forecasts, demographers have to know a great deal more than now about the social and economic determinants of fertility.

A. have to know a great deal more than now about the social and economic
B. have to know a great deal more than they do now about the social and economical
C. would have to know a great deal more than they do now about the social and economical
D. would have to know a great deal more than they do now about the social and economic
E. would have to know a great deal more than now about the social and economical

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Q33:
A company plans to develop a prototype weeding machine that uses cutting blades with optical sensors and microprocessors that distinguish weeds from crop plants by differences in shade of color. The inventor of the machine claims that it will reduce labor costs by virtually eliminating the need for manual weeding.

Which of the following is a consideration in favor of the company’s implementing its plan to develop the prototype?

A. There is a considerable degree of variation in shade of color between weeds of different species.
B. The shade of color of some plants tends to change appreciably over the course of their growing season.
C. When crops are weeded manually, overall size and leaf shape are taken into account in distinguishing crop plants from weeds.
D. Selection and genetic manipulation allow plants of virtually any species to be economically bred to have a distinctive shade of color without altering their other characteristics.
E. Farm laborers who are responsible for the manual weeding of crops carry out other agricultural duties at times in the growing season when extensive weeding is not necessary.

------------------------------------------------------------------------------------------------------------
Q34:
Although people in France consume fatty foods at a rate comparable to the United States, their death rates from heart disease are far lower in France.

A. people in France consume fatty foods at a rate comparable to the United States, their
B. people in France and the United States consume fatty foods at about the same rate, the
C. fatty foods are consumed by people in France at a comparable rate to the United States’s, their
D. the rate of fatty foods consumed in France and the United States is about the same, the
E. the rate of people consuming fatty foods is about the same in France and the United States, the

------------------------------------------------------------------------------------------------------------
Q35 to Q37:
In corporate purchasing,
competitive scrutiny is typically
limited to suppliers of items that are
Line directly related to end products.
(5) With “indirect” purchases (such as
computers, advertising, and legal
services), which are not directly
related to production, corporations
often favor “supplier partnerships”
(10) (arrangements in which the
purchaser forgoes the right to
pursue alternative suppliers), which
can inappropriately shelter suppliers
from rigorous competitive scrutiny
(15) that might afford the purchaser
economic leverage. There are two
independent variables—availability
of alternatives and ease of changing
suppliers—that companies should
(20) use to evaluate the feasibility of
subjecting suppliers of indirect
purchases to competitive scrutiny.
This can create four possible
situations.
(25) In Type 1 situations, there are
many alternatives and change is
relatively easy. Open pursuit of
alternatives—by frequent com-
petitive bidding, if possible—will
(30) likely yield the best results. In
Type 2 situations, where there
are many alternatives but change
is difficult—as for providers of
employee health-care benefits—it
(35) is important to continuously test
the market and use the results to
secure concessions from existing
suppliers. Alternatives provide a
credible threat to suppliers, even if
(40) the ability to switch is constrained.
In Type 3 situations, there ate few
alternatives, but the ability to switch
without difficulty creates a threat that
companies can use to negotiate
(45) concessions from existing suppliers.
In Type 4 situations, where there
are few alternatives and change
is difficult, partnerships may be
unavoidable.
--------------------------------------------------------------------------------
Q35:
Which of the following best describes the relation of the second paragraph to the first?

A. The second paragraph offers proof of an assertion made in the first paragraph.
B. The second paragraph provides an explanation for the occurrence of a situation described in the first paragraph.
C. The second paragraph discusses the application of a strategy proposed in the first paragraph.
D. The second paragraph examines the scope of a problem presented in the first paragraph.
E. The second paragraph discusses the contradictions inherent in a relationship described in the first paragraph.

--------------------------------------------------------------------------------
Q36:
Which of the following can be inferred about supplier partnerships, as they are described in the passage?

A. They cannot be sustained unless the goods or services provided are available from a large number of suppliers.
B. They can result in purchasers paying more for goods and services than they would in a competitive-bidding situation.
C. They typically are instituted at the urging of the supplier rather than the purchaser.
D. They are not feasible when the goods or services provided are directly related to the purchasers’ end products.
E. They are least appropriate when the purchasers’ ability to change suppliers is limited.

--------------------------------------------------------------------------------
Q37:
According to the passage, which of the following factors distinguishes an indirect purchase from other purchases?

A. The ability of the purchasing company to subject potential suppliers of the purchased item to competitive scrutiny
B. The number of suppliers of the purchased item available to the purchasing company
C. The methods of negotiation that are available to the purchasing company
D. The relationship of the purchased item to the purchasing company’s end product
E. The degree of importance of the purchased item in the purchasing company’s business operations

------------------------------------------------------------------------------------------------------------
Q38:
Retail sales rose 8/10 of 1 percent in August, intensifying expectations that personal spending in the July-September quarter more than doubled that of the 1.4 percent growth rate in personal spending for the previous quarter.

