1. If \(\alpha\) and \(\beta\) are the roots of \(x^{2} + 7x + 3 = 0\) and \(\frac{2\alpha}{3 - 4\alpha}, \frac{2\beta}{3 - 4\beta}\) are the roots of \(ax^{2} + bx + c = 0\) and GCD of \(a, b, c\) is 1, then \(a + b + c =\)

1. 1. 11
2. 2. 0
3. 3. 243
4. 4. 81

2. If \(\alpha, \beta\) are the roots of \(x^{2} + bx + c = 0\), \(\gamma, \delta\) are the roots of \(x^{2} + b_{1}x + c_{1} = 0\) and \(\gamma < \alpha < \delta < \beta\), then \((c - c_{1})^{2} =\)

1. 1. \((b_{1} - b)(bc_{1} - b_{1}c)\)
2. 2. 1
3. 3. \((b - b_{1})^{2}\)
4. 4. \((c - c_{1})(b_{1}c - b_{1}c_{1})\)

3. If \(\alpha_{1}, \alpha_{2}\) are the roots of \(x^{2} + ax + 1 = 0\) and \(\alpha_{3}, \alpha_{4}\) are the roots of \(x^{2} + bx + 1 = 0\), then \((\alpha_{1} + \alpha_{3})(\alpha_{2} + \alpha_{3})(\alpha_{1} + \alpha_{4})(\alpha_{2} + \alpha_{4}) =\)

1. 1. \(3a^{2} - b^{2}\)
2. 2. \(a^{2} - 3b^{2}\)
3. 3. \((a - b)^{2}\)
4. 4. \((b + a)^{2}\)

4. The roots of the equation \(|x^{2} - x - 6| = x + 2\) are

1. 1. -2, 1, 4
2. 2. 0, 2, 4
3. 3. 0, 1, 4
4. 4. -2, 2, 4

5. If \(x\) is complex, the expression \(\frac{x^{2} + 34x - 71}{x^{2} + 2x - 7}\) takes all values which lie in the interval \((a, b)\), find the values of \(a\) and \(b\)

1. 1. \(a = -1, b = 1\)
2. 2. \(a = 1, b = -1\)
3. 3. \(a = 5, b = 9\)
4. 4. \(a = 9, b = 5\)

6. If the roots of the equation \(ax^{2} + ax + c = 0\) are in the ratio \(p: q\), then \(\sqrt{\frac{p}{q}} + \sqrt{\frac{q}{p}} =\)

1. 1. \(\sqrt{\frac{a^{2}}{c}}\)
2. 2. \(\sqrt{\frac{a}{2c}}\)
3. 3. \(\sqrt{\frac{a}{c}}\)
4. 4. \(\sqrt{\frac{a^{2}}{2c}}\)

7. If the sum of the roots of the quadratic equation is 1 and sum of the square of the roots is 13, then find that equation

1. 1. \(x^{2} + x - 6 = 0\)
2. 2. \(x^{2} - x + 6 = 0\)
3. 3. \(x^{2} - x - 6 = 0\)
4. 4. \(x^{2} + x + 6 = 0\)

8. If the roots of the given equation \((\cos p - 1)x^{2} + (\cos p)x + \sin p = 0\) are real, then

1. 1. \(p\in (-\pi, 0)\)
2. 2. \(p\in (-\frac{\pi}{2}, \frac{\pi}{2})\)
3. 3. \(p\in (0, \pi)\)
4. 4. \(p\in (0, 2\pi)\)

9. For how many values \(a \in C\), the equations \(x^{2} - 8x + 7 = 0\) and \(x^{2} - 2ax + 49 = 0\) have a common root?

1. 1. 1
2. 2. 3
3. 3. 2
4. 4. 0

10. If \(a, b, c\) are in arithmetic progression (A.P), then the roots of the equation \(ax^{2} - 2bx + c = 0\) are

1. 1. \(1, \frac{c}{a}\)
2. 2. \(-\frac{1}{a}, -c\)
3. 3. \(-1, -\frac{c}{a}\)
4. 4. \(-2, -\frac{c}{2a}\)

11. Solve the equation \(3^{x^{2} - x} = 25 - 4^{x^{2} - x}\)

1. 1. -1
2. 2. 2
3. 3. Both -1 and 2
4. 4. No solution

12. If \(2 + 4i\) is a root of \(x^{2} + bx + c = 0\) with \(b, c \in R\), then \((b, c) =\)

1. 1. \((-4, 20)\)
2. 2. \((4, 20)\)
3. 3. \((4, -20)\)
4. 4. \((-4, -20)\)

13. Given: \(\alpha + \beta = \frac{-q}{p}, \alpha\beta = \frac{r}{p}\). If \(p, q, r\) are in A.P and \(\frac{1}{\alpha} + \frac{1}{\beta} = 4\), then \(|\alpha - \beta| =\)

1. 1. \(\frac{2\sqrt{13}}{9}\)
2. 2. \(\frac{2\sqrt{13}}{3}\)
3. 3. \(\frac{4\sqrt{13}}{9}\)
4. 4. \(\frac{4\sqrt{13}}{3}\)

14. Solve \((8 - t)^{2} < (t^{2} - 3t - 10)\)

1. 1. \(\left(\frac{74}{13}, 8\right)\)
2. 2. \(\left(\frac{74}{13}, \infty\right)\)
3. 3. \((8, \infty)\)
4. 4. \([8, \infty)\)

15. If \(\alpha, \beta\) are the roots of \(x^{2} + px + q = 0\), then the values of \(\alpha^{3} + \beta^{3}\) and \(\alpha^{4} + \alpha^{2}\beta^{2} + \beta^{4}\) are respectively

