MARCH, 1920. sides No. 1.-(a) How many s 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. (a) Distinguish between 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. No. 4. (a) Theorem: Upon a given straight line, to construct the segment of a circle which shall contain a Wt. 12). given angle. (b) Theorem: A B is a fixed chord of a circle, and P is any point in either arc. Show that the bisector of the angle 4 P B intersects the oppo ite 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 the given triangle and having a given perimeter. No. 6.-Theorem: Two rectangles having equal altitudes are to each 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: (1o) a sector of a circle, (2o) a segment of a circle. (Wt. 12.) (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. (e) Prove geometrically the statement of the algebraic formula (a+b)(a+2b)=a3+3ab+263. No. 3. (a) When is an angle inseribed in a circle? (Wt. 12.) (b) Theorem: An inscribed angle is measured by one-half its intercepted are. (e) 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, X Y, 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 X Y.) 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: A D bisects the angle 4 of the triangle A B C, meets B C in D and the circumscribed circle of the triangle in 0. 1°. Prove that the triangles A D C and A BO are similar. 2o. Prove that A B.A CAD2+BD. DC. MARCH, 1922. No. 1.-(a) Distinguish between similar, equivalent and equal magnitudes in geometry. (Wt. 10.) (b) Each angle of a regular polygon is 157°.5. How many sides has the polygon? No. 2.-Theorem: The bisecter of an angle is the locus of all points within the angle equally distant from (Wt. 10.) its sides. No. 3.-Problem: To construct all the common tangents to two given nonintersecting circles. (Wt. 10.) No. 4. (a) Theorem: The angle between two chords which intersect within a circumference is measured (Wt. 12.) by (Complete statement of theorem and prove it.) (b) Exercise: An are contains 16°; at its extremities tangents are drawn. What kind of a triangle do they form with the chord, and how large is each angle of the triangle? No. 5.-Problem: To find the locus of the centroid (or intersection of the medians) of a triangle whose base (Wt. 10.) is a and whose vertical angle is A. (Hint: Through the centroid draw parallels to the sides that form the angle A. Show that the triangle thus formed has a constant base.) 1 a A No. 6. (a) Problem: Find the lengths of the tangents drawn from a point to a circle whose radius is 13 (Wt. 12.) cm. when the chord joining the points of tangency is 24 cm. (b) Problem: Given an inch scale. To construct segments equal to T, √2, √3, dễ, √5. (Complete statement and prove.) No. 7.-(a) Theorem: The area of a triangle is equal to (Wt. 12.) (b) Theorem: The areas of similar triangles are to each other as the squares on homologous sides. No. 8. (a) Define similar polygons. (Wt. 12.) (b) To construct a polygon similar to abcde an equivalent to ABCDE. No. 9.-Exercise: Given a regular octagon inscribed in a circle whose radius is one. Compute the actual (Wt. 12.) error and percentage of error made in taking the area of the octagon for the area of the circle. Extra.-Substitute for No. Exercise: Given a triangle whose sides are 3, 5, and 7. Find the length of the median and of the angle-bisector drawn from the vertex opposite to the side 7. MARCH, 1923. No. 1.-(a) Define: a parallelogram; a rhombus. (Wt. 10.) (b) Theorem: The bisectors of the exterior angles of a rhombus inclose a rectangle. No. 2.-Theorem: The perpendiculars from the vertices of a triangle to the opposite sides meet in a point. (Wt. 10.) No. 3.-Problem: To find the locus of the intersection of the diagonals of the parallelogram formed by draw(Wt. 10.) ing lines from any point in the base of a triangle parallel to the other two sides. No. 4.-(a) Theorem: An in-cribed angle is measured by half its intercepted arc. (Wt. 12.) (b) Exercise: What angle is formed by a secant drawn through the center of a circle and a tangent, if one of the arcs intercepted is 29°? No. 5.-Problem: To construct a triangle, given a base 4 B equal to four inches; an angle opposite the base, (Wt. 10.) C, equal to 30°; and the point D, where the bisecter of the angle 'cuts the base, at a distance one inch from A. No. 6.-Theorem: The common internal tangents to two unequal nonintersecting circles intersect on the (Wt. 12.) line joining their centers. No. 7.-Exercise: Find the area of a circle inscribed in a square whose side is 40 feet. (Give answer to (Wt. 12.) nearest tenth of a foot.) Theorem: Four times the sum of the s uares of the medians of any triangle is equivalent to three times the sum of the squares of the sides. 