Problems

Shew how by projection from a vertex, intersecting straight lines can be transformed into intersecting straight lines with the same angles between them as before. Prove that if two conics touch at \(A\) and \(B\), then the two polars of any point \(P\) with respect to the conics intersect on \(AB\).

Prove that the mid-point of a chord of the ellipse \((\frac{x^2}{a^2}+\frac{y^2}{b^2}=1)\), which is of fixed length \(l\), lies on the curve \[ 4c^2l^2 \left(\frac{x^2}{a^4}+\frac{y^2}{b^4}\right) \left(\frac{x^2}{a^2}+\frac{y^2}{b^2}-1\right) + l^2 \left(\frac{x^2}{a^2}+\frac{y^2}{b^2}\right) = 0. \]

Explain what is meant by the equation of a point in tangential coordinates. If \(P=l\alpha+m\beta-p=0\) is the equation of the point \((\alpha, \beta)\), and \(Q=0\) refers similarly to the point \((\alpha', \beta')\), and if \(\Sigma = Al^2+Bm^2+Cp^2+2Hlm+2Glp+2Fmp=0\) is a conic, interpret:

1. [(1)] \(P^2 - \lambda Q = 0\),
2. [(2)] \(\Sigma - \lambda P.Q = 0\),
3. [(3)] \(\Sigma - \lambda P^2 = 0\).

(i) Find the real roots of the equation \[ x^8+1+(x+1)^8 = 2(x^2+x+1)^4. \] (ii) Eliminate \(x, y, z\) from the equations: \[ \frac{y}{z} + \frac{z}{y} + \frac{x}{z} = a, \] \[ \frac{z}{x} + \frac{x}{z} + \frac{y}{x} = b, \] \[ \left(\frac{x}{y}+\frac{y}{z}\right)\left(\frac{y}{z}+\frac{z}{x}\right)\left(\frac{z}{x}+\frac{x}{y}\right) = c. \]

A sequence of terms \(u_0, u_1, u_2, \dots u_n, \dots\) is such that any three consecutive terms are connected by the relation \[ 6u_{n+1}-5u_n+u_{n-1}=0. \] If \(u_0=1, u_1=\frac{1}{2}\), find an expression for \(u_n\), and shew that the infinite series \(u_0+u_1+u_2+\dots\) converges to the sum unity.

(i) If \(\alpha, \beta, \gamma\) are the roots of the equation \(x^3+px+q=0\), find the equation whose roots are \[ \frac{1}{\alpha}+\frac{1}{\beta}, \frac{1}{\beta}+\frac{1}{\gamma}, \frac{1}{\gamma}+\frac{1}{\alpha}. \] (ii) Find the condition that the equation \(x^4+2px^3+2rx+pr=0\) has a pair of equal roots, and if the condition is satisfied, solve the equation completely.

(i) From the identity \(2\log(1-x) = \log(1-2x+x^2)\), or otherwise, prove that \[ 2^n - n2^{n-2} + \frac{n(n-3)}{1.2}2^{n-4} - \frac{n(n-4)(n-5)}{1.2.3}2^{n-6}+\dots=2. \] (ii) Sum to \(n\) terms the series \[ \frac{1}{a} + \frac{1.2}{a(a+1)} + \frac{1.2.3}{a(a+1)(a+2)} + \dots. \] If this series converges, find also the sum to infinity, and investigate the condition of convergence.

It is required to find two numbers, each of two digits, such that the first number is equal to the product of the digits of the second number, and is also less by 100 than twice the second number. Shew that there are two solutions, and find them.

(i) Solve the equation \[ \tan 3\theta = \tan\theta + \tan 2\theta. \] (ii) Eliminate \(\theta\) from the equations \[ \sin 3(\frac{1}{4}\pi+\theta)+3\sin(\frac{1}{4}\pi+\theta) = 2a, \] \[ \sin 3(\frac{1}{4}\pi-\theta)+3\sin(\frac{1}{4}\pi-\theta) = 2b. \]

The centre of three concentric circles is \(O\). \(ON\) is drawn perpendicular to a straight line which cuts the circles in three points \(A, B, C\) respectively, all lying on the same side of \(N\). Prove that the area of the triangle formed by the tangents at \(A, B, C\) is \[ \frac{BC.CA.AB}{2ON}. \]