Higher Mathematics.

Junior and Senior.

TRIGONOMETRY; STATICS.

1. Give the trigonometrical definition of an angle. Explain the methods of measuring angles by degrees and grades. The difference of two angles is 5 degrees, and their sum is 50 grades ; find the angles.

2. Define the trigonometrical ratios of an angle. Express all the rest in terms of the secant. Find all the ratios of an angle whose secant is 5.

3. Compare the trigonometrical ratio of the following angles :

(i) A and (180° - A)

(ii) A and (— A)

(iii) A and (180° + A)

(iv) A and (90°.+ A)

4. Trace the changes in sign and magnitude of

cose + 1 cose-1'

as varies from 0 to 360°.

5. Find the values of the trigonometrical ratios for angles of 45° and 135°.

6. Find an expression for all the angles which have a given sine, and also for those which have a given cosine.

Show that all the angles which have the same sine as a are included in the formula (2n + 4) π ± (1⁄2 −

2

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7. How are forces measured, and why can we represent them by straight lines? State and illustrate the principle of the transmission of force.

8. Explain the term resultant force. Show that two forces acting at a point have a resultant; find in what direction it acts when the forces are equal.

9. Enunciate and prove the parallelogram of forces.

10. What is the triangle of forces? If P, Q, R be three forces balancing at a point 0, show that—

11. Find the resultant of any number of forces acting in one plane at a point by referring them to two straight lines at right angles drawn through the point. Hence deduce the conditions of equilibrium.

12. If two forces P and Q act at such an angle that P = R, their resultant, show that if P be doubled, the new resultant will be at right angles to Q.

Higher Mathematics.

Women.

STATICS, DYNAMICS, ASTRONOMY.

1. Describe the screw, and find the relation between P and W when there is equilibrium.

2. What is mechanical advantage?

Illustrate the phrase

that what is gained in power is lost in velocity.

3. A weightless lever in the form of an arc of a circle subtending an angle 2a at its centre, having two weights P and Q hung from its ends, rests with its convexity downwards on a horizontal plane; find the inclination of the chord PQ to the horizon when there is equilibrium.

4. If the accelerating effect of gravity be represented by 9,660, a yard being the linear unit, find the unit of time.

5. A body falling vertically describes 112 feet in a certain second; for how long previously had it been falling? (g = 32).

6. ABC is a triangle, and D, E, F the points where the inscribed circle meets the sides BC, CA, AB respectively. Show that if a ball (elasticity e) be projected from D to strike AC in E, and then rebound to F, then AE= e. CE, and if the ball return to D, AB = e. AC.

7. What is meant by Precession of the Equinoxes? how does it affect the apparent position of a star and the real position of the pole of the equator?

8. Compare the effects of aberration and of annual parallax on the apparent places of stars, showing that the position of the earth for maximum aberration is that of minimum parallax, and vice versa...

9. Define the following terms as applied to planets:—Superior, inferior, conjunction, opposition, elongation, node, phase.

Which signs must be taken when A lies between 450° and 630°?

+ that is nearly the circular measure of 3°.

4. The distances of the centre of the circle inscribed in a triangle from the centres of the three escribed circles are re

5. The tangent at P to an ellipse (foci S and S') is equally inclined to SP and S'P.

What does this theorem become when the ellipse assumes its limiting form of a parabola?

6. If from any point T tangents TQ, TQ', be drawn to a parabola, the point T is equidistant from the diameters through Q and Q'.

7. What is the parameter of a diameter of a parabola? Show that its length is four times the focal distance of the vertex of the diameter.

8. Express the equation of a line in terms of the intercepts a, b, it cuts off on the axes. Given two fixed points A and B one on each of the axes, if A' and B' be taken on the axes so that OA' + OB' = OA + OB; find the locus of the intersection of AB', A'B.

9. Find the equation of the line joining the point (2, 3) to the intersection of the lines, 2x + 3y + 1 = 0, 3x-4y = 5.

10. Define the terms radical axis, and radical centre as applied to circles.