1. 1. \((3pq - p^{3}), (p^{4} - 3p^{2}q + 3q^{2})\)
2. 2. \(-p(3q - p^{2}), (p^{2} - q)(p^{2} + 3q)\)
3. 3. \((pq - 4), (p^{4} - q^{4})\)
4. 4. \((3pq - p^{3}), (p^{2} - q)(p^{2} - 3q)\)

16. The number of solutions for the equation \(x^{2} - 5|x| + 6 = 0\) is

1. 1. 4
2. 2. 3
3. 3. 2
4. 4. 1

17. For which value of 'k', the roots of equation \(2x^{2} + 5x + k = 0\) are rational?

1. 1. \(\frac{25}{8}\)
2. 2. \(\frac{25}{4}\)
3. 3. \(\frac{25}{2}\)
4. 4. \(\frac{25}{16}\)

18. The polynomial \(x^{2} - 6x + 12 \in \mathbb{Q}[x]\) is

1. 1. Irreducible over \(Q\)
2. 2. reducible over \(Q\)
3. 3. Irreducible over \(C\)
4. 4. Zero polynomial

19. If the equations \(2x^{2} - 3bx + 4c = 0\) and \(3x^{2} - 4x + 5 = 0\) have a common root, then \(\frac{a + b}{b + c}\) is equal to (with \(a, b, c \in R\))

1. 1. \(\frac{1}{2}\)
2. 2. \(\frac{3}{35}\)
3. 3. \(\frac{34}{31}\)
4. 4. \(\frac{29}{23}\)

20. Assertion (A): \(3x^{2} - 16x + 4 > -16\) is satisfied for some values of real x in \(\left(0, \frac{10}{3}\right)\). Reason (R): \(ax^{2} + bx + c\) and \(a\) will have the same sign for some values of \(x \in R\) when \(b^{2} - 4ac > 0\)

1. 1. (A) is true, (R) is true and (R) is the correct explanation for (A)
2. 2. (A) is true, (R) is true but (R) is not the correct explanation for (A)
3. 3. (A) is true, but (R) is false
4. 4. (A) is false, but (R) is true

21. If the roots of the quadratic equation \(ax^{2} + bx + c = 0\) are imaginary, then for all real values of \(x\), the minimum value of the expression \(3a^{2}x^{2} + 6abx + 2b^{2}\) is

1. 1. \(< 4ab\)
2. 2. \(> 4ac\)
3. 3. \(= 4ac\)
4. 4. \(= 4ab\)

22. The equation \(\sin^{4}x - (k + 3)\sin^{2}x - k - 4 = 0\) has a solution if

1. 1. \(k > 4\)
2. 2. \(-4 \leq k \leq -3\)
3. 3. k is any positive integer
4. 4. \(k = 0\)

23. The curves \(y = x^{2} + 9x + 20\) and \(y = x^{2} + bx + c\) intersect the X-axis at the points \((\alpha_{i}, 0), (i = 1, 2, 3, 4)\). If \(\alpha_{1} < \alpha_{2} < \alpha_{3} < \alpha_{4}\) be such that \(|\alpha_{1} - \alpha_{3}| = |\alpha_{2} - \alpha_{4}| = 8\), then the sum of all possible values of \(b\) and \(c\) is

1. 1. 186
2. 2. 159
3. 3. 216
4. 4. 214

24. If \(\frac{x^{2} + ax + 3}{x^{2} + x + 1}\) takes real values for all real values of \(x\), then \(a\) lies in the interval

1. 1. \((-2 - \sqrt{11}, \sqrt{11} - 2)\)
2. 2. (4,3)
3. 3. \((-2 + \sqrt{2}, 2 + \sqrt{2})\)
4. 4. (-1,0)

25. Let S be the set of all possible integral values of \(\lambda\) in the interval (-3,7) for which the roots of the quadratic equation \(\lambda x^{2} + 13x + 7 = 0\) are all rational numbers. Then the sum of the elements in S is

1. 1. 4
2. 2. 2
3. 3. 3
4. 4. 1

26. \(\alpha\) is the maximum value of \(1 - 2x - 5x^{2}\) and \(\beta\) is the minimum value of \(x^{2} - 2x + r\). If \(5\alpha x^{2} + \beta x + 6 > 0\) for all real values \(x\), then the interval in which r lies is

1. 1. (0,5)
2. 2. \((-5, \infty)\)
3. 3. \((-\infty, 7)\)
4. 4. \((-11, 13)\)

27. The minimum value of \(\frac{9 \cdot 3^{2x} + 6 \cdot 3^{x} + 4}{9 \cdot 3^{2x} - 6 \cdot 3^{x} + 4}\) is

1. 1. -1
2. 2. \(\frac{1}{2}\)
3. 3. \(\frac{1}{4}\)
4. 4. \(\frac{1}{3}\)

28. \(p\) and \(q\) are the roots of the equation \(x^{2} + 7x + 3 = 0\). If \(\frac{3p}{1 - 2p}, \frac{3q}{1 - 2q}\) are the roots of \(lx^{2} + mx + n = 0\) and the greatest common divisor of \(l, m, n\) is 1, then \(l - m + n =\)

1. 1. 11
2. 2. -3
3. 3. -1
4. 4. 12

29. If the quadratic equations \(3x^{2} - 7x + 2 = 0\) and \(kx^{2} + 7x - 3 = 0\) have a common root then the positive value of \(k\) is

1. 1. 6
2. 2. \(\frac{11}{4}\)
3. 3. 4
4. 4. \(\frac{7}{2}\)

30. If \(\alpha, \beta\) are the roots of \(ax^{2} + bx + c = 0\), then \(\left(\frac{\alpha}{a\beta + b}\right)^{2} - \left(\frac{\beta}{a\alpha + b}\right)^{2} =\)

1. 1. 0
2. 2. 1
3. 3. \((a + b)^{2}\)
4. 4. \((a - b)^{2}\)

