Could You Pass This Entrance Test?
High School Entrance Examination For Mathematics (China, 1955)
Part I (7 points each)
Part II (18 points each)
- Find the equation whose roots are the square of the roots of x2-3x-1 = 0.
- Find an expression for the interior angles of an isosceles triangle whose side is four times the length of the base.
- 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?
- 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?
- 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<
- 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.
- Solve for x: cos 2x = cos x + sin x.
- 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.
Source: F. Swetz's Mathematics Education in China: Its Growth and Development, MIT Press, 1974, pp. 148-149