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Chapter 3: Pair of Linear Equations in Two Variables

NCERT Solutions for CBSE Class 10 Mathematics — 205 solved questions with detailed explanations.

205

Questions

6

Topics

Important Formulas

Unique solution: a₁/a₂ ≠ b₁/b₂
No solution: a₁/a₂ = b₁/b₂ ≠ c₁/c₂
Infinite solutions: a₁/a₂ = b₁/b₂ = c₁/c₂
Cross-multiplication: x/(b₁c₂−b₂c₁) = y/(c₁a₂−c₂a₁) = 1/(a₁b₂−a₂b₁)

Solved Questions

Q1. If a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the system is:

Difficulty: Easy · Topic: Consistency of System

Solution

This gives parallel lines → no solution → inconsistent.

Q2. For what k will 2x+3y=7, (k−1)x+(k+2)y=3k have infinitely many solutions?

Difficulty: Easy-Medium · Topic: Consistency of System

Solution

2/(k−1) = 3/(k+2) → 2k+4 = 3k−3 → k = 7. Verify third ratio: 7/21 = 1/3 = 2/6 ✓.

Q3. Solve: 3x+4y=10, 2x−2y=2.

Difficulty: Easy-Medium · Topic: Elimination Method

Solution

Multiply eq.2 by 2: 4x−4y=4. Add to eq.1: 7x=14 → x=2. Then y=1.

Q4. 3x−5y=7 and 6x−10y=12 represents:

Difficulty: Easy-Medium · Topic: Consistency of System

Solution

3/6=1/2, −5/−10=1/2, 7/12≠1/2 → parallel → no solution.

Q5. The graphs x=a and y=b intersect at:

Difficulty: Easy-Medium · Topic: Graphical Method of Solution

Solution

x=a is vertical, y=b is horizontal. They meet at (a,b).

Q6. Solve graphically: x+y=5, 2x−y=1. Find area of triangle formed with y-axis.

Difficulty: Easy-Medium · Topic: Graphical Method of Solution

Solution

Solving: 3x=6→x=2, y=3. y-intercepts: (0,5) and (0,−1). Base=6, height=2. Area=½×6×2=6.

Q7. Find the discriminant of \(1x^2 + 3x + 4 = 0\) and state the nature of its roots.

A) 25
B) 7
C) 1
D) -7 (No real roots)

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (3)^2 - 4(1)(4)\)

\(D = -7 (No real roots)\)

Q8. Find the discriminant of \(4x^2 + -8x + 6 = 0\) and state the nature of its roots.

A) 32
B) 160
C) 16
D) -32 (No real roots)

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-8)^2 - 4(4)(6)\)

\(D = -32 (No real roots)\)

Q9. Find the discriminant of \(2x^2 + 3x + 6 = 0\) and state the nature of its roots.

A) 39
B) -15
C) 57
D) -39 (No real roots)

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (3)^2 - 4(2)(6)\)

\(D = -39 (No real roots)\)

Q10. Find the discriminant of \(1x^2 + 6x + 6 = 0\) and state the nature of its roots.

A) 60
B) 24
C) -12
D) 12 (Two distinct real roots)

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (6)^2 - 4(1)(6)\)

\(D = 12 (Two distinct real roots)\)

Q11. Find the discriminant of \(4x^2 + 1x + -1 = 0\) and state the nature of its roots.

A) -17
B) -15
C) 9
D) 17 (Two distinct real roots)

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (1)^2 - 4(4)(-1)\)

\(D = 17 (Two distinct real roots)\)

Q12. Find the discriminant of \(4x^2 + 1x + 8 = 0\) and state the nature of its roots.

A) 127
B) -127 (No real roots)
C) -63
D) 129

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (1)^2 - 4(4)(8)\)

\(D = -127 (No real roots)\)

Q13. Find the discriminant of \(2x^2 + -5x + 0 = 0\) and state the nature of its roots.

A) 25 (Two distinct real roots)
B) -25
C) N/A (1)
D) 25

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-5)^2 - 4(2)(0)\)

\(D = 25 (Two distinct real roots)\)

Q14. Find the discriminant of \(1x^2 + 10x + -7 = 0\) and state the nature of its roots.

