No. 2. (a) Factor 1-64x3-x+64; also 10-82-78. (Wt. 8.) (b) Given H -825 (T-un). Find H when 7-400, w=0. 7, g=32. 2, and v=80. No. 3.-If n be an odd integer, prove (Wt.) 10.) (a) that a+b is exactly divisible by a+b. (b) that a"-b" is exactly divisible by a-b, but not by a+b. No. 4.-Two runners are practicing on a circular track 126 yards in circumference. When running in (Wt. 12.) opposite directions they meet every 6 seconds. When running in the same direction, the faster passes the slower every 126 seconds. How many minutes does it take each to run a mile? No. 5. (a) Simplify (1+√2)3-(1−√−2)3. (Wt. 12.) (b) Solve 2√5x=√3+√7x+2. No. 6. (a) What is the area of a square whose diagonal is 3 feet longer than a side? (Wt. 12.) (b) Solve the equations. No. 7.-(a) Determine the value of c so that the roots of the equation x2+3cx+2c+5=0 shall be equal and (Wt. 14.) find the corresponding equal roots. (b) For what value of c will the roots of the same equation be real? Imaginary? Numerically equal with contrary signs? No. 8.-A man sold some horses all at the same price and received $1,980. If the number of horses had (Wt. 12.) been one greater, and the price per horse $15 cheaper, he would have received the same amount of money. How many horses were there and what was the price per horse? No. 9. (a) Write the general term in the expansion of (a+b)" by the Binomial Formula. (Wt. 12.) Extra. (b) How many terms will there be in the expansion of (2-k)? Write the 3rd term. (a) Find the sum of the numbers 1, 2, 3, 4, etc., to n inclusive. (b) In a group of points every point is connected with every other point by a straight line. There are 190 straight lines. How many points are there? MARCH, 1925 No. 1. (a) Find the highest common factor of a3+3a2+8a+6 and a1+3a1+9a2+8a+6. (Wt. 12.) No. 5.-Solve √+4+√2x+6=√7x+14. Prove your answers are correct. (Wt. 10.) No. 6. A certain train leaves A for B, distant 216 miles; 3 hours later another train leaves A to travel (Wt. 12.) over the same route; the second train travels 8 miles per hour faster than the first, and arrives at B 45 minutes behind the first. Find the time each train takes to travel over the route. No. 9.-There are placed in a straight line upon a lawn 50 eggs 3 feet distant from each other. A person (Wt. 12.) is required to pick them up, one by one, and carry them to a basket in the line of the eggs and 3 feet from the first egg, while a runner, starting from the basket, touches a goal and returns. At what distance ought the goal to be placed that both men may have the same distance to pass over? Extra. Substitute for No. Write the general or rth term of the expansion of (a+b)" by the Binomial Formula. (b) A merchant bought wheat at the price of $2.40 for 4 bushels, corn at the price of $1.60 for 7 bushels, and barley at the price of $1.10 for 3 bushels. He spent $546.90. The cost of the wheat exceeded that of the corn by $80; the cost of the corn, that of the barley by $85.10. How many bushels of wheat, corn, and barley did he buy? No. 4.-(a) Find the lowest common multiple of (Wt. 10) (b) Simplify 1+ a b+c No. 5.-(a) How many rods of fencing will be required to enclose a plantation containing 8000 acres in the (Wt. 12) form of a rectangle if it is twice as long as it is wide? (b) Solve the following equation 16-1=0 (by factoring). No. 6. (a) Solve the equation (Wt. 12) √x+2+ √2x+2=1 and test your answers. (b) The hands of a clock are at right angles to each other at 3 o'clock. When are they next at right angles to each other? No. 7.-The fore wheel of a carriage makes 6 revolutions more than the hind wheel in going 120 yards; II (Wt. 10) the circumference of the fore wheel be increased by one-fourth of its present size and the cir cumference of the hind wheel by one-fifth of its present size, the 6 revolutions will be changed to 4. Find the circumference of each wheel in both cases. No. 8.-A cask contains 360 gallons of wine; a certain quantity is drawn off and an equal quantity of water (Wt. 12) is put in; from this mixture the same quantity as before is drawn and 84 gallons in addition; on replacing the drawn liquid with water it is found that the barrel contains equal quantities of wine and water. How many gallons were drawn the first time? No. 9.-The sum of three numbers in arithmetical progression is 6. If 1, 2, 5 are added to the numbers, the (Wt. 12) three resulting numbers are in geometrical progression. Find the numbers. EXTRA.-Substitute for No. (a) Determine for what values of m the equation 2mr2+(5m+2)x+(4m+1)=0 shall have equal roots and find the corresponding equal roots. (b) For what values of m will the roots of the above equation be real? Imaginary? Numerically equal with contrary signs? 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: 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 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 A. Show that the triangle thus formed has a constant base.) a Α 4. 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 ì, √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. 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 inscribed 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 AB equal to 4 inches; an angle opposite the base, (Wt. 10.) C, equal to 30°; and the point D, where the bisector of the angle C 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 squares of the medians of any triangle is equivalent to three times the sum of the squares of the sides. No.8.-Problem: To construct a square whose area shall be 2/3 the area of the square whose side is 5 inches. (Wt. 12.) (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 (Wt. 12.) area of the 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. MARCH, 1924 No. 1.