A conic is inscribed in a triangle \(ABC\), and \(D\) is its point of contact with \(BC\). The tangent parallel to \(BC\) meets \(AB\) and \(AC\) in \(L\) and \(M\) respectively and touches the conic at \(N\). By considering the cross-ratios of the ranges produced by the four tangents \(BC, CA, AB\) and \(LM\) on \(BC\) and \(LM\), prove that \(BC:DC = LM:LN\). Hence, or otherwise, prove that if an ellipse inscribed in a triangle has its centre at the circumcentre then the three altitudes are normals to it.
Show that the polar equation of a conic referred to a focus as pole may be put in the form \[ l/r = 1+e\cos\theta, \] and find the equation of the tangent at the point \(\theta=\alpha\). A conic has foci \(S\) and \(H\), and the tangents from a general point \(R\) touch the conic at \(P\) and \(Q\). Prove that \(RS\) and \(RH\) bisect the angles \(PSQ\) and \(PHQ\) respectively.
Find the condition, or conditions, that the general equation of the second degree \[ ax^2+2hxy+by^2+2gx+2fy+c=0 \] should represent (i) a parabola, (ii) a pair of straight lines, and (iii) a pair of coincident straight lines. If the coordinate axes are rectangular, find also the condition for the equation to represent a rectangular hyperbola, and use the condition to prove the theorem that if a rectangular hyperbola circumscribes a triangle it passes through the orthocentre.
Show by comparison with the identity \(4\cos^3\alpha - 3\cos\alpha - \cos 3\alpha = 0\) that the cubic equation \(x^3-3qx-r=0\) can be solved in terms of cosines provided that \(4q^3 > r^2\). If \(\alpha\) is defined by the equation \(\cos 3\alpha = r/2q^{3/2}\), show that \(2q^{1/2}\cos\alpha\) is a root, and find the other two roots. Use the method to solve the equation \[ x^3 - 6x^2 + 6x + 8 = 0. \]
If two triangles \(ABC\) and \(A_1B_1C_1\) are of equal area, prove that \[ \sum_{a,b,c} a^2 \cot A_1 = \sum_{a_1,b_1,c_1} a_1^2 \cot A. \]
(a) Find the limit, as \(x\) tends to zero, of (i) \((b^x - a^x)/x\) where \(a\) and \(b\) are positive; (ii) \(x \sin x / \log \cos x\). (b) If \(n\) is a positive integer, show that \[ \left(1+\frac{1}{n}\right)^n \le 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \dots + \frac{1}{n!} < 3. \]
Prove that the equation \(x^5+5x+3=0\) has only one real root. Calculate this root correct to 3 decimal places.
Find the indefinite integrals
The co-ordinates \((x, y)\) of a point on a simple closed plane curve are expressed in terms of a parameter \(t\). Show that the area enclosed by the curve is given by \[ \frac{1}{2} \int \left(x \frac{dy}{dt} - y \frac{dx}{dt}\right) dt \] taken between suitable limits. Sketch the curve \(x=a(\cos^3 t + \sin^3 t)\); \(y=a(\sin^3 t - \cos^3 t)\), and find its area.
A family of curves is given by the equation \(y = \cos x + \lambda \cos 3x\), where \(\lambda\) is a positive parameter. Examine the maximum and minimum points of these curves, showing that they fall into two classes, (a) those whose \(x\)-coordinate depends on \(\lambda\), and (b) those whose \(x\)-coordinate is independent of \(\lambda\). Show that the points in class (a) lie on the curve \[ 3(3-4\sin^2 x)(\cos x - y) = \cos 3x \] for all relevant values of \(\lambda\).