(i) Prove that for real values of the \(a\)'s and \(b\)'s \[ (a_1^2+a_2^2+\dots+a_n^2)(b_1^2+b_2^2+\dots+b_n^2) \ge (a_1b_1+a_2b_2+\dots+a_nb_n)^2. \] (ii) Without assuming the cosine series, prove (by differentiation or otherwise) that for \(-1 \le x \le +1\) \[ 1 - \frac{x^2}{2!} + \frac{x^4}{4!} \ge \cos x \ge 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!}. \]
(i) Find the sum of the infinite series \[ 1 + \frac{2^2}{1!} + \frac{3^2}{2!} + \frac{4^2}{3!} + \dots. \] (ii) Find the sum of \(n\) terms of the series \[ \frac{1}{3.7.11} + \frac{5}{7.11.15} + \frac{9}{11.15.19} + \dots. \]
Shew that if \(l_1, m_1, n_1; l_2, m_2, n_2; l_3, m_3, n_3\) are real quantities satisfying relations \begin{align*} l_r^2+m_r^2+n_r^2 &= 1 \quad (r=1,2,3), \\ l_p l_q + m_p m_q + n_p n_q &= 0 \quad (\{\begin{smallmatrix}p\\q\end{smallmatrix}\}=1,2,3; p \ne q), \end{align*} then \[ \Sigma l_r^2 = \Sigma m_r^2 = \Sigma n_r^2 = 1; \quad \Sigma l_r m_r = \Sigma m_r n_r = \Sigma n_r l_r = 0, \] and \[ \begin{vmatrix} l_1 & m_1 & n_1 \\ l_2 & m_2 & n_2 \\ l_3 & m_3 & n_3 \end{vmatrix} = \pm 1. \]
If \(f(x)\) is an algebraic function, shew that between two consecutive real roots of the equation \(f'(x)=0\) there can at most be only one real root of \(f(x)=0\). Prove that the necessary and sufficient condition for the reality of all three roots of the cubic \(x^3+3px+q=0\) is that \(4p^3+q^2<0\). Discuss the case \(4p^3+q^2=0\). Shew that \(x^5+3x^2+x+2=0\) has only one real root, and locate it between two consecutive integers.
If \(r, R\) denote the radii of the inscribed and circumscribed circles of triangle \(ABC\), the centres of the circles being \(I\) and \(O\) respectively, shew that \[ r = R(\cos A + \cos B + \cos C - 1), \] \[ OI^2 = R^2\left(1-8\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}\right). \]
A chord is drawn to cut a circle of radius \(a\) so that the smaller segment is one-sixth of the total area. Shew that if the distance \(p\) of the chord from the centre be \(a\cos\theta\), \(\theta\) is given in radians by \[ \sin\theta\cos\theta - \theta + \frac{\pi}{6} = 0. \] Prove that this equation has a root between \(\frac{\pi}{4}\) and \(\frac{\pi}{3}\) and (by simple interpolation or otherwise) deduce an approximate value for \(p\).
Integrate: \[ \int_{-1}^1 \frac{x+1}{(x+3)(\sqrt{x+2})} dx, \quad \int_0^1 \frac{x^3+4x^2+x-1}{(x^2+1)(x+1)^2} dx, \quad \int \sin^{-1}x.dx. \] By reduction formula or otherwise, find an expression for \(\int x^n e^x.dx\).
Trace the curve \(x^4+ax^2y-ay^3=0\), determining the turning points. Using polar coordinates or otherwise, calculate the area of a loop of the curve \(\left[\frac{4}{105}a^2\right]\).
Prove the formula \(\rho=r\frac{dr}{dp}\) for the radius of curvature of a curve given in terms of \(p\) the perpendicular on to the tangent from the origin, and \(r\) the radius vector. Obtain the \((p,r)\) equation of a parabola referred to its focus as origin in the form \(a.r=p^2\), and deduce that if the parabola is made to roll without slipping on a fixed straight line, its focus describes a curve whose radius of curvature is equal to the focal radius of the parabola at the corresponding instantaneous point of contact.
If \(X, Y, x, y\) are real quantities connected by the complex relation \[ Z = X+iY = f(x+iy) = f(z), \] where \(i=\sqrt{-1}\), shew that \[ J = \frac{\partial X}{\partial x}\frac{\partial Y}{\partial y} - \frac{\partial X}{\partial y}\frac{\partial Y}{\partial x} = (|f'(z)|)^2. \] Prove that if \(V\) is a function of \(x,y\), then \[ J\left(\frac{\partial^2V}{\partial X^2} + \frac{\partial^2V}{\partial Y^2}\right) = \frac{\partial^2V}{\partial x^2} + \frac{\partial^2V}{\partial y^2}. \]