A. that personal spending in the July-September quarter more than doubled that of
B. that personal spending in the July-September quarter would more than double
C. of personal spending in the July-September quarter, that it more than doubled
D. of personal spending in the July-September quarter, more than doubling that of
E. of personal spending in the July-September quarter, that it would more than double that of

------------------------------------------------------------------------------------------------------------
Q39:
Which of the following most logically completes the passage below?

Heavy rains during Centralia’s corn planting season prevented some farmers there from planting corn. It is now the planting season for soybeans, another of Centralia’s principal crops, and those fields originally intended for corn are dry enough for planting. Nonetheless, even though soybean prices are unusually high at present, the farmers will leave most of these fields empty rather than plant them with soybeans, since ‗‗‗‗‗‗.

A. the extensive rains have led to an increase in the price of corn
B. some Centralian farmers anticipate serious financial losses due to the extremely wet spring planting season
C. chemicals that were used to prepare the fields for corn planting would stunt the growth of soybeans
D. the majority of Centralia’s corn farmers were able to plant corn as they had intended, despite the wet planting season
E. many Centralian farmers grow both corn and soybeans

------------------------------------------------------------------------------------------------------------
Q40:
Until mow, only injectable vaccines against influenza have been available. Parents are reluctant to subject children to the pain of injections, but adults, who are at risk of serious complications from influenza, are commonly vaccinated. A new influenza vaccine, administered painlessly in a nasal spray, is effective for children. However, since children seldom develop serious complications from influenza, no significant public health benefit would result from widespread vaccination of children using the nasal spray.

Which of the following is an assumption on which the argument depends?

A. Any person who has received the injectable vaccine can safely receive the nasal-spray vaccine as well.
B. The new vaccine uses the same mechanism to ward off influenza as jnjectable vaccines do.
C. The injectable vaccine is affordable for all adults.
D. Adults do not contract influenza primarily from children who have influenza.
E. The nasal spray vaccine is mot effective when administered to adults.

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Q41:
Leaching, the recovery of copper from the drainage water of mines, as a method of the extraction of minerals, it was well established as early as the eighteenth century, but until about 25 years ago miners did not realize that bacteria take an active part in the process.

A. as a method of the extraction of minerals, it was well established
B. as a method of the extraction of minerals well established
C. was a well-established method of mineral extraction
D. was a well-established method of extracting mineral that was
E. had been a method of mineral extraction, well established

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Math Tricks

2007-12-21T12:23:00.000+05:30

04/10/2002
-------------
THEORY OF EQUATIONS:
-----------------------------

(1) If an equation (i:e f(x)=0 ) contains all positive co-efficients of any powers of x , it has no positive roots then.
eg: x^4+3x^2+2x+6=0 has no positive roots .

(2) For an equation , if all the even powers of x have some sign coefficients and all the odd powers of x have the opposite sign coefficients , then it has no negative roots .

(3)Summarising DESCARTES RULE OF SIGNS:

For an equation f(x)=0 , the maximum number of positive roots it can have is the number of sign changes in f(x) ; and the maximum number of negative roots it can have is the number of sign changes in f(-x) .
Hence the remaining are the minimum number of imaginary roots of the equation(Since we also know that the index of the maximum power of x is the number of roots of an equation.)

(4) Complex roots occur in pairs, hence if one of the roots of an equation is 2+3i , another has to be 2-3i and if there are three possible roots of the equation , we can conclude that the last root is real . This real roots could be found out by finding the sum of the roots of the equation and subtracting (2+3i)+(2-3i)=4 from that sum. (More about finding sum and products of roots next time )

------------------------------------------
07/10/2002 THEORY OF EQUATIONS
------------------------------------------

(1) For a cubic equation ax^3+bx^2+cx+d=o

sum of the roots = - b/a
sum of the product of the roots taken two at a time = c/a
product of the roots = -d/a

(2) For a biquadratic equation ax^4+bx^3+cx^2+dx+e = 0

sum of the roots = - b/a
sum of the product of the roots taken three at a time = c/a
sum of the product of the roots taken two at a time = -d/a
product of the roots = e/a

(3) If an equation f(x)= 0 has only odd powers of x and all these have the same sign coefficients or if f(x) = 0 has only odd powers of x and all these have the same sign
coefficients then the equation has no real roots in each case(except for x=0 in the second case.

(4) Besides Complex roots , even irrational roots occur in pairs. Hence if 2+root(3) is a root , then even 2-root(3) is a root .
(All these are very useful in finding number of positive , negative , real ,complex etc roots of an equation )

Today's Section:
----------------
08/10/2002
----------------

(1) If for two numbers x+y=k(=constant), then their PRODUCT is MAXIMUM if
x=y(=k/2). The maximum product is then (k^2)/4 .