75437°-24-3 No. 8.-Problem: To const Fuct a square whose area shall be 2/3 the area of the square whose side is 5 (Wt. 12.) inches. (Explain construction clearly, no proof required.) No. 9.-Theorem: The area of a regular hexagon inscribed in a circle is a mean proportional between the area of the (Wt. 12.) inscribed and circumscribed equilateral triangles. Extra.-Substitute for No. Exercise: The sides about the right angle of a right-angled triangle are 3 meters and 4 meters. 1° The hypotenuse and corresponding altitude. 2° The projections of the given sides on the hypotenuse. 3° The radii of the inscribed and circumscribed circles. 4° The segments of the hypotenuse determined by the bisector of the right angle. 5° The length of the bisector of the right angle. 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 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: 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. (c) A transitive verb with its object. 2. 3. 4. (b) 5.-Define, and give an example of, each of the following grammatical terms: Possessive case (of a pronoun). Comparative degree (of an adverb). Intransitive verb. Passive voice. Prepositional phrase. 20.-Write a co:npound 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. 40.-Parse the words in italics in the following sentence: Wt. There is nothing that more betrays a base, ungenerous spirit, than the giring 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. 3. is 4. nothing 3. more 3. betrays Wt. 3. base 3. than 3. giving 2. man's 15.-Write sentence containing the following: The preterite (or past) tense (active voice) first person plural of the verb "strike." The perfect tense (active voice) third person plural of the verb “sell.” The future tense (active voice) second person plural of the verb "hang." 5. The future tense (passive voice) third person singular of the verb "do." 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." 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. 4.-(b) Indicate the number of the following nouns by underlining once, those that are singular; twice, those that are plural; thrice, those that may be either: 3. 4. 5. 6. 7. 5. women 5. nephew 7. data 8. cherubim 7. lord 8. boar 3. (b) Indicate the gender of the following nouns by underlining once, those that are masculine; twice, those that are feminine; thrice, those that are neuter; and four times those that are common gender: 1. knowledge 2. mare 3. monk 4. buck 5. servant 6. goose 15. With regard to the sentences below state the following: (a) Whether simple, complex or compound; (b) if complex, the kind or nature of subordinate clause relationship; (c) if compound, the kind or nature of relationship between the coordinate clauses. We waited until the tide came in. Though I admire his courage, I have little confidence in his integrity. Knowledge comes, but wisdom lingers. That honesty is the best policy is a mistaken conception of honesty. 20.-Write sentences containing the following: The preterite (or past) tense (active voice) third person singular of the verb " The perfect tense (active voice) second person plural of the verb “lay." 'dare." The pluperfect (or past perfect) tense (active voice) first person singular of the verb "lic." The pre ent participle of the verb "swim." The perfect infinitive of the verb "sing.' The perfect participle of the verb "drink." The future tense (passive voice) of the verb "choose." The pluperfect (or past perfect) tense (passive voice) of the verb "find.” 5.-(a) Rewrite the following sentences, correcting all errors: Neither he nor she are here. Every one opened their window. He don't seem to appreciate those kind of songs. 5.-(b) Fill in the blank spaces in the following sentences, using properly one of the words bracketed after the sentence: The formal statement of the teachings and rules......set forth in the constitution [is, are]. This money comes from Boyle.. handed [who, whom]. you know is very liberal [who, whom]. ..he had every reason to believe the police had caught red 10.-Punctuate and capitalize the following: a glass he said hoarsely and then paused and repeated it more clearly a glass for christmas surely not and why not cried the dealer why not a glass 30.-Parse each word in the following sentence: 'From the mountain pass came a joyous sound of some person whistling." 10.-Classify in parallel columns the following pronouns as personal, demonstrative, relative, and indefinite: this, each, who, that, what, any, she, all, we, himself, whatever, those, their, who, it. 10.