31. The maximum of \(\left\{x \in R \mid \sqrt{x + 2} > \sqrt{8 - x^{2}}\right\} =\)

1. 1. 2
2. 2. \(\sqrt{2} + 1\)
3. 3. 3
4. 4. \(2\sqrt{2}\)

32. If \(x\) is real, then the maximum and minimum values of \(\frac{x^{2} + 14x + 9}{x^{2} + 2x + 3}\) are respectively

1. 1. 4, -5
2. 2. 5, -4
3. 3. 9, 3
4. 4. 24, 6

33. When R is the set of all real numbers, \(\left\{x \in R \mid \frac{\sqrt{12 - x - x^{2}}}{x + 10} \leq \frac{\sqrt{12 - x - x^{2}}}{2x + 9}\right\} =\)

1. 1. \((-4, 1] \cup \{3\}\)
2. 2. \([-4, 1]\)
3. 3. \([-4, 1] \cup \{3\}\)
4. 4. \(\phi\), the empty set

34. If \((x^{2} + 5x + 5)^{x + 5} = 1\), then the number of integers satisfying this equation is

1. 1. 2
2. 2. 3
3. 3. 4
4. 4. 5

35. If \(1 + x^{2} = \sqrt{3} x\), then \(\sum_{n=1}^{24}\left(x^{n} - \frac{1}{x^{n}}\right)^{2}\) is equal to

1. 1. 48
2. 2. -48
3. 3. -24
4. 4. 24

36. If \(\alpha, \beta\) are the roots of \(11x^{2} + 12x - 13 = 0\), then \(\frac{1}{\alpha^{2}} + \frac{1}{\beta^{2}} = ?\) (approximately close to)

1. 1. 4.54
2. 2. 3.54
3. 3. 2.54
4. 4. 1.54

37. If 'a' is a positive integer such that roots of the equation \(7x^{2} - 13x + a = 0\) are rational numbers, then the smallest possible value of 'a' is

1. 1. 5
2. 2. 6
3. 3. 7
4. 4. 8

38. If one root of the equation \(ix^{2} - 2(i + 1)x + (2 - i) = 0\) is \((2 - i)\), then the other root is

1. 1. \(-i\)
2. 2. \(2 + i\)
3. 3. \(i\)
4. 4. \(2 - i\)

39. If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(x^{2} + x + 1 = 0\), then the equation whose roots are \(\alpha^{2021}, \beta^{2021}\) is given by

1. 1. \(x^{2} - x + 1 = 0\)
2. 2. \(x^{2} + x - 1 = 0\)
3. 3. \(x^{2} - x - 1 = 0\)
4. 4. \(x^{2} + x + 1 = 0\)

40. If \(f(10 - x) = 3x^{2} + 4x - 5\) & \(f(x) = px^{2} + qx + r\), then \(p + q + r =\)

1. 1. 272
2. 2. 274
3. 3. 275
4. 4. 273

41. For \(a \neq b\), if the equations \(x^{2} + ax + b = 0\) & \(x^{2} + bx + a = 0\) have a common root, then the value of \(a + b =\)

1. 1. \(-1\)
2. 2. 0
3. 3. 1
4. 4. 2

42. Let a, b, c be positive real numbers. If \(x^{2} - bx = \frac{m - 1}{m + 1}\) has two roots which are numerically equal but opposite in sign, then the value of 'm' is

1. 1. c
2. 2. \(\frac{1}{c}\)
3. 3. \(\frac{a + b}{a - b}\)
4. 4. \(\frac{a - b}{a + b}\)

43. For the equation \(x^{2} - 5|x| - 14 = 0\)

1. 1. All roots are real
2. 2. All the roots are imaginary
3. 3. Two roots are real
4. 4. No real roots

44. The number of real roots of the equation \(\left(\frac{x^{2} + 1}{x^{3}}\right)^{3} + \frac{x^{2} + 1}{3x} = 0 (x \neq 0)\) is

1. 1. 1
2. 2. 0
3. 3. 2
4. 4. 3

45. If one of the roots of the equation \(x^{2} + px + q = 0\) is equal to the square of the other, then

1. 1. \(p(q^{2} - 3p) = q(p - 1)\)
2. 2. \(p(3p - q^{2}) = p(p + 1)\)
3. 3. \(p(3q - p^{2}) = q(q - 1)\)
4. 4. \(p(3q - p^{2}) = q(q + 1)\)

46. The equations \(x^{2} - ax + b = 0\) and \(x^{2} + bx - a = 0\) have a common root, then

1. 1. \(a = b\)
2. 2. \(a + b = 1\)
3. 3. \(a + b = 0\) or \(a - b = 1\)
4. 4. \(a - b = 2\)

47. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^{2} - x + 1 = 0\), then \(\alpha^{2009} + \beta^{2009} =\)

1. 1. \(-2\)
2. 2. \(-1\)
3. 3. 1
4. 4. 2

48. Which of the following condition imply that roots of the equation \(\left(\frac{1}{4}\right)x^{2} + bx + c = 0\) are integers?

1. 1. \(b^{2} - c > 0\)
2. 2. \(b\) & \(c\) are even integers
3. 3. \(b^{2} - c\) is the square of an integer and b is an integer
4. 4. \(b\) & \(c\) are integers

49. Let m and n be two integers such that \(0 \leq m \leq 10\) and \(0 \leq n \leq 10\). Then the number of ordered pairs (m, n) such that \(x^{2} + mx + n = 0\) has real roots is

1. 1. 71
2. 2. 73
3. 3. 75
4. 4. 72

50. If \(x^{2} + px + 1\) is a factor of \(ax^{3} + bx + c\), then

1. 1. \(a^{2} + c^{2} = -ab\)
2. 2. \(a^{2} - c^{2} = -ab\)
3. 3. \(a^{2} - c^{2} = ab\)
4. 4. \(a^{2} + c^{2} = ab\)

