8. Using the same axes of graph plot the equations 2x-y-4=0 y=5+2x-x2 and from the figure estimate the values of x and y which satisfy both equations. 9. When a certain number containing two digits is divided by the sum of its digits the quotient is 7 and the remainder is 6; the product of the digits exceeds three times the digit in the unit's place by 6. Find the number. PLANE GEOMETRY 10. Prove that a radius perpendicular to a chord of a circle bisects the chord and the arc which it subtends. 11. Prove that a circle can be circumscribed about any regular polygon. *12. Prove that the diagonals of a parallelogram which is not a rectangle are unequal. 13. Given a parallelogram with sides 2 and 3 inches, one of whose angles is 60°. A circle of radius 1 inch rolls around outside the parallelogram, always touching it. Draw accurately the locus of the center of the circle and compute the length of the locus. 14. A point, P, is 9 inches from the center, O, of a circle whose radius is 6 inches. Tangents, PA and PB, are drawn meeting the circle at A and B. Find the length of that segment of OP which is included between the chord AB and the circle. 1923 ADVANCED MATHEMATICS Thursday, September 20 2-5 p.m. For candidates who have submitted for admission school records which include one or more of the following branches of advanced mathematics: solid geometry, trigonometry, advanced algebra. New Plan candidates will omit all questions marked with an asterisk (*). Answers to nine questions constitute a full paper. If more than nine questions are attempted and not crossed out, probably only the first nine will be read. Old Plan candidates will answer all questions in any subject which they are offering. 1. Solve: PART I. ELEMENTARY x+2y+4z=4 4x+3z=1 2x-4y+5%-3 2. Simplify, without extracting roots, 21/8—101/2+2a−6(2o+√2−2−1)+ 5+31/2 3. A boy goes 15 miles at a certain rate. Returning by the same route, he goes 5 miles at double this rate, but then has to finish at one mile an hour less than his original rate. The return trip takes 1 hour and 40 minutes longer than the other. Find the original rate. 4. Prove that the bisector of an interior angle of a triangle divides the opposite side of the triangle into segments which are proportional to the adjacent sides. 5. A trapezoid is circumscribed about a circle. Prove that the line joining the mid points of the non-parallel sides is equal in length to one-fourth the perimeter of the trapezoid. PART II. ADVANCED SOLID GEOMETRY 6. Prove that if two planes are perpendicular to each other, a straight line drawn in one of them perpendicular to the line of intersection is perpendicular to the other plane. 7. Prove that the sum of the angles of a spherical triangle lies between 180° and 540°. 8. A regular pyramid with square base and altitude 8 inches, is inscribed in a sphere of radius 6 inches. Calculate the volume of the pyramid. 9. A regular hexagon is inscribed in a circle whose radius is 10 inches. The whole figure is revolved about one of the long diagonals of the hexagon as an axis. Find the volume of the solid generated by that portion of the figure lying between the hexagon and the circle. Find also the area of the surface generated by the hexagon. *11. Derive the formula for tan (x+y) in terms of tan x and tan y. 12. Prove the identity tan 2x (cos x csc x—tan x) = 2 sin x csc x. 13. Obtain the remaining angles of the triangle A=18° 33′ or A=18.55° a=10.32, b=15.23. 14. Without tables find all values of x between 0° and 360° which satisfy the equation has one real positive root and at least two imaginary roots. 17. Locate the real roots of the following equation between two successive integers and calculate the positive root correct to the nearest hundredth. x3x2-4x-2=0. 18. Factor the following polynomial, given that one of the factors is x-1+2i: 2x4x35x2+13x+5. 19. A football coach has on his squad 12 men who can play any of the seven positions in the line, and 10 other men who can play any of the four back-field positions. In how many ways can he pick eleven men to form a team? PHYSICS 1923 9 a.m.-12 m. Friday, June 22 Answer ten questions as indicated below. No extra credit will be given for more than ten questions. Indicate clearly your reasoning in each problem and state the units in which each answer is expressed. Number and letter each answer to correspond with the questions selected. PART I (Answer all questions in this part.) 1. a) How would you construct a simple mercury barometer ? b) What is measured by means of a barometer? c) State two practical purposes for which barometers are used. 2. a) Define acceleration. b) A body starting from rest and moving with uniformly accelerated motion acquired a velocity of 60 feet per second in 5 seconds. Find 1) the acceleration; 2) the average velocity during the 5 seconds; 3) the total distance traversed in 5 seconds. 3. a) A meter stick, the center of gravity of which is at the 51 centimeter mark, is supported at the 45 centimeter mark and balanced by weights of 200 grams and 500 grams suspended at the 5 centimeter and 59 centimeter marks respectively. What is the weight of the meter stick? b) If the above system is suspended from a spring balance, what is the reading of the balance? 4. One hundred grams of ice at -10° C. are dropped into 120 grams of water at 50° C. contained in a cup the heat capacity of which is negligible. How much ice remains after a constant temperature has been reached? The specific heat of ice=0.5. 5. a) Explain, using a diagram, the construction and action of a common form of galvanometer. b) How could a galvanometer be adapted for measuring volts? c) How could a galvanometer be adapted for measuring amperes? 6. Describe an experimental method for finding the number of vibrations per second made by a tuning fork. 7. a) A standard candle furnished the same intensity of light upon a screen 2 feet distant as another source of light furnished when at a distance of 8 feet. What was the candle-power of the second light? b) How great was the intensity of illumination of the screen in foot-candles when one of these lights shines upon it? PART II (Answer three questions from this part.) 8. A liter of air weighs 1.29 grams and a liter of helium 0.18 gram. If a small rubberized silk balloon weighing 10 grams is filled with 40 liters of helium, what is its lifting power? 9. A steel bar of uniform cross-section is 20 feet long and weighs 100 pounds. How much work is required to lift it from a horizontal to a vertical position, one end being kept always on the ground? 10. At the beginning of the compression stroke of a certain gas engine its cylinder contained 42 cubic inches of gasoline vapor and air at a pressure of 15 pounds per square inch and at a temperature of 40° C. At the end of the compression stroke the volume was 6 cubic inches and the pressure 225 pounds per square inch. What was the temperature of the gas? 11. Describe an experimental method of determining the practical efficiency and the cost of operating an electrical device and state how the numerical results are secured. 12. What horsepower is required to drive a generator, of 85 per cent efficiency, which supplies 120 incandescent lamps, each of 220 ohms resistance, connected in parallel at 110 volts? 13. Assuming that a bomb explodes immediately upon striking the ground, calculate the time that elapses from the instant it is released 1000 feet above the ground until the sound of the explosion returns to that level. 14. a) State clearly what are always the relative positions of object, plane mirror and image. b) State the law of reflection by which the positions are determined. c) Show by a diagram where a man lying on his back on the floor in front of a plane mirror tilted forward at an angle of 45° would see himself. Represent the man by an arrow and suppose his length to make a right angle with the edge of the mirror which rests on the floor. 15. Two converging lenses are used as a compound microscope. Show by means of a diagram the positions of lenses and object; also the image and principal focus of each lens. |