| A1. ABCD is a
square side 2a with vertices in that order. It
rotates in the first quadrant with A remaining on
the positive x-axis and B on the positive y-axis.
Find the locus of its center. |
| A2. a and b
are unequal reals. What is the remainder when the
polynomial p(x) is divided (x - a)2(x -
b). |
| A3. Does
∑n≥0 n! kn/(n +
1)n converge or diverge for k = 19/7?
|
| A4. Let C be
the family of conics (2y + x)2 = a(y +
x). Find C', the family of conics which are
orthogonal to C. At what angle do the curves of
the two families meet at the origin? |
| A5. C is a
circle radius a whose center lies a distance b
from the coplanar line L. C is rotated through π
about L to form a solid whose center of gravity
lies on its surface. Find b/a. |
| A6. P is a
plane and H is the half-space on one side of P. K
is a fixed circle in P. C is a circle in P which
cuts K at an angle α. Let C have center O and
radius r. f(C) is the point in H on the normal to
P through O and a distance r from O. Show that the
locus of f(C) is a one-sheet hyperboloid and that
it has two families of rulings in it. |
| B1. S is a
solid square side 2a. It lies in the quadrant x ≥
0, y ≥ 0, and it is free to move around provided a
vertex remains on the x-axis and an adjacent
vertex on the y-axis. P is a point of S. Show that
the locus of P is part of a conic. For what P does
the locus degenerate? |
| B2. Let
Pa be the parabola y =
a3x2/3 + a2x/2 -
2a. Find the locus of the vertices of
Pa, and the envelope of Pa.
Sketch the envelope and two Pa.
|
| B3. f(x, y)
and g(x, y) satisfy the differential equation
f1(x, y) g2(x, y) -
f2(x, y) g1(x, y) = 1 (*).
Taking r = f(x, y) and y as independent variables,
and x = h(r, y), g(x, y) = k(r, y), show that
k2(r, y) = h1(r, y).
Integrate and hence obtain a solution to (*). What
other solutions does (*) have? |
| B4. A
particle moves in a circle through the origin
under the influence of a force a/rk
towards the origin (where r is its distance from
the origin). Find k. |
| B5. Let f(x)
= x/(1 + x6sin2x). Sketch
the curve y = f(x) and show that
∫0∞ f(x) dx exists.
|