Problems

Find the rationalized form of \(x^{1/r}+y^{1/r}+z^{1/r}=0\) in the cases \(r=3\) and \(4\).

If \((1+x+x^2)^n = 1+c_1x+c_2x^2+\dots+c_{2n}x^{2n}\), where \(n\) is a positive integer, prove that

1. \(c_n = 1-c_1^2+c_2^2-c_3^2+\dots+c_{2n}^2\),
2. \(c_r = {}_nC_r + \frac{r-1}{1!} {}_nC_{r-1} + \frac{(r-2)(r-3)}{2!} {}_nC_{r-2} + \dots\),

and find the last term, where \({}_nC_s\) is the coefficient of \(x^s\) in \((1+x)^n\).

Prove Wilson's theorem that if \(n\) is a prime number \(1+(n-1)!\) is divisible by \(n\). If \(n\) and \(n+2\) are both prime numbers, prove that \((n-2)\{(n-1)!\}-2\) is divisible by \(n(n+2)\).

If \(f'(x)\) is positive shew that \(f(x)\) is increasing. Prove that \(2x+x\cos x-3\sin x > 0\) if \(0 < x < \frac{\pi}{2}\).

If \(\{\cosh^{-1}(1+x)\}^2 = a_0+a_1x+a_2x^2+\dots\), find the values of \(a_0, a_1, \dots\) and shew that \[ (n+1)(2n+1)a_{n+1} + n^2 a_n = 0. \]

If \(x=r\cos\theta, y=r\sin\theta\), find \(\frac{\partial x}{\partial r}, \frac{\partial r}{\partial x}\) and interpret the results geometrically. \(R\), the radius of the circumscribed circle of a triangle \(ABC\), is expressed in terms of \(a, b\) and \(C\); find \(\frac{\partial R}{\partial a}\) and prove that \(\frac{\partial R}{\partial a} = R\cot A \cos B \text{cosec } C\).

Find the equation of the normal at any point of the curve given by \[ x/a = 3\cos t - 2\cos^3 t, \quad y/a = 3\sin t - 2\sin^3 t, \] and also find the equation of its evolute.

Integrate:

1. \(\int \frac{(x+1)dx}{(x+2)\sqrt{x^2+4}}\),
2. \(\int_0^{\pi/2} x^2\cos x dx\),
3. \(\int \frac{\sin x dx}{\sin(x-a)\sin(x-b)}\),
4. \(\int_0^{\pi} \frac{\sin^2 x dx}{a^2-2ab\cos x + b^2} \quad (a>b>0)\)

Find the areas of the curves

1. \(a^2(y-x)^2 = (a+x)^3(a-x)\),
2. \((2x^2+3y^2)^3 = a^2xy^4\).

Solve the equations:

1. \(x^4+1+(x+1)^4=2(x^2+x+1)^2\),
2. \(x\sqrt{1-y^2}-y\sqrt{1-x^2} = xy - \sqrt{1-x^2}.\sqrt{1-y^2} = \frac{1}{2}\).