(2) If for two numbers x*y=k(=constant), then their SUM is MINIMUM if
x=y(=root(k)). The minimum sum is then 2*root(k) .

(3) |x| + |y| >= |x+y| (|| stands for absolute value or modulus )
(Useful in solving some inequations)

(4) Product of any two numbers = Product of their HCF and LCM .
Hence product of two numbers = LCM of the numbers if they are prime to each other .

1) For any regular polygon , the sum of the exterior angles is equal to 360 degrees
hence measure of any external angle is equal to 360/n. ( where n is the number of sides)

(2) If any parallelogram can be inscribed in a circle , it must be a rectangle.

(3) If a trapezium can be inscribed in a circle it must be an isosceles trapezium (i:e oblique sies equal).

(4) For an isosceles trapezium , sum of a pair of opposite sides is equal in length to the sum of the other pair of opposite sides .(i:e AB+CD = AD+BC , taken in order) .

(5) Area of a regular hexagon : root(3)*3/2*(side)*(side)

1) For any 2 numbers a>b

a>AM>GM>HM>b (where AM, GM ,HM stand for arithmetic, geometric , harmonic menasa respectively)

(2) (GM)^2 = AM * HM

(3) For three positive numbers a, b ,c

(a+b+c) * (1/a+1/b+1/c)>=9

(4) For any positive integer n

2<= (1+1/n)^n <=3

(5) a^2+b^2+c^2 >= ab+bc+ca
If a=b=c , then the equality holds in the above.

(6) a^4+b^4+c^4+d^4 >=4abcd

(7) (n!)^2 > n^n (! for factorial)

--------------------------
This is for 21/10/2002
--------------------------

(1) If a+b+c+d=constant , then the product a^p * b^q * c^r * d^s will be maximum
if a/p = b/q = c/r = d/s .

(2)Consider the two equations

a1x+b1y=c1
a2x+b2y=c2

Then ,
If a1/a2 = b1/b2 = c1/c2 , then we have infinite solutions for these equations.
If a1/a2 = b1/b2 <> c1/c2 , then we have no solution for these equations.(<> means not equal to )
If a1/a2 <> b1/b2 , then we have a unique solutions for these equations..

(3) For any quadrilateral whose diagonals intersect at right angles , the area of the quadrilateral is
0.5*d1*d2, where d1,d2 are the lenghts of the diagonals.

(4) Problems on clocks can be tackled as assuming two runners going round a circle , one 12 times as fast as the other . That is ,
the minute hand describes 6 degrees /minute
the hour hand describes 1/2 degrees /minute .

Thus the minute hand describes 5(1/2) degrees more than the hour hand per minute .

(5) The hour and the minute hand meet each other after every 65(5/11) minutes after being together at midnight.
(This can be derived from the above) .

1)If n is even , n(n+1)(n+2) is divisible by 24

(2)If n is any integer , n^2 + 4 is not divisible by 4

(3)Given the coordinates (a,b) (c,d) (e,f) (g,h) of a parallelogram , the coordinates of the meeting point of the diagonals can be found out by solving for
[(a+e)/2,(b+f)/2] =[ (c+g)/2 , (d+h)/2]

(4)Area of a triangle
1/2*base*altitude = 1/2*a*b*sinC = 1/2*b*c*sinA = 1/2*c*a*sinB = root(s*(s-a)*(s-b)*(s-c)) where s=a+b+c/2
=a*b*c/(4*R) where R is the CIRCUMRADIUS of the triangle = r*s ,where r is the inradius of the triangle .

(5)In any triangle
a=b*CosC + c*CosB
b=c*CosA + a*CosC
c=a*CosB + b*CosA

(6)If a1/b1 = a2/b2 = a3/b3 = .............. , then each ratio is equal to
(k1*a1+ k2*a2+k3*a3+..............) / (k1*b1+ k2*b2+k3*b3+..............) , which is also equal to
(a1+a2+a3+............./b1+b2+b3+..........)

(7)In any triangle
a/SinA = b/SinB =c/SinC=2R , where R is the circumradius

cosC = (a^2 + b^2 - c^2)/2ab

sin2A = 2 sinA * cosA
cos2A = cos^2(A) - sin^2 (A)

1) x^n -a^n = (x-a)(x^(n-1) + x^(n-2) + .......+ a^(n-1) ) ......Very useful for finding multiples .For example (17-14=3 will be a multiple of 17^3 - 14^3)

(2) e^x = 1 + (x)/1! + (x^2)/2! + (x^3)/3! + ........to infinity
(2a) 2 < e < 3

(3) log(1+x) = x - (x^2)/2 + (x^3)/3 - (x^4)/4 .........to infinity [ Note the alternating sign . .Also note that the ogarithm is with respect to base e ]

(4) In a GP the product of any two terms equidistant from a term is always constant .

