1. Determine the nature of the roots of the following equations:
a. \( x^2 - 12x + 40 = 0 \)
b. \( x^2 - 4x - 3 = 0 \)
c. \( 2x^2 - 12x + 18 = 0 \)
d. \( 4x^2 + 8x - 5 = 0 \)
e. \( x^2 - 16 = 0 \)
2. For what value of \( p \) will the equation \( 5x^2 - px + 45 = 0 \) have equal roots?
3. Find the value of \( k \) so that the equation:
a. \( x^2 + (k + 2)x + 2k = 0 \) has equal roots.
b. \( x^2 - (2k - 1)x - (k - 1) = 0 \) has equal roots.
4. If the equation \( (1 + m^2)x^2 + 2mcx + c^2 - a^2 = 0 \) has equal roots, show that \( c = a^2(1 + m^2) \).
5. Show that the roots of the equation \( (a^2 - bc)x^2 + 2(b - ca)x + (c^2 - ab) = 0 \) will be equal if either \( b = 0 \) or \( a^3 + b^3 + c^3 - 3abc = 0 \).
6. Prove that the roots of the equation \( (x - a)(x - b) + (x - b)(x - c) + (x - c)(x - a) = 0 \) are real. Also, prove that the roots are equal if \( a = b = c \).
7. If the roots of the equation \( (a^2 + b^2)x^2 - 2(ac + bd)x + (c^2 + d^2) = 0 \) are equal, then prove that \( \frac{a}{b} = \frac{c}{d} \).
8. If \( a, b, c \) are rational and \( a + b + c = 0 \), show that the roots of \( (b + c - a)x^2 + (c + a - b)x + (a + b - c) = 0 \) are rational.
9.
a. Prove that the roots of the equation \( (x - a)(x - b) = k^2 \) are real for all values of \( k \).
b. Show that the roots of the quadratic equation \( (b - c)x^2 + 2(c - a)x + (a - b) = 0 \) are always real.
c. Show that the roots of the equation \( x^2 + (2m - 1)x + m^2 = 0 \) are real if \( m \leq \frac{1}{4} \).
d. Show that the roots of the equation \( x^2 + 4bx + (a^2 + 2b)^2 = 0 \) are imaginary.
e. If the roots of the equation \( qx^2 + 2px + 2q = 0 \) are real and unequal, prove that the roots of the equation \( (p + q)x^2 + 2qx + (p - q) = 0 \) are imaginary.
10. If the roots of the equation \( a(b - c)x^2 + b(c - a)x + c(a - b) = 0 \) are equal, then show that \( a, b, c \) are in H.P. (i.e., \( b(a + c) = 2ac \)).