51. Let S be the set of all quadratic equations of the form \(x^{2} + bx + c = 0\) where \(b, c \in \{1, 2, 3, 4, 5, 6\}\). If an equation is selected at random from S, then the probability that the equation has real roots is

1. 1. \(\frac{9}{12}\)
2. 2. \(\frac{9}{36}\)
3. 3. \(\frac{19}{36}\)
4. 4. \(\frac{7}{36}\)

52. The smallest negative integer satisfying both the quadratic inequalities \(x^{2} < 4x + 77\) & \(x^{2} > 4\) is

1. 1. -3
2. 2. -6
3. 3. -2
4. 4. -7

53. If the roots of equation \(x^{2} - 2cx + ab = 0\) are real and unequal, then the roots of \(x^{2} - 2(a + b)x + a^{2} + b^{2} + 2c^{2} = 0\) are

1. 1. Real and unequal
2. 2. Imaginary
3. 3. Irrational & unequal
4. 4. Real and equal

54. If \(\frac{\alpha}{\alpha + 1}\) and \(\frac{\beta}{\beta + 1}\) are the roots of the quadratic equation \(x^{2} + 7x + 3 = 0\), then the equation having roots \(\alpha\) and \(\beta\) is

1. 1. \(3x^{2} - x - 3 = 0\)
2. 2. \(11x^{2} + 13x + 3 = 0\)
3. 3. \(13x^{2} + 11x + 13 = 0\)
4. 4. \(11x^{2} + 3x + 13 = 0\)

55. If \(y = \frac{x^{2} + 14x + 9}{x^{2} + 2x + 3}\) \(\forall x \in R\), then the interval of maximum length in which \(y\) lies is

1. 1. \([-5, 4]\)
2. 2. \([-4, 5]\)
3. 3. \(\left[\frac{1}{3}, 3\right]\)
4. 4. \(\left[\frac{-1}{3}, 3\right]\)

56. If \(x^{2} - 5x - 14 > 0 \Rightarrow x\) lie outside \([\alpha, \beta]\), then \(\frac{\alpha}{\beta} =\)

1. 1. \(-2\)
2. 2. \(-7\)
3. 3. \(\frac{2}{7}\)
4. 4. \(\frac{7}{2}\)

57. For \(x \in R \setminus \{-6\}\), the value of \(\frac{(x + 2)(x + 5)}{(x + 6)}\) does not lie in the interval

1. 1. \([-9, -1]\)
2. 2. \([-5, -2]\)
3. 3. \((-5, -2)\)
4. 4. \((-9, -1)\)

58. If \(x = 2 + 2^{2/3} + 2^{1/3}\), then \(x^{3} - 6x^{2} + 6x =\)

1. 1. 3
2. 2. 2
3. 3. 1
4. 4. 0

59. \(f(x) = ax^{2} - bx - a\) is a quadratic expression. If K is the least real number such that \(f(x) \leq K \forall x \in R\), then

1. 1. \(K = 0\)
2. 2. \(K < -2\)
3. 3. \(K > 0\)
4. 4. \(-1 < K < 0\)

60. Assertion(A): The maximum value of \(-x^{2} + 3x + 1\) is \(\frac{11}{4}\). Reason(R): If \(a < 0\), the maximum value of \(ax^{2} + bx + c\) exists at \(x = \frac{-b}{2a}\)

1. 1. (A) is true, (R) is true and (R) is the correct explanation for (A)
2. 2. (A) is true, (R) is true but (R) is not the correct explanation for (A)
3. 3. (A) is true but (R) is false
4. 4. (A) is false but (R) is true

61. If \(f(x) \equiv x^{2} + ax + 2 = 0\) and \(g(x) \equiv x^{2} + 2x + a = 0\) have only one real common root, then sum of the roots of \(f(x) + g(x) = 0\) is

1. 1. \(-\frac{1}{2}\)
2. 2. 0
3. 3. \(\frac{1}{2}\)
4. 4. 1

62. Suppose \(\alpha\) is minimum value of \(x^{2} + bx + 5\) and \(\beta\) is maximum value of \(-x^{2} + ax + 5\). If \([\alpha, \beta]\) is the interval of maximum length for \(x\) in which \(x^{2} - 10x + 24 \leq 0\), then \(a^{2}b^{2} =\)

1. 1. 25
2. 2. 16
3. 3. 4
4. 4. 18

63. If the minimum value of the quadratic expression \(x^{2} + 5x - 2\) is M and it exists at a, then \(\frac{M}{a} =\)

1. 1. 3.5
2. 2. \(\frac{33}{5}\)
3. 3. 2.5
4. 4. -0.25

64. For \(\forall x \in R\) the minimum value \(\frac{1}{3}\) and the maximum value 3 of \(\frac{x^{2} + x + 1}{x^{2} - x + 1}\) exist at \(l\) & \(m\) respectively, then \(l + m =\)

1. 1. -22
2. 2. 0
3. 3. 17
4. 4. -7

65. If 2 and 3 are the two roots of the equation \(2x^{3} + mx^{2} - 13x + n = 0\), then the values of m, n are respectively

1. 1. -5, -30
2. 2. -5, 30
3. 3. 5, 30
4. 4. 5, -30

66. If \(f(x) = ax^{2} + bx + c\) for some \(a, b, c \in R\) with \(a + b + c = 3\) and \(f(x + y) = f(x) + f(y) + xy \forall x, y \in \mathbb{R}\), then \(\sum_{n=1}^{10} f(n) =\)

