Lac mihi non aestate novum, non frigore defit; 3. Consumptis his forte aliis, ut vertere morsus Sen., VIL ARITHMETIC. [Give all the work. Reduce each answer to its simplest form.] 1. What part of is 1*1? * i 2. What is the cost of a pile of wood whose dimensions are 2,1.9, and 42.5 metres, at $ 2 per stere? 3. Find, by logarithms, the third power of the fourth , iat x .01 r00t 0f -317T 4. A and B gain in business $5,040, of which A is to have ten per cent more than B. What is the share of each? 5. If 2 cubic inches of iron weigh as much as 15 cubic inches of water, and a cubic foot of water weigh 1000 ounces, find the weight, in tons, of a cubic yard of iron. 6. If 12 pipes, each delivering 12 gallons a minute, fill a cistern in 3 hours 24 minutes, how many pipes, each delivering 16 gallons a minute, will fill a cistern 6 times as large in 6 hours 48 minutes? 7. How many kilometres make a mile? 8. How many bags, each containing 2 bu. 1 pk. 3 qt, will be required to hold 111 bu. 2 pk. 4 qt. of grain? 9. What is the compound interest of $ 1 for 143 years, allowing it to double once in 11 yr. 11 mo.? ALGEBRA. [Give the whole work.] 1. Find the greatest common divisor and the least common multiple of (243aW + l) and (81a864—' 1), by resolving each expression into factors. 2. Solve the equation x-\-a x — a 1 1 1 x — a x-\-a x — a x* — a1 x -\- a What is the value of x, if 6a + 7 = 0? 3. Divide —^— by 'and express the result without fractional or negative exponents. 4. Solve the equations 2x— y = 21, 2za + ya = 153. 5. A person buys some cloth for $ 90. If he had got two yards more for the same sum, the price per yard would have been fifty cents less. How much did he buy, and at what price per yard? 6. Find (a — b)13 by the Binomial Theorem. ADVANCED ALGEBRA. [Give the whole work.] 1. Solve the equations x* — y* = 215, x* + xy -4- y* = 43. 2. A certain number consists of three digits, in arithmetical progression. If it be divided by the sum of the digits, the quotient is 48; but if 198 be subtracted from it, the digits are inverted. Find the number. 3. Prove the formula for the sum of a geometric progression, in terms of a, r, and n. 4. The first term of a geometric progression is 512, the last term is 162, and the sum is 1562. Find the whole series. Find also what the sum of this series would be, if continued to infinity. 5. Solve the equation V (z + 4) — V # — V + f )• 6. Simplify/^+ ^U(^±12-^Y v J\a—b^a+b) \a* — b1 a' + b1} 7. Find the greatest common divisor of 2x>— llx* — 9 and 4s5 + lis4 + 81. PLANE GEOMETRY. 1. When are two polygons said to be similar? What are similar arcs? similar segments? 2. If a triangle has two sides equal, what is it called? Prove what is true of the angles opposite the equal sides. 3. If, in any triangle, a line be drawn parallel to the base, it will divide the other two sides proportionally. Prove. 4. At a given point in the circumference of a circle a tangent to the circle is drawn. What is the measure of the angle between the tangent and a chord drawn from the point of contact? Prove. What will this angle be if the chord passes through the centre of the circle? 5. Prove that the perimeters of regular polygons, of the same number of sides, are to each other as the radii of the circumscribed circles. State, without proving, what the ratio of the areas of the polygons is 6. Find the area of the circle in which a square, each side of which is V 8 inches long, can be inscribed; and then find the radius of a second circle which shall be nine times as large as the first. SOLID GEOMETRY. 1. Define the following terms: prism; right prism; pentagonal prism; altitude of a zone; spherical sector; 2. Given two planes perpendicular to each other, and a line in one of them perpendicular to their common intersection; prove that the line is perpendicular to the other plane. 3. How may the frustum of a right cone be generated? How is its convex surface found? Give proof. 4. The altitude of a given right cone is ten inches: how far from the vertex of the cone must two planes be passed, parallel to the base of the cone, in order to divide the lateral surface into three equal parts. 5. Prove that, if two spherical triangles on the same sphere, or on equal spheres, are equiangular with respect to each other, they are also equilateral with respect to each other. If the radius of one sphere is three times as great as that of another, what will be the ratio of the sides of two mutually equiangular spherical triangles, one on one sphere and the other on the other? ANALYTIC GEOMETRY. [Give the whole work.] 1. What angle does the line y + 4a; + 2 = 0 make with 2y + 8z = 0? with4#=a;? with 5y + 3a; = l? 2. Which of the four lines in the previous question pass through the origin, and which do not? Prove. 3. The general equation of a circle referred to rectangular axes is (y — n)1 -f- (x — m)1 = r'. At what points is the circle whose radius is 4. Deduce formulas for passing from a rectangular to a polar system. [Denote the polar coordinates by p, cp; the coordinates of the pole with reference to the rectangular system by m, n; the angle which the initial line makes with x by a.] 5. The equation of the tangent to a circle is xx' + yy' = P. Lines are drawn through (7, 1) tangent to the circle z?-\-yi= 25. Find the points of tangency. 6. What is meant by the parameter of a curve? What is the parameter of y1 — 2px? Prove. Of a'y* + b*x* = a^b1? Prove. 7. Explain in full one method of drawing a tangent to a parabola at a given point of the parabola. 8. Find whether the line iy — 3x = 0 intersects the hyperbola 5y* — 2a;2 + 15=0, or its conjugate. What is the tangent of the angle which the asymptotes of this curve make with the axis of x? |