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1. The curve with equation
x3 — 2x2y+2y3—y2x—4y2+2xy−x+2y=0
breaks up into three lines, of which one is x-2y. What are the equations of the other two? What are the coordinates of the vertices of the corresponding triangle; find its area.
x2 y2 2. In the hyperbola -1=0 a diameter with slope u is given. What is undera2 b2 stood by a diameter conjugate to this diameter and how is the slope μ of this diameter derived? How large are the semi-diameters corresponding to these two directions? 3. Discuss and sketch the curve with equation
4. Given a circle of radius r and a line which is at a distance a from the middle point of the circle. A second circle touches the line in its intersection with the line drawn through the middle point of the first circle perpendicular to the line; the common tangents of the circles touch the second circle in points, the locus of which is required. (The given line is to be taken as x-axis, and the foot of the perpendicular as origin of coordinates.)
An ellipse is projected horizontally as a circle of 10 cm. diameter, vertically as a line (30° toward the right with the base line). The horizontal trace of a given plane makes an angle of 60°, the vertical trace of 45°, both toward the right with the base line. Construct the shadows of the ellipse on these planes. (Both projections of rays of light coming from the left make angles of 45° with the base line.)
The two projections of the shadows are to be constructed independently of one another; at any given point construct the tangent and find the nature of the curve in order that its conjugate diameters can be determined.
1. A locomotive weighing 20,000 kilograms moves on a track 1 meters wide and its center of gravity is 13 meters above the rails; what is the greatest velocity that the locomotive may attain in order that, on a curve of 80 meters radius, it shall not leave the rails? What is the maximum velocity, if the outer rail be so raised that the plane of the rails is inclined to the horizontal plane with an angle of 5°?
2. For determining the temperature of a smelting furnace, a platinum sphere of 100 grams is put in it and then thrown into a mixing calorimeter which contains 800 grams of water at 10° C. What is the temperature of the furnace if the brass calorimeter tub weighed 250 grams and the final temperature reached 14.8°? (Specific heat of brass, 0.0926; of platinum, 0.0326.)
3. Two biconvex lenses with focal lengths f1=4cm. and f2=6cm. are arranged from left to right, such that the distance of their optical middle point amounts to d=1 cm.; the thickness of the lenses may be neglected. To the left of the first lens is a luminous substance AB 1 cm. high. Construct the picture of the object which is thrown through the pair of lenses and also determine the distance b1 of the picture from the second lens. How great is the common focal distance ƒ of the system of lenses counted from the second lens; and what advantage is there in such a combination of lenses over a simple lens with the same focal distance?
The following mathematical papers were set in Tokyo for the twenty-fifth examination for teachers' licenses, in 1911.
PRELIMINARY EXAMINATION QUESTIONS.
ARITHMETIC (3 hours).
1. Find three fractions, A, B, and C equal to §,, and fr, respectively, such that A's denominator is equal to B's numerator, and B's denominator to C's numerator. Find the simplest forms of such three fractions.
2. A certain company, dividing its capital in the ratio of 3:5:7, carried on its business in three divisions. At a semiannual settlement it was found that the first division had made 2,600 yen, and the second had earned 8 per cent a year on its capital, but that the third had suffered a loss of 5 per cent a year of its capital. However, the net result was found to be a gain of 6 per cent a year on the total capital. What was the amount of the capital?
3. A steamer, bound for a certain port, had its engine damaged when one-fifth of its voyage had been completed. As it had to reduce its speed by 10 knots for the rest of its course, the average speed was found to be less than the first by 4 knots. What was the initial speed?
4. By evaporating 600 grams of water containing 3 per cent of salt, one containing 5 per cent of salt was to be obtained. It was found, however, that 70 per cent of the water had already evaporated. How much water containing 3 per cent of salt must be added in order to obtain the solution of required strength?
5. Of a cylindrical vessel holding one shō, the height and diameter of which are equal, find the height to the hundredth place.
3. Cut a triangle and a rectangle, having equal bases on a straight line, by another straight line [parallel to it] 1 so that the sum of the areas cut out between the parallel lines shall be equal to the area of the triangle. Find the distance between the parallel lines.
4. In how many different ways can 10 balls be arranged in a straight line, provided that 2 special balls must in all cases be placed so as to occupy alternate positions? 2 5. Let a1, a2, Az, be an arithmetic progression, and b1, b2, b3, ... be a geometric having all its terms positive. Prove that a, is not greater than b1, a,b, and a b2.
These words do not occur in the original.
It is not clear what is meant by "alternate positions."
