Let \[ \Delta = \begin{vmatrix} x^2-y^2-z^2-t^2 & -2xy & 2xz & -2xt \\ 2xy & x^2-y^2-z^2-t^2 & 2xt & 2xz \\ -2xz & 2xt & x^2-y^2-z^2-t^2 & -2xy \\ 2xt & -2xz & 2xy & x^2-y^2-z^2-t^2 \end{vmatrix}. \] By expressing \(\Delta\) as the square of another determinant \(D\), and forming the square of \(D\) in a different way, or otherwise, prove that \(\Delta = (x^2+y^2+z^2+t^2)^4\).
If \(p, q\) and \(x\) are integers, and \(4q-p^2\) is a perfect square, prove that \(p\) is even and that \(y=x^2+px+q\) can be expressed as the sum of two perfect squares. Prove also that if \(p^2-4q\) is a perfect square and \(p\) is even, then \(y\) can be expressed as the difference of two perfect squares. What happens in this case when \(p\) is odd?
Prove that, if \(y=\tan^{-1}x\), then \[ u = \frac{d^n y}{dx^n} = (n-1)! \cos^n y \cos\left[ny+\frac{1}{2}(n-1)\pi\right] \] for every positive integer \(n\). Deduce, or prove otherwise, that \(u\) satisfies the differential equation \[ (1+x^2)\frac{d^2u}{dx^2} + 2(n+1)x\frac{du}{dx} + n(n+1)u=0. \]
Let \(y^2=f(x)\) be the equation of a curve symmetrical about the \(x\)-axis. Corresponding to each point \(P(x_0, y_0)\) of the curve, we construct the circle of centre \((x_0,0)\) and radius \(|y_0|\). Show that, if this family of circles has an envelope, it can be represented by the equations \begin{align*} x &= t - \frac{1}{2}f'(t), \\ y^2 &= f(t) - \frac{1}{4}[f'(t)]^2. \end{align*} If the circle of the family corresponding to a point \(P\) of the given curve touches the envelope at a point \(Q\), prove that the subnormal of the given curve at \(P\) is equal to the subnormal of the envelope at \(Q\). [It may be assumed that the function \(f(x)\) has finite derivatives of any required order. The subnormal at a point \(P\) of a curve is the segment of the \(x\)-axis cut off by the ordinate of \(P\) and the normal to the curve at \(P\).]
Prove that, if an equation of the second degree (with real coefficients) \[ S=ax^2+2hxy+by^2+2gx+2fy+c=0 \] represents two straight lines, then the value of the determinant \[ \Delta = \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix} \] is zero. Prove conversely that, if \(\Delta=0\), then the equation \(S=0\) does represent two straight lines (possibly ``coincident''). Prove that the point common to the two lines, assumed distinct and not parallel, can be expressed in any of the equivalent alternative forms \((A/G, H/G), (H/F, B/F), (G/C, F/C)\), where \[ A=bc-f^2, \quad F=gh-af, \quad \text{etc.} \] Deduce, or find otherwise, conditions for the lines to ``coincide.'' Prove that, if the lines are real, distinct and not parallel, then \(A, B, C\) are non-positive and at least one of them is negative. Determine conversely whether these conditions (or a lesser number selected from them) ensure that the equation \(S=0\) represents two distinct real lines when \(\Delta\) is zero.
Prove that the equation of a straight line may be expressed, in terms of rectangular Cartesian coordinates, in the form \[ x \cos\alpha + y \sin\alpha - p = 0. \] Find also the length of the perpendicular from the point \((x_1, y_1)\) to the line. The three lines \[ l_i = x \cos\alpha_i + y \sin\alpha_i - p_i = 0 \quad (i=1,2,3) \] form a triangle \(ABC\). Prove that the coordinates of the incentre and the three centres of the escribed circles can be found from the equations \[ l_1 = \pm l_2 = \pm l_3. \] Prove also that the equation of any conic through these four centres can be expressed in the form \[ \lambda l_1^2 + \mu l_2^2 + \nu l_3^2 = 0, \] where \(\lambda, \mu, \nu\) are constants such that \[ \lambda+\mu+\nu=0. \] Deduce, or prove otherwise, that the equation of the circumcircle of the triangle \(ABC\) is \[ l_2 l_3 \sin(\alpha_2-\alpha_3) + l_3 l_1 \sin(\alpha_3-\alpha_1) + l_1 l_2 \sin(\alpha_1-\alpha_2)=0. \]
A uniform rod is placed with one end on a rough horizontal plane and the other end against a rough vertical face of a box standing on the same horizontal plane. The vertical plane containing the rod is perpendicular to the vertical face of the box against which the rod rests. A string is attached to the mid-point of the rod and is pulled vertically downwards by a gradually increasing force. Find under what conditions the box will topple over.
