7. It takes a man as long to scull a mile up a stream as a mile and quarter down the stream. He sculls down the stream for five minutes, and then drifts for five minutes, and so on alternately for 40 minutes, when he finds that he has travelled 23 miles. Find the velocity of the current. 9. ABCD is a field such that the straight lines joining opposite corners A, C and B, D meet at right angles at F; FA, FB, FC, FD measuring 83, 97, 125, and 228 yards respectively. Find the area of the field in acres, roods, &c. 10. How many square feet of metal will be required to construct a rectangular tank (open at the top) 12 feet long, 10 broad, and 8 deep? 11. It is desired to cover a piece of ground 80 feet square by a pyramidal tent 30 feet in perpendicular height. Find the cost of the requisite quantity of canvas at 4d. per square yard. 12. A solid metal sphere, 6 inches in diameter, is formed into a tube 10 inches in external diameter and 4 inches in length. Find the thickness of the tube. 13. Water is poured into a cylindrical reservoir 20 feet in diameter at the rate of 400 gallons a minute. Assuming a gallon of water to measure 2771 cubic inches, find the rate at which the surface of the water rises in the reservoir. MATHEMATICS. (2.) 1. If a straight line be bisected, and produced to any point, the square of the whole line thus produced, and the square of the part of it produced, are together double of the square of half the line bisected, and of the square of the line made up of the half and the part produced. 2. In a triangle ABC, if the side AB be produced to D, and AC to E, so that AD = twice AB, and CE = half of AC: show that the triangle ADE = three times the triangle ABC. 3. In a circle the angle in a semicircle is a right angle: but the angle in a segment greater than a semicircle is less than a right angle and the angle in a segment less than a semicircle is greater than a right angle. 4. What condition must be fulfilled by a quadrilateral figure (1) if it be capable of having a circle described about it, (2) if it be capable of having a circle inscribed within it ? 5. Describe a circle about a given triangle. Also describe a circle which shall pass through two given points and touch a given straight line. 6. Give and comment upon Euclid's definition of ratio and proportion, and explain the terms triplicate ratio, componendo, and ex æquali. 7. Prove that similar triangles are to one another in the duplicate ratio of their homologous sides. How many 8. What is meant by the circular measure of an angle? degrees, minutes, and seconds are there in an angle whose circular measure is? 5 9. Trace the changes of sign of sine as e increases through four 2 right angles, and show that its equivalent 2 sin 00 COS has always the same sign. 10. Find the values of sin 30° and sin 18°; and prove that a value of which satisfies the equation 2 tan 0 = cos o lies between 18° and 30°. Find the general values of e from the equation 12. Given two sides and an angle opposite to one of them, determine the triangle. When is there more than one solution? Illustrate by a geometrical construction. 7b, and A = 6° 37' 24", find the other angles, having given log 230103, L tan 3° 18′ 42′′ = L tan 8° 14' = 9.1604569, L tan 8° 13′ 50′′ 8.7624080, 14. If the sides of a triangle be in arithmetical progression, the perpendicular on the mean side from the opposite angle, and the radius of the circle which touches the mean side and the two other sides produced, are each three times the radius of the inscribed circle. 15. Find the area of a regular polygon inscribed in a circle, and hence the area of the circle itself. |