GEOMETRY (3 hours).
1. Let two circles touch internally at A. From any point P in the circumference of the external circle draw a tangent P M to the internal one, and prove that P A:P M is constant.
2. Given a vertical angle, the radius of the inscribed circle, and the area, construct the triangle.
3. The vertex A of the rectangle ABCD is a fixed point, and B and D are on the circumference of a fixed circle. Find the locus of the point C.
4. Find the limit of the position of a point, the ratio of whose distances from two fixed points is less than a given ratio.
5. Of a quadrilateral whose four vertices are not all in one plane, three are fixed and one moves along a straight line. Find the locus of the intersection of the lines joining the middle points of its opposite sides.
FINAL EXAMINATION QUESTIONS.
ARITHMETIC, Algebra, and GEOMETRY (written).
Part I (3 hours).
1. The sum of a certain irreducible fraction and its reciprocal is equal to 138794
Find the irreducible fraction.
3. If a, b, p, and q be real, prove that the following equation has real roots,
4. Solve the following inequality, x−b>√a(a−2x), where a and b are positive, and√represents the positive square root.
5. Prove that the following three equalities are consistent with one another,
1. If rectangles ABDE, ACFG be externally constructed on the two sides AB and AC of the right angle A of a right-angled triangle ABC, prove that the straight lines BF and CD intersect with each other on the perpendicular from A to the hypotenuse BC.
2. Draw a circle with its center on a straight line passing through the center of a given circle, intersecting this circle at right angles and passing through a given point. 3. Of a triangle ABC, the vertex A is a fixed point on an edge of a trihedral angle and the other two vertices B and C move respectively along two other edges. Find the locus of the center of gravity of the triangle.
A boat is rowed over a certain distance, when there is no tide, in 24 hours. the tide, however, the same distance can be rowed over in 15 hours. Against the tide the boat can be made to go over 32 knots1 in 2 hours. Find the speed, accordingly, when it is rowed with the tide.
Solve the following simultaneous equations,
3 (1+k) x−(3+k)y=3+k.
Draw a straight line meeting two straight lines not in the same plane and normal to a given plane.
4. If the length of three bisectors of the three angles, A, B, C of a triangle ABC be respectively equal to p, q, r, prove that
5. Having given one angle, the perimeter, and the radius of the circumscribed circle of a triangle; solve the triangle.
Application (3 hours).
When the three sides of a triangle are known to be respectively,
compute the three angles and the area.
ANALYTIC GEOMETRY (3 hours).
1. Given a point (1, 1) and a straight line 3x+4y-6=0, the axes being rectangular. Form the equation of the curve of the second degree, having the point and the straight line for its corresponding focus and directrix and 5 for its eccentricity, and reduce it to the standard form.
Such is the original.
2. Let N be the point of intersection of the normal at any point M on an ellipse and its major axis. Prove that the orthogonal projection of MN on the line passing through M and one of the foci is constant.
3. Prove that the four vertices of a parallelogram circumscribing an ellipse and its two foci are on the same equilateral hyperbola.
4. Given an ellipse and a circle concentric with each other, the radius of the circle being equal to the sum of half the major axis and half the minor axis of the ellipse. Prove that the locus of the point of intersection of the two normals to the ellipse at the points at which two tangents are drawn to the ellipse from any point on the circle is a circle.
5. Let N be the point of contact at which a tangent is drawn from the center M of a fixed circle to the circumscribed circle of a triangle self-conjugate with respect to the fixed circle. Prove that MN is constant.
DIFFERENTIAL AND INTEGRAL CALCULUS (4 hours).
1. If ƒ′(x). ❤ (x)−ƒ (x). ø′ (x)=0 within the interval a ≤ x ≤b, and ƒ (a) f(b)=0, then prove that (x) will become zero within the given interval at least once. Heref'(x) and ø′ (x) are continuous within the given limits.
2. Let Y be the point at which the line passing through any point X on the diagonal AC of a parallelogram ABCD and the vertex B intersects the side AD or its extension. Find the minimum of the sum of the areas of the two triangles AX Y and BXC.
3. Take z as the function of two independent variables x and y; substitute
x=r sin @ cos,
y=r sin 9 sin,1.
z=r cos 0
in-√√1+ (dz)*+ (dz)2; taking e and as independent variables eliminate x, y, z.
5. Find the whole length of the space curve represented by the equations
a and b being positive.
6. Take x=(u,v) and y=0(u,v), and change the variables of integration in
SS dxdy from
dxdy from x,y to u,v.
In the report this equation is given as y=r sin .