(5) For a cyclic quadrilateral , area = root( (s-a) * (s-b) * (s-c) * (s-d) ) , where s=(a+b+c+d)/2

(6) For a cyclic quadrilateral , the measure of an external angle is equal to the measure of the internal opposite angle.

(7) (m+n)! is divisible by m! * n! .

_________________
"I have miles to go before I sleep

-------------
02/11/2002
-------------

(1) If a quadrilateral circumscribes a circle , the sum of a pair of opposite sides is equal to the sum of the other pair .

(2)The sum of an infinite GP = a/(1-r) , where a and r are resp. the first term and common ratio of the GP .

(3)The equation whose roots are the reciprocal of the roots of the equation
ax^2+bx+c is cx^2+bx+a

(4) The coordinates of the centroid of a triangle with vertices (a,b) (c,d) (e,f)
is((a+c+e)/3 , (b+d+f)/3) .

(5) The ratio of the radii of the circumcircle and incircle of an equilateral triangle is 2:1 .

(6) Area of a parallelogram = base * height

(7)APPOLLONIUS THEOREM:

In a triangle , if AD be the median to the side BC , then
AB^2 + AC^2 = 2(AD^2 + BD^2) or 2(AD^2 + DC^2) .

_________________

1) for similar cones , ratio of radii = ratio of their bases.

(2) The HCF and LCM of two nos. are equal when they are equal .

(3) Volume of a pyramid = 1/3 * base area * height

(4) In an isosceles triangle , the perpendicular from the vertex to the base or the angular bisector from vertex to base bisects the base.

(5) In any triangle the angular bisector of an angle bisects the base in the ratio of the
other two sides.

(6) the quadrilateral formed by joining the angular bisectors of another quadrilateral is
always a rectangle.

(7) Roots of x^2+x+1=0 are 1,w,w^2 where 1+w+w^2=0 and w^3=1

( |a|+|b| = |a+b| if a*b>=0
else |a|+|b| >= |a+b|

(9) 2<= (1+1/n)^n <=3

(10) WINE and WATER formula:

If Q be the volume of a vessel
q qty of a mixture of water and wine be removed each time from a mixture
n be the number of times this operation be done
and A be the final qty of wine in the mixture

then ,
A/Q = (1-q/Q)^n

(11) Area of a hexagon = root(3) * 3 * (side)^2

(12) (1+x)^n ~ (1+nx) if x<<<1

(13) Some pythagorean triplets:

3,4,5 (3^2=4+5)
5,12,13 (5^2=12+13)
7,24,25 (7^2=24+25)
8,15,17 (8^2 / 2 = 15+17 )
9,40,41 (9^2=40+41)
11,60,61 (11^2=60+61)
12,35,37 (12^2 / 2 = 35+37)
16,63,65 (16^2 /2 = 63+65)
20,21,29(EXCEPTION)

(14) Appolonius theorem could be applied to the 4 triangles formed in a parallelogram.

(15) Area of a trapezium = 1/2 * (sum of parallel sids) * height = median * height
where median is the line joining the midpoints of the oblique sides.

(16) when a three digit number is reversed and the difference of these two numbers is taken , the middle number is always 9 and the sum of the other two numbers is always 9 .

(17) ANy function of the type y=f(x)=(ax-b)/(bx-a) is always of the form x=f(y) .

(1 Let W be any point inside a rectangle ABCD .
Then
WD^2 + WB^2 = WC^2 + WA^2

(19) Let a be the side of an equilateral triangle . then if three circles be drawn inside
this triangle touching each other then each's radius = a/(2*(root(3)+1))

(20) Let 'x' be certain base in which the representation of a number is 'abcd' , then the decimal value of this number is a*x^3 + b*x^2 + c*x + d

5) For a cyclic quadrilateral , area = root( s* (s-a) * (s-b) * (s-c) * (s-d) ) , where s=(a+b+c+d)/2

Here are some neat shortcuts on Simple/Compound Interest.

Shortcut #1:
-------------
We all know the traditional formula to compute interest...
CI = P*(1+R/100)^N - P

The calculation get very tedious when N>2 (more than 2 years). The method suggested below is elegant way to get CI/Amount after 'N' years.

You need to recall the good ol' Pascal's Triange in following way:
Code:

Number of Years (N)
-------------------
1 1
2 1 2 1
3 1 3 3 1
4 1 4 6 4 1
. 1 .... .... ... ... 1

Example: P = 1000, R=10 %, and N=3 years. What is CI & Amount?

Step 1: 10% of 1000 = 100, Again 10% of 100 = 10 and 10% of 10 = 1
We did this three times b'cos N=3.

