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1. Two sides of one triangle are respectively equal to two sides of another triangle, but the angles included by these sides are not equal. What is true of the third sides? State and prove. State the converse theorem. Is it true ?
2. The area of a triangle. State and prove.
3. Prove that the areas of two rectangles are proportional to the products of their bases by their altitudes.
4. The radius of a given circle is ten inches; what is the radius of a circle having twice the area of the given circle of a circle having one half the area of the given circle ?
5. State and prove the Pythagorean theorem.
6. Given the base, the altitude, and one of the angles at the base of a triangle, to construct the triangle.
7. Prove that two triangles are similar, if an angle of one equals an angle of the other, and the sides which include these angles are proportional. 8. A perpendicular drawn from any point of a semi-circum
the diameter is a mean proportional between what? State and prove.
1. Two planes are perpendicular to each other, and a straight line is drawn in one of them perpendicular to their intersection ; prove that this straight line is perpendicular to the other plane.
2. Two planes are perpendicular to each other, and through any point of one is drawn a straight line perpendicular to the other : prove that this straight line lies wholly in the first plane.
3. Prove that if a solid angle is formed by three plane angles, the sum of either two of these angles is greater than the third.
4. Prove that sections of a pyramid made by parallel planes are similar polygons whose areas are proportional to the squares of their distances from the vertex.
5. Prove that two pyramids which have equal bases and altitudes are equivalent. Why not say equal ?
6. Prove that a triangular pyramid is a third part of a triangular prism of the same base and altitude. Deduce from this a rule for finding the volume of any pyramid or cone.
7. How large a part of the surface of a sphere is covered by a spherical triangle whose angles are 90°, 150°, 132° ?
8. What is a regular polyhedron? How many are there ? Give their names and a brief description of each.
[Ask for Trigonometric Tables.]
1. What are Rectangular Co-ordinates ? Polar Co-ordinates ?
2. Lay down a few points of, and then draw the curves represented by, these equations:
(i.) 7.2c2 — 16 ya=112,
cos What are these curves ?
3. The centre of a circle is at the point (-2, 0) and its radius=5; what is its equation?
4. Define the Ellipse, Parabola, Hyperbola.
5. From its definition deduce the rectangular equation of the parabola.
6. Given the equation of a parabola yü= 6x; what is the
ya distance from the origin to the focus ? Transform this equation to a set of axes through the focus. What does the new equation represent? Transform it to polar co-ordinates. Illustrate by a diagram. 7. Is the point (2, 1) on the straight line x - 3y +1=0?
8. Find the equation of a straight line passing through (2, 1) and perpendicular to the line x - 3y+1=0. Draw both lines.
9. In what point do the straight lines x - 3y+1=0 and x + 7y +11=0 intersect ?
10. Find the angle between the two straight lines given in the last question.
Course II. and Advanced Standing.
1. The cosine of an angle in the first quadrant is 0.7. Find, either by formulæ or by tables, the sine of half that angle.
2. What is the sine of 240° ? The cosine of 300° ? The tangent of 225o ? The secant of 150° ?
3. One angle of a plane triangle is 64° 18', and the other angles are equal. The greatest side is 10. Solve the triangle.
4. Find the trigonometric functions of (270° — y).
5. Prove that the sides of a plane triangle are proportional to the sines of the opposite angles.
6. Obtain, from fundamental formula,
cot y cot x + 1 cot (x - y)=
7. Two sides of a plane triangle are 4, 6, and the included angle is 38° 54'.
Solve the triangle. 8. One side of a plane triangle is double another, and the third side equals one half the sum of the other two. Find the largest 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 Trial Scene, in the Merchant of Venice ;