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1. Prove that the intersections of two parallel planes with a third plane are parallel planes.
2. Prove that the sum of the line angles that compose a solid angle is less than four right angles.
3. What is the frustum of a pyramid? Show how to find the convex surface of a regular pyramid. Prove that the surface of a right circular cone is equal to the product of the slant height multiplied by the circumference of a section drawn midway between the bases.
4. Given the radius of a sphere: write a formula for its surface and one for its volume.
5. What is the segment of a sphere? Explain how to find the volume of a segment of a sphere having two bases, one each side of the centre.
6. Given a spherical triangle, to draw its polar triangle. What relations exist between the sides and angles of a spherical triangle and those of its polar triangle? State and prove.
7. Given a spherical triangle, to draw another symmetrical with it on the same sphere. Prove that two symmetrical triangles on the same sphere have the same area.
8. What is a regular polyhedron? How many are there? Give their names, and a brief description of each.
Course II. and Advanced Standing.
[Ask for a Table of Natural Cosines.]
1. To find the equation of a straight line that passes through two given points.
2. Find the equation of a line that passes through the origin and the point (-3, 2).
3. Find the equation of a line which passes through the point (2,1) and makes an angle of 45° with the line x-2y+3=0.
4. Establish formulas for changing rectangular into polar coordinates.
5. Write down the equation of a circle having a radius = 7 and its centre at (3, — 4).
6. What curves do these equations represent?
What are the polar equations of these curves? Sketch one of these curves from its rectangular equation, and the other from its polar equation. Find the foci. Find the parameter of each curve, and draw it.
7. Which of the points (4, 2), (3, — 34), (3, 34), is on the x2 y2 curve + 25 16
1. Find the equation of the tangent and
that of the normal at this point. Find also the lengths of the subtangent and subnormal.
8. How do you find the points where two curves intersect? As an example take these two curves: y2 = 4x and x2+6x+ y2 24. What are these curves? Draw them.
Course II. and Advanced Standing.
1. The sine of an angle x is greater than the sine of another angle y, both angles being in the second quadrant. Compare the other trigonometric functions of these angles (cosine with cosine, etc.), stating which in each set is numerically the larger. Prove your results, either by formulæ or by a diagram.
2. Obtain, from fundamental formulæ, the trigonometric functions of (360° — y). Given the functions of (180° —— y), how can those of (180°+y) be obtained ?
3. Solve the right triangle in which one angle is 74° 18', and the hypothenuse is .01.
4. What angle in the third quadrant has a cosine equal to the sine of 330° ?
5. Obtain, from fundamental formulæ,
6. Obtain, from the second member of the equation in the previous question, an equally simple expression in terms of the cotangents of x and y.
7. Find the smallest angle in the triangle whose sides are 1236, 1342, 1729.
8. Obtain the formulæ necessary for the complete solution of an oblique triangle, in which are given two sides and the included angle.
A short English composition is required, correct in spelling, punctuation, grammar, and expression. Thirty lines will be sufficient. Make at least two paragraphs.
The story of the Caskets, in the Merchant of Venice;
Or, The story of Shakespeare's Tempest;
Or, The story of Rebecca, in Scott's Ivanhoe.