Step 2:
Now Amount after 3 years = 1 * 1000 + 3 * 100 + 3 * 10 + 1 * 1 = Rs.1331/-
The coefficents - 1,3,3,1 are lifted from the pascal's triangle above.

Step 3:
CI after 3 years = 3*100 + 3*10 + 3*1 = Rs.331/- (leaving out first term in step 2)

If N =2, we would have had, Amt = 1 * 1000 + 2 * 100 + 1 * 10 = Rs. 1210/-
CI = 2 * 100 + 1* 10 = Rs. 210/-

This method is extendable for any 'N' and it avoids calculations involving higher powers on 'N' altogether!

A variant to this short cut can be applied to find depreciating value of some property. (Example, A property worth 100,000 depreciates by 10% every year, find its value after 'N' years).

Shortcut #2:
-------------
(i) When interest is calculated as CI, the number of years for the Amount to double (two times the principal) can be found with this following formula:
P * N ~ 72 (approximately equal to).
Exampe, if R=6% p.a. then it takes roughly 12 years for the Principal to double itself.

Note: This is just a approximate formula (when R takes large values, the error % in formula increases).

(ii) When interest is calculated as SI, number of years for amt to double can be found as:
N * R = 100 . BTW this formula is exact!

Adding to what 'Peebs' said, this shortcut does work for any P/N/R.

Basically if you look closely at this method, what I had posted is actually derived from the Binomial expansion of the polynomial -- (1+r/100)^n but in a more "edible" format digestable by us!

BTW herez one shortcut on recurring decimals to fractions ...Its more easier to explain with an example..

Eg: 2.384384384 ....

Step 1: since the 3 digits '384' is recurring part, multiply 2.384 by 1000 = so we get 2384.

Next '2' is the non recurring part in the recurring decimal so subtract 2 from 2384 = 2382.

If it had been 2.3848484.., we would have had 2384 - 23 = 2361. Had it been 2.384444.. NR would be 2384 - 238 = 2146 and so on.

We now find denominator part .......

Step 3: In step 1 we multiplied 2.384384... by 1000 to get 2384, so put that first.

Step 4: next since all digits of the decimal part of recurring decimal is recurring, subtract 1 from step 3. Had the recurring decimal been 2.3848484, we need to subtract 10. If it had been 2.3844444, we needed to have subtracted 100 ..and so on...

Hence here, DR = 1000 - 1 = 999

Hence fraction of the Recurring decimal is 2382/999!!

Some more examples ....

1.56787878 ... = (15678 - 156) / (10000 - 100) = 15522/9900

23.67898989... = (236789 - 2367) / (10000 - 100) = 234422/9900

124.454545... = (12445 - 124) / (100 - 1) = 12321/99

Maths Shortcut

2007-12-21T12:22:00.001+05:30

Numbers

1) 2^2n-1 is always divisible by 3
2^2n-1 = (3-1)^2n -1
= 3M +1 -1
= 3M, thus divisible by 3

2) What is the sum of the divisors of 2^5.3^7.5^3.7^2?
ANS : (2^6-1)(3^8-1)(5^4-1)(7^3-1)/2.4.6
Funda : if a number 'n' is represented as
a^x * b^y * c^z ....
where, {a,b,c,.. } are prime numbers then
Quote:
(a) the total number of factors is (x+1)(y+1)(z+1) ....
(b) the total number of relatively prime numbers less than the number is n * (1-1/a) * (1-1/b) * (1-1/c)....
(c) the sum of relatively prime numbers less than the number is n/2 * n * (1-1/a) * (1-1/b) * (1-1/c)....
(d) the sum of factors of the number is {a^(x+1)} * {b^(y+1)} * ...../(x*y*...)

3) what is the highest power of 10 in 203!ANS : express 10 as product of primes; 10 = 2*5

divide 203 with 2 and 5 individually
203/2 = 101
101/2 = 50
50/2 = 25
25/2 = 12
12/2 = 6
6/2 = 3
3/2 = 1
thus power of 2 in 203! is, 101 + 50 + 25 + 12 + 6 + 3 + 1 = 198

divide 203 with 5
203/5 = 40
40/5 = 8
8/5 = 1

thus power of 5 in 203! is, 49

so the power of 10 in 203! factorial is 49

4) x + y + z = 7 and xy + yz + zx = 10, then what is the maximum value of x? ( CAT 2002 has similar question )
ANS: 49-20 = 29, now if one of the y,z is zero, then the sum of other 2 squares shud be equal to 29, which means, x can take a max value at 5

5) In how many ways can 2310 be expressed as a product of 3 factors?
ANS: 2310 = 2*3*5*7*11
When a number can be expressed as a product of n distinct primes,
then it can be expressed as a product of 3 numbers in (3^(n+1) + 1)/2 ways

6) In how many ways, 729 can be expressed as a difference of 2 squares?
ANS: 729 = a^2 - b^2
= (a-b)(a+b),
since 729 = 3^5,
total ways of getting 729 are, 1*729, 3*243, 9*81, 27*27.
So 4 ways
Funda is that, all four ways of expressing can be used to findout distinct a,b values,
for example take 9*81
now since 9*81 = (a-b)(a+b) by solving the system a-b = 9 and a+b = 81 we can have 45,36 as soln.

