0 1 2 √ 5 and find its value to two places of decimals. No. 5.-A boy runs a mile race in 6 minutes; he does the last lap, one quarter of a mile, at a rate 3 miles (Wt. 10.) an hour faster than his rate on the first three laps. Find his pace for the last lap. No. 6.-A man sells a certain number of articles at $3 each above cost price, and receives $150 from the sale. (Wt. 12.) He sells the remainder, seven more in number than the first set, at $2 each below cost price, and receives $170. Find the total number of articles. No. 8.--The sum of three numbers in arithmetical progression is 6. If 1, 2, and 5 are added to the num(Wt. 10.) bers the three resulting numbers are in geometrical progression. Find the numbers. No. 9.-1f a field were made a feet longer and b feet wider, its area would be increased by 7 square feet; (Wt. 12.) if its length were made e fect less, and its width feet less, its area would be decreased by n square feet. Find its dimensions. Extra.-Substitute for No.. (a) Factor: 1°. 9x1-37x24·4. 2°. 35x+12c-35. (b) At what time between one and two o'clock is the long hand of a clock exactly one minute in advance of the short hand. 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: MARCH, 1917. No. 1.--Theorem: If two triangles have two sides of the one equal, respectively, to two sides of the other, (Wt. 10.) but the included angle of the first greater than the included angle of the secen-1, then the third side of the first is greater than the third side of the second. No. 2.-Theorem: The perpendiculars from the vertices of a triangle to the opposite sides meet in a point. (Wt. 10.) No. 3.--(a) Theorem: In a right angled triangle, the bisector of the right angle bisects the angle contained (Wt. 12.) by the altitude and median to the hypothenuse. (b) Theorem: The bisectors of the angles of a rectangle inclose a square. No. 4.-(a) Theorem: If any two chords be drawn through a fixed point within a circle, the product of (Wt. 12.) the segments of one chord is equal to the product of the segments of the other. (3) Problem: To find the locus of a point which divides all chords of a given circle drawn through it into segments whose product is equal to a given square. No. 5.-Froblem: To draw the common tangents to two intersecting circles. (Wt. 12.) No. 6.-Theorem: The square on the hypothenuse of a right angled triangle is equal to the sum of the squares (Wt. 12.) on the other two sides. No. 7.-(a) Theorem: If a circle be inscribed in a triangle ABC, the length of the tangent from the vertex (Wt. 10.) A to the circle equals one-half the perimeter of the triangle minus the side opposite A. (b) Exercise: How high is a monument which casts a shadow 192 feet long, when a vertical shaft 5 feet high casts a shadow 3 feet long. No. 8.--(a) Problem: To construct a triangle given the base, ab, the altitude, ap, drawn to one of the other (Wt. 12.) sides, and the angle at the vertex equal to 30°. (b) Problem: On the side ab, to construct a rectangle equivalent to the sum of the squares on pq and rs. No. 9. (a) Theorem: The area of a circle equals half the product of its radius and circumference. (Wt. 10.) (b) Exercise: The side of a square inscribed in a circle is 10 meters. Find the area of the circle to two places of decimals. Extra.-Substitute for No. Exercise: The area of a regular dodecagon inscribed in a circle being 3888 square meters, find the area of the regular dodecagon circumscribed about the same circle. MARCH 1918. No. 1.-(a) Theorem: The sum of the angles of a polygon is equal to two right angles taken as many times, (Wt. 12.) less two, as the polygon has sides. (b) Exercise: How many sides has a polygon if the sum of the interior angles is 8 times the sum of the exterior angles?. No. 2.-(a) Theorem: The tangents drawn from a point to a circle are equal, and make equal angles with (Wt. 10.) the line joining the point to the center of the circle. (b) Theorem: The median (the line joining the middle points of the nonparallel sides) of a circumscribed trapezeid is equal to one-fourth of its perimeter. No. 3. (a) Problem: To construct a triangle, having given two sides a and b and the median to the third (Wt. 10.) side, m. (b) Problem: To construct a triangle, having given one side, s, and the medians to the other two sides, m and n. (Hint: Where do the medians intersect?) No. 4.-Problem: To find the locus of the intersection of the diagonals of the parallelogram formed by (Wt. 10.) drawing lines from any point in the base of a triangle parallel to the other two sides. No. 5.-(a) Problem: To construct a square which shall have a given ratio to a given square. (Wt. 12.) (b) Problem: To divide a triangle into two equivalent parts by drawing a line parallel to the base. No. 6.-Theorem: The square on the sum of two lines is equivalent to the sum of the squares on the lines (Wt. 10.) increased by twice the rectangle of the lines. No. 7.