Four rods, jointed at their extremities, form a quadrilateral \(ABCD\). Points \(E, F\) on \(AB, BC\) respectively and points \(G, H\) on \(BC, CD\) respectively are joined by taut strings with tensions \(T, T'\) respectively. \(K\) is a point on \(EF\) produced, and \(E'F'K\) cuts \(AB, BC\) at \(E', F'\) respectively. By transferring the tension \(T\) acting at \(F\) on \(BC\) to \(K\) and then resolving it along \(KF'\) and \(KB\), and then transferring the component in \(KF'\) to act at \(F'\), show that the taut string \(EF\) may be replaced by a taut string \(E'F'\) without altering any reaction except that at \(B\). Deduce that we may transfer the strings \(EF, GH\), while still attached to the same rods, until they coincide with the diagonals without altering any reactions except those at \(B\) and at \(C\). Prove that when the strings are along the diagonals \(AC, BD\) then the tensions \(U, U'\) in them satisfy the equation \[ U\left(\frac{1}{AO}+\frac{1}{OC}\right) = U'\left(\frac{1}{BO}+\frac{1}{OD}\right), \] where \(O\) is the point of intersection of the diagonals. Deduce that the tensions \(T, T'\) in the strings \(EF, GH\) satisfy the equations \[ \lambda T=U, \quad \lambda' T'=U', \] where \(\lambda\) is the ratio of the perpendiculars from \(B\) on to \(EF\) and \(AC\), and \(\lambda'\) is the ratio of the perpendiculars from \(C\) on to \(GH, BD\).
A particle of unit mass is attached to one end \(A\) of an elastic thread of natural length \(l\) and modulus \(\lambda n^2\), in a medium the resistance of which to the motion of the particle is \(2k\) times the speed. The other end \(B\) of the thread is fixed and the particle is held at a distance \(h\) below the fixed point. Show that when the particle is released its motion is given by the equation \[ \ddot{x}+2k\dot{x}+n^2x=0, \] as long as the string does not become slack, where \(x\) is the displacement of the particle from the equilibrium position. Find the subsequent motion in the cases: \[ \text{(i) } n^2 < k^2, \quad \text{(ii) } n^2=k^2, \quad \text{(iii) } n^2 > k^2, \] and discuss their physical significance. Develop in the same manner the case when the end \(B\) is forced to execute simple harmonic motion of period \(\dfrac{2\pi}{p}\).
\(OX, OY\) are fixed lines at right angles to each other; \(OX_t, OY_t\) are lines at right angles to each other which rotate about the fixed point \(O\). At time \(t\) the angle \(XOX_t\) is \(\theta\). A moving point \(P\) has coordinates \((\xi, \eta)\) referred to \(OX_t, OY_t\) as axes. Show that the absolute velocity of \(P\) at time \(t\) has components: \begin{align*} \dot{\xi}\cos\theta - \dot{\eta}\sin\theta - \xi\sin\theta.\dot{\theta} - \eta\cos\theta.\dot{\theta} & \quad \text{along } OX, \\ \dot{\xi}\sin\theta + \dot{\eta}\cos\theta + \xi\cos\theta.\dot{\theta} - \eta\sin\theta.\dot{\theta} & \quad \text{along } OY. \end{align*} Deduce that if at time \(t\) the angle \(\theta\) is zero then the absolute acceleration of \(P\) at time \(t\) has components: \begin{align*} \ddot{\xi} - 2\dot{\eta}\dot{\theta} - \xi\dot{\theta}^2 - \eta\ddot{\theta} & \quad \text{along } OX_t, \\ \ddot{\eta} + 2\dot{\xi}\dot{\theta} - \eta\dot{\theta}^2 + \xi\ddot{\theta} & \quad \text{along } OY_t. \end{align*} Deduce that the components of acceleration of \(P\) along and perpendicular to \(OP\) are \(\ddot{r}-r\dot{\theta}^2, 2\dot{r}\dot{\theta}+r\ddot{\theta}\) respectively, where \(r\) is the length of \(OP\) and \(\theta\) is the angle \(XOP\).