MATHEMATICS II.

FACULTY OF PRACTICAL SCIENCE.

1. (a) Develop the normal form of the linear equation. (b) For the line 3x+2y=6, determine

1st. Its intercepts with the coordinate axes.

2nd. Its slope.

3rd. Its distance from the origin.

4th. Its distance from the point (1, 8).

2. Find the equations of the tangents to the circle 2+y2-25 at the points of intersection of the circle with the line x-3y+5=0.

3. (a) The equation of the outer curve of a roof truss is y=√15x. What is the nature of the curve, and what is the rise of truss, if the span is 30 feet?

(b) Give a practical method, with all necessary measurements, for laying out an ellipse having axes 20 in. and 12 in.

4. A thin elliptic plate having axes 12 in. and 8 in. has a six inch diameter hole bored symmetrically through its centre. If the plate is divided in two along the minor axis, find the centre of gravity of each half.

5. (a) Prove the sine formula for a spheric triangle. (b) Find the polar of (i) the cosine formula, and

6. From a point on the equator a ship starts along a great circle N. 42° 23′ E., and goes 991 miles. Find (a) its direction at the end of the course, (b) its change in longitude, (c) its latitude.

7. At lat. 44° 18′ N. the altitude of a star east of the local meridian is found to be 47° 20′; and in tables its declination is given as 16° 30′ N., and its right ascension as 19h 43m 228. What is the time of observation by a siderial clock?

8. For lat. 48° 30′ N., find the azimuth of a circumpolar star when at greatest elongation east, and the approximate time by a mean time clock. Given a=21h 27m,8=70° 2′ N., and the corrected right ascension of the mean sun = 1 16m

228.

FACULTY OF PRACTICAL SCIENCE.

MATHEMATCS III.

1. (a) From fundamental principles develop the derivative of " with regard to r.

(b) Give a graphical proof that the derivative of sin(r) is cos(x).

2. Differentiate with regard to r the following,

(i) log x cos x. (ii) (√1+sin x2.) (iii) x-√√1-x=

(iv) sin x cos3x.

(v) e"+e ̄*. (vi) tan-15 x.

3. A point moves so that the distance travelled s is equal to Vt where t is the time in seconds to go any distance s. Show that the acceleration is negative and is proportional to the cube of the velocity.

4. A weight of 100 lbs. is raised by means of a lever fixed at one end which weighs 5 lbs. per foot of length. The weight is 2 ft. from the end of the lever, and at the free end is applied a force of F lbs. What is the length of lever

which makes F a minimum?

5. Expand log(1-x) in terms of x.

6. (a) Write out the results of these integrals:

i. S√x-2`dx. ii. Ssin x dx.

iii.

dx

√25+x2 Se10x dx.

(b) Reduce, giving the work, the following integrals:

7. Find the area bounded by the curve y=x2-2x, and the r axis.

8. Find the equation of the curve through the origin whose subtangent is always of length a.

9. Prove that the cable of a uniformly loaded suspension bridge hangs in the form of a parabola. Find the tension at the mid-point in terms of the span, sag, and loading.

FACULTY OF APPLIED SCIENCE.

PHYSICS I A.

1. A street car motor is geared directly to the axle of the driving wheels so that the angular speed of the wheels is one-quarter that of the motor. If the driving wheels are 20 inches in diameter, if car and load be 6 tons, if friction be 15 lbs. per ton, and if the car be moving up a 2% grade at constant speed, find

and

(a) the force exerted on the rail by the driving wheels,

(b) the torque (in pound-feet) exerted by the motor.

2. A 3 kilogram block rests on a level table. A 2 gram bullet is shot into it with a velocity of 30,000 centimetres per second in a direction making 30° with the vertical. If the coefficient of friction between block and table be 0.1, find

(a) how far the block is driven along the table, and (b) the energy converted into heat in the impact.

3. An engine at 30 miles per hour takes water from a trough between the rails (by lowering a spout into it). If the stream taken be 2 inches by 18 inches in section, and if it be raised 6 feet into the tank, find the horse-power absorbed, taking account of the kinetic energy of the water in the tank but not of that in the trough.

4. A forty-pound picture rests on two nails in the same horizontal line. Its upper edge is held 6 inches from the wall by a cord 24 inches long that runs from the middle of the upper edge to a nail in the wall. If the picture be 42

inches square, find the tension of the reaction at the nails.

cord and the total