-Define and illustrate the following: (Wt. 12.) 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. No. 2.-Theorem: Of two oblique lines drawn from the same point in a perpendicular, cutting off unequal (Wt. 10.) distances from the foot of the perpendicular, the more remote is the greater. No. 3.-Problem: (Wt. 12.) (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. No. 4.-Theorem: Two triangles having an angle of one equal to an angle of the other are to each other as the (Wt. 12.) products of the sides including the equal angles. No 5.-Problem: To draw in a given circle a chord of given length parallel to a given right line. (Wt. 9.) No. 6.-Theorem: The lines which join the middle points of the consecutive sides of a quadrilateral form a (Wt. 12.) parallelogram equivalent to one-half the quadrilateral, and having a perimeter equal to the sum of the diagonals of the quadrilateral. No. 7.-A rectangle and a circle have equal perimeters. Find the difference in their areas if the radius of (Wt. 12.) the circle is 9 inches, and the width of the rectangle is three-fourths its length. No. 8.-Theorem: The angle between two secants intersecting without the circumference; the angle be (Wt. 12.) tween a secant and a tangent, and the angle between two tangents are each measured by one-half the difference of the intercepted arcs. No. 9.-Problem: To construct a circumference which shall be tangent to a given straight line, CD, and (Wt. 9.) shall 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. MARCH, 1925 No. 1.-Define: Equilateral polygons, equiangular polygons, regular polygons. Draw an equilateral hexa(Wt. 10.) gon which is not regular. Draw an equiangular hexagon which is not regular. Prove that an equilateral triangle is regular. No. 2. (a) Theorem: The lines joining the mid-points of the opposite sides of a quadrilateral bisect each (Wt. 12.) other. (b) How many sides has an equiangular polygon, if the sum of 5 of its angles is 8 right angles? No. 3.-Theorem: The angle contained by the bisector of the right angle and the median drawn to the (Wt. 10.) hypotenuse of a right triangle is equal to one-half the difference of the two acute angles of the triangle. No. 4.-Theorem: If in the same circle or in equal circles two chords are unequal, the shorter is at the (Wt. 10.) greater distance from the center. No. 5.-Problem: To construct a common external tangent to two given nonintersecting circles. (Wt. 12.) No. 6.-Problem: Given the line CD and the points A and B. To find the point X on the line CD such (Wt. 12.) that AX + XB is less than AY+ YB where I'is any other point on the line CD. No. 7.-(a) Theorem: Two triangles are similar if their sides are respectively proportional. (Wt. 12.) (b) Exercise: The sides of a triangle are 9, 15, and 8, and its area is 40 square feet. Find the area of a similar triangle whose smallest side is 16. No. 8.-Theorem: If Pis any point on the circumference of a circle whose diameter is AB, the sum of the (Wt. 10.) squares on PA and PBis constant. No. 9.-Exercise: Find the perimeter of a regular hexagon circumscribed about a circle whose radius is (Wt. 12.) 15inches and the area of the part of the hexagon that is outside the circle. Extra.-Substitute for No. Problem: To construct a square equivalent to three-fifths of the square on the line AB, No. 1.--(a) Theorem: If a series of parallel lines cut off equal segments on one transversal, they cut off (Wt. 12.) 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. No. 2.-Problem: To inscribe within a given circle three equal circles, each of which shall be externally (Wt. 12.) tangent to the other two and internally tangent to the given circle. No. 3.-Theorem: The feet of two altitudes of a triangle are equidistant from the mid-point of the third (Wt. 10.) side. No. 4.-Theorem: If two circles are tangent internally, all the chords of the greater drawn from the point (Wt. 10.) of contact are divided proportionally by the circumference of the smaller. No. 5.-Problem: Given a series of triangles having the base a and the vertical angle A, to find the locus (Wt. 12.) of the center of their inscribed circles. (Hint: Prove that angle subtended by a at center is constant.) a I Α No. 6.-Problem: To construct a triangle given the angles and the perimeter. (Wt. 10.) No. 7.-Theorem: The square on the hypotenuse of a right triangle is equivalent to the sum of the squares (Wt. 12.) on the other two sides. No. 8.-Theorem: In a parallelogram the sum of the squares of the four sides is equivalent to the sum of (Wt. 10.) the squares of the diagonals. No. 9. (a) Problem: To inscribe a regular dodecagon in a given circle. (Wt. 12.) (b) Exercise: Find the length of the side of a regular dodecagon inscribed in a circle of radius 2 inches. Find what proportion the area of the polygon is of the area of the circle. EXTRA.-Substitute for No. (a) Theorem: If, through a point outside a given circle, a tangent and a secant are drawn to the circle, the length of the tangent is the mean proportional between the whole secant and its external segment. (b) Problem: To construct a circle passing through two given points and tangent to a given straight line. 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: No. Wt. MARCH, 1922 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; 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. 2. 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 10.-Write sentences containing the following: The comparative of "many." The superlative of "little." The plural of "valley." The feminine of "wizard." An abstract noun. A copulative verb. A compound pronoun. A predicate adjective. An indirect object. |