(a) Define a geometrical progression and give an example.

(b) What distance will an elastic ball travel before coming to rest if it be dropped from a height of 15 feet and if after each fall it rebounds to the height from which it fell?

Solve:

3

12

A and B are walking toward each other along a road represented by the axis of X. A's distance from the origin of coordinates y hours after 12 o'clock is given by the equation 3x-2y=-2. B's distance from the same origin y hours after 12 o'clock is given by the equation @+y=6. Show graphically how to find when A and B will be at the same distance from the origin. How far will they be from the origin at this time? How far apart will A and B be at 12 o'clock (i. e. when y=0)?

Several persons hired an automobile for $12, but three of them failed to pay their share and as a result each of the others had to advance 20 cents more. How many persons were in the party?

(a) Solve: √2-7x=-52.

(b) (−1+√−3)'=

?

What integral values may k have in the equation 4☛2−2(k+3)x+k2=0 such that the values of a shall be real?

The fore wheel of a carriage makes 28 revolutions more than the rear wheel in going 560 yards, but if the circumference of each wheel be increased by 2 feet, the difference would be only 20 revolutions. Find the circumference of each wheel.

with positive exponents.

A weight suspended by a string is pulled aside on a circular are to a point A. When released it swings forward to the opposite side of the verti cal to a point B; it swings back on the arc to Ĉ, then forward to D, etc. If the first swing (that is, the circular are from A to B) is 6 inches long and each succeeding swing is five-sixths as long as the one just preceding it, how far will the weight travel before coming to rest? EXTRA. Substitute for No.

Solve for and y:

y2 98

บ x 15

Wt.

12

10

In determining a "locus" what two facts must be proven? Problem: Find the locus of points equidistant from two intersecting lines. Theorem The medians of a triangle meet in a point which is two-thirds the distance from each vertex to the middle of the opposite side.

:

(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 are of 7 inches.

12

3

Problem: To inscribe a square in the given triangle A B C, one side of
the square lying along B C.
A

12

10

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.)

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. 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 area to the square on the line a.

a

12

(a) Problem: To inscribe a regular hexagon in a circle.
(Take radius of circle equal to two inches.)

(b) Exercise: Find the area of the circle outside the hexagon.
Extra.-Substitute for No.

Theorem: Given any three circles with centers at 0, 0', and O', tangent
externally two and two, at the points A, B, and C, respectively. A and
C lie 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 opposite the first side is greater than the angle opposite the
second side.
(b) Exercise: One of the base angles of a triangle is double the other
and the exterior angle at the vertex is 105°. Find the angles of
the triangle.

Problem: To construct a circle tangent to three intersecting straight lines
but which is not inscribed in the triangle formed by those lines.
Problem: To construct all the common tangents to two nonequal non-
intersecting 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, re-
spectively. 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, r, of the inscribed circle, to construct the triangle.
(Hint: Use theorem stated in question No. 6 to find locus of center
of inscribed circle.

7

T

1

a

(a) Theorem: In a right triangle a perpendicular from the vertex of the right angle to the hypothenuse cuts the hypothenuse into seg ments which are proportional to the squares on the sides adjacen to the right angle.

(b) Problem: Given a square containing 16 square inches, to construct a
square containing one-third of 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.

Problem: In an inscribed quadrilateral, ABCD, the product of the diag
onals is equal to the sum of the products of the opposite sides.
(Hint: Draw a line from B to the point F on AC such that 4 BFC L
BAD and find the similar triangles.)

MARCH, 1929

(a) Define: (1) parallelogram, (2) rhombus, (3) trapezoid.

(b) Theorem: The lines joining the mid-points of the adjacent sides of a rhombus inclose a rectangle.

Problem: To construct a triangle, given two sides and the median to the third side.

Theorem: Of two straight lines drawn from the same point in a perpendicular 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.

Extra. Substitute for No.

Problem: To construct a right triangle, given the hypotenuse and the difference of the other two sides.

MARCH, 1930

Theorem: If CD is the perpendicular from C to the side AB of the triangle ABC, and CE the bisector of the angle C, the angle DCE equals one-half the difference of the angles A and B.

Theorem: The altitudes of a triangle meet in a point.

Problem: Upon a given straight line to describe a segment of a circle which shall contain a given angle.