1. 1. 330
2. 2. 255
3. 3. 165
4. 4. 190

67. The number of positive real roots of the equation \(3^{x+1} + 3^{-x+1} = 10\) is

1. 1. 3
2. 2. 2
3. 3. 1
4. 4. Infinitely many

68. The number of real roots of the equation \(\sqrt{\frac{x}{1 - x}} + \sqrt{\frac{1 - x}{x}} = \frac{13}{6}\) is

1. 1. 1
2. 2. 2
3. 3. 3
4. 4. 4

69. If \(4^{x} - 3^{x - 1/2} = 3^{x + 1/2} - 2^{2x - 1}\) then the value of \(x\) is

1. 1. 7/2
2. 2. 5/2
3. 3. 1/2
4. 4. 3/2

70. If \(f(f(0)) = 0\), where \(f(x) = x^{2} + ax + b\), \(b \neq 0\), then \(a + b =\)

1. 1. 2
2. 2. 1
3. 3. -1
4. 4. -2

71. The sum of the real roots of the equation \(|x - 2|^{2} + |x - 2| - 2 = 0\) is

1. 1. 4
2. 2. 4
3. 3. 2
4. 4. -2

72. If the difference between the roots of \(x^{2} + ax + b = 0\) and that of the roots of \(x^{2} + bx + a = 0\) is same and \(a \neq b\), then

1. 1. \(a - b - 4 = 0\)
2. 2. \(a - b + 4 = 0\)
3. 3. \(a + b + 4 = 0\)
4. 4. \(a + b - 4 = 0\)

73. For what values of \(a \in Z\), the quadratic expression \((x + a)(x + 1991) + 1\) can be factorised as \((x + b)(x + c)\), where \(b, c \in Z\)?

1. 1. 1990
2. 2. 1989
3. 3. 1991
4. 4. 1992

74. If \(S = \{m \in R : x^{2} - 2(1 - 3m)x + 7(3 + 2m) = 0 \text{ has distinct roots}\}\), then the number of elements in S is

1. 1. 2
2. 2. 3
3. 3. 4
4. 4. Infinite

75. The sum of the real roots of the equation \(x^{4} - 2x^{3} + x - 380 = 0\) is

1. 1. -1
2. 2. 0
3. 3. 1
4. 4. 2

76. If \(x = -5 + 2\sqrt{-4}\), then the value of \(x^{4} + 9x^{3} + 35x^{2} - x + 4\) is

1. 1. 80
2. 2. 160
3. 3. -160
4. 4. -80

77. \(\alpha, \beta\) are the roots of \(x^{2} - 10x - 8 = 0\) with \(\alpha > \beta\). If \(a_{n} = \alpha^{n} - \beta^{n}\) for \(n \in N\), then the value of \(\frac{a_{10} - 8a_{8}}{5a_{9}}\) is

1. 1. -3
2. 2. 3
3. 3. -2
4. 4. 2

78. The number of real values of m so that the equation \(x^{2} + (2m + 1)x + m = 0\) has equal roots is

1. 1. 1
2. 2. 0
3. 3. 2
4. 4. 3

79. If \(f(x) = ax^{2} + bx + c\) satisfies \(f(1) + 2f(2) = 0\) and \(2f(1) + f(2) = 0\), then \(3a + b =\)

1. 1. 2
2. 2. -1
3. 3. 0
4. 4. 1

80. The sum of squares of roots of the equation \(x^{3} + x^{3} - 2 = 0\) is

1. 1. 82
2. 2. 65
3. 3. 50
4. 4. 37

81. If a, b, c, d are real numbers such that \(a < b < c < d\), then the roots of the equation \((x - a)(x - c) + 2(x - b)(x - d) = 0\) are

1. 1. Real & need not be distinct
2. 2. Real and distinct
3. 3. Non-real and distinct
4. 4. Non-real and need not be distinct

82. If one root of the quadratic equation \(ax^{2} + bx + c = 0\) is equal to the \(n^{th}\) power of the other, then \((ac^{n})^{1/(n+1)} + (a^{n}c)^{1/(n+1)} =\)

1. 1. -2b
2. 2. -b
3. 3. b-1
4. 4. b+1

83. The range of the function \(f(x) = \frac{x^{2} + x + 1}{x^{2} - x + 1}\) is

1. 1. \(\left[\frac{1}{3}, 3\right]\)
2. 2. \(\left[\frac{1}{2}, 2\right]\)
3. 3. \(\left[-\frac{1}{2}, -\frac{1}{4}\right]\)
4. 4. \(\left[-\frac{1}{2}, 2\right]\)

84. Which of the following quadratic equations whose real roots \(x_{1}, x_{2}\) satisfy the conditions \(x_{1}^{2} + x_{2}^{2} = 5\), \(3(x_{1}^{5} + x_{2}^{5}) = 11(x_{1}^{3} + x_{2}^{3})\)

1. 1. \(x^{2} \pm 3x + 2 = 0\)
2. 2. \(x^{2} \pm 3x + 11 = 0\)
3. 3. \(x^{2} \pm 5x + 2 = 0\)
4. 4. \(x^{2} \pm 5x + 11 = 0\)

85. If \(\alpha, \beta\) are the roots of \(ax^{2} + bx + c = 0\), then the quadratic equation whose roots are \(\sqrt{5}\alpha, \sqrt{5}\beta\) is

1. 1. \(ax^{2} + \sqrt{5}bx + 5c = 0\)
2. 2. \(ax^{2} + \sqrt{5}bx + \sqrt{5}c = 0\)
3. 3. \(ax^{2} + 5bx + \sqrt{5}c = 0\)
4. 4. \(ax^{2} + 5bx + 5c = 0\)

86. If \(a^{2} + b^{2} + c^{2} = 1\), \(a, b, c \in \mathbb{R}\), then the set of extreme values of \(ab + bc + ca\) is

1. 1. \(\left\{\frac{1}{2}, 2\right\}\)
2. 2. \(\{-1, 2\}\)
3. 3. \(\left\{-1, \frac{1}{2}\right\}\)
4. 4. \(\left\{\frac{-1}{2}, 1\right\}\)