-Theorem: AB CD is a given square. E, F, G and Hare the middle points of AB, BC, CD, and DA, (Wt. 12.) respectively. Draw the lines A F and BG and CH and DE. Prove that these lines form a square equivalent to 1/5 of the square AB CD. No. 8.-Theorem: Given the side of a regular polygon inscribed in a circle whose radius is unity, to find the (Wt. 12.) side of a similar circumscribed polygon. No. 9.-(a) Theorem: The radius of an escribed circle of an equilateral triangle is 3 times the radius of the (Wt. 12.) inscribed circle. (b) Theorem: The perimeter of the inscribed equilateral triangle is 3/4 of the perimeter of the circumscribed regular hexagon. MARCH, 1919. No. 1.-(a) Theorem: If two sides of a triangle are unequal, the angles opposite are unequal, and the greater (Wt. 10.) angle is opposite the greater side. (Direct proof required.) (b) How many sides has a regular polygon whose angle is 165 31? No. 2.-Theorem: The line joining the mid-points of the opposite sides, and the line joining the mid-points (Wt. 10.) of the diagonals of a quadrilateral, meet in a point. No. 3.-Theorem: In the same circle or in equal circles, two central angles have the same ratio as their in(Wt. 12.) tercepted arcs (whether these be commensurable or not). No. 4.-Theorem: If two circles intersect, any two parallel straight lines drawn through the points of inter(Wt. 10.) section and terminated by the circumferences are equal. No. 5.-Problem: To construct a triangle given the base, a, the angle opposite, 0, and the altitude, p, upon (Wt. 10.) one of the remaining sides. No. 6. Theorem: If from a point without a circle a tangent and secant are drawn, the tangent is the mean (Wt. 12.) proportional between the whole secant and its external segment. Problem: To construct a circle through two given points a and b and tangent to the given line mn. No. 7.-Problem: To divide a given line in extreme and mean ratio. (Wt. 12.) No. 8.-Theorem: The sum of the squares of the four sides of a quadrilateral is equal to the sum of the (Wt. 12.) squares of the diagonals plus four times the square of the line joining the mid-points of the diagonals. No. 9. (a) Problem: To inscribe a regular hexagon in a given circle. (Wt. 12.) (b) Exercise: Find to 3 places of decimals the area of a circle that circumscribes an equilateral triangle whose side is 4.3 meters. MARCH, 1920. No. 1.—(a) How many sides has a polygon the sum of whose angles is 14 right angles. (Wt. 12.) (b) Theorem: If two angles of a triangle are unequal, their sides opposite are unequal, and the greater side is opposite the greater angle. No. 2.-Theorem: The sum of the perpendiculars drawn from any point within an equilateral triangle to (Wt. 10.) the three sides is constant. No. 3.—(g) Distinguish 1etween similar, equivalent, and equal magnitudes, illustrating by figures. (Wt. 12.) (b) Problem: A ladder slides down between a vertical wall and a horizontal floor. Find the locus of its mid-point. Theorem: Upon a given straight line, to construct the segment of a circle which shall contain a given angle. Theorem: AB is a fixed chord of a circle, and P is any point in either arc. Show that the bisector of the angle APB intersects the opposite are in the same point for all positions of P. No. 5. (a) Theorem: Two triangles are similar if their sides are respectively proportional. (Wt. 12.) (b) Problem: To construct a triangle similar to a given triangle and having a given perimeter. No. 6.-Theorem: Two rectangles having equal altitudes are to cach other as their bases (whether these be (Wt. 12.) commensurable or incommensurable). No. 7.-Theorem: Let an equilateral triangle be inscribed in a circle. If the arcs subtended by its sides be (Wt. 10.) bisected, the chords joining these points of bisection will be trisected at the points where they cross the sides of the original triangle. No. 8.-Theorem: The sum of the squares of the sides of a parallelogram is equal to the sum of the squares (Wt. 10.) of the diagonals. No. 9. (a) Define a regular polygon. (Wt. 10.) (b) Theorem: Given a side and the radius of an inscribed regular polygon, to find a formula for the length of a side of a similar circumscribed polygon. (c) Example: How many revolutions does a bicycle wheel 28 inches in diameter make in going 1 mile? How many square inches are there in the area of the wheel? MARCH, 1921. No. 1.-(a) Define and illustrate: (1°) a sector of a circle, (2°) a segment of a circle. (c) The cross-section of a gutter is a semicircle whose diameter is 1 foot. A stream of water run- No. 2. (a) Theorem: The difference of the squares described on two straight lines is equivalent to the (Wt. 12.) rectangle whose sides are respectively the sum and the difference of the two straight lines. (b) State the above theorem by an algebraic formula. (c) Prove geometrically the statement of the algebraic formula (a+b)(a+2b)=a2+3ab+2b3. No. 3. (a) When is an angle inscribed in a circle? (Wt. 12.) (b) Theorem: An inscribed angle is measured by one-half its intercepted are. (c) Prove that, if the angle subtended by a chord is 150°, the chord is equal to the radius. No. 4.