Problems

Prove that the value of a determinant is unaltered by adding to each element of one column the same multiple of the corresponding element of another column. Evaluate \[ \begin{vmatrix} 12 & 34 & 56 \\ 14 & 26 & 35 \\ 16 & 18 & 14 \end{vmatrix}. \]

Prove that if \(A\) and \(B\) are acute angles while \(A+B\) is obtuse, \[ \cos(A+B) = \cos A\cos B - \sin A\sin B. \] Prove that \[ \sin(\alpha+\theta)\cos^3(\alpha-\theta) - \sin(\alpha-\theta)\cos^3(\alpha+\theta) = \sin 2\theta\{1+\cos 2\alpha\sin(\alpha+\theta)\sin(\alpha-\theta)\}. \]

If \(a, b, c\) are the sides and \(A, B, C\) the angles of a triangle prove, ab initio,

1. \((a+b)\tan\frac{1}{2}(A-B) = (a-b)\cot\frac{1}{2}C\),
2. \((a+b)(1-\cos C) = c(\cos A+\cos B)\).

Find an expression for all the angles which have the same cosine as a given angle. Prove \[ \cos n\phi - \cos n\theta = 2^{n-1} \prod_0^{n-1} \left\{\cos\phi - \cos\left(\theta+\frac{2r\pi}{n}\right)\right\}. \] Deduce \[ \cos\alpha = 2^{n-1} \prod_0^{n-1} \sin \frac{2\alpha+(2r+1)\pi}{2n}. \]

Sum the series

1. \(\cos\alpha+\cos(\alpha+2\beta)+\cos(\alpha+4\beta)+\dots+\cos(\alpha+2\overline{n-1}\beta)\),
2. \(\sin\theta+\dfrac{1}{2!}\sin 2\theta+\dfrac{1}{3!}\sin 3\theta+\dots\) to infinity.

Prove that the equation \[ Ax^2+Ay^2+2Gx+2Fy+C=0 \] represents a circle. Find the coordinates of its centre, its radius and the length of the tangent to the circle from the point \(h, k\). Find the locus of the centre of a circle which cuts \[ x^2+y^2+2Gx+2Fy+C=0 \quad \text{and} \quad x^2+y^2+2gx+2fy+c=0 \] orthogonally.

Shew that the coordinates of any point on a hyperbola can be expressed as \(a\sec\theta, b\tan\theta\); and further that the normal at this point has for its equation \[ ax\cos\theta+by\cot\theta=a^2+b^2. \] Hence prove that four normals can be drawn from any point to a hyperbola, and that the sum of the angles corresponding to the feet of these normals is an odd multiple of \(180^\circ\).

Prove that the equation \[ l=r(1+e\cos\theta) \] represents a conic whose focus is the pole. Find the equation of the tangent at the point whose vectorial angle is \(\alpha\). Shew that the pole of a chord, which subtends a constant angle at the focus, lies on a conic having the same focus and directrix.

Prove that the straight lines \[ ax^2+2hxy+by^2=0 \] are conjugate diameters of the conic \[ Ax^2+2Hxy+By^2=1 \] if \[ Ab+Ba=2Hh. \] Hence shew that the asymptotes are the double rays of the involution formed by pairs of conjugate diameters of a conic.

Prove that the moment of the resultant of a system of forces, acting in one plane on a rigid body, about a point in the plane, is equal to the algebraic sum of the moments of the forces. A circular tray of radius \(a\) stands on a single circular foot of radius \(b\). If \(w\) is the whole weight of the tray and its support, find how far from the centre a weight \(W\) can be placed without the tray falling over.