(a) Factor: z1—2(ba—c3) x2+b1—2b2c2+c1. A merchant maintained himself for three years at an expense of $1,500 a year, and each year increased that part of his stock that was not so expended by one-third of it. At the end of the third year his original stock was doubled. What was his original stock? A person invests $10,000 in 3 per cent bonds, $16,500 in 3 per cents, and has an income from both investments of $1,056.25. If his investments had been $2,750 more in the 3 per cents and less in the 3 per cents, his income would have been 62 cents greater. Find the price of each kind of bonds. Find the value of m for which the two roots of 2mx2 + x2 — 6mx-6x+6m+1= 0 are equal and find these equal roots. Between what limits of m are the roots real? A and B travel from P to Q, 14 miles, at uniform rates, B taking 20 minutes longer than A to perform the journey. On the return, each travels 1 mile an hour faster, and B now takes 15 minutes longer than A. Find the rates of traveling. A number consists of three digits in geometrical progression. The sum of the digits is 13; and if 792 is added to the number, the digits of the units and hundreds places will be interchanged. Find the number. Extra. Substitute for No. Solve x2-2xy= 5 19. Plane geometry.-Candidates will be required to give accurate definitions of the terms used in plane geometry, to demonstrate any proposition of plane geometry as given in the ordinary textbooks, and to solve simple geometrical problems, either by a construction or by an application of algebra. The following sets of questions were used at recent examinations: 4 12 5 9 6 12 7 12 8 Define and illustrate the following: 1°. A trapezoid. 2°. The limit of a variable. 3°. A sector of a circle. 4°. A mean proportional. 5. Similar polygons. 6°. The apothem of a regular polygon. Theorem: Of two oblique lines drawn from the same point in a perpendicular, cutting off unequal distances from the foot of the perpendicular, the more remote is the greater. Problem: (a) To bisect a given angle. (b) Upon a given straight line to construct a segment of a circle which shall contain a given angle. Theorem: Two triangles having an angle of one equal to an angle of the other are to each other as the products of the sides including the equal angles. Problem: To draw in a given circle a chord of given length parallel to a given right line. Theorem: The lines which join the middle points of the consecutive sides of a quadrilateral form a parallelogram equivalent to one-half the quadrilateral, and having a perimeter equal to the sum of the diagonals of the quadri. lateral. A rectangle and a circle have equal perimeters. Find the difference in their areas if the radius of the circle is 9 inches, and the width of the rectangle is three-fourths its length. 12 Theorem: The angle between two secants intersecting without the circumference; the angle between a secant and a tangent, and the angle between two tangents are each measured by one-half the difference of the intercepted arcs. 9 9 Problem: To construct a circumference which shall be tangent to a given straight line, CD, and shail pass through two points, A and B, on the same side of the line CD, but not equally distant from it. Extra. Substitute for No. Problem: To inscribe a regular decagon in a given circle. 1 10 2 12 3 10 Draw Define Equilateral polygons, equiangular polygons, regular polygons. (b) How many sides has an equiangular polygon, if the sum of 5 of its angle Theorem: The angle contained by the bisector of the right angle and the median drawn to the hypotenuse of a right triangle is equal to one-half the difference of the two acute angles of the triangle. 4 10 Theorem: If in the same circle or in equal circles two chords are unequal, the shorter is at the greater distance from the center. 222 12 5 6 12 Problem: To construct a common external tangent to two given nonintersecting circles. Problem: Given the line CD and the points A and B. To find the point X on the line CD such that AX+XB is less than AY+ YB where Y is any other point on the line CD. *B (a) Theorem: Two triangles are similar if their sides are respectively pro- Exercise: Find the perimeter of a regular hexagon circumscribed about a Extra. Substitute for No. Problem: To construct a square equivalent to three-fifths of the square on the line AB. (a) Theorem: If a series of parallel lines cut off equal segments on one transversal, they cut off equal segments on any other transversal. (b) Exercise: If the vertex angle of an isosceles triangle is 40°, find the angle included between the altitudes drawn from the extremities of the base to the opposite sides. Problem: To inscribe within a given circle three equal circles, each of which shall be externally tangent to the other two and internally tangent to the given circle. Theorem The feet of two altitudes of a triangle are equidistant from the mid-point of the third side. Theorem: If two circles are tangent internally, all the chords of the greater drawn from the point of contact are divided proportionally by the circumference of the smaller. (Hint: Prove Problem: Given a series of triangles having the base a and the vertical angle Problem: To construct a triangle given the angles and the perimeter. 12 Theorem The square on the hypotenuse of a right triangle is equivalent to the sum of the squares on the other two sides. Theorem: In a parallelogram the sum of the squares of the four sides is (a) Problem: To inscribe a regular dodecagon in a given circle. EXTRA.-Substitute for No. (a) Theorem: If, through a point outside a given circle, a tangent and a No. Wt. 1 MARCH, 1927 12 Define a "locus of a point." In determining a "locus what two facts must be proven? Theorem: The medians of a triangle meet in a point which is two-thirds the (a) Theorem: An angle formed by two intersecting chords of a circle is measured by -- (Complete the statement and deduce the theorem.) (b) Exercise Find the number of inches in the circumference of a circle in which a central angle of 72° intercepts an arc of 7 inches. Problem: To inscribe a square in the given triangle A B C, one side of the square lying along B C. 5 12 Theorem: In any triangle, the product of two sides is equal to the square of the bisector of their included sides, increased by the product of the segments of the third side. (Hint: Circumscribe a circle about the triangle and prolong the bisector.) 