A) -128
B) 114
C) 128 (Two distinct real roots)
D) 72

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (10)^2 - 4(1)(-7)\)

\(D = 128 (Two distinct real roots)\)

Q15. Find the discriminant of \(5x^2 + -8x + -7 = 0\) and state the nature of its roots.

A) -76
B) 204 (Two distinct real roots)
C) 134
D) -204

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-8)^2 - 4(5)(-7)\)

\(D = 204 (Two distinct real roots)\)

Q16. Find the discriminant of \(3x^2 + 3x + 10 = 0\) and state the nature of its roots.

A) 111
B) -111 (No real roots)
C) -51
D) 129

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (3)^2 - 4(3)(10)\)

\(D = -111 (No real roots)\)

Q17. Find the discriminant of \(5x^2 + 1x + -2 = 0\) and state the nature of its roots.

A) 21
B) 41 (Two distinct real roots)
C) -39
D) -41

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (1)^2 - 4(5)(-2)\)

\(D = 41 (Two distinct real roots)\)

Q18. Find the discriminant of \(2x^2 + -5x + -6 = 0\) and state the nature of its roots.

A) 49
B) -73
C) -23
D) 73 (Two distinct real roots)

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-5)^2 - 4(2)(-6)\)

\(D = 73 (Two distinct real roots)\)

Q19. Find the discriminant of \(5x^2 + -7x + 1 = 0\) and state the nature of its roots.

A) 69
B) -29
C) 39
D) 29 (Two distinct real roots)

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-7)^2 - 4(5)(1)\)

\(D = 29 (Two distinct real roots)\)

Q20. Find the discriminant of \(2x^2 + 4x + 3 = 0\) and state the nature of its roots.

A) 40
B) 4
C) -8 (No real roots)
D) 8

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (4)^2 - 4(2)(3)\)

\(D = -8 (No real roots)\)

Q21. Find the discriminant of \(1x^2 + -1x + 7 = 0\) and state the nature of its roots.

A) 27
B) -27 (No real roots)
C) -13
D) 29

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-1)^2 - 4(1)(7)\)

\(D = -27 (No real roots)\)

Q22. Find the discriminant of \(4x^2 + 0x + -7 = 0\) and state the nature of its roots.

A) -112
B) N/A (1)
C) 112 (Two distinct real roots)
D) 56

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (0)^2 - 4(4)(-7)\)

\(D = 112 (Two distinct real roots)\)

Q23. Find the discriminant of \(2x^2 + -2x + 9 = 0\) and state the nature of its roots.

A) -68 (No real roots)
B) -32
C) 68
D) 76

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-2)^2 - 4(2)(9)\)

\(D = -68 (No real roots)\)

Q24. Find the discriminant of \(1x^2 + -9x + 9 = 0\) and state the nature of its roots.

A) 63
B) -45
C) 45 (Two distinct real roots)
D) 117

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-9)^2 - 4(1)(9)\)

\(D = 45 (Two distinct real roots)\)

Q25. Find the discriminant of \(3x^2 + -4x + 0 = 0\) and state the nature of its roots.

A) 16 (Two distinct real roots)
B) -16
C) N/A (1)
D) 16

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-4)^2 - 4(3)(0)\)

\(D = 16 (Two distinct real roots)\)

Q26. Find the discriminant of \(4x^2 + -2x + -4 = 0\) and state the nature of its roots.

A) -60
B) 36
C) 68 (Two distinct real roots)
D) -68

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-2)^2 - 4(4)(-4)\)

\(D = 68 (Two distinct real roots)\)

Q27. Find the discriminant of \(1x^2 + -8x + 10 = 0\) and state the nature of its roots.

A) -24
B) 104
C) 44
D) 24 (Two distinct real roots)

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-8)^2 - 4(1)(10)\)

\(D = 24 (Two distinct real roots)\)

Q28. Find the discriminant of \(4x^2 + 5x + 10 = 0\) and state the nature of its roots.