-1. "I wandered lonely as a cloud 2. That floats on high o'er vales and hills 3. When all at once I saw a crowd, 4. A host of golden daffodils; 5. Beside the lake, beneath the trees, 6. Fluttering and dancing in the breeze."-Wordsworth. From the lines above, pick out the following constructions, indicating in each instance the number of the line, and writing out the word: Two participles used as descriptive adjectives; Five common nouns in the singular number; One descriptive adjective; One relative pronoun; Two verbs in the indicative mood; One relative adverb (or conjunctive adverb). 10.-Define and give an example of each of the following: (a) Common noun; (b) indefinite pronoun; (c) adverb; (d) neater gender; (e) passive voice. 10. Write sentences containing the following: (a) The present subjunctive passive first person singular of allow. (b) The superlative of little. (c) A relative pronoun in the possessive case. (d) An adverb of degree. (e) A subordinate conjunction. (f) A demonstrative pronoun. (g) The plural of chimney. (h) An interrogative adverb. (1) Two interjections. (j) A prepositional phrase used as an adjective. No. Wt. 5. 6. 7. 10.-Give the second person plural passive voice of the verb "take" through all the tenses of the int cative and potential moods. 20.-(a) Write a simple sentence containing a compound subject. (b) Write a complex sentence containing a noun clause used as object of a preposition. (e) Write a complex sentence containing an adverbial clause of time. (d) Write a complex sentence containing an adverbial clause of concession. (e) Write a compound-complex sentence. (f) Write a compound sentence containing two intransitive verbs. (g) Write a compound sentence containing two verbs in the active voice. (h) Write a complex interrogative sentence containing an adjective clause modifying a proton 30.-Parse the italicized words in the following sentenee: That man is truly happy who has solved the question of how he begins, and not of how he rac, of what he wants, and not of what he has. Wt. .2 the .2 he .2 not .2 of .2 question .2 what .2 has MARCH, 1922. No. Wt. 1. 10.-1. Do you know the blackened timber-do you know that racing stream 2. With the raw, right-angled log-jam at the end: 2. 3. And the bar of sun-warmed shingle where a man may bask and dream 4. To the click of shod canoe-poles round the bend? 5. It is there that we are going with our rods and reels and traces, 6. To a silent, smoky Indian that we know 7. To a couch of new-pulled hemlock with the starlight on our faces, 8. For the Red Gods call us out and we must go. In the above sentences pick out the following grammatical constructions. (Indicate the number of the line and write the word or words which answer the question,) Interrogative mood. Compound adjective. Proper noun. Descriptive adjective. Possessive pronoun. Demonstrative adjective. 20.--Write a simple sentence containing an interrogative pronoun. Write a sentence containing a preposition with a compound object. Write a sentence containing an intransitive verb. Write a sentence containing an adjective clause. Write a sentence containing a relative pronoun, Write a compound sentence with the verbs of both clauses in the passive voice. Write a simple sentence containing a compound object. Write a complex sentence having a noun clause as its subject. Write a sentence containing an adjective in the superlative degree. Write a sentence containing a perfect participle. 3. 15.-Write sentences containing the following: The present tense, passive voice of the verb "persuade." The present tense, progressive, active voice of the verb "call." The perfect tense, active voice of the verb "swim." The pluperfect tense, active voice of the verb "eat." The preterite (or past) tense, passive voice of the verb "meet." The present participle of the verb "lay." The perfect infinitive of the verb "invite." The future perfect tense, active voice of the verb "know." The future tense, passive voice of the verb "sing." The perfect participle of the verb "sit." 4. 10. In the passage below underscore once all nouns; twice all pronouns; and thrice all verbs: 5 "Daughters of Time, the hypocritic Days, Muffled and dumb like barefoot dervishes, And marching single in an endless file, Bring diadems and faggots in their hands. To each they offer gifts after his will, Bread, kingdoms, stars, and sky that holds them all I, in my pleached garden, watched the pomp, Forgot my morning wishes, hastily Took a few herbs and apples, and the Day The comparative of "many." The superlative of "little." The plural of "valley." The feminine of "wizard." An abstract noun, Parse the italicized words in the following sentence: 20.-" We can not know all the best thoughts and sayings of the Greeks unless we know what they |