7) How many times the digit 0 will appear from 1 to 10000
ANS: In 2 digit numbers : 9,
In 3 digit numbers : 18 + 162 = 180,
In 4 digit numbers : 2187 + 486 + 27 = 2700,
total = 9 + 180 + 2700 + 4 = 2893

8 ) What is the sum of all irreducible factors between 10 and 20 with denominator as 3?
ANS :
sum = 10.33 + 10.66 + 11.33 + 11.66 + 12.33 + 12.66 + 13.33 + 13.66…….
= 21 + 23 + ……
= 300

9) if n = 1+x where x is the product of 4 consecutive number then n is,
1) an odd number,
2) is a perfect square
SOLN : (1) is clearly evident
(2) let the 4 numbers be n-2,n-1,n and n+1 then by multing the whole thing and adding 1 we will have a perfect square

10) When 987 and 643 are divided by same number 'n' the reminder is also same, what is that number if the number is a odd prime number?
ANS : since both leave the same reminder, let the reminder be 'r',
then, 987 = an + r
and 643 = bn + r and thus
987 - 643 is divisible by 'r' and
987 - 643 = 344 = 86 * 4 = 43 * 8 and thus the prime is 43
hence 'r' is 43

11) when a number is divided by 11,7,4 the reminders are 5,6,3 respectively. what would be the reminders when the same number is divided by 4,7,11 respectively?
ANS : whenever such problem is given,
we need to write the numbers in top row and rems in the bottom row like this

11 7 4
| \ \
5 6 3

( coudnt express here properly )
now the number is of the form, LCM ( 11,7,4 ) + 11*(3*7 + 6) + 5
that is 302 + LCM(11,7,4) and thus the rems when the same number is divided by 4,7,11 respectively are,

302 mod 4 = 2
75 mod 7 = 5
10 mod 11 = 10

12) a^n - b^n is always divisible by a-b

13) if a+b+c = 0 then a^3 + b^3 + c^3 = 3abc
EXAMPLE: 40^3-17^3-23^3 is divisble by
since 40-23-17 = 0, 40^3-17^3-23^3 = 3*40*23*17 and thus, the number is divisible by 3,5,8,17,23 etc.

14) There is a seller of cigerette and match boxes who sits in the narrow lanes of cochin. He prices the cigerattes at 85 p, but found that there are no takers. So he reduced the price of cigarette and managed to sell all the cigerattes, realising Rs. 77.28 in all. What is the number of cigerattes?

a) 49
b) 81
c) 84
d) 92

ANS : (d)
since 77.28 = 92 * 84, and since price of cigarette is less than 85, we have (d) as answer
Quote:

i have given this question to make the funda clear

15) What does 100 stand for if 5 X 6 = 33
ANS : 81
SOLN : this is a number system question,
30 in decimal system is 33 in some base 'n', by solving we will have n as 9
and thus, 100 will be 9^2 = 81

16) In any number system 121 is a perfect square,
SOLN: let the base be 'n'
then 121 can be written as n^2 + 2*n + 1 = (n+1)^2
hence proved

17) Most of you ppl know these, anyways, just in case
Quote:
(a) sum of first 'n' natural numbers - n*(n+1)/2
(b) sum of the squares of first 'n' natural numbers - n*(n+1)*(2n+1)/6
(c) sum of the cubes of first 'n' natural numbers - n^2*(n+1)^2/4
(d) total number of primes between 1 and 100 - 25

18 ) See Attachment to know how to find LCM, GCF of Fractions
Quote:
CAT 2002 has 2 questions on the above simple concept

19) Converting Recurring Decimals to Fractions

let the number x be 0.23434343434........

thus 1000 x = 234.3434343434......
and 10 x = 2.3434343434.........
thus, 990 x = 232
and hence, x = 232/990

20) Reminder Funda

(a) (a + b + c) % n = (a%n + b%n + c%n) %n
EXAMPLE: The reminders when 3 numbers 1221, 1331, 1441 are divided by certain number 9 are 6, 8, 1 respectively. What would be the reminder when you divide 3993 with

9? ( never seen such question though )
the reminder would be (6 + 8 + 1) % 9 = 6

(b) (a*b*c) % n = (a%n * b%n * c%n) %n
EXAMPLE: What is the remainder left when 1073 * 1079 * 1087 is divided by 119 ? ( seen this kinda questions alot )
1073 % 119 = ?
since 1190 is divisible by 119, 1073 mod 119 is 2
and thus, "the remainder left when 1073 * 1079 * 1087 is divided by 119 " is 2*8*16 mod 119 and that is 256 mod 119 and that is (238 + 18 ) mod 119 and that is 18

Glossary : % stands for reminder operation

find the number of zeroes in 1^1* 2^2* 3^3* 4^4.............. 98^98* 99^99* 100^100

the expresion can be rewritten as (100!)^100 / 0!* 1!* 2!* 3!....99!

