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Could You Pass This Entrance Test?

High School Entrance Examination For Mathematics (China, 1955)

Part I (7 points each)

  1. Find the equation whose roots are the square of the roots of x2-3x-1 = 0.
  2. Find an expression for the interior angles of an isosceles triangle whose side is four times the length of the base.
  3. Given a square right pyramid whose base has a side of length a. If the angle the pyramid's face makes with the base is 45o, what is the altitude of the given solid?
  4. Two planes intersect in space. From a point in each plane a normal is constructed. The two normals to the given plane are coplaner and meet. What is the angle of their intersection called?
Part II (18 points each)
  1. Find the coefficients b, c, and d such that x2+bx+cx+d can:
    a. be divided evenly by x-1
    b. be divided by x-3 with remainder 2
    c. be divided by x+2 and x-2 and have the same remainder in both cases<
  2. Given the triangle ABC, circumscribe a circle about the triangle, from a point D on side AC draw a line perpendicular to side AB and extend it to intersect the extension of BC at F. The constructed line intersects the circle at G. Prove: (EG)2 = EF * ED.
  3. Solve for x: cos 2x = cos x + sin x.
  4. Given a triangle with a perimeter of 12 ft. and area 6 square feet. Prove: The given triangle is a right triangle and that the length of its sides are 3, 4, and 5 feet.

Solution:

Source: F. Swetz's Mathematics Education in China: Its Growth and Development, MIT Press, 1974, pp. 148-149