A) 185
B) -55
C) -135 (No real roots)
D) 135

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (5)^2 - 4(4)(10)\)

\(D = -135 (No real roots)\)

Q29. Find the discriminant of \(2x^2 + -8x + -7 = 0\) and state the nature of its roots.

A) 120 (Two distinct real roots)
B) 92
C) 8
D) -120

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-8)^2 - 4(2)(-7)\)

\(D = 120 (Two distinct real roots)\)

Q30. Find the discriminant of \(4x^2 + 4x + 0 = 0\) and state the nature of its roots.

A) 16
B) N/A (1)
C) 16 (Two distinct real roots)
D) -16

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (4)^2 - 4(4)(0)\)

\(D = 16 (Two distinct real roots)\)

Q31. Find the discriminant of \(2x^2 + 2x + -1 = 0\) and state the nature of its roots.

A) 12 (Two distinct real roots)
B) 8
C) -12
D) -4

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (2)^2 - 4(2)(-1)\)

\(D = 12 (Two distinct real roots)\)

Q32. Find the discriminant of \(2x^2 + 9x + 1 = 0\) and state the nature of its roots.

A) 73 (Two distinct real roots)
B) -73
C) 89
D) 77

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (9)^2 - 4(2)(1)\)

\(D = 73 (Two distinct real roots)\)

Q33. Find the discriminant of \(4x^2 + -5x + -2 = 0\) and state the nature of its roots.

A) -57
B) -7
C) 41
D) 57 (Two distinct real roots)

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-5)^2 - 4(4)(-2)\)

\(D = 57 (Two distinct real roots)\)

Q34. Find the discriminant of \(4x^2 + -8x + -2 = 0\) and state the nature of its roots.

A) 96 (Two distinct real roots)
B) -96
C) 32
D) 80

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-8)^2 - 4(4)(-2)\)

\(D = 96 (Two distinct real roots)\)

Q35. Find the discriminant of \(3x^2 + -10x + 0 = 0\) and state the nature of its roots.

A) 100 (Two distinct real roots)
B) 100
C) N/A (1)
D) -100

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-10)^2 - 4(3)(0)\)

\(D = 100 (Two distinct real roots)\)

Q36. Find the discriminant of \(2x^2 + -9x + 6 = 0\) and state the nature of its roots.

A) -33
B) 129
C) 33 (Two distinct real roots)
D) 57

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-9)^2 - 4(2)(6)\)

\(D = 33 (Two distinct real roots)\)

Q37. Find the discriminant of \(4x^2 + 5x + -2 = 0\) and state the nature of its roots.

A) -57
B) -7
C) 41
D) 57 (Two distinct real roots)

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (5)^2 - 4(4)(-2)\)

\(D = 57 (Two distinct real roots)\)

Q38. Find the discriminant of \(4x^2 + -4x + -9 = 0\) and state the nature of its roots.

A) -128
B) 160 (Two distinct real roots)
C) -160
D) 88

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-4)^2 - 4(4)(-9)\)

\(D = 160 (Two distinct real roots)\)

Q39. Find the discriminant of \(4x^2 + 6x + 3 = 0\) and state the nature of its roots.

A) 84
B) -12 (No real roots)
C) 12
D) N/A (1)

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (6)^2 - 4(4)(3)\)

\(D = -12 (No real roots)\)

Q40. Find the discriminant of \(1x^2 + -9x + 5 = 0\) and state the nature of its roots.

A) -61
B) 101
C) 61 (Two distinct real roots)
D) 71

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-9)^2 - 4(1)(5)\)

\(D = 61 (Two distinct real roots)\)

Q41. Find the discriminant of \(2x^2 + -3x + -6 = 0\) and state the nature of its roots.

A) -39
B) -57
C) 33
D) 57 (Two distinct real roots)

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-3)^2 - 4(2)(-6)\)

\(D = 57 (Two distinct real roots)\)

Q42. Find the discriminant of \(3x^2 + -9x + 2 = 0\) and state the nature of its roots.