87. If \(x^{2} + px + 1\) is a factor of \(ax^{3} + bx + c\), then

1. 1. \(a^{2} + c^{2} = ab + 3\)
2. 2. \(a^{2} - c^{2} = ab\)
3. 3. \(a^{2} - c^{2} = -ab\)
4. 4. \(a^{2} + c^{2} = ab\)

88. The quadratic equation whose sum of the roots is 11 and sum of squares of the roots is 61 is

1. 1. \(x^{2} + 11x - 30 = 0\)
2. 2. \(x^{2} + 11x + 30 = 0\)
3. 3. \(x^{2} - 11x - 30 = 0\)
4. 4. \(x^{2} - 11x + 30 = 0\)

89. The number of pairs of consecutive positive even integers such that the sum of their squares is 290 is

1. 1. 0
2. 2. 1
3. 3. 2
4. 4. 3

90. The range of the function \(f(x) = \frac{x}{x^{2} - 5x + 9}\) is

1. 1. \(\left[\frac{1}{11}, 1\right]\)
2. 2. \(\left[\frac{-1}{11}, 1\right]\)
3. 3. \(\left[-1, \frac{-1}{11}\right]\)
4. 4. \(\left[-1, \frac{1}{11}\right]\)

91. If \(\alpha, \beta\) are the roots of the equation \(2x^{2} + 6x + k = 0\), then the maximum value of \(\left[\frac{\alpha}{\beta} + \frac{\beta}{\alpha}\right]\) is

1. 1. 0
2. 2. 1
3. 3. -1
4. 4. -2

92. If \(A = \{x \in R \mid \sqrt{x^{2} - 8x + 15} \in R\}\) and \(B = \left\{x \in R \mid \frac{x - 3}{2x - 5} < \frac{x - 6}{2x - 11}\right\}\), then \(A \cap B =\)

1. 1. \(\phi\)
2. 2. \(\left(\frac{5}{2}, 3\right] \cup \left[\frac{5}{2}, 11\right)\)
3. 3. \(\left(\frac{5}{2}, \frac{21}{4}\right)\)
4. 4. \(\left(\frac{5}{2}, \frac{11}{2}\right)\)

93. If the extreme value of \(3x - 2x^{2} + 1\) is k then the set of all real values of \(x\) for which \(kx^{2} + 2x + 1 > 0\) is

1. 1. \(\left(\frac{1}{2}, 1\right)\)
2. 2. \(\left(-\infty, \frac{1}{2}\right) \cup (1, \infty)\)
3. 3. \(\left(-\infty, \infty\right)\)
4. 4. \(\left(-\infty, \frac{17}{8}\right)\)

94. If the quadratic equations \(x^{2} - 7x + 3c = 0\) and \(x^{2} + x - 5c = 0\) have a common root, then for non-zero real value of c the sign of the expression \(x^{2} - 3x + c\) is

1. 1. negative for all \(x \in R\)
2. 2. positive for all \(x \in (1, 3)\)
3. 3. negative for all \(x \in (1, 3)\)
4. 4. positive for all \(x \in R\)

95. Let \(f(x) = \frac{6x^{2} - 18x + 21}{6x^{2} - 18x + 17}\). If m is the maximum value of \(f(x)\) and \(f(x) > n \forall x \in R\). Then \(14m - 7n =\)

1. 1. -1
2. 2. 23
3. 3. 35
4. 4. 42

96. If \(\alpha, \beta\) are the roots of the equation \(x^{2} - 2\sqrt{3}x + 4 = 0\), then \(\alpha^{6} + \beta^{6} =\)

1. 1. 128
2. 2. -64
3. 3. 64
4. 4. -128

97. When \(b = 17\), it is found that the roots of the equation \(x^{2} + bx + c = 0\) are -2 and -15. If \(\alpha, \beta\) are the roots of the same equation when \(b = 13\), then \(|\alpha - \beta| =\)

1. 1. 7
2. 2. 13
3. 3. 17
4. 4. 30

98. Let \(x\) be the real number. Match the following:

1. 1. A-IV, B-III, C-II, D-V
2. 2. A-IV, B-V, C-II, D-III
3. 3. A-IV, B-III, C-V, D-II
4. 4. A-III, B-V, C-IV, D-I

99. If \(\alpha, \beta\) are the roots of a quadratic equation \(x^{2} + bx + c = 0\) such that \(\alpha^{2} + \beta^{2} = 5\) and \(\alpha^{3} + \beta^{3} = 9\), then \(b + c =\)

1. 1. -5
2. 2. -1
3. 3. 1
4. 4. 5

100. The set of all real values of the expression \(\frac{x^{2} - x + 2}{x^{2} + x - 2}\) for all \(x \in \mathbb{R} - \{-2, 1\}\) is

1. 1. (-2, 3)
2. 2. \(\left[\frac{7}{9}, \infty\right)\)
3. 3. \((-\infty, -1] \cup \left[\frac{7}{9}, \infty\right)\)
4. 4. \((-\infty, -1]\)

101. Statement (I): The set of solutions of \(|x|^{2} - 4|x| + 3 < 0\) is the interval \((-3, 3)\). Statement (II): If \(x < 3\) or \(x > 5\) then \(x^{2} - 8x + 15 > 0\). Which of the above statements is(are) true?

