1) The value of \(a\) for which the sum of the squares of the roots of the equation \(x^2 - (a - 2)x - a - 1 = 0\) assumes the least value is:

• (a) 1
• (b) 0
• (c) 3
• (d) 2
• (e) None of These

2) If the roots of the equation \(x^2 - bx + c = 0\) are two consecutive integers, then \(b^2 - 4c\) equals:

• (a) -2
• (b) 3
• (c) 2
• (d) 1
• (e) None of These

3) If both the roots of the quadratic equation \(x^2 - 2kx + k^2 + k - 5 = 0\) are less than 5, then \(k\) lies in the interval:

• (a) (5, 6]
• (b) (6, ∞)
• (c) \((-\infty, 4)\)
• (d) [4, 5]
• (e) None of These

4) Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation:

• (a) \(x^2 + 18x + 16 = 0\)
• (b) \(x^2 - 18x + 16 = 0\)
• (c) \(x^2 + 18x - 16 = 0\)
• (d) \(x^2 - 18x - 16 = 0\)
• (e) None of These

5) If \((1 - p)\) is a root of the quadratic equation \(x^2 + px + (1 - p) = 0\), then the roots are:

• (a) 0, 1
• (b) -1, 1
• (c) 0, -1
• (d) -1, 2
• (e) None of These

6) If one root of the equation \(x^2 + px + 12 = 0\) is 4, while the equation \(x^2 + px + q = 0\) has equal roots, then the value of \(q\) is:

• (a) \(\frac{49}{4}\)
• (b) 12
• (c) 3
• (d) 4
• (e) None of These

7) The number of real solutions of the equation \(x^2 - 3|x| + 2 = 0\) is:

• (a) 2
• (b) 4
• (c) 1
• (d) 3
• (e) None of These

8) The value of \(a\) for which one root of the quadratic equation \((a^2 - 5a + 3)x^2 + (3a - 1)x + 2 = 0\) is twice as large as the other is:

• (a) \(\frac{2}{3}\)
• (b) \(-\frac{2}{3}\)
• (c) \(\frac{1}{3}\)
• (d) \(-\frac{1}{3}\)
• (e) None of These

9) If roots of the equation \(x^2 - 5x + 16 = 0\) are \(\alpha, \beta\) and the roots of the equation \(x^2 + px + q = 0\) are \(\alpha^2 + \beta^2\) and \(\frac{\alpha\beta}{2}\), then:

• (a) \(p = 1,\ q = -56\)
• (b) \(p = -1,\ q = -56\)
• (c) \(p = 1,\ q = 56\)
• (d) \(p = -1,\ q = 56\)
• (e) None of These

10) If \(\alpha\) and \(\beta\) are the roots of the equation \((x - a)(x - b) = c\), where \(c \ne 0\), then the roots of the equation \((x - \alpha)(x - \beta) = c\) are:

• (a) \(a,\ c\)
• (b) \(b,\ c\)
• (c) \(a,\ b\)
• (d) \(a + b,\ b + c\)
• (e) None of These