A) -57
B) 105
C) 69
D) 57 (Two distinct real roots)

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-9)^2 - 4(3)(2)\)

\(D = 57 (Two distinct real roots)\)

Q43. Find the discriminant of \(1x^2 + -9x + -3 = 0\) and state the nature of its roots.

A) 87
B) 69
C) 93 (Two distinct real roots)
D) -93

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-9)^2 - 4(1)(-3)\)

\(D = 93 (Two distinct real roots)\)

Q44. Find the discriminant of \(2x^2 + -4x + 3 = 0\) and state the nature of its roots.

A) 4
B) -8 (No real roots)
C) 8
D) 40

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-4)^2 - 4(2)(3)\)

\(D = -8 (No real roots)\)

Q45. Find the discriminant of \(2x^2 + 4x + -2 = 0\) and state the nature of its roots.

A) 0
B) 32 (Two distinct real roots)
C) 24
D) -32

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (4)^2 - 4(2)(-2)\)

\(D = 32 (Two distinct real roots)\)

Q46. Find the discriminant of \(4x^2 + -8x + -1 = 0\) and state the nature of its roots.

A) 80 (Two distinct real roots)
B) 48
C) -80
D) 72

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-8)^2 - 4(4)(-1)\)

\(D = 80 (Two distinct real roots)\)

Q47. Find the discriminant of \(5x^2 + 9x + -1 = 0\) and state the nature of its roots.

A) -101
B) 101 (Two distinct real roots)
C) 61
D) 91

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (9)^2 - 4(5)(-1)\)

\(D = 101 (Two distinct real roots)\)

Q48. Find the discriminant of \(2x^2 + 0x + -8 = 0\) and state the nature of its roots.

A) 64 (Two distinct real roots)
B) 32
C) N/A (1)
D) -64

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (0)^2 - 4(2)(-8)\)

\(D = 64 (Two distinct real roots)\)

Q49. Find the discriminant of \(4x^2 + -5x + -1 = 0\) and state the nature of its roots.

A) 9
B) 41 (Two distinct real roots)
C) 33
D) -41

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-5)^2 - 4(4)(-1)\)

\(D = 41 (Two distinct real roots)\)

Q50. Find the discriminant of \(1x^2 + -8x + 2 = 0\) and state the nature of its roots.

A) 72
B) 56 (Two distinct real roots)
C) 60
D) -56

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-8)^2 - 4(1)(2)\)

\(D = 56 (Two distinct real roots)\)

Q51. Find the discriminant of \(4x^2 + 10x + 6 = 0\) and state the nature of its roots.

A) 52
B) 4 (Two distinct real roots)
C) 196
D) -4

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (10)^2 - 4(4)(6)\)

\(D = 4 (Two distinct real roots)\)

Q52. Find the discriminant of \(5x^2 + 3x + -8 = 0\) and state the nature of its roots.

A) -169
B) 169 (Two distinct real roots)
C) 89
D) -151

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (3)^2 - 4(5)(-8)\)

\(D = 169 (Two distinct real roots)\)

Q53. Find the discriminant of \(4x^2 + -9x + 5 = 0\) and state the nature of its roots.

A) 41
B) -1
C) 1 (Two distinct real roots)
D) 161

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-9)^2 - 4(4)(5)\)

\(D = 1 (Two distinct real roots)\)

Q54. Find the discriminant of \(5x^2 + -8x + 4 = 0\) and state the nature of its roots.

A) 16
B) -16 (No real roots)
C) 24
D) 144

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-8)^2 - 4(5)(4)\)

\(D = -16 (No real roots)\)

Q55. Find the discriminant of \(4x^2 + -8x + -7 = 0\) and state the nature of its roots.

A) -176
B) -48
C) 176 (Two distinct real roots)
D) 120

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-8)^2 - 4(4)(-7)\)

\(D = 176 (Two distinct real roots)\)

Q56. Find the discriminant of \(3x^2 + 6x + 6 = 0\) and state the nature of its roots.

A) 0
B) 36
C) -36 (No real roots)
D) 108

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (6)^2 - 4(3)(6)\)

\(D = -36 (No real roots)\)

Q57. Find the discriminant of \(2x^2 + -4x + 8 = 0\) and state the nature of its roots.