Now the numerator has 2400 zeros

the formular for finding number of zeros in n! is

[n/5]+[n/5^2]...[n/5^r]
where r is such that 5^r<=n<5^(r+1)

and [..] is the grestest integer function

for the numerator find the number of zeros using the above formulae..

for 0!...4! number of zeros ..0
5!...9!.number os zeros ..1
9!...14!... 2
15!..19!..................3
20!..24!..................4!
now at 25! the series makes a jump to 6
25!...29!.................6
30!...34!.................7
this goes on and again makes a jump at 50!
and then at 75!

so the number of zeros is...

5(1+2....19) + 25+ 50+ 75

the last 3 terms 25 50 and 75 are because of the jumps..

this gives numerator has 1100 zeros

now total number of zeros in expression is no of zeros in denominator - no of zeros in numerator
2400 - 1100

the Answer 1300

Quadratic Equation Tips

2007-12-21T12:20:00.001+05:30

Quadratic Equations and Functions

1. Introduction

A quadratic equation in one unknown is an equation of the form , where . When we solve a linear equation, we may transpose the terms and leave the unknown on one side of the equation. However, this is often not the case for a quadratic equation. There are other methods to solve a quadratic equation, e.g. by factorization, by completing the square and by the quadratic formula. Furthermore, for a linear equation in the form of , where , there is always a solution , which is a real number. On the contrary, a quadratic equation may have two real roots, one double root or no real roots.

2. Solving quadratic equations

There are several methods to solve a quadratic equation. Some quadratic expressions can be factorized and then the equation is easy to solve.

Theorem 2.1.

If pq = 0, then p = 0 or q = 0.

From theorem 2.1, if a quadratic equation can be factorized to the form , then we have px + q = 0 or rx + s = 0, which gives or respectively.

Example 2.1.

Solve .

Solution.

We try to factorize . This can be written as .

Thus we have

Example 2.2.

Solve .

Solution.

This method is useful only when the factorization is easy to do. Many quadratic expressions, like and , cannot be factorized into the form , where p, q, r, s are rational numbers. There are two other methods to solve quadratic equations and these two can solve all quadratic equations. These two methods are completing the square and using the quadratic formula.

The method of completing the square is to change the equation from the form to . This can be done by dividing the whole equation by a and then we have

and thus and . If q is positive, then . If q = 0, then x + p = 0 and hence x = –p. If q is negative, since the square of all real numbers is non-negative, the equation has no real roots. The result still holds, but the roots will be complex.

Example 2.3.

Solve .

Solution.

Example 2.4.

Solve .

Solution.

From the method of completing the square, we can derive the quadratic formula.

Thus we have theorem 2.2.

Theorem 2.2.

If and , the roots of this equation is given by

This formula can be used to solve all quadratic equations in the form with . Furthermore, by observing the expression in the square root, i.e. , we can know if the equation has real roots or not. More about this will be discussed in the next section.

Example 2.5.

Solve .

Solution.

We put a = 1, b = –1 and c = –12 to the quadratic formula. Then we have

Example 2.6.

Solve .

Solution.

We put a = 1, b = –2 and c = 2 to the quadratic formula. Then we have

Ø is not a real number and it is defined that . Thus the answer above can be written as . i is an imaginary number and are unreal solutions to the equation.

Examples 2.7 and 2.8 are further examples.

Example 2.7.

Solve .

Solution.

Let , then the original equation becomes

Since for real x, the solution is rejected. Then we have , which gives .

Example 2.8.

Solve .

Solution.

Note that the equation is equivalent to .

Let , then the equation becomes

Exercises

1. Solve .

2. Solve .

3. Solve .

4. Solve .

3. Sum and product of roots

Theorem 3.1.

Let a and b be the roots of a quadratic equation with , then

(1) and

(2) .

Proof. From the quadratic formula, we have

and

Q.E.D.

Ø Alternatively, we can find these two relations by comparing coefficients. Since a and b are roots to the equation, we have

The original equation is

and by comparing coefficients, we have and .

Ø Note that the relation of sum and product of roots with the coefficients always holds, no matter if the roots are real or not.

Example 3.1.

Let a and b be the roots of and a > b. Find the values of , and a – b.

Solution.

From theorem 3.1, we have

Thus .

Note that

Since a > b, we have .

Theorem 3.2.

