Define a determinant, and prove from your definition that if two rows or two columns of a determinant are equal the determinant vanishes. \par Find the factors of \[ \begin{vmatrix} a & b & c \\ b+c & c+a & a+b \\ a^3 - abc & b^3 - abc & c^3 - abc \end{vmatrix}. \]
Through a given point inside a parallelogram construct a straight line which shall divide the area of the parallelogram into two parts as unequal as possible.
Deduce from the triangle of forces that the resultant of two parallel forces is, in general, a third force whose moment about any point is the sum of the moments of the two forces about the point. \par A uniform straight rod of length \(3a\) is hinged freely at one end to a wall, and is supported in a horizontal position by a knife-edge distant \(2a\) from the wall. Find where the bending moment is greatest.
(i) Define an involution pencil and prove that the pairs of tangents from a fixed point to conics touching four fixed lines form an involution pencil. \par (ii) A fixed conic meets corresponding rays of an involution pencil, whose vertex O is not on the conic, in P, P' and Q, Q'. Prove that the lines PQ, P'Q', PQ', P'Q touch a fixed conic which also touches the double lines of the pencil.
The equation \(x^2 + ax + b = 0\) has real roots \(\alpha, \beta\). Form the quadratic equation with roots \(\alpha^2 - k^2, \beta^2 - k^2\), and show that \(\alpha, \beta\) are outside the interval \((-k, k)\) if \[ (b + k^2)^2 > k^2a^2 > 2k^2 (b + k^2). \] Find the conditions that the roots of the equation \[ x^4 + px^3 + qx^2 + px + 1 = 0 \] may be all real and unequal.
A, B, C, D, E are five points in space, no four lying in the same plane. From each of the five points, the three lines are drawn which meet the pairs of opposite edges of the tetrahedron formed by the remaining four points. Prove that of this set of fifteen lines, the three lines which meet AB all do so at the same point and are coplanar.
The crane ABCD is built up from freely hinged light rods, and is hinged to the horizontal ground at A and B. A weight W is suspended by a light chain passing over a small frictionless pulley at C to B. Find, by means of a force diagram or otherwise, the tensions or thrusts in the rods. \par [A diagram is shown with points A and B on a horizontal line. The structure ABCD is above this line. Angles are given as \(\angle DAB = 45^\circ\), \(\angle ADC = 45^\circ\), \(\angle CBA = 30^\circ\), and \(\angle BCD=30^\circ\). A weight W hangs from C.]
Prove that through a given point there are two conics confocal with a given ellipse, one being an ellipse and the other a hyperbola, and that they intersect at right angles. \par Find the locus of the pole of a given straight line with respect to a system of confocal conics. \par Prove that the tangents to a conic from a given point T make equal angles with the tangent at T to either of the confocal conics through T.
(i) Prove that an approximate solution of the equation \[ xe^{x-1} + x - 2 = \epsilon, \] where \(\epsilon\) is small, is \[ x = 1 + \tfrac{1}{3}\epsilon - \tfrac{1}{18}\epsilon^2. \] (ii) Assuming that \(\sin^{-1} x\) may be expanded in ascending powers of \(x\), find the first three non-zero terms in the expansion. The expansion of \(\sin x\) in powers of \(x\) may be assumed, if required.
A point A lies in the plane of a circle S and outside the circle. Find the loci of the centres of circles which pass through A and