A) 48
B) 80
C) -48 (No real roots)
D) -16

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-4)^2 - 4(2)(8)\)

\(D = -48 (No real roots)\)

Q58. Find the discriminant of \(3x^2 + -8x + -7 = 0\) and state the nature of its roots.

A) 106
B) -20
C) -148
D) 148 (Two distinct real roots)

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-8)^2 - 4(3)(-7)\)

\(D = 148 (Two distinct real roots)\)

Q59. Find the discriminant of \(2x^2 + -9x + -4 = 0\) and state the nature of its roots.

A) -113
B) 49
C) 97
D) 113 (Two distinct real roots)

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-9)^2 - 4(2)(-4)\)

\(D = 113 (Two distinct real roots)\)

Q60. Find the discriminant of \(4x^2 + 6x + 10 = 0\) and state the nature of its roots.

A) 196
B) -124 (No real roots)
C) -44
D) 124

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (6)^2 - 4(4)(10)\)

\(D = -124 (No real roots)\)

Q61. Find the discriminant of \(3x^2 + 4x + -6 = 0\) and state the nature of its roots.

A) 88 (Two distinct real roots)
B) 52
C) -88
D) -56

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (4)^2 - 4(3)(-6)\)

\(D = 88 (Two distinct real roots)\)

Q62. Find the discriminant of \(5x^2 + 7x + 4 = 0\) and state the nature of its roots.

A) -31 (No real roots)
B) 9
C) 129
D) 31

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (7)^2 - 4(5)(4)\)

\(D = -31 (No real roots)\)

Q63. Find the discriminant of \(3x^2 + 10x + -10 = 0\) and state the nature of its roots.

A) -220
B) 160
C) 220 (Two distinct real roots)
D) -20

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (10)^2 - 4(3)(-10)\)

\(D = 220 (Two distinct real roots)\)

Q64. Find the discriminant of \(2x^2 + 6x + -10 = 0\) and state the nature of its roots.

A) -116
B) -44
C) 76
D) 116 (Two distinct real roots)

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (6)^2 - 4(2)(-10)\)

\(D = 116 (Two distinct real roots)\)

Q65. Find the discriminant of \(1x^2 + -4x + 1 = 0\) and state the nature of its roots.

A) 12 (Two distinct real roots)
B) -12
C) 20
D) 14

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-4)^2 - 4(1)(1)\)

\(D = 12 (Two distinct real roots)\)

Q66. Find the discriminant of \(5x^2 + 7x + 9 = 0\) and state the nature of its roots.

A) -131 (No real roots)
B) 131
C) -41
D) 229

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (7)^2 - 4(5)(9)\)

\(D = -131 (No real roots)\)

Q67. Find the discriminant of \(2x^2 + -3x + -10 = 0\) and state the nature of its roots.

A) 49
B) -89
C) 89 (Two distinct real roots)
D) -71

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-3)^2 - 4(2)(-10)\)

\(D = 89 (Two distinct real roots)\)

Q68. Find the discriminant of \(3x^2 + -3x + -4 = 0\) and state the nature of its roots.

A) 57 (Two distinct real roots)
B) 33
C) -57
D) -39

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-3)^2 - 4(3)(-4)\)

\(D = 57 (Two distinct real roots)\)

Q69. Find the discriminant of \(5x^2 + -2x + 0 = 0\) and state the nature of its roots.

A) 4 (Two distinct real roots)
B) -4
C) N/A (1)
D) 4

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-2)^2 - 4(5)(0)\)

\(D = 4 (Two distinct real roots)\)

Q70. Find the discriminant of \(3x^2 + -7x + 9 = 0\) and state the nature of its roots.

A) 59
B) -5
C) 157
D) -59 (No real roots)

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-7)^2 - 4(3)(9)\)

\(D = -59 (No real roots)\)

Q71. Find the discriminant of \(1x^2 + -2x + 6 = 0\) and state the nature of its roots.

A) -8
B) -20 (No real roots)
C) 20
D) 28

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-2)^2 - 4(1)(6)\)

\(D = -20 (No real roots)\)

Q72. Find the discriminant of \(4x^2 + -8x + -5 = 0\) and state the nature of its roots.

