Problems

Prove the formulae (i) \(4AR=abc\), (ii) \(16Q^2R^2 = (\alpha\beta+\gamma\delta)(\alpha\gamma+\delta\beta)(\alpha\delta+\beta\gamma)\), when \(R\) is the radius of a circle, \(a,b,c\) and \(\alpha, \beta, \gamma, \delta\) the sides in order of a triangle and quadrilateral inscribed in the circle, and \(A\) and \(Q\) the areas of the triangle and quadrilateral respectively.

Prove by induction or otherwise that if \(zx=1\), \[ x^n \frac{d^ny}{(n-1)!\,dx^n} = (-)^n \sum_{r=1}^n \frac{n!}{r!\,(n-r)!\,(r-1)!} z^r \frac{d^ry}{dz^r}. \]

The pressure \(p\), volume \(v\), temperature \(T\), and energy \(u\) of a substance are connected by two relations, so that each can be expressed as a function of any two of the others. Prove that \[ \left(\frac{\partial u}{\partial T}\right)_v + \left(\frac{\partial u}{\partial v}\right)_T \left(\frac{\partial v}{\partial T}\right)_p = \left(\frac{\partial u}{\partial T}\right)_p, \] when the suffix denotes the variable kept constant.

Find the values of \[ \int \frac{x^2\,dx}{\sqrt{1-x^2}}, \quad \int \frac{x^3\,dx}{\sqrt{1-x^2}}, \quad \int \frac{dx}{(1+e\cos x)^2}. \]

A continuous function \(\phi(x)\) is such that \[ \phi(x) = 2\int_0^1 (x+y)\phi(y)\,dy. \] Show that \(\phi(x)=0\). Solve the equation \[ u(x) = x^2 + \int_0^1 xyu(y)\,dy. \]

Prove that a continuous function attains its upper bound in an interval. Discuss the continuity of the function \[ \log(1+x)\sin\frac{1}{x} \] in the interval \((0,1)\).

Defining an ellipse as the locus of a point \(P\) which moves so that the sum of its distances from two fixed points \(S, H\) is a constant \(2a\), prove that the tangent at \(P\) to the ellipse is equally inclined to \(SP, HP\). Points \(A, B\) are taken on the tangent at \(P\) such that the lengths \(AP, BP\) are respectively equal to the lengths \(SP, HP\) and such that the angles \(APS, BPH\) are acute. The lines \(AS, BH\) when produced intersect in \(C\), and \(O\) is the circumcentre of the triangle \(SCH\). Shew that the points \(O, S, H, P\) are concyclic, and that the radius of the circle on which they lie is equal to the circumradius of the triangle \(SCH\) provided the area of the triangle \(ABC\) is \(a^2\sqrt{3}\) and the length \(SH\) lies between \(a\) and \(2a\).

Prove that a circle can be projected orthogonally into an ellipse, and give examples of properties of an ellipse derived by this method from corresponding properties of a circle.

A, B, C, D are the vertices of a tetrahedron in which the straight line joining \(A\) to the orthocentre of the triangle \(BCD\) is perpendicular to the plane \(BCD\). Prove that \[ AB^2+CD^2=AC^2+BD^2=AD^2+BC^2. \] Determine whether the converse of this theorem is also true.

Obtain the condition that the equation \[ ax^2+2hxy+by^2+2gx+2fy+c=0 \] shall represent some pair of straight lines \(l_1, l_2\). If also the equation \(a_1x^2+2h_1xy+b_1y^2+2g_1x+2f_1y+c_1=0\) represents a pair of straight lines \(l_2, l_3\) (so that \(l_2\) is a line common to the two pairs), shew that \begin{align*} (a_1c-ac_1)^2 &= 4(g_1c-gc_1)(a_1g-ag_1), \\ \text{and} \quad (b_1c-bc_1)^2 &= 4(f_1c-fc_1)(b_1f-bf_1). \end{align*} Shew further that the coordinates of the point of intersection of \(l_1\) and \(l_3\) are \[ \left( \frac{(bc_1-cb_1)(ca_1-ac_1)}{2(ab_1-ba_1)(cg_1-gc_1)}, \frac{(bc_1-cb_1)(ca_1-ac_1)}{2(ab_1-ba_1)(fc_1-cf_1)} \right). \]