If \[ (B,C) = B_1C_2-B_2C_1, \text{ etc.,} \] show that \[ (B,C)(A,D)+(C,A)(B,D)+(A,B)(C,D) = 0. \] If there are four relations \(A_i a_j + B_i b_j + C_i c_j + D_i d_j = 0\) for \(i=1,2; j=1,2\), show that \[ \frac{(B,C)}{(a,d)} = \frac{(C,A)}{(b,d)} = \frac{(A,B)}{(c,d)} = \frac{(A,D)}{(b,c)} = \frac{(B,D)}{(c,a)} = \frac{(C,D)}{(a,b)}. \]
Show that if \(u_n>0\) and \(\frac{u_{n+1}}{u_n} < \rho < 1\), then \(\sum_{n=1}^\infty u_n\) is convergent.
Show that \(\sum n^p r^n\) and \(\sum n! r^{n^2}\) are convergent if \(0
If \[ \cos\theta_1+2\cos\theta_2+3\cos\theta_3=0 \] and \[ \sin\theta_1+2\sin\theta_2+3\sin\theta_3=0, \] then prove that \[ \cos(3\theta_1)+8\cos(3\theta_2)+27\cos(3\theta_3) = 18\cos(\theta_1+\theta_2+\theta_3) \] and \[ \cos(2\theta_1-\theta_2-\theta_3)+8\cos(2\theta_2-\theta_3-\theta_1)+27\cos(2\theta_3-\theta_1-\theta_2) = 18. \]
Solution: Let \(z_i = \cos \theta_i + i \sin \theta_i\), then \(z_1 + 2z_2 + 3z_3 = 0\) and so \(z_1^3+8z_2^3+27z_3^3-3\cdot2\cdot3 z_1z_2z_3 = 0\), so taking real parts we have: \begin{align*} && 0 &= \cos(3\theta_1) + 8\cos(3 \theta_2) +27\cos(3\theta_3)-18\cos(\theta_1+\theta_2+\theta_3) \end{align*} and dividing by \(z_1z_2z_3\) we have the other result
Find the values of \(x\) for which \((x-a)^l (x-b)^m (x-c)^n\) has maxima or minima. \(a, b, c\) are real and \(l, m, n\) are integers. Determine which of these values give maxima and which minima (i) when \(l, m, n\) are all even, (ii) when \(l, m, n\) are all odd.
Show that for any ellipse \[ \frac{x^2}{a^2+\lambda} + \frac{y^2}{b^2+\lambda} = 1, \text{ where } \lambda \text{ is a parameter,} \] \[ \left(x+y\frac{dy}{dx}\right)\left(x-y\frac{dx}{dy}\right) = a^2-b^2.\] Show that the system of curves cutting the above system at right angles satisfies the same differential equation.
Show how to find \(\int \frac{ax^2+2bx+c}{(Ax^2+2Bx+C)^2}dx\). Find the condition that it should be a rational function.
Find a formula of reduction for \(\int x^m (\log x)^n dx\) and evaluate the integral between the limits 0, 1 when \(m \ge 0\) and \(n\) is a positive integer. Hence or otherwise find a formula of reduction for \(\int \theta^n \cos m\theta.d\theta\).
Evaluate \(\int_0^1 t^{\alpha-1}(1-t)^\beta dt\), where \(\alpha>0\). If \(S\) be the area bounded by the curve \[ \left(\frac{x}{a}\right)^{2/p} + \left(\frac{y}{b}\right)^{2/q} = 1, \text{ where } p \text{ and } q \text{ are positive integers,} \] show that \[ 4ab-S < \frac{2ab}{p}.\]
If \(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{a+b+c}\), prove that for all integral values of \(n\) \[ \frac{1}{a^{2n+1}} + \frac{1}{b^{2n+1}} + \frac{1}{c^{2n+1}} = \frac{1}{(a+b+c)^{2n+1}}. \] If \(\frac{x}{p}+\frac{y}{q}=1\) and \(\frac{x^2}{p}+\frac{y^2}{q} = \frac{pq}{p+q}\), prove that for all positive integral values of \(n\) \[ \frac{x^{n+1}}{p} + \frac{y^{n+1}}{q} = \left(\frac{pq}{p+q}\right)^n. \]
If \(a, b, c, k\), and \(p\) are real quantities, find the necessary and sufficient conditions that \((ax^2+2bx+c)\) may be positive for all real values of \(x\). If these conditions are satisfied, prove that \[ ax^2+2bx+c > k(x-p)^2 \] for all real values of \(x\) if \[ k < \frac{ac-b^2}{ap^2+2bp+c}. \]