A) -16
B) 144 (Two distinct real roots)
C) -144
D) 104

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-8)^2 - 4(4)(-5)\)

\(D = 144 (Two distinct real roots)\)

Q73. Find the discriminant of \(4x^2 + 10x + 0 = 0\) and state the nature of its roots.

A) 100
B) N/A (1)
C) 100 (Two distinct real roots)
D) -100

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (10)^2 - 4(4)(0)\)

\(D = 100 (Two distinct real roots)\)

Q74. Find the discriminant of \(5x^2 + -4x + 10 = 0\) and state the nature of its roots.

A) -84
B) 216
C) 184
D) -184 (No real roots)

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-4)^2 - 4(5)(10)\)

\(D = -184 (No real roots)\)

Q75. Find the discriminant of \(2x^2 + -9x + 5 = 0\) and state the nature of its roots.

A) 41 (Two distinct real roots)
B) -41
C) 61
D) 121

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-9)^2 - 4(2)(5)\)

\(D = 41 (Two distinct real roots)\)

Q76. Find the discriminant of \(2x^2 + -10x + 5 = 0\) and state the nature of its roots.

A) 80
B) 140
C) -60
D) 60 (Two distinct real roots)

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-10)^2 - 4(2)(5)\)

\(D = 60 (Two distinct real roots)\)

Q77. Find the discriminant of \(5x^2 + -1x + 7 = 0\) and state the nature of its roots.

A) -69
B) -139 (No real roots)
C) 141
D) 139

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-1)^2 - 4(5)(7)\)

\(D = -139 (No real roots)\)

Q78. Find the discriminant of \(2x^2 + 10x + -8 = 0\) and state the nature of its roots.

A) 164 (Two distinct real roots)
B) 36
C) 132
D) -164

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (10)^2 - 4(2)(-8)\)

\(D = 164 (Two distinct real roots)\)

Q79. Find the discriminant of \(3x^2 + 8x + -6 = 0\) and state the nature of its roots.

A) -8
B) 100
C) 136 (Two distinct real roots)
D) -136

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (8)^2 - 4(3)(-6)\)

\(D = 136 (Two distinct real roots)\)

Q80. Find the discriminant of \(1x^2 + -8x + -4 = 0\) and state the nature of its roots.

A) -80
B) 80 (Two distinct real roots)
C) 72
D) 48

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-8)^2 - 4(1)(-4)\)

\(D = 80 (Two distinct real roots)\)

Q81. Find the discriminant of \(2x^2 + 8x + -10 = 0\) and state the nature of its roots.

A) 144 (Two distinct real roots)
B) -144
C) -16
D) 104

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (8)^2 - 4(2)(-10)\)

\(D = 144 (Two distinct real roots)\)

Q82. Find the discriminant of \(5x^2 + -4x + -8 = 0\) and state the nature of its roots.

A) -176
B) -144
C) 176 (Two distinct real roots)
D) 96

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-4)^2 - 4(5)(-8)\)

\(D = 176 (Two distinct real roots)\)

Q83. Find the discriminant of \(5x^2 + 9x + 2 = 0\) and state the nature of its roots.

A) 121
B) 61
C) 41 (Two distinct real roots)
D) -41

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (9)^2 - 4(5)(2)\)

\(D = 41 (Two distinct real roots)\)

Q84. Find the discriminant of \(1x^2 + -4x + -7 = 0\) and state the nature of its roots.

A) -12
B) 44 (Two distinct real roots)
C) 30
D) -44

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-4)^2 - 4(1)(-7)\)

\(D = 44 (Two distinct real roots)\)

Q85. Find the discriminant of \(5x^2 + -10x + 6 = 0\) and state the nature of its roots.

A) 20
B) 40
C) -20 (No real roots)
D) 220

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-10)^2 - 4(5)(6)\)

\(D = -20 (No real roots)\)

Q86. Find the discriminant of \(5x^2 + 8x + 6 = 0\) and state the nature of its roots.