-Prove the following theorem: (Wt. 10.) Two triangles having an angle of the one equal to an angle of the other are to each other as the products of the sides including the equal angles. No. 5.-A shore line, XY, is a straight line: B and C are two buoys fixed not far from the shore. Show (Wt. 10.) how to determine at what point on the shore the straight line B C subtends the greatest angle. (AID: Construct a circle through B and C and tangent to XY.) No. 6. To inscribe a regular hexagon in a given circle and find its area. (Wt. 12.) No. 7.-Prove that if two circles intersect, their common chord bisects their common tangent. (Wt. 10.) No. 8.-Problem: To construct a triangle having given the angles and the radius of the circumscribed circle. (Wt. 12.) No. 9. Two parallel tangents are drawn to a circle, and a third tangent is drawn intersecting these two in (Wt. 10.) A and B, and tangent to the circle at C. Prove that the product of the segments A Cand CB is equal to the square of the radius of the circle. Extra. Substitute for No.... Theorem: AD bisects the angle A of the triangle ABC, meets BC in D and the circumscribed circle of the triangle in 0. 1°. Prove that the triangles ADC and ABO are similar. 2. Prove that AB. AC-AD2+BD. DC. 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 thy corresponding masculine and feminine gender nouns; to give and apply the ordinare 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 connects, what each adjective and adverb qualifies or limits, 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 grammatical errors. 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: 16 2. 20. Write a simple sentence containing an adverb of manner. Write a simple sentence containing an intransitive verb. 3. Write a complex sentence containing an adjective clause. Write a complex sentence containing an adverbial clause of place. Write a compound sentence containing a preposition with a compound object. Write a complex sentence containing an adverbial clause of time. Write a sentence containing a noun (or substantive) clause used as the subject of the sentence. Write a sentence containing an adjective phrase and an adverbial phrase. Write a sentence containing a verb in the passive voice. 10.- Punctuate and capitalize the following: you know i never approved of it pursued utterson ruthlessly disregarding the fresh topic my will yes certainly i know that said the doctor a trifle sharply you have told me so well i tell you so again eontinued the lawyer i have been learning something of young hyde the large handsome face of dr jekyll grew pale to the very lips and there came a blackness about his eyes i do not care to hear more said he this is a matter i thought we had agreed to drop what i heard was abominable said utterson. it can make no change you do not understand my position returned the doctor with a certain incoherency of manner i am painfully situated utterson my position is very strange a very strange one it is one of those affairs that cannot be mended by talking. In parsing give no rules, declensions, comparisons or principal parts of verbs, but in other respects parse fully. Be careful to give, in each example, the name of the part of speech. Give the following information:" Verbs: Regular or irregular, active or passive, transitive or intransitive, subject, object, mood, tense, number, person. Nouns: Kind, number, person, case, construction. Pronouns: Kind, antecedent, gender, number, person, case, construction. Adverbs: Kind, degree, word qualified. Adjective: Kind, degree, word qualified. Directions: (a) Give the governing word of each objective case. (b) State precisely what each conjunction and each relative pronoun connects. (c) State between what words each preposition shows the relation. NOTE. The words must be parsed in the precise form in which they are given in the vertical column; that is, two or more numbers must not be parsed together, but each number separately, and whenever possible without the aid of words supplied or understood. Omission of any one of the above requirements wil be taken to indicate ignorance." Intelligible abbreviations are allowed. No. Wt. 3.-I I could not find a better proof of what I said the other day, that the sincere man was by nature the obedient man; that only in a world of heroes was there loyal oledience to the heroic.-Carlyle. 4.-find 3.-better 3.-proof 3.-what 3.-that 3.-sincere 5. 6. The present infinitive of the verb "draw". The pluperfect (or past perfect) tense (active voice) of the verb "slay". 30. (a) Write the plurals of the following words: 1. cargo 2. zero 3. pony 5. Ox 6. valley 20.-(b) Of the following nouns, underline once those that are singular; twice those that are plural; thrice those that may be either. 