Two gear-wheels are mounted on parallel axles. Their radii are \(a\) and \(2a\), and their moments of inertia about their axles are respectively \(I\) and \(16I\). The smaller wheel is at rest, and the larger is rotating freely with angular velocity \(\omega\). The spindles of the wheels are moved parallel to their lengths to make the wheels engage. Prove that their angular velocities become \(-\frac{4}{5}\omega\) and \(\frac{2}{5}\omega\).
If \(x+y+z+t=0\), prove that
If \(a, b, c\) are unequal non-zero numbers, solve the simultaneous equations \begin{align*} x+y+z &= a+b+c, \\ \frac{x}{a} + \frac{y}{b} + \frac{z}{c} &= 1, \\ \frac{x}{a^3} + \frac{y}{b^3} + \frac{z}{c^3} &= 0, \end{align*} distinguishing the various cases that may arise.
Prove that, if the roots of the equation \[ x^n - \binom{n}{1}p_1 x^{n-1} + \dots + (-)^r \binom{n}{r} p_r x^{n-r} + \dots + (-)^n p_n = 0, \] where \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\) and \(p_n \ne 0\), are all real and positive, so are the roots of the equation \[ x^{n-1} - \binom{n-1}{1}p_1 x^{n-2} + \dots + (-)^r \binom{n-1}{r} p_r x^{n-r-1} + \dots + (-)^{n-1} p_{n-1} = 0. \] Deduce that \[ p_{r-1}p_{r+1} < p_r^2 \quad (1< r< n; p_0=1), \] stating when equality occurs. Prove also that \[ p_1 \ge p_2^{1/2} \ge p_3^{1/3} \ge \dots \ge p_n^{1/n}. \]
Find the equation whose roots are less by 2 than the squares of the roots of \[ x^3+qx+r=0. \] Examine the particular case \[ x^3-3x+1=0, \] and interpret the result.
\(P\) is a point inside a triangle \(ABC\), at distances \(a', b', c'\) from \(A, B, C\) respectively; the angles \(BPC, CPA, APB\) are \(A', B', C'\). By producing \(AP, BP, CP\) to meet the circumcircle of the triangle again, or otherwise, prove that \[ \frac{aa'}{\sin(A'-A)} = \frac{bb'}{\sin(B'-B)} = \frac{cc'}{\sin(C'-C)}. \]
Prove that \(\tan^2(\pi/11)\) is a root of the equation \[ t^5 - 55t^4 + 330t^3 - 462t^2 + 165t - 11 = 0. \] What are the other roots? Hence, by expressing \[ u = \tan(3\pi/11) + 4\sin(2\pi/11) \] as a rational function of \(\tan(\pi/11)\), or otherwise, prove that \(u\) is equal to \(\sqrt{11}\).
If \(y^2 = ax^2+2bx+c\), prove that \[ y^3 \frac{d^2y}{dx^2} = ac-b^2. \] Prove that, if \(n\) is a positive integer, \[ y^{2n+1} \frac{d^{2n}}{dx^{2n}}(y^{2n-1}) = 1^2 \cdot 3^2 \cdot 5^2 \dots (2n-1)^2 (ac-b^2)^n. \]
Discuss the maxima and minima of the function \[ \sin mx \csc x, \] where \(m\) is a positive integer, finding how many there are between \(-\frac{1}{2}\pi\) and \(\frac{1}{2}\pi\). Deduce that \[ \sin^2 mx \le m^2 \sin^2 x. \]
Integrate \[ \frac{1}{(6x^2-7x+2)\sqrt{(x^2+x+1)}}, \quad \frac{1}{(a+b\tan\theta)^2}. \] Prove that, if \(m\) is a number not less than 2, and \(n\) is a positive integer, \[ \int_0^{\frac{1}{2}\pi} \sin^m x \cos 2nx dx = k_{m,n} \int_0^{\frac{1}{2}\pi} \sin^{m-2}x \cos 2nx dx, \] evaluating the number \(k_{m,n}\). Find the value of \[ \int_0^{\frac{1}{2}\pi} \sin^8 x \cos 2nx dx. \]