Problems

Solve the equations:

1. [(i)] \(x+y=(1+xy)\sin\alpha, \quad x-y=(1-xy)\sin\beta\).
2. [(ii)] \(\frac{ax+by+cz}{x} = \frac{bx+cy+az}{y} = \frac{cx+ay+bz}{z}\).

If \(2s=a+b+c\), shew that \[ \begin{vmatrix} a^2 & (s-a)^2 & (s-a)^2 \\ (s-b)^2 & b^2 & (s-b)^2 \\ (s-c)^2 & (s-c)^2 & c^2 \end{vmatrix} = 2s^3(s-a)(s-b)(s-c). \]

If \(f(x) = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots+\frac{x^n}{n!}+\dots\) prove that \[ f(x) \times f(y) = f(x+y). \] Find the sum of the series \[ \sum_{n=1}^{n=\infty} \frac{n^4}{(n+1)!}. \]

Prove the law of formation of successive convergents to the continued fraction \[ \frac{1}{a_1+} \frac{1}{a_2+} \frac{1}{a_3+\dots}. \] Prove that the difference between the continued fraction and the \(n\)th convergent is less than \(1/q_nq_{n+1}\) and greater than \(a_{n+2}/q_nq_{n+2}\).

Prove that the number of prime numbers is infinite. Prove that \((2n+1)^5-2n-1\) is divisible by 240.

Draw the graphs of \(\cot x\) and \(e^x\sin x\). Find the tangents of the angles which satisfy the equation \[ (2n-1)\cos\theta + (n+2)\sin\theta = 2n+1. \]

Prove that

1. [(i)] \(1-\cos^2\alpha-\cos^2\beta-\cos^2\gamma+2\cos\alpha\cos\beta\cos\gamma = 4\sin\sigma\sin(\sigma-\alpha)\sin(\sigma-\beta)\sin(\sigma-\gamma)\), where \(2\sigma=\alpha+\beta+\gamma\).
2. [(ii)] \(\Sigma \sin 2\alpha \sin^2(\beta+\gamma) = \sin 2\alpha\sin 2\beta\sin 2\gamma+2\sin(\beta+\gamma)\sin(\gamma+\alpha)\sin(\alpha+\beta)\).

Prove that the distance between the orthocentre of a triangle \(ABC\) and the centre of the circumscribed circle is \(R^2(1-8\cos A\cos B\cos C)\), where \(R\) is the radius of the circumcircle. Prove that the sum of the squares of the distances of \(A, B, C\) from the centre of the nine-points circle is \(R^2\left(\frac{11}{4}+2\cos A\cos B\cos C\right)\).

Find all the values of \((\cos q\theta+i\sin q\theta)^{p/q}\). Sum the series to infinity \[ 1+\cos\theta\tan\theta + \frac{1}{2!}\cos 2\theta\tan^2\theta + \dots + \frac{1}{n!}\cos n\theta\tan^n\theta+\dots. \]

Find the \(n\) real quadratic factors of \(x^{2n+1}+1\), where \(n\) is a positive integer. Prove that \[ \frac{1}{(1+x)^5+(1-x)^5} = \frac{A_1}{x^2+\tan^2\frac{\pi}{10}} + \frac{A_2}{x^2+\tan^2\frac{3\pi}{10}}, \] and find the values of \(A_1\) and \(A_2\).