A plank rests across a cylindrical barrel on flat ground and initially has one end on the ground. A man walks up the plank. DIscuss qualitatively the apossible resulting motions on the assumption that no slipping occurs. [For example, consider different ratios of masses and dimensions and consider the man taking short or long steps and moving irregularly.]
If \(a, b, c\) are three constants, all different, show that the equations \begin{align*} x+y+z &= 3(a+b+c), \\ ax+by+cz &= (a+b+c)^2, \\ xyz &= ayz+bzx+cxy, \end{align*} have in general only one solution in which \(x, y, z\) are unequal, and find this solution.
If \(f(x)=0\) is an algebraic equation of integral degree, show that the sum of the \(m\)th powers of its roots is the coefficient of \(x^{-m}\) in the expansion of \(xf'(x)/f(x)\). Find the sum of the cubes of the roots of the quartic equation \[ x^4+x^3-2x^2-7=0. \]
Solution: Notice that \begin{align*} \frac{xf'(x)}{f(x)} &= x \cdot \frac{\left ( \prod (x-\alpha_i)^{k_i} \right)'}{\prod (x-\alpha_i)^{k_i}} \\ &= x \cdot \frac{\sum_i \prod_{j \neq i} (x-\alpha_j)^{k_j} k_i(x-\alpha_i)^{k_i-1}}{\prod (x-\alpha_i)^{k_i}} \\ &= x \sum_i \frac{k_i}{(x-\alpha_i)} \\ &= \sum_i \frac{k_i}{1 - \frac{\alpha_i}{x}} \\ &= \sum_i \sum_{m=0}^\infty k_i \left ( \frac{\alpha_i}{x} \right)^m \\ &= \sum_{m=0}^\infty \underbrace{\left ( \sum_{i} k_i \alpha_i^m \right)}_{\text{sum of the }m{\text{th powers of roots}}} x^{-m} \end{align*} We are looking for the coefficient of \(x^{-3}\) in the expansion of: \begin{align*} \frac{xf'(x)}{f(x)} &= \frac{x(4x^3+3x^2-4x)}{ x^4+x^3-2x^2-7} \\ &= \frac{4x^4+3x^3-4x^2}{x^4+x^3-2x^2-7} \\ &= \frac{4+3x^{-1}-4x^{-2}}{1+x^{-1}-2x^{-2}-7x^{-4}} \\ &= (4+3x^{-1}-4x^{-2})(1 -(x^{-1}-2x^{-2}-7x^{-4})+(x^{-1}-2x^{-2}-7x^{-4})^2-(x^{-1}-2x^{-2}-7x^{-4})^3 + \cdots) \\ &= (4+3x^{-1}-4x^{-2})(1 -(x^{-1}-2x^{-2})+(x^{-1}-2x^{-2})^2-(x^{-1})^3) + o(x^{-4}) \\ &= (4+3x^{-1}-4x^{-2})(1 -x^{-1}+2x^{-2}+x^{-2}-4x^{-3}-x^{-3}) + o(x^{-4})\\ &= (4+3x^{-1}-4x^{-2})(1 -x^{-1}+3x^{-2}-5x^{-3}) + o(x^{-4})\\ &= \cdots + (4+9-20)x^{-3} + \cdots \end{align*} Threfore the sum of the cubes is \(-7\).
Prove that the arithmetic mean of a set of unequal positive quantities is greater than their geometric mean. Hence establish that if \(n\) is a positive integer
Explain briefly the rule for multiplying two determinants together. Show that \[ \begin{vmatrix} b^2+c^2+1 & c^2+1 & b^2+1 & b+c \\ c^2+1 & c^2+a^2+1 & a^2+1 & c+a \\ b^2+1 & a^2+1 & a^2+b^2+1 & a+b \\ b+c & c+a & a+b & 3 \end{vmatrix} \] is the square of a certain determinant, and hence obtain its value.
Prove that if for a polynomial \(f(x)\) of degree \(n\) with real coefficients the values of \(f(x)\) and all its derivatives are positive for \(x=x_0\), then no root of \(f(x)=0\) can exceed \(x_0\). Prove also the truth of this statement for the equations \(f^{(r)}(x)=0\) obtained by differentiating \(f(x)\) \(r\) times. Consider the case \(f(x)=x^4-2x^3-3x^2-15x-3\) and show that all the roots of \(f(x)=0\) are less than 4.
The polynomial \(f_n(x)\) is defined as \(\dfrac{d^n}{dx^n}(x^2-1)^n\). Prove that all the roots of the equation \(f_n(x)=0\) are real and distinct and lie between \(\pm 1\). Prove also that \(\int_{-1}^1 f_n(x)f_m(x)dx=0\) if \(m \neq n\), and find its value when \(m=n\).
Define \(\log_e x\) for \(x>0\). Prove that for \(x>1\): \[ x^2-x > x\log_e x > x-1 \quad \text{and} \quad x^2-1 > 2x\log_e x > 4(x-1)-2\log_e x. \]
Evaluate the following integrals: \[ \int_{\pi/4}^{3\pi/4} \frac{dx}{2\cos^2 x+1}; \quad \int_0^\infty \frac{dx}{(1+x^2)^n}, \text{where } n \text{ is a positive integer}; \quad \int_0^\infty \frac{dx}{1+x^3}. \]
Discuss the general nature of the plane curve whose polar equation is \(r = \dfrac{a}{\theta^2-1}\) for values of \(\theta>1\). Prove that one of the bisectors of the angle between the radius vector and the normal is inclined at an angle \(\tan^{-1}\theta\) to the radius vector, and find an expression for the length of arc from the origin to the given point.