# Morning Session ## Problem 1 A square of side $2 a$, lying always in the first quadrant of the $X Y$ plane, moves so that two consecutive vertices are always on the $X$ - and $Y$-axes respectively. Find the locus of the midpoint of the square. ## Problem 2 If a polynomial $f(x)$ is divided by $(x-a)^2(x-b)$, where $a \neq b$, derive a formula for the remainder. ## Problem 3 Is the following series convergent or divergent? $1+\frac{1}{2} \cdot \frac{19}{7}+\frac{2 !}{3^2}\left(\frac{19}{7}\right)^2+\frac{3 !}{4^3}\left(\frac{19}{7}\right)^3+\frac{4 !}{5^4}\left(\frac{19}{7}\right)^4+\cdots $ ## Problem 4 Find the orthogonal trajectories of the family of conics $(x+2 y)^2$ $=a(x+y)$. At what angle do the curves of one family cut the curves of the other family at the origin? ## Problem 5 A circle of radius $a$ is revolved through $180^{\circ}$ about a line in its plane, distant $b$ from the center of the circle, where $b>a$. For what value of the ratio $b / a$ does the center of gravity of the solid thus generated lie on the surface of the solid? ## Problem 6 Any circle in the $X Y$ (horizontal) plane is "represented" by a point on the vertical line through the center of the circle and at a distance "above" the plane of the circle equal to the radius of the circle. Show that the locus of the representations of all the circles which cut a fixed circle at a constant angle is a (portion of a) one-sheeted hyperboloid. By consideration of suitable families of circles in the plane, demonstrate the existence of two families of rulings on the hyperboloid. # Afternoon Session ## Problem 7 A square of side $2 a$, lying always in the first quadrant of the $X Y$ plane, moves so that two consecutive vertices are always on the $X$ - and $Y$-axes respectively. Prove that a point within or on the boundary of the square will in general describe a (portion of a) conic. For what points of the square does this locus degenerate? ## Problem 8 For the family of parabolas $y=\frac{a^3 x^2}{3}+\frac{a^2 x}{2}-2 a$ > [!i] find the locus of vertices, > [!ii] find the envelope, > [!iii] sketch the envelope and two typical curves of the family. ## Problem 9 Given $ \begin{aligned} & x=\phi(u, v) \\ & y=\psi(u, v) \end{aligned} $ where $\phi$ and $\psi$ are solutions of the partial differential equation $\frac{\partial \phi}{\partial u} \frac{\partial \psi}{\partial v}-\frac{\partial \phi}{\partial v} \frac{\partial \psi}{\partial u}=1\tag{1}$ By assuming that $x$ and $v$ are the independent variables, show that (1) may be transformed to $ \frac{\partial y}{\partial v}=\frac{\partial u}{\partial x}\tag{2} $ Integrate (2), and show how this effects in general the solution of (1). What other solutions does (1) possess? ## Problem 10 A particle moves under a central force inversely proportional to the $k$ th power of the distance. If the particle describes a circle (the central force proceeding from a point on the circumference of the circle), find $k$. ## Problem 11 Sketch the curve $y=\frac{x}{1+x^6 \sin ^2 x}$ and show that $\int_0^{\infty} \frac{x d x}{1+x^6 \sin ^2 x}$ exists.