No. 2.-Examine the following statements and say what you can as to the truth or falsity of cach (i. e. for (Wt. 12.) what values of 1, if any, the statement is true): (1) 3(2x+7)+x=7(x+3). (2) 3-[1-2(3x-4)]=(x+2)2+x-13. No. 3.-Solve graphically the following pairs of equations: (Wt. 12.) 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 $470. 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 numbers (Wt. 10.) the three resulting numbers are in geometrical progression. Find the numbers. No. 9.-If a field were made a feet longer and b feet wider, its area would be increased by m square feet; if (Wt. 12.) its length were made c feet less, and its width d feet less, its area would be decreased by n square feet. Find its dimensions. Extra-Substitute for No.. (a) Factor: 1°. 91-3712+4. 2°. 36+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. No. 1.-Factor 4-10+9. (Wt. 12.) MARCH, 1922. Find the value of (x−a)3+ (x—b)3+(x−c)3—3(x—a)(r—b)(x−c) when 31=a+b+c. No. 2. (a) Divide z−2+1 ̄¡ by x3-x ̄3 ̧ (Wt. 12.) No. 4.-One man and two boys can do in 12 days a piece of work which would be done in 6 days by 3 men (Wt. 10.) and one boy. How long would it take 1 man to do it? No. 6. For what values of k are the roots of the equation 12+k(x+1)+3=0 equal? Real? Complex? (Wt. 10.) Find the roots when they are equal. No. 7.-A, walking 3 miles per hour, leaves a starting point shortly after 5 o'clock when the minute hand (Wt. 12.) of the clock just makes a right angle with the hour hand. B, walking 34 miles per hour, starts from the same point more than a half hour later, when the minute hand is again making a right angle with the hour hand. When did A and B start? How far has A gone when B overtakes him? No. 9.-A rock pushed over a cliff falls approximately 16 feet during the first second, 48 feet the next, 80 feet (Wt. 12.) the next, and so on. It strikes the water at the base of the cliff in 10 seconds. Find how far it falls during the 10th second. How high is the cliff? How many seconds has it been falling when it passes a point q feet below the top of the cliff? Extra.-Substitute for No. (a) How long will an up and a down train take to pass each other, each being 44 yards long and each traveling 30 miles per hour? (b) A man spent $90.00 for cigars. If he had gotten 1 box more for the money, cach box would have been $1.25 cheaper. How many boxes did he buy? 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, 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) (Wt. 10.) Theorem: The tangents drawn from a point to a circle are equal, and make equal angles with the line joining the point to the center of the circle. (b) Theorem: The median (theline joining the middle points of the nonparallel sides) of a circumscribed trapezoid 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?) 8 L n 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° 36'? 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.) scetion 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. a Р 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. 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. (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: 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 arc 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 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.—(2) 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.) (b) What proportion of a circle is a sector whose angle is 60°? (c) The cross-section of a gutter is a semicircle whose diameter is 1 foot. A stream of water running through the gutter has a width of 6 inches at the surface. Find the area of the crosssection of the stream. 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)==a2+3ab+2b2. 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 arc. (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, 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 AC and 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. 1o. Prove that the triangles ADC and ABO are similar. 2o. Prove that AB. AC=AD1+BD . DC. MARCH, 1922. No. 1.-(a) Distinguish between similar, equivalent and equal magnitudes in geometry. (Wt. 10.) No. 2.-Theorem: The bisector 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 arc 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 4. Show that the triangle thus formed has a constant base.) I 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 √ī, √2, √3, √4, √5. No. 7. (a) Theorem: The area of a triangle is equal to (Wt. 12.) (Complete statement and prove.) (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 and 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. 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: (b) 3.-Define and give an example of each of the following grammatical terms: 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 present infinitive of the verb "pay." 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 are to be held responsi ble. 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. (4) 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. 4. 15.-Write sentence 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 "sce." The future perfect tense (passive voice) second person singular of the verb "tell.” The present infinitive of the verb "contain." The pluperfect (or past perfect) tense (passive voice) third person singular of the verb "fling." 5. 10.-Rewrite the following sentences, correcting all errors: No. Wt. 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 dogs were laying close by the fire, just where they laid when we first seen them. Any one of a dozen boys are able to pass this examination. MARCH, 1920. 5. eyetooth 1. pliers 2. children 3. stiletto 4-(b) indicate the number of the following nouns by underlining 2. 4-(a) Write the feminine form of the following words: 5. women 5. nephew 7. it 8. foot once, those that are singular; 7. data 8. cherubim 7. lord 8. boar 3.-(0) 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 |