1. 1. Statement I is true, but Statement II is false
2. 2. Statement II is true, but Statement I is false
3. 3. Both statement I and Statement II are true
4. 4. Both statement I and Statement II are false

102. If \(6x - x^{2} + 12\) attains its extreme value \(\beta\) at \(x = \alpha\), then \(\beta =\)

1. 1. \(7\alpha\)
2. 2. \(5\alpha\)
3. 3. \(3\alpha\)
4. 4. \(\alpha\)

103. Let \(\alpha\) be a common root of the equations \(x^{3} - 2x - 25\lambda = 0\), \(3x^{3} - 8x - \frac{175}{3}\lambda = 0\) and \(\lambda > 0\). Then \(\lambda =\)

1. 1. \(\frac{3}{\sqrt{5}}\)
2. 2. \(\frac{\sqrt{3}}{5\sqrt{5}}\)
3. 3. \(\frac{3}{5\sqrt{5}}\)
4. 4. \(\frac{3\sqrt{5}}{5}\)

104. If the values of k for which the equation \(x^{2} + 2(k + 2)x + 6k + 7 = 0\) has equal roots are \(k_{1}\) and \(k_{2}\), then \(k_{1}^{2} + k_{2}^{2} =\)

1. 1. 8
2. 2. 9
3. 3. 10
4. 4. 12

105. If \((3 + 2\sqrt{2})^{x^{2} - 4} + (3 - 2\sqrt{2})^{x^{2} - 4} = 6\), then \(x^{4} + x^{2} + 5 =\)

1. 1. -30
2. 2. -35
3. 3. 30
4. 4. 35

106. If the equation \(x^{4} + ax^{3} + bx^{2} + cx + d = 0\) has three equal roots, then that root is

1. 1. \(\frac{6c - ab}{8b - 3a^{2}}\)
2. 2. \(\frac{ab - 6c}{8b + 3a^{2}}\)
3. 3. \(\frac{6c - ab}{3a^{2} - 4b}\)
4. 4. \(\frac{6c - ab}{3a^{2} - 8b}\)

107. \(\alpha\) and \(\beta\) are the roots of the equation \(x^{2} - ax + b = 0\). If \(\alpha^{2} + \beta^{2}\) and \(\alpha^{3} + \beta^{3}\) are the roots of the equation \(Ax^{2} + Bx + C = 0\), then C =

1. 1. \(a^{5} - 5a^{3}b + 6ab^{2}\)
2. 2. \(a^{5} + 5a^{3}b - 6ab^{2}\)
3. 3. \(a^{5} - 5a^{3}b - 6ab^{2}\)
4. 4. \(a^{5} + 5a^{3}b + 6ab^{2}\)

108. The minimum value of \(f(x) = \frac{x^{2} - 2x + 3}{x^{2} - 4x + 7}\) is

1. 1. \(1 + \frac{1}{\sqrt{3}}\)
2. 2. \(\frac{3 - \sqrt{3}}{3}\)
3. 3. \(2 - \frac{1}{\sqrt{3}}\)
4. 4. \(3 - \frac{1}{\sqrt{3}}\)

109. If \(\cot x \cot y = a\) and \(x + y = \frac{\pi}{6}\), then the quadratic equation satisfying \(\cot x\) and \(\cot y\) is

1. 1. \(t^{2} + (1 - a)\sqrt{3}t + a = 0\)
2. 2. \(\sqrt{3}t^{2} + (1 - a)t + a\sqrt{3} = 0\)
3. 3. \(\sqrt{3}t^{2} + (a - 1)t + a\sqrt{3} = 0\)
4. 4. \(t^{2} + (a - 1)\sqrt{3}t + a = 0\)

110. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^{2} + x + 1 = 0\), then the quadratic equation whose roots are \(\alpha^{2023}\) and \(\beta^{1012}\) is

1. 1. \(x^{2} + x + 1 = 0\)
2. 2. \(x^{2} - x + 1 = 0\)
3. 3. \(x^{2} - x + 2 = 0\)
4. 4. \(x^{2} + x + 2 = 0\)

111. If \(\alpha\) and \(\beta\) are the roots of the equation \(ax^{2} + bx + c = 0\), then the equation whose roots are \(\alpha + \beta\) and \(\frac{1}{\alpha} + \frac{1}{\beta}\) is

1. 1. \(acx^{2} - (ab + bc)x + b^{2} = 0\)
2. 2. \(acx^{2} + (ab + bc)x - b^{2} = 0\)
3. 3. \(acx^{2} + (ab + bc)x + b^{2} = 0\)
4. 4. \(acx^{2} - (ab + bc)x - b^{2} = 0\)

112. If c and d are the roots of \(x^{2} + ax + b = 0\), then a root of \(x^{2} + (4c + a)x + (b + 2ac + 4c^{2}) = 0\) is

1. 1. d+2c
2. 2. d+c
3. 3. d-c
4. 4. d-2c

113. The set \(\left\{x \in R : 16(2^{x}) > 16^{\frac{-1}{x}}\right\} =\)

1. 1. \(\{x \in R : x > 0\}\)
2. 2. \(\{x \in R : x < 0\}\)
3. 3. R
4. 4. \(\{x \in R : x > 2\}\)

114. The set \(\{x \in R : 4 + 11x - 3x^{2} > 0\}\) is the interval

1. 1. \(\left(-\frac{1}{3}, 4\right)\)
2. 2. \(\left(\frac{1}{3}, 4\right)\)
3. 3. \(\left(-4, \frac{1}{3}\right)\)
4. 4. \(\left(-4, -\frac{1}{3}\right)\)

115. For \(x \in R\), the minimum value of \(\frac{x^{2} + 2x + 5}{x^{2} + 4x + 10}\) is

1. 1. \(\frac{1}{2}\)
2. 2. \(\frac{4}{3}\)
3. 3. \(\frac{3}{4}\)
4. 4. \(-\frac{1}{2}\)

116. If \(\alpha\) and \(\beta\) are the roots of the equation \(2^{6x} - 3(2^{3x+2}) + 32 = 0\) with \(\beta < 1\), then \(2\alpha + 3\beta =\)

1. 1. -3
2. 2. -4
3. 3. 3
4. 4. 4

117. If \(\alpha, \beta\) and \(\gamma\) are the roots of the equation \(x^{3} - ax^{2} + bx - c = 0\), then \(\alpha^{-2} + \beta^{-2} + \gamma^{-2} =\)

