$4 HIGHER BRANCHES OF MATHEMATICS. SECTION I. 1. Find x from the equation x2 12x=- 35. A draper bought a piece of silk for £16 4s., and the number of shillings which he paid per yard was the number of yards. How much did he buy? 2. Find x and y from the equations xy=4, x2 + y2 = 40. There is a rectangular field whose length exceeds. its breadth by 16 yards, and it contains 960 yards: find its dimensions. 3. A fast passenger train starts at 20 minutes past 12; it overtakes a luggage train which travels 15 miles an hour, and after having gone 15 miles further overtakes a slow passenger train which travels 20 miles an hour. Another fast train, which travels at the same rate as the first, starts from the same station at 2 o'cloek of that day, overtakes the luggage train, and, after having gone 65 miles further overtakes the slow train also, and finds that it has then travelled 120 miles. Required the rate at which the fast trains travel. SECTION II. 1. Prove the formula for finding the sum of an arithmetical series. The first term of an arithmetical progression is 1 ; the common difference, 1; the sum of the series, 36. Required, the number of terms. 2. When are quantities said to be proportional? And when is one quantity said to vary as another? Prove that, if abcd; then a ab::c:c~ d. 3. Find the number of different combinations that may be made of n different things, taking r of them together. SECTION III. 1. 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. 2. Describe a circle in a given triangle. 3. Similar triangles are to each other in the duplicate ratio of their homologous (or corresponding) sides, Prove this, and indicate the steps of the proof of the corresponding proposition for all similar figures. SECTION IV. What 2 1. Define the sine and the tangent of an angle. are the values of sin. 90°, tan. 180°, and tan. 45°. Prove that cos. 2 A = 2 cos. 2A 1=1 sin. 2A. 2. Having given one side of a right-angled triangle and the angle adjacent to that side, show by what calculations the other parts of the triangle may be obtained. 3. Write down the expression for the cosine of an angle of a triangle in terms of the sides; and prove the expression for the sine of an angle, and for the area of the triangle, in terms of the sides. What is the logarithm of a number? What are the properties of logarithms on which the utility of logarithmic tables depends? SECTION V. 1. Prove the expressions for the circumference and area of a circle in terms of its radius. 2. Show that the solid content of any cone, or pyramid, is found by multiplying the area of its base by one-third its height. SECTION 1. 1. "Find x from the equation x2 35....(a) 12 x = "A draper bought a piece of silk for £16 4s., and the number of shillings which he paid per yard was the number of yards. How much did he buy ?" (b) (a)....x2. -12x=- -35; completing the square we get x2-12x+3636-35=1 per yard in shillings. But by the question this is 66 324 X 9 = 81 X 9 x = 9 × 3 = 27 2. (a) Find x and y from the equations x - y = 4, x2 + y2 = 40. (b) "There is a rectangular field whose length exceeds its breadth by 16 yards, and it contains 960 yards; find its dimensions." (a) Squaring the first we get x2-2xy + y2 = 16; subtracting the second 2 xy == 24 .. 2xy = 24 x2 + 2 xy + y2 x + y And x y 2 y (b) Let x = Add the second. 40+ 24 = 64 = 12 and x 6 = 4 and y = 2 breadth, whence x + 16 = length. But x(x+16) = 960 x2+16x 960 x2+16x+ 64 = 1024 x+8=+32 The plus sign being taken, 24 is the breadth, and 241640, the length. 3. "A fast passenger train starts at 20 minutes past 12; it overtakes a luggage train, which travels 15 miles an hour, and after having gone 15 miles further overtakes a slow passenger train which travels 20 miles an hour. Another fast train, which travels at the same rate as the first, starts from the same station at 2 o'clock of that day, overtakes the luggage train, and, after having gone 65 miles further overtakes the slow train also, and finds that it has then travelled 120 miles. Required the rate at which the fast trains travel." Let x=rate per hour of the fast trains: now the former starts one hour and two-thirds before the second; therefore that time will elapse before the second comes to the point at which the first overtook the luggage train. We shall deduce our data from the consideration of this as a starting point. Let us first find, in terms of x, how far the slow passenger train was in advance when the luggage train was overtaken. Let y that distance in miles. The space travelled over by the first train after passing the luggage, and before overtaking the passenger train, was 15 miles: therefore y miles, plus the miles travelled from the time the fast train passed the luggage to the time of coming up with the passenger train, are equal to 15 miles. Now the first train travelled x miles per hour, and the slow 20 miles. Hence, the sum of the following series equals 15 miles. 20 S = y + y X + y x (20)+ &c. ad infinitum By the summation of this series, we obtain Now, the luggage train travels 15 miles per hour. Hence, by the time the second fast train arrives at the point at which the first overtook the luggage train, this the distance of the fast train's point of junction with these two trains from the termination of two respective series similar to the preceding: But from the question the difference of the distance represented by these expressions equals 65 miles. |