Find necessary and sufficient conditions that the expression \(ax^2 + 2bx + c\) should be positive for all real values of \(x\). Determine the range of values of \(k\) for which the roots of the equation \[ k (x^2 + 2x + 3) = 4x + 2 \] are real and unequal.
Find the relation between the coefficients of the equation \(x^4 + px^3 + qx^2 + rx + s = 0\), when the product of two of the roots is equal to the product of the other two roots. Verify that this condition is satisfied by the equation \(x^4 + 9x^3 + 24x^2 + 18x + 4 = 0\), and find the four roots of the equation.
Write down, without proof, in the form of a determinant the product of the two determinants \[ \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}, \quad \begin{vmatrix} x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \\ x_3 & y_3 & z_3 \end{vmatrix}. \] If \begin{align*} ax_1^2 + by_1^2 + cz_1^2 = ax_2^2 + by_2^2 + cz_2^2 &= ax_3^2 + by_3^2 + cz_3^2 = d, \\ ax_2x_3 + by_2y_3 + cz_2z_3 = ax_3x_1 + by_3y_1 + cz_3z_1 &= ax_1x_2 + by_1y_2 + cz_1z_2 = f, \end{vmatrix} \end{align*} prove that \[ \begin{vmatrix} x_1 & y_1 & z_1 \\ x_2 & y_2 & z_2 \\ x_3 & y_3 & z_3 \end{vmatrix}^2 = (d-f)\{(d+2f)/abc\}^{\frac{1}{2}}. \]
The internal bisector of the angle \(A\) of a triangle \(ABC\) meets \(BC\) in \(D\); prove that \[ AD = \frac{2bc}{b+c} \cos \frac{A}{2}. \] \(A_1A_2\dots A_n\) is a regular polygon, whose centre is \(O\). If \(OA_2\) meets \(A_1A_3\) in \(P_1\), \(OA_3\) meets \(P_1A_4\) in \(P_2\), \(OA_4\) meets \(P_2A_5\) in \(P_3\), and so on, prove that \[ OP_r = a (1-2 \cos 2\alpha) (2 \cos 2\alpha)^r / \{4 \sin^2\alpha - (2 \cos 2\alpha)^{r+1}\}, \] where \(\alpha = \pi/n\) and \(OA_1 = a\). If \(n=6\), prove that \(OP_r = a/(r+1)\).
Prove De Moivre's Theorem for an integral index, positive or negative. Find all the roots of the equation \((x+i)^n + (x-i)^n=0\).
Prove that
The altitude of a triangle is to be determined from its base \(a\) and its two base angles \(B, C\). If the same small error \(\theta\) is made in the measurement of each of the base angles and a small error \(\alpha\) is also made in the measurement of the base, prove that the resulting error in the altitude is negligible, if \[ \alpha \sin B \sin C \sin(B+C) + a\theta (\sin^2 B + \sin^2 C) = 0. \]
Find an expression for the radius of curvature at any point of the curve given by \(x=f(t), y=\phi(t)\), where \(t\) is a variable parameter. Prove that the radius of curvature at the point \((x,y)\) of the curve \(x^{\frac{2}{3}} + y^{\frac{2}{3}} = a^{\frac{2}{3}}\) is \(3(axy)^{\frac{1}{3}}\). Prove also that the line joining the origin to the centre of curvature is divided in the ratio of \(1:3\) by the corresponding tangent, and that the evolute is a similar curve of double the linear dimensions of the given curve.
State the necessary and sufficient conditions that \(f(x)\) should have a maximum value, when \(x=x_0\). A triangle \(ABC\) has the vertex \(A\) fixed and the vertices \(B, C\) lie on a circle, whose centre is \(O\); prove that, if the area of the triangle is a maximum or a minimum, \(OB \cos 2\theta - OA \cos\theta = 0\), where \(\theta\) is the angle \(AOB\). Determine the value of \(\theta\) which makes the area a maximum, and discuss the case of \(\theta=0\).
Integrate \[ \int \frac{dx}{\sin^3 x}, \quad \int \frac{dx}{1+e\cos x} \quad (e<1). \] Find a reduction formula for \(\int (x^2+a^2)^n dx\), and evaluate \(\int_0^1 (x^2+4)^{\frac{5}{2}} dx\).