Problems

Define the curvature at a point of a curve and obtain its value when the equation of the curve is given by \(x=\phi(t), y=\psi(t)\). Prove that \[ \frac{1}{\rho^2} = \left(\frac{d^2x}{ds^2}\right)^2 + \left(\frac{d^2y}{ds^2}\right)^2. \]

Evaluate the integrals

1. [(i)] \(\int \frac{dx}{\sqrt{x^2-a^2}}\),
2. [(ii)] \(\int \cos^3 x \, dx\),
3. [(iii)] \(\int_0^\pi \frac{d\theta}{1+e\cos\theta} \quad (e<1)\).

Find a formula of reduction for \[ \int_0^{\pi/2} \cos^m\theta \sin^n\theta \, d\theta. \]

Trace the curve \(r=2+3\cos 2\theta\), and find the area of a loop.

Given two polynomials \(A\) and \(B\), with no common factor, show that it is always possible to find a pair of polynomials \(P\) and \(Q\), of degrees less than those of \(B\) and \(A\) respectively, such that \(AP+BQ=1\). If \(A=2x^3-4x^2+2x-3\), and \(B=x^2-2x+3\), find \(P\) and \(Q\).

Prove the Binomial Theorem for a positive integral exponent. If \(c_r\) is the coefficient of \(x^r\) in the expansion of \((1-x)^n\), prove that \[ 1+c_1+c_2+\dots+c_r = \frac{n+1}{n}\left(1-\frac{1}{c_r-c_{r+1}}\right), \] and that \[ 1+\frac{1}{c_1}+\frac{1}{c_2}+\dots+\frac{1}{c_r} = \frac{n+1}{n+2}\left\{1-\frac{1}{c_{r+1}-c_r}\right\}. \]

Solve the simultaneous equations \begin{align*} x^2 - yz &= a^2 \\ y^2 - zx &= b^2 \\ z^2 - xy &= c^2 \end{align*} If \(p\) is small compared with \(n\), show that an approximation to a root of the equation \(x^{n+p} = a^n\) is \[ x = a\left(1-\frac{p}{n}\log_e a\right), \] and continue the approximation as far as the term involving \(p^2\).

If \(n\) is a prime number, prove that \(n-1+1\) is divisible by \(n\). Prove that the number formed by adding one to the sum of the squares of three consecutive odd numbers is divisible by 12, but is not divisible by either 24 or 60.

Establish the law of formation of successive convergents to a continued fraction. Prove that the product of the first \(n\) convergents to the fraction \[ \frac{12}{7-} \frac{12}{7-} \frac{12}{7-} \dots \text{ is } \frac{12^n}{4^{n+1}-3^{n+1}}. \]

If \(\alpha, \beta, \gamma, \delta\) are the angles of a plane quadrilateral, prove that \[ \cos 2\alpha + \cos 2\beta + \cos 2\gamma + \cos 2\delta = 4 \cos(\alpha+\beta) \cos(\alpha+\gamma) \cos(\alpha+\delta). \] Prove that the real solutions of the equation \[ \tan^2 x \tan \frac{x}{2} = 1 \] satisfy the equation \[ \cos 2x = 2-\sqrt{5}. \]

Prove the formula \[ \cos(A+B) = \cos A \cos B - \sin A \sin B \] for all real values of \(A\) and \(B\). If \[ \tan(\theta-\alpha)+\tan(\theta-\beta) = \tan(\phi-\alpha)+\tan(\phi-\beta) = 2\tan\gamma, \] and \((\theta-\phi)\) is not a multiple of \(\pi\), prove that \[ \alpha+\beta+\gamma-\theta-\phi \] is an odd multiple of \(\frac{\pi}{2}\).