A) 56
B) -56 (No real roots)
C) 184
D) 4

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (8)^2 - 4(5)(6)\)

\(D = -56 (No real roots)\)

Q87. Find the discriminant of \(1x^2 + -2x + -10 = 0\) and state the nature of its roots.

A) -44
B) -36
C) 24
D) 44 (Two distinct real roots)

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-2)^2 - 4(1)(-10)\)

\(D = 44 (Two distinct real roots)\)

Q88. Find the discriminant of \(5x^2 + -10x + -5 = 0\) and state the nature of its roots.

A) -200
B) 0
C) 150
D) 200 (Two distinct real roots)

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-10)^2 - 4(5)(-5)\)

\(D = 200 (Two distinct real roots)\)

Q89. Find the discriminant of \(1x^2 + -1x + 10 = 0\) and state the nature of its roots.

A) 41
B) 39
C) -39 (No real roots)
D) -19

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-1)^2 - 4(1)(10)\)

\(D = -39 (No real roots)\)

Q90. Find the discriminant of \(2x^2 + 3x + -10 = 0\) and state the nature of its roots.

A) 89 (Two distinct real roots)
B) -71
C) 49
D) -89

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (3)^2 - 4(2)(-10)\)

\(D = 89 (Two distinct real roots)\)

Q91. Find the discriminant of \(3x^2 + 9x + -6 = 0\) and state the nature of its roots.

A) 117
B) -153
C) 9
D) 153 (Two distinct real roots)

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (9)^2 - 4(3)(-6)\)

\(D = 153 (Two distinct real roots)\)

Q92. Find the discriminant of \(1x^2 + 8x + -7 = 0\) and state the nature of its roots.

A) -92
B) 78
C) 36
D) 92 (Two distinct real roots)

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (8)^2 - 4(1)(-7)\)

\(D = 92 (Two distinct real roots)\)

Q93. Find the discriminant of \(5x^2 + -8x + 9 = 0\) and state the nature of its roots.

A) -26
B) 116
C) -116 (No real roots)
D) 244

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-8)^2 - 4(5)(9)\)

\(D = -116 (No real roots)\)

Q94. Find the discriminant of \(5x^2 + -3x + 1 = 0\) and state the nature of its roots.

A) 29
B) -11 (No real roots)
C) 11
D) -1

Difficulty: Easy-Medium · Topic: Discriminant and nature of roots

Solution

\(D = b^2 - 4ac = (-3)^2 - 4(5)(1)\)

\(D = -11 (No real roots)\)

Q95. A fraction becomes 9/11 when 2 is added to both numerator and denominator. It becomes 5/6 when 3 is added to both. Find the fraction.

Difficulty: Medium · Topic: Substitution Method

Solution

Let fraction = x/y. 11(x+2)=9(y+2) → 11x−9y=−4. 6(x+3)=5(y+3) → 6x−5y=−3.

Solve: multiply eq.1 by 5, eq.2 by 9: 55x−45y=−20, 54x−45y=−27. Subtract: x=7. Then y=9.

Q96. Sum of a two-digit number and its reverse is 66. Digits differ by 2. Find the number.

Difficulty: Medium · Topic: Elimination Method

Solution

Let digits x,y. 11(x+y)=66 → x+y=6. |x−y|=2. Solving: (4,2) or (2,4). Numbers: 42 or 24.

Q97. Solve: 1/(3x+y)+1/(3x−y)=3/4, 1/(2(3x+y))−1/(2(3x−y))=−1/8.

Difficulty: Medium · Topic: Equations Reducible to Linear Equations

Solution

Let u=1/(3x+y), v=1/(3x−y). u+v=3/4, (u−v)/2=−1/8 → u−v=−1/4.

u=1/4→3x+y=4. v=1/2→3x−y=2. Add: 6x=6→x=1, y=1.

Q98. A boat goes 30 km upstream and 44 km downstream in 10 hours. It goes 40 km upstream and 55 km downstream in 13 hours. Find speeds.

Difficulty: Medium · Topic: Substitution Method

Solution

Let u=1/(x−y), v=1/(x+y). 30u+44v=10, 40u+55v=13.