Problem: Through a given point without a given circle to draw a secant whose internal and external segments shall be equal.

Problem: To construct a circle passing through the two given points a and band tangent to the given line pq.

(a) Define similar triangles.

(b) Theorem: Two triangles are similar if their sides are respectively proportional.

Problem: To construct a polygon similar to a given polygon a bed and having a given ratio to it, viz, as m is to n.

b

Theorem: The area of an inscribed regular hexagon is three-fourths of the area of the circumscribed regular hexagon.

Exercise: Two chords of a circle AB and CD are perpendicular to each other and intersect at 0. Prove that the sum of the squares on the segments AO, OB, CO, and OD, equals the square on the diameter of the circle.

Extra. Substitute for No.

(a) Theorem: The bisectors of the angles of a rectangle inclose a square.

(b) Exercise: If the sides of the above rectangle be 10 and 14 inches, respectively, find the area of the square.

MARCH, 1931

Theorem: If two triangles have two sides of one respectively equal to two
sides of the other, and the included angles unequal, the triangle which
has the greater included angle has the greater third side.
Problem: To find the locus of a point which moves so that the ratio of
its distances from two fixed points, A and B, is constant and equal to
m [in the construction makes m= 21.

Theorem: If, through a point outside a circle, a tangent and a secant are
drawn to the circle, the length of the tangent is a mean proportional
between the whole secant and its external segment.

Problem: A right line segment, AB, is produced to a point C such that AB= ACX BC. To construct the point C. [Hint: Theorem of question No. 3 above may be useful.]

Problem: To find the locus of the centers of the inscribed circles of all right triangles having line AB as a common hypotenuse.

B

Problem: To construct a triangle given two of its sides, b, c, and the median me to the third side

1

Ma

b

C

Theorem: Given two chords of a circle, AB and CD, perpendicular to each other and intersecting at 0.

Then OA2+OB2+OC2+OD2=(the diameter)2.

(a) Theorem: The area of a triangle is equal to one-half the product of
its perimeter by the radius of the inscribed circle.
(b) If an arc of 45° on one circumference is equal to an are of 60 on
another, what is the ratio of the areas of the circles?

Exercise: If the radius of a circle be one inch and the side of a given
regular inscribed polygon be m inches, find the length of the side of 1
regular inscribed polygon which has twice as many sides as the given
polygon.

EXTRA. Substitute for No.

Exercise: The legs of a right angled triangle are 3 and 4 inches respec tively. Find (to hundredths of an inch) :

(a) The hypotenuse and corresponding altitude.

(b) The projections of the given sides on the hypotenuse.

(c) The radius of the inscribed circle.

[Hint: Use statement in question No. 8.]

(d) The radius of the circumscribed circle.

(e) The segments of the hypotenuse fixed by the point of contact of the inscribed circle.

3. ENGLISH GRAMMAR

MARCH, 1927

(a) Write a simple sentence containing a compound subject.
(b) Write a simple sentence containing a compound predicate.
(c) Write a complex sentence containing an adjectival clause.

(d) Write a complex sentence containing an adverbial clause of manner.
(e) Write a sentence containing a preposition with a compound object.
(f) Write a sentence containing an adverbial clause of time.

(g) Write a sentence containing a noun clause used as a subject of the

sentence.

(h) Write a complex periodic sentence containing an adverbial clause of place.

(1) Write a compound sentence expressing alternation.

() Write a sentence containing an adjectival phrase and an adverbial phrase.

Rewrite the following sentences correctly:

(a) They hung the man which committed the murder.

(b) There ain't any doubt but what he done those kind of things.

(c) Who can this letter be from?

(d) He must have came after we went home.

(e) He hadn't ought to act so foolish.

(f) It is me that he fears.

(g) He only offered me three dollars a dozen for these here sort of apples.

(h) He asked if either of the men could identify their ow

clothing.

(i) Neither he or I are to be held responsible.

(j) If it wasn't for you and I, he might have been hurt bad.

Write sentences containing the following:

(a) An auxiliary verb.

(b) The comparative of "soon."
(c) The superlative of good."
(d) The plural of lily."
(e) The masculine of witch."
(f) An adverb of degree.
(g) The plural of "valley."
(h) The comparative of "lazy."
(4) The plural of "shelf."

(j) The superlative of "feeble."