Putnam 1939

1. Find the length of the curve $y^{2}=x^{3}$ from the origin to the point where the tangent makes an angle of $45^{\circ}$ with the $x$-axis.

2. A point $P$ is taken on the curve $y=x^{3}$. The tangent at $P$ meets the curve again at $Q$. Prove that the slope of the curve at $Q$ is four times the slope at $P$.

\begin{array}{l} \text { 3. Find the cubic equation whose roots are the cubes of the roots of }\ x^{3}+a x^{2}+b x+c=0 \end{array}

4. Find the equations of the two straight lines each of which cuts all the four straight lines $$x=1, y=0 ; y=1, z=0 ; z=1, x=0 ; x=y=-6 z$$

5. Take either (i) or (ii). (i) Solve the system of differential equations $$\begin{array}{l} \frac{d x}{d t}=x+y-3 \\ \frac{d y}{d t}=-2 x+3 y+1 \end{array}$$ subject to the conditions $x=y=0$ for $t=0$. (page 101) (ii) A heavy particle is attached to the end $A$ of a light $\operatorname{rod} A B$ of length $a$. The rod is hinged at $B$ so that it can turn freely in a vertical plane. The rod is balanced in the vertical position above the hinge and then slightly disturbed. Prove that the time taken to pass from the horizontal position to the lowest position is $$\sqrt{\frac{a}{g}} \log _{e}(1+\sqrt{2})$$

6. Take either (i) or (ii). (i) A circle of radius $a$ rolls on the inner side of the circumference of a circle of radius $3 a$. Find the area contained within the closed curve generated by a point on the circumference of the rolling circle. (page 105) (ii) A shell strikes an airplane flying at a height $h$ above the ground. It is known that the shell was fired from a gun on the ground with a muzzle velocity of magnitude $V$, but the position of the gun and its angle of elevation are both unknown. Deduce that the gun is situated within a circle whose center lies directly below the airplane and whose radius is $$\frac{V}{g} \sqrt{V^{2}-2 g h}$$ (Neglect the resistance of the atmosphere.)

8. From the vertex $(0, c)$ of the catenary $$y=c \cosh \frac{x}{c}$$ a line $L$ is drawn perpendicular to the tangent to the catenary at a point $P$. Prove that the length of $L$ intercepted by the axes is equal to the ordinate $y$ of the point $P$.

9. Evaluate the definite integrals (i) $\int_{1}^{3} \frac{d x}{\sqrt{(x-1)(3-x)}}$, (ii) $\int_{1}^{\infty} \frac{d x}{e^{x+1}+e^{3-x}}$.

10. Given the power-series $$a_{0}+a_{1} x+a_{2} x^{2}+\cdots$$ in which $$a_{n}=\left(n^{2}+1\right) 3^{n}$$ show that there is a relation of the form $$a_{n}+p a_{n+1}+q a_{n+2}+r a_{n+3}=0$$ in which $p, q, r$ are constants independent of $n$. Find these constants and the sum of the power-series.

11. Find the equation of the parabola which touches the $x$-axis at the point $(1,0)$ and the $y$-axis at the point $(0,2)$. Find the equation of the axis of the parabola and the coordinates of its vertex.

12. Take either (i) or (ii). (i) Prove that $$\int_{1}^{a}[x] f^{\prime}(x) d x=[a] f(a)-\{f(1)+\cdots+f([a])\}$$ where $a$ is greater than 1 and where [ $x$ ] denotes the greatest of the integers not exceeding $x$. Obtain a corresponding expression for $$\int_{1}^{a}\left[x^{2}\right] f^{\prime}(x) d x$$ (ii) A particle moves on a straight line, the only force acting on it being a resistance proportional to the velocity. If it started with a velocity of $1,000 \mathrm{ft}$. per sec. and had a velocity of 900 ft . per sec. when it had travelled $1,200 \mathrm{ft}$., calculate to the nearest hundredth of a second the time it took to travel this distance.

13. Take either (i) or (ii). (i) Let $f(x)$ be defined for $a \leq x \leq b$. Assuming appropriate properties of continuity and derivability, prove for $a<x<b$ that $$\frac{\frac{f(x)-f(a)}{x-a}-\frac{f(b)-f(a)}{b-a}}{x-b}=\frac{1}{2} f^{\prime \prime}(\xi)$$ where $\boldsymbol{\xi}$ is some number between $\boldsymbol{a}$ and $\boldsymbol{b}$. (page 123) (ii) Calculate the mutual gravitational attraction of two uniform rods, each of mass $m$ and length $2 a$, placed parallel to one another and perpendicular to the line joining their centers at a distance $b$ apart.

In your answer let $a$ approach zero, and comment on the form of the result.