Given the sum of roots and product of roots, we can form a quadratic equation whose roots have the required sum and product and this equation is

Example 3.2.

Let a and b be roots of . Form a quadratic equation, with integer coefficients, whose roots are and .

Solution.

We form the quadratic equation by finding the sum and product of roots of the required equation.

For ,

For the required equation,

and

Thus the required equation is

Exercises.

1. Given a and b are the roots of . Find

(a) a + b

(b) ab

(c)

(d)

2. Given a and b are the roots of . Find an equation with integer coefficients whose roots are and .

4. Discriminant and nature of roots

As we can see from the quadratic formula, the term inside the square root sign, i.e. , determines the nature of roots.

Definition 4.1.

For a quadratic equation , where , the discriminant, usually denoted as D, is defined as

The value of the discriminant tells us the number of real roots of the quadratic equation.

Theorem 4.1.

For a quadratic equation , where , and its discriminant is ,

(1) if D > 0, the equation has two distinct real roots.

(2) if D = 0, the equation has one double real root.

(3) if D <>

Example 4.1.

Find the number of real root(s) for

(1)

(2)

(3)

Solution.

We first calculate the value of the discriminant of the equation and then we can determine the number of real root(s).

For (1), , thus there are two distinct real roots.

For (2), , thus there is no real root.

For (3), , thus there is one real root.

Example 4.2.

Find the range of values of k such that has real root(s).

Solution.

For the equation to have real root(s), .

Thus we have

Exercises.

1. Find the range of values of k such that has two distinct real roots.

5. Quadratic functions and their graphs

A quadratic function is a function in the form , where . The graph of a quadratic function is a parabola. If a > 0, the parabola opens upward; if a <>

Figure 1: The graph of , where a > 0

Figure 2: The graph of , where a <>

For a parabola, it has a vertex and an axis of symmetry. The coordinates of the former and the equation of the latter can be found from by completing the square.

If a is positive, since for all real x, a and b, f(x) attains its minimum when . If a is negative, since , this gives and hence f(x) attains its maximum when .

For both cases, the vertex is and the axis of symmetry is . The vertex corresponds to the maximum or minimum value of the function.

Example 5.1.

Find the maximum value or minimum value, if exist, of and .

Solution.

Thus the minimum value of f(x) is 5.

Thus the maximum value of g(x) is .

Exercises.

1. For each of the following functions, find the maximum or minimum value, if exists. Find also the value of x when the maximum or minimum is attained.

(a)

(b)

The graph of a quadratic function may cut the x-axis at two points, one point or it may not intersect with the x-axis, depending on the sign of the discriminant. For , the value of the discriminant tells us the number of real roots of the equation , i.e. the number of intersection point(s) of the graph of f(x) and x = 0.

If D > 0, the equation has two real roots and thus the parabola of cuts the x-axis at two distinct points. This is shown in Figure 3.

Figure 3: The graph of , with D > 0

If D = 0, the parabola touches the x-axis, for has only one real root, as shown in Figure 4.

Figure 4: The graph of , with D = 0

If D < style="position: relative; top: 5pt;"> has no real root and hence the graph of f(x) does not intersect with the x-axis. The whole parabola lies above the x-axis (for a > 0, as shown in Figure 5) or below the x-axis (for a <>

Figure 5: The graph of , with D <>a > 0

Figure 6: The graph of , with D <>a <>

Exercises

2. Show that the graph of must cut the x-axis at at least one point for all real k.

6. Solving a linear equation and a quadratic equation

Given a system of two equations, one being linear and the other being quadratic, in the form

we can solve it by substituting the linear equation to the quadratic one to give

Then we can solve for x and hence y.

The number of real solutions to the system of equations can be determined by finding the discriminant of the quadratic equation .

Furthermore, these two equations can be solved by drawing the graphs of the straight line and the parabola . The point(s) of intersection, if any, give(s) the solution. The discriminant also tells the number of intersecting point(s).

Example 6.1.

Find the equation of straight line that passes through (1, 1) and is tangent to the parabola . Find also the point of tangency.

Solution.

Let the equation of the required straight line be . Since (1, 1) lies on this line, we have

Putting the equation of the straight line to that of the parabola, we get

Since the straight line is tangent to the parabola, there is only one point of intersection and hence

For m = –5, c = 1 – (–5) = 6 and so the equation of the straight line required is y = –5x + 6.

For m = 3, c = 1 – 3 = –2 and thus the equation required is y = 3x – 2.

Exercises

7. Solutions to Selected Exercise

Solving quadratic equations

1. x = 8 or x = 2

3. x = 100

4. x = 3

Sum and product of roots

1. (a) –3

(b)

(d)

Discriminant and nature of roots

1. k > 3 or k < –2

Quadratic functions and their graphs

1. (a) The minimum value is –8 when x = –2.

(b) The maximum value is 7 when x = 1.