10 Problem: Given a circle with chord AB; to draw through a given point P, in the arc subtended by AB, a chord which shall be bisected by AB. 6 12 Exercise: In a triangle the side a 21, b=10, and c=17. 1. Find the length of the projection of b on a. 2. Find the length of the altitude drawn to the side a 3. Find the area of the triangle. Problem: To construct a triangle similar to the triangle PQR and equal in 8 10 12 (a) Problem: To inscribe a regular hexagon in a circle. (b) Exercise: Find the area of the circle outside the hexagon. Theorem: Given any three circles with centers at O, O', and O", tangent externally two and two, at the points A, B, and C, respectively. A and C lle on the circle 0. If BC and BA be prolonged to cut the circle O at M and N, respectively, the line MN is a diameter of the circle 0. MARCH, 1928 (a) Theorem: If one side of a triangle is greater than a second side, the angle Problem: To construct a circle tangent to three intersecting straight lines but Problem: To construct all the common tangents to two nonequal nonintersecting circles. Problem: To find a point P in the arc subtended by the chord AB such that the chord PA is to PB as 2 is to 3. Exercise: The two parallel sides of a trapezoid are 12 and 4 inches, respectively. A nonparallel side 6 inches long makes an angle of 60° with the 12-inch base. If the two nonparallel sides be prolonged to intersect, find the area of the trapezoid and the area of the triangle above it. Theorem If lines be drawn from the center of the inscribed circle to the extremities of the base of a triangle, the angle so formed equals 90° plus one-half the angle of the triangle opposite the base. Problem Given one side of a triangle, a, the angle opposite, A, and the radius, 1 a T (a) Theorem: In a right triangle a perpendicular from the vertex of the right angle to the hypothenuse cuts the hypothenuse into segments which are proportional to the squares on the sides adjacent to the right angle. (b) Problem: Given a square containing 16 square inches, to construct a square containing one-third this area. (a) Problem: To inscribe a regular hexagon in a given circle. (b) Exercise: The radius of a circle is 8 inches. If the area of the inscribed or circumscribed hexagon be taken for the area of the circle, find the percentage of error in each case. Extra-Substitute for No. 1 12 2 3 Problem: In an inscribed quadrilateral, ABCD, the product of the diagonals is (Hint: Draw a line from B to the point F on AC such that MARCH, 1929 (a) Define: (1) parallelogram, (2) rhombus, (3) trapezoid. BFC= (b) Theorem: The lines joining the mid-points of the adjacent sides of a rhombus inclose a rectangle. 12 Problem: To construct a triangle, given two sides and the median to the third side. 10 Theorem: Of two straight lines drawn from the same point in a perpendicu lar to a given line and cutting off unequal segments from the foot of the perpendicular, the more remote is the greater. Problem: To construct a circle through the point P and tangent to the two intersecting lines AB and CD. Theorem: In an obtuse triangle the square of the side opposite the obtuse angle is equal to the sum of the squares of the other two sides, increased by twice the product of one of those sides and the projection of the other side upon it. Theorem: A line parallel to the bases of a trapezoid, passing through the intersection of the diagonals, and terminating in the nonparallel sides, is bisected by the diagonals. Problem: To construct a triangle similar to a given triangle and having a given perimeter. Exercise: Find the length of the bisector of one of the angles of the triangle whose sides are 8, 10, 12. (a) Define a segment of a circle. (b) Exercise: In a circle of 4 inches radius, draw a chord which subtends a right angle at the center of the circle. Find to hundredths of a square inch the areas of the two segments having this chord for a base. Problem: To construct a right triangle, given the hypotenuse and the difference of the other two sides. Extra. Substitute for No. 20. English grammar.-Candidates must have a good knowledge of English grammar; they must be able to define the terms used therein; to define the parts of speech; to give inflections, including declension, conjugation, and comparison; to give the corresponding masculine and feminine gender nouns; to give and apply the ordinary rules of syntax. They must be able to parse correctly any ordinary sentence, giving the subject of each verb, the governing word of each objective case, the word for which each pronoun stands or to which it refers, the words between which each preposition shows the relation, precisely what each conjunction and each relative pronoun connect, what each adjective and adverb qualify or limit, the construction of each infinitive, and generally to show a good knowledge of the function of each word in the sentence. They must be able to correct in sentences or extracts any ordinary errors of grammar. It is not required that any particular textbook shall be followed; but the definitions, parsing, and corrections must be in accordance with good usage and common sense. The following sets of questions were used at recent examinations: MARCH, 1925 Speak! Who art thou, that on designs unknown, Opposite each word, as it appears below, state what part of speech it is, and Speak who art thou that on designs While others sleep thus range the camps alone 4 Write sentences containing the following: (a) The past perfect tense, active voice, of "stand." (b) The perfect participle of "climb." (c) The present infinitive of "look." (d) The present tense, passive voice, subjunctive mood of "think," (e) The imperative mood of "remain." Honor thy father and thy mother, that thy days may be long upon the land which the Lord thy God hath given thee. "I come, great duke, for justice." "You shall have it.' Parse the following verbs, taken from the above quotations, giving (1) kind of verb, (2) mood, (3) tense, (4) person, (5) number, (6) subject, and objects or predicate nominative, if any: (a) Honor. (b) may be (c) hath given (e) shall have 10 Write a sentence containing each of the following grammatical constructions, and underline the word which fulfills the requirement: (a) A relative pronoun in the objective case. (b) A pronoun in the possessive case. (c) A proper noun. |