A curve is given by the equation \[ ax+by+cx^2+dxy+ey^2=0. \] Find the values \(y', y''\) and \(y'''\) of \(\dfrac{dy}{dx}, \dfrac{d^2y}{dx^2}\) and \(\dfrac{d^3y}{dx^3}\) respectively, at the origin. Substitute \(y=y'x + \dfrac{y''x^2}{2} + \dfrac{y'''x^3}{6} + \eta x^3\) in the above equation and shew that \(\eta\) tends to zero as \(x\) tends to zero. Find the limit of \(\dfrac{\eta}{x}\) as \(x\) tends to zero.
Shew that there are three points of inflexion on the curve \[ y = \frac{x}{x^2+x+1}. \] Shew that these three points of inflexion lie on a line.
Give an account of the application of the differential calculus to the investigation of the maxima and minima of a function of a single variable, explaining how to distinguish between maxima and minima. Investigate completely the maxima and minima of the distance of a variable point \(P\) on an ellipse from a fixed point \(Q\) on the major axis.
Shew that \[ \lim_{x\to 0} \frac{\cos(\sin x) + \sin(1-\cos x) - 1}{x^4} = -\frac{1}{6}. \]
Integrate \[ x^2\sqrt{(1+x^2)}, \quad \frac{\cos^2 2x}{\sin^4 x \cos^2 x}, \quad x^m(\log x)^2. \]
(i) Evaluate \[ \int_0^\infty \frac{dx}{(x^2+a^2)(x^2+b^2)(x^2+c^2)}, \] where \(a, b\) and \(c\) are positive. \item[(ii)] Find a reduction formula for the integral \[ \int_0^{\frac{\pi}{4}} \frac{dx}{\cos^n x}, \] and evaluate the integral for the cases \(n=1, 2\).
\(P\) is any point on the circumcircle of a triangle \(ABC\). \(PL, PM, PN\) are drawn perpendicular to the sides of the triangle, produced if necessary. Prove that \(LMN\) is a straight line. If \(PQ\) is a diameter of the circle, and \(QF, QG, QH\) are drawn perpendicular to the sides of the triangle, prove that the straight lines \(FGH\) and \(LMN\) are at right angles.
\(PQ\) is any chord of a parabola. Any line parallel to the axis of the parabola meets \(PQ\) in \(E\), the curve in \(R\), and the tangent at \(P\) in \(F\). Prove that \(\dfrac{FR}{EP} = \dfrac{ER}{EQ}\).
The tangent to an ellipse at any point \(P\) meets a given tangent in \(T\). From a focus \(S\) a line is drawn perpendicular to \(ST\), meeting the tangent at \(P\) in \(Q\). Prove that the locus of \(Q\) is a straight line that touches the ellipse.
\(S\) and \(H\) are the foci of a hyperbola. The tangent at \(P\) meets an asymptote in \(T\). Prove that the angle between that asymptote and \(HP\) is twice the angle \(STP\).