Problems

Find from first principles the differential coefficient of \(\cos^{-1}x\). Find the \(n\)th differential coefficients of \(1/(1+x^2)\) and \(\sin^3 x\).

The radius \(R\) of the circumcircle of the triangle \(ABC\) is expressed in terms of \(a,b\) and \(C\); find \(\frac{\partial R}{\partial a}\) and prove that \[ \frac{\partial R}{\partial C} = R \frac{\cos A \cos B}{\sin C}. \]

Prove that if \(\phi\) is the angle between the radius vector and the tangent at any point of a curve \(\tan\phi = r \frac{d\theta}{dr}\). Find \(\phi\) at the point \((r, \theta)\) on the curve \(r^2=a^2\cos 2\theta\) and prove that if \(p\) is the perpendicular from the origin on the tangent at the point \(pa^2=r^3\). Find also the equation of the first positive pedal.

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.