Problems

Find the real roots of the equations

1. \(x^3 - 15x + 30 = 0\);
2. \(xy(x+y) = 12x+3y\), \\ \(xy(4x+y-xy) = 12(x+y-3)\).

Sum to infinity the series

1. \(1 - \frac{3}{4} + \frac{3 \cdot 5}{4 \cdot 8} - \frac{3 \cdot 5 \cdot 7}{4 \cdot 8 \cdot 12} + \dots\),
2. \(\frac{x}{1 \cdot 2} + \frac{x^2}{2 \cdot 3} + \frac{x^3}{3 \cdot 4} + \frac{x^4}{4 \cdot 5} + \dots\),

and state the condition of convergence of the second series.

(i) Prove that all the roots of the equation \[ x^4 - 14x^2 + 24x = k \] are real if \(8 < k < 11\). (ii) Prove that if the product of two roots of the equation \[ x^4 - ax^3 + bx^2 - cx + d = 0 \] is equal to the product of the other two, \[ a^2 d = c^2. \]

Prove that there are 462 ways in which 12 similar coins can be distributed among 6 different persons, so that every person gets one coin at least.

Find a general formula for all the positive integers which, when divided by 5, 6, 7, will leave remainders 1, 2, 3 respectively, and show that 206 is the least of them.

(i) Find all the real roots of the equation \[ \tan^2 x + \tan^2 2x = 10. \] (ii) Eliminate \(\theta\) from the equations \begin{align*} x(1+\sin^2\theta - \cos\theta) - y\sin\theta(1+\cos\theta) &= c(1+\cos\theta), \\ y(1+\cos^2\theta) - x\sin\theta\cos\theta &= c\sin\theta. \end{align*}

For a triangle \(ABC\), \(R\) is the radius of the circumscribed circle, and \(r_1\) the radius of the escribed circle that touches \(BC\). If a circle is drawn to touch the circumscribed circle, and to touch also the sides \(AB, AC\) produced, prove that its radius \(\rho = r_1 \sec^2\frac{A}{2}\). If \(AB=AC\), and \(\rho=R\), prove that \(\sin\frac{A}{2} = \frac{1}{3}\).

\(ABC\) is an acute angled triangle, \(D,E,F\) are the middle points of the sides \(BC, CA, AB\) respectively, and \(O\) is the circumcentre. On \(OE, OF\) produced points \(Q,R\) respectively are taken so that the angles \(CQA, ARB\) are supplementary. Prove that \(DQ, DR\) are perpendicular.

A circle passing through the foci of a hyperbola cuts one asymptote in \(Q\) and the other in \(Q'\). Show that \(QQ'\) either touches the hyperbola or is parallel to the major axis.

Explain what is meant by the statement that a curve \(U\) is the polar reciprocal of a second curve \(V\) with respect to a conic \(S\). Prove that \(V\) is then the polar reciprocal of \(U\). Reciprocate the theorem that rectangular hyperbolas which pass through three fixed points pass also through a fourth fixed point, (a) when \(S\) is unrestricted, (b) when \(S\) is a circle.