Problems

Find the maximum and minimum values of \(\cos\theta + \cos(z - \theta)\), where \(z\) is fixed and \(\theta\) is variable. Hence, or otherwise, show that, if \(A\), \(B\), \(C\) are the angles of any triangle, then $$\cos A + 2(\cos B + \cos C) \leq 2.$$

The coordinates of any point on a curve are given by \(x = \phi(t)\), \(y = \psi(t)\), where \(t\) is a parameter; prove that the equation of the tangent is $$\begin{vmatrix} x & \phi(t) & \phi'(t) \\ y & \psi(t) & \psi'(t) \\ 1 & f(t) & f'(t) \end{vmatrix} = 0.$$ Prove that the condition that the tangents at the points of the curve $$x = at/(t^2 + bt^2 + ct + d), \quad y = a/(t^2 + bt^2 + ct + d),$$ whose parameters are \(t_1\), \(t_2\), \(t_3\) may be concurrent is $$3(t_2 t_3 + t_3 t_1 + t_1 t_2) + 2b(t_1 + t_2 + t_3) + b^2 = 0.$$

Specify the loci in the complex plane given by $$|z - 1| = a|z + 1| + b,$$ when \((a, b)\) take the values \((1, 0)\), \((1, 1)\), \((0, 3)\), \((3, 0)\), \((1, 3)\).

If the substitutions \(x = \frac{1}{2}(u^2 - v^2)\), \(y = uv\) transform \(f(x, y)\) into \(F(u, v)\), show that $$u^2 \frac{\partial F}{\partial v} - v \frac{\partial F}{\partial u} = 2\left(x \frac{\partial f}{\partial y} - y \frac{\partial f}{\partial x}\right)$$ and that $$\frac{\partial^2 F}{\partial u^2} + \frac{\partial^2 F}{\partial v^2} = (u^2 + v^2)\left(\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}\right).$$

The sides \(a\), \(b\), \(c\) of a triangle are measured with a possible small percentage error \(\epsilon\) and the area is calculated. Prove that the possible percentage error in the area is approximately \(2\epsilon\) or \(2\cot B \cot C\) according as the triangle is acute-angled or obtuse-angled at \(A\).

Obtain indefinite integrals of the functions

1. [(i)] \(\frac{x^2}{1-x}\),
2. [(ii)] \(\frac{\tan^3 x}{1 - \tan x}\),

with respect to \(x\). The following ranges of integration are proposed for the two functions respectively

1. [(i)] \((-2, 0)\), \((0, 2)\);
2. [(ii)] \((0, \frac{1}{4}\pi)\), \((\frac{1}{4}\pi, \frac{1}{3}\pi)\), \((\frac{2}{3}\pi, \pi)\).

State in each of the five cases whether the integration is possible, justifying your answers and giving values when they exist.

Prove (using tables if you wish) that $$1 < \int_0^{1\pi} \sqrt{\sin x} \, dx < \sqrt{1 \cdot 25}.$$

Prove that a loop of the curve \(r = 2a\cos k\theta\) (\(k > 1\)) has the same area and perimeter as an ellipse with semi-axes \(a\) and \(a/k\). If \(k = 1 + \epsilon\), where \(\epsilon\) is small, obtain an approximate expression for the perimeter of the loop in powers of \(\epsilon\) including the term in \(\epsilon^2\).

It is given that the equation $$x^2(1-x) \frac{d^2y}{dx^2} + Py = 0,$$ where \(P\), \(Q\) are functions of \(x\), is satisfied by \(y = x^2\) and \(y = x^3\). Find \(P\), \(Q\). With these values of \(P\), \(Q\), what condition must be satisfied by the numerical coefficients \(A\), \(B\) if the equation is satisfied by $$y = Ax^2 + Bx^3$$ for all values of \(x\)?

James. \(\pi\) is the most important constant in mathematics. John. No, \(e\) is. Continue the discussion.