1. 1. \(\frac{b^{2} - 3ac}{c^{2}}\)
2. 2. \(\frac{b^{2} - ac}{c^{2}}\)
3. 3. \(\frac{b^{2} - 2ac}{c^{2}}\)
4. 4. \(\frac{b^{2} - 4ac}{c^{2}}\)

118. If the roots of the equation \(3x^{2} + 4kx + 3 = 0\) are non-real, then k lies in the interval

1. 1. \(\left[-2, -\frac{3}{2}\right]\)
2. 2. \(\left[\frac{3}{2}, 2\right]\)
3. 3. \(\left(-\frac{3}{2}, \frac{3}{2}\right)\)
4. 4. \((2, 3)\)

119. If \(\csc\theta\) and \(\cot\theta\) are the roots of \(cx^{2} + bx + a = 0 (bc \neq 0)\), then \(b^{2}(b^{2} - 4ac) =\)

1. 1. \(-2c^{4}\)
2. 2. \(2c^{4}\)
3. 3. \(-c^{4}\)
4. 4. \(c^{4}\)

120. The sum of the fourth powers of the roots of the equation \(16x^{2} - 10x + 1 = 0\) is

1. 1. \(\frac{257}{4096}\)
2. 2. \(\frac{257}{2048}\)
3. 3. \(\frac{257}{1024}\)
4. 4. \(\frac{257}{512}\)

121. The number of elements in the set \(S = \{x \in Z : x^{2} - 7x + 6 \leq 0 \text{ and } x^{2} - 3x > 0\}\) is

1. 1. \(\infty\)
2. 2. 2
3. 3. 3
4. 4. 4

122. If one root of the equation \(4x^{2} - 2x + k - 4 = 0\) is the reciprocal of the other, then the value of \(k\) is

1. 1. -8
2. 2. 8
3. 3. -4
4. 4. 4

123. If \((x - 2)\) is a common factor of the expressions \(x^{2} + ax + b\) and \(x^{2} + cx + d\), then \(\frac{b - d}{c - a} =\)

1. 1. 1
2. 2. 2
3. 3. 3
4. 4. 4

124. The set of all values of \(x\) which satisfy both the inequations \(x^{2} - 1 \leq 0\) and \(x^{2} - x - 2 \geq 0\) simultaneously is

1. 1. (-1, 2)
2. 2. (-1, 1)
3. 3. (-2, -1)
4. 4. {-1}

125. For all real values of \(x\), the minimum value of \(\frac{1 - x + x^{2}}{1 + x + x^{2}}\) is

1. 1. 0
2. 2. \(\frac{1}{3}\)
3. 3. 1
4. 4. 3

126. The quadratic equations \(x^{2} - 6x + a = 0\) and \(x^{2} - cx + 6 = 0\) have one root in common. If the other roots of the first and second equations are integers and are in the ratio 4:3, then their common root is

1. 1. 4
2. 2. 3
3. 3. 2
4. 4. 1

127. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^{2} + 2x + 2 = 0\), then \(\alpha^{15} + \beta^{15} =\)

1. 1. -512
2. 2. -256
3. 3. 256
4. 4. 512

128. If \(x^{2} + 3x - 2k = 0\) and \(x^{2} - 2x - 7k = 0\) have a non-zero common root, then the positive root of the equation \(kx^{2} + (k + 2)x - (k + 1) = 0\) is

1. 1. 2
2. 2. 5
3. 3. 3
4. 4. 3

129. The values of \(x^{2} - 2x + 1\) do not lie in the interval

1. 1. \(\left(-\frac{4}{5}, 0\right)\)
2. 2. \(\left(-\infty, -\frac{4}{5}\right)\)
3. 3. \((0, \infty)\)
4. 4. \(\left(\frac{4}{5}, \infty\right)\)

130. If \(x^{2} + 2px - 2p + 8 > 0\) for all real values of \(x\), then the set of all possible values of \(p\) is

1. 1. (2, 4)
2. 2. \((-\infty, -4)\)
3. 3. \((2, \infty)\)
4. 4. \((-4, 2)\)

131. If \(R - (\alpha, \beta)\) is the range of \(\frac{x + 3}{(x - 1)(x + 2)}\), then the sum of the intercepts of the line \(\alpha x + \beta y + 1 = 0\) on the coordinate axes is

1. 1. -8
2. 2. 10
3. 3. 8
4. 4. 9

132. The quadratic equation whose roots are \(\sin^{2}18^{\circ}\) and \(\cos^{2}36^{\circ}\) is

1. 1. \(16x^{2} - 12x - 1 = 0\)
2. 2. \(16x^{2} - 12x + 4 = 0\)
3. 3. \(16x^{2} - 12x + 1 = 0\)
4. 4. \(16x^{2} + 12x + 1 = 0\)

133. If \(\alpha, \beta, \gamma, \delta\) are the roots of the equation \(x^{4} + x^{2} + 1 = 0\) such that \(\alpha + \beta = -1\), \(\gamma + \delta = 1\), \(\alpha^{2} = \beta\) and \(\gamma^{2} = -\delta\), then \(\alpha^{2023} + \beta^{2023} + \gamma^{2022} + \delta^{2022} =\)

1. 1. 1
2. 2. 0
3. 3. \(1 + 3\omega\)
4. 4. \(\omega - 2\omega^{2}\)

134. Let the equations \(ax^{2} - 7x + c = 0\) and \(ax^{2} + 5x - c = 0\) have a common root and \(ac \neq 0\). If 3 is a root of \(ax^{2} - 7x + c = 0\) other than the common root, then the common root of the given equations is

1. 1. 3
2. 2. 1/2
3. 3. 2
4. 4. 1/3

135. The set of all values of \(x\) for which inequalities \(x^{2} - 7x + 10 \geq 0\) and \(2x + 3 - x^{2} > 0\) hold simultaneously is

1. 1. \((-\infty, 2]\)
2. 2. \((3, \infty)\)
3. 3. \((-1, 2]\)
4. 4. [2, 3]