Prove the identity \[ \begin{vmatrix} x_1 & y_1 & a & b \\ \lambda_1 x_2 & x_2 & y_2 & c \\ \lambda_1 \lambda_2 x_3 & \lambda_2 x_3 & x_3 & y_3 \\ \lambda_1 \lambda_2 \lambda_3 x_4 & \lambda_2 \lambda_3 x_4 & \lambda_3 x_4 & x_4 \end{vmatrix} = (x_1 - \lambda_1 y_1)(x_2 - \lambda_2 y_2)(x_3 - \lambda_3 y_3)x_4. \] Hence, or otherwise, prove that \[ \begin{vmatrix} a_1b_1 & a_1b_2 & a_1b_3 & a_1b_4 \\ a_1b_2 & a_2b_2 & a_2b_3 & a_2b_4 \\ a_1b_3 & a_2b_3 & a_3b_3 & a_3b_4 \\ a_1b_4 & a_2b_4 & a_3b_4 & a_4b_4 \end{vmatrix} = a_1 b_4 (a_2b_1 - a_1b_2)(a_3b_2-a_2b_3)(a_4b_3-a_3b_4), \] and evaluate \[ \begin{vmatrix} 1 & 1 & 1 & 1 \\ b_1 & a_1 & a_1 & a_1 \\ b_1 & b_2 & a_2 & a_2 \\ b_1 & b_2 & b_3 & a_3 \end{vmatrix}. \]
A man on a hill observes that three vertical towers standing on a horizontal plane subtend equal angles at his eye, and that the angles of depression of their bases are \(\alpha_1, \alpha_2, \alpha_3\); prove that, if \(c_1, c_2, c_3\) are the heights of the towers, \[ \frac{\sin(\alpha_2-\alpha_3)}{c_1\sin\alpha_1} + \frac{\sin(\alpha_3-\alpha_1)}{c_2\sin\alpha_2} + \frac{\sin(\alpha_1-\alpha_2)}{c_3\sin\alpha_3} = 0. \]
If the angles \(\theta_1, \theta_2, \dots, \theta_n\) all lie between \(0\) and \(\frac{1}{2}\pi\), and \(\theta_1+\theta_2+\dots+\theta_n=\alpha\), where \(\alpha\) is fixed, show that \[ \sin\theta_1+\sin\theta_2+\dots+\sin\theta_n \] attains its maximum value when all the angles \(\theta_r\) are equal. State and prove the corresponding result for \[ \tan\theta_1+\tan\theta_2+\dots+\tan\theta_n. \]
A leaf of a book is of width \(a\) in. and height \(b\) in., where \(3a \le 2\sqrt{2}b\); the lower corner of the leaf is folded over so that the corner just reaches the inner edge of the page. Find the minimum length of the resulting crease. Explain why the condition \(3a \le 2\sqrt{2}b\) has to be imposed.
Sketch roughly the curve \[ y^2(a^2+x^2) = x^2(a^2-x^2), \] and find the area of one of its loops.
Find the relation between \(p\) and \(\alpha\) in order that the straight line \[ x\cos\alpha+y\sin\alpha=p \] should cut the circles \[ (x-a)^2+y^2=b^2, \quad (x+a)^2+y^2=c^2, \] in chords of equal length. Prove that the envelope of the lines satisfying this condition is a parabola, and find its equation.
Solve the equations \begin{align*} x+y+z &= 3, \\ x^2+y^2+z^2+2z &= 9, \\ xyz+xy &= -2. \end{align*}
(i) Show that, if \begin{align*} x^3+px+q &= 0, \\ x^3+rx+s &= 0 \end{align*} have a common root, then \[ (q-s)^3 = (ps-qr)(p-r)^2. \] (ii) If \(\alpha, \beta, \gamma\) are the roots of \(x^3+px+q=0\), find the equation with roots \(\alpha^3, \beta^3, \gamma^3\).
If \(x_1, \dots, x_n; y_1, \dots, y_n\) are real numbers, prove that \[ (x_1^2 + \dots + x_n^2)(y_1^2 + \dots + y_n^2) \ge (x_1y_1 + \dots + x_ny_n)^2, \] and state under what conditions the equality sign holds. If \[ C_r = \frac{n!}{r!(n-r)!}, \] prove that \[ \sqrt{C_1} + \sqrt{C_2} + \dots + \sqrt{C_n} \le \sqrt{\{n(2^n-1)\}}. \]
The circumference of a circle, centre \(O\) and radius \(a\), is divided into \(2n+1\) equal arcs by points \(A_1, A_2, \dots, A_{2n+1}\), where \(n \ge 1\). Starting from \(A_1\), with the pencil never leaving the paper and moving always in an anticlockwise direction round \(O\), a sequence of chords is drawn which subtend at \(O\) successively the angles \(\theta, 2\theta, \dots, n\theta, \theta, 2\theta, \dots, n\theta, \theta, 2\theta, \dots\), where \((2n+1)\theta=2\pi\). Show that, if this construction is continued for sufficiently long, a polygonal line is drawn which begins and ends at \(A_1\) and contains each chord \(A_iA_j\) (\(1\le i < j \le 2n+1\)) exactly once; and find the total length of this line.