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Pass the Test? But...No Algebra?

Entrance Examination In Mathematics
(Fuh Tan University, China, 1917)

Geometry

  1. Define: axiom, locus, postulate, perpendicular bisector and tangent.
  2. State the cases in which two triangles are congruent.
  3. Prove: The diagonals of a parallelogram bisect each other.
  4. Prove: The lines joining the midpoints of the consecutive sides of any quadrilateral form a parallelogram.
  5. Prove: The tangents to a circle drawn from an external point are equal, and make equal angles with the line joining the point to the center.
  6. The diameter AB and the chord DC (of a circle) are prolonged until they meet at E. Prove EA > EC and EB < ED.
Solid Geometry
  1. Prove that a spherical angle is measured by the arc of the great circle described from its vertex as a pole and inclined between its sides (produce if necessary).
  2. The sides of a spherical triangle are 80o, 74o and 128o. The radius of the sphere is 14 ft. Find the area of the polar triangle.
  3. Prove that the sum of the sides of a spherical polygon is less than 360o.
  4. A plane divides the surface of a sphere of radius R into two zones such that the surface of the greater is the mean proportional between the entire surface and the surface of the smaller. Find the distance of the plane from the center of the sphere.
  5. Find the area of the surface and the volume of a sphere if the diameter is 3 ft. 6 inches.
  6. Prove that two mutually equilateral triangles on the same sphere or equal spheres are mutually equiangular and are equal or symmetrical.
Trigonometry
  1. Prove that: (1 + tan x) (tan 2x) = sec 2x.
  2. If sin x = -5/13 and x is in the fourth quadrant, find the other functions of x.
  3. Show that tan 3x = (3 tan x - tan3x)/(1 - 3 tan2x).
  4. If Y = sin-1(1/3) find tan Y.
  5. Compute the value of sin 90o - b cos 360o + (a-b) cos 180o.
  6. Find the value of x by logarithms if x = [(25243 x 0.0534)/(0.0063 x 275200)]1/4

Solution:

Source: Fuh Tan University Catalogue: 1917-1918, p. 49.