40.-(d) Ofth following nouns underline once those that are masculine: twice those that are feminine; thrice those that are neuter; and four times those that are common gender. 3. freedom 5. goose 7. child 4. parent 6. goat 8. executrix 1. servant 40.-(e) Write the possessive form singular and plural of the following words: (b) 5.-Deane, and give an example of, each of the following grammatical terms: Intransitive verb. Neuter gender. 7. family 17 2. 20.-Write a complex sentence containing an adverbial clause of purpose. Write a compound sentence containing two clauses related by contrast. Write a complex sentence containing an adjective clause. Write a complex sentence containing an adverbial clause of result. Write a compound sentence containing a preposition with a compound object. Write a complex sentence containing an adverbial clause of time. Write a sentence containing a noun (or substantive) clause used as the subject of the sentence. Write a complex sentence containing an adverbial clause of concession. Write a sentence containing a verb in the passive voice. 3. 10.-Punctuate and capitalize the following: i cant pretend that i shall ever like him said the lawyer i dont ask that pleaded jekyll laying his hand upon the others arm i ask only for justice i ask you to help him only for my sake when i am no longer here utterson heaved an irrepressible sigh well he said i promise. 4. 30.-Parse the words in italics in the following sentence: A spirit goes out of the man, which outlives the most untimely ending. All who have meant good work with their whole hearts, have done good work, although they may die before they have the time to sign it. Wt. 3. out 4. which 3. untimely 3. who 3. with 5. 15.-Write sentences containing the following: Wt. 3. their 3. done 2. may 3. sign The preterite (or past) tense (active voice) first person plural of the verb "win." The future tense (active voice) second person plural of the verb "assist." The future tense (passive voice) third person singular of the verb "sew." The present participle of the verb "sing." The perfect participle of the verb "knock." The perfect infinitive of the verb "hang." The future perfect tense (active voice) second person singular of the verb "tell." The pluperfect (or past perfect) tense (active voice) third person singular of the verb "fall," 6. 10.-Rewrite the following sentences, correcting all errors: No. Wt. The sniper shot at him and I, but we were both used to these kind of experiences. told him that he hadn't ought to have done it and that neither Jim or I are to be held responsible. He let the logs lay where they had been laying since we first seen them. I don't know who you was talking about when I came in, but none were smiling. Either of we boys, who have graduated from High School, are able to pass this examination. MARCH, 1919. 1(a) 10.-Write a sentence containing: (a) A verb in the infinitive mood. (b) A conjunction expressing concession. (c) A relative pronoun. (d) The past participle of a verb. (e) A transitive verb with its object. (b) 5.-Define, and give an example of, each of the following grammatical terms: Possessive case (of a pronoun). Comparative degree (of an adverb). Passive voice. Intransitive verb. Prepositional phrase. 2. 20.-Write a compound sentence containing two clauses related by contrast. Write a complex sentence containing an adverbial clause of purpose. Write a sentence containing a noun (or substantive) clause used as the subject of the sentence. Write a complex sentence containing an adverbial clause of condition. 3. 40.-Parse the words in italics in the following sentence: There is nothing that more betrays a base, ungenerous spirit, than the giving of secret stabs to a man's reputation. Hence I am much troubled when I see the talents of humor and ridicule in the possession of an ill-natured man. Wt. 3. is 4. nothing 3. more 3. betrays Wt. 3. than 3. giving 4. 15.-Write sentences containing the following: The preterite (or past) tense (active voice) first person plural of the verb "strike." The future tense (active voice) second person plural of the verb "hang." The future tense (passive voice) third person singular of the verb "do." The present participle of the verb "admit." The perfect participle of the verb "fly." The perfect infinitive of the verb "see." The future perfect tense (passive voice) second person singular of the verb "tell." The pluperfect (or past perfect) tense (passive voice) third person singular of the verb "fling." 5. 10.-Rewrite the following sentences, correcting all errors: Between you and I, I think the sick man is feeling very good. "You hadn't ought to do it," I said, and added, "Neither Harry or I will take the blame for the consequences." The dogs were laying close by the fire, just where they laid when we first seen them. I'm sure he don't know who we were talking about. Any one of a dozen boys are able to pass this examination. |