Solve the equations \begin{align*} \lambda x + 2y + z &= 2\lambda, \\ 2x + \lambda y + z &= -2, \\ x+y+z &= -1, \end{align*} when \(\lambda\) is not equal to 2 or to 0. \par Shew that the equations have no solution when \(\lambda = 2\), and that they have an infinite number of solutions when \(\lambda = 0\).
The roots of the equation \(x^3 + px + q = 0\) are \(\alpha, \beta, \gamma\), and \(\omega\) is a complex number such that \(\omega^3=1\). If \[ \phi = \alpha + \omega\beta + \omega^2\gamma, \quad \psi = \alpha + \omega^2\beta + \omega\gamma, \] find the values of \(\alpha, \beta, \gamma\) in terms of \(\phi\) and \(\psi\). \par Shew that \[ \phi\psi = -3p, \quad \phi^3+\psi^3 = -27q, \quad (\phi^3-\psi^3)^2 = (27q)^2 + 108p^3. \]
If all the numbers \(a_i, b_i\) and \(c_i\) are positive, and if \(m\) is a positive integer, shew that \[ \left\{ \frac{a_1^m c_1 + a_2^m c_2 + \dots + a_n^m c_n}{b_1^m c_1 + b_2^m c_2 + \dots + b_n^m c_n} \right\}^{1/m} \] lies between the greatest and the least of the numbers \[ \frac{a_1}{b_1}, \frac{a_2}{b_2}, \dots, \frac{a_n}{b_n}. \]
(i) Shew that, if \(x > 0\), then \(x^{1/n} \to 1\) as \(n \to \infty\). \par (ii) Shew that, if \(a>0\) and \(a_n \to a\), then \(\sqrt{a_n} \to \sqrt{a}\).
Define the modulus \(|z|\) of the complex number \(z\). \par Shew that \(|z_1+z_2| \leq |z_1|+|z_2|\), and give the geometrical interpretation. \par Shew that, if \(z \neq 1\), then \(\left| \frac{1}{1-z} \right| \leq \frac{1}{|1-|z||}\). \par Shew that, if \(|z-1|+|z+1| = 2a\), where \(a>1\), then \(|z| \leq a\).
The numbers \(u_1, u_2, u_3, \dots\) are connected by the relation \(u_n - 2u_{n+1}\cos\theta + u_{n+2}=0\). \(n=1, 2, \dots\), and \(\theta\) is not an integral multiple of \(\pi\). Shew that \(u_n = A \cos n\theta + B \sin n\theta\), and express \(A, B\) in terms of \(u_1, u_2\), and \(\theta\).
Discuss the maxima and minima of the function \(\frac{(x-a)(x-b)}{x}\) when \(a
If \(x=f(t), y=g(t)\), express \(\frac{dy}{dx}, \frac{dx}{dy}, \frac{d^2y}{dx^2}, \frac{d^2x}{dy^2}\) in terms of \(\frac{dx}{dt}, \frac{dy}{dt}, \frac{d^2x}{dt^2}, \frac{d^2y}{dt^2}\). \par Shew that \[ \left(\frac{dx}{dy}\right)^3 \frac{d^3y}{dx^3} + \left(\frac{dy}{dx}\right)^3 \frac{d^3x}{dy^3} + 3 \frac{d^2y}{dx^2}\frac{d^2x}{dy^2} = 0. \]
Evaluate
Obtain a reduction formula for \(\int (\sin x)^m dx\) and use it to evaluate \[ \int_0^{\pi/2} (\sin x)^{2n} dx \quad \text{and} \quad \int_0^{\pi/2} (\sin x)^{2n+1} dx. \] By multiplying the inequality \((1-\sin x)^2 \geq 0\) by \((\sin x)^m\), with suitable values of \(m\), and integrating between \(0\) and \(\frac{1}{2}\pi\), shew that \[ \left\{ \frac{2n(2n+1)}{4n+1}\pi \right\}^{\frac{1}{2}} < \frac{2 \cdot 4 \dots 2n}{1 \cdot 3 \dots (2n-1)} < \left\{ \frac{(4n+3)(2n+1)}{n+1} \frac{\pi}{8} \right\}^{\frac{1}{2}}. \] (Note: The second inequality in the original paper seems to have a typo. It has (2n(2n+1)). I have corrected it to (2n+1) based on standard forms of Wallis' inequality bounds.)