Solve: v=1/11→x+y=11, u=1/5→x−y=5. x=8, y=3.

Q99. Solve using cross-multiplication: x+2y+1=0, 2x−3y−12=0.

Difficulty: Medium · Topic: Cross-Multiplication Method

Solution

x/(2×(−12)−(−3)×1) = y/(1×2−(−12)×1) = 1/(1×(−3)−2×2)

x/(−21) = y/14 = 1/(−7). x=3, y=−2.

Q100. 5 pencils + 7 pens = Rs 50; 7 pencils + 5 pens = Rs 46. Find cost of each.

Difficulty: Medium · Topic: Substitution Method

Solution

5x+7y=50, 7x+5y=46. Add: 12(x+y)=96→x+y=8. Subtract: −2(x−y)=4→x−y=−2. x=3, y=5.

Q101. Father's age is 3 times son's age. 5 years ago it was 4 times. Find ages.

Difficulty: Medium · Topic: Elimination Method

Solution

y=3x. y−5=4(x−5)→3x−5=4x−20→x=15. y=45.

Q102. Value of k for unique solution of kx−y=2, 6x−2y=3:

Difficulty: Medium · Topic: Consistency of System

Solution

Unique: a₁/a₂ ≠ b₁/b₂. k/6 ≠ 1/2 → k ≠ 3.

Q103. The sum of the numerator and denominator of a fraction is 12. If the denominator is increased by 3, the fraction becomes 1/2. Find the fraction.

Difficulty: Medium · Topic: Substitution Method

Solution

x+y=12, x/(y+3)=1/2 → 2x=y+3. From first: y=12−x → 2x=15−x → 3x=15 → x=5, y=7. Fraction = 5/7.

Q104. Find the roots of the equation \(x^2 + 8x + -8 = 0\).

A) 1.8, -17.8
B) 0.9, -8.9
C) 8.9, -0.9
D) -1.17, -6.83

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 8x + -8 = 0\), here \(a=1, b=8, c=-8\).

Discriminant \(D = b^2 - 4ac = 8^2 - 4(1)(-8)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(0.9, -8.9\)

Q105. Find the roots of the equation \(x^2 + 3x + 14 = 0\).

A) No real roots
B) N/A (1)
C) N/A (2)
D) 2.53, -5.53

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 3x + 14 = 0\), here \(a=1, b=3, c=14\).

Discriminant \(D = b^2 - 4ac = 3^2 - 4(1)(14)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(No real roots\)

Q106. Find the roots of the equation \(x^2 + 7x + -2 = 0\).

A) 0.55, -14.55
B) -0.3, -6.7
C) 7.27, -0.27
D) 0.27, -7.27

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 7x + -2 = 0\), here \(a=1, b=7, c=-2\).

Discriminant \(D = b^2 - 4ac = 7^2 - 4(1)(-2)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(0.27, -7.27\)

Q107. Find the roots of the equation \(x^2 + 12x + 17 = 0\).

A) -1.64, -10.36
B) -3.28, -20.72
C) 1.28, -13.28
D) 10.36, 1.64

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 12x + 17 = 0\), here \(a=1, b=12, c=17\).

Discriminant \(D = b^2 - 4ac = 12^2 - 4(1)(17)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(-1.64, -10.36\)

Q108. Find the roots of the equation \(x^2 + 2x + 16 = 0\).

A) N/A (2)
B) 3.12, -5.12
C) No real roots
D) N/A (1)

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 2x + 16 = 0\), here \(a=1, b=2, c=16\).

Discriminant \(D = b^2 - 4ac = 2^2 - 4(1)(16)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(No real roots\)

Q109. Find the roots of the equation \(x^2 + 11x + -4 = 0\).

A) -0.38, -10.62
B) 11.35, -0.35
C) 0.7, -22.7
D) 0.35, -11.35

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 11x + -4 = 0\), here \(a=1, b=11, c=-4\).

Discriminant \(D = b^2 - 4ac = 11^2 - 4(1)(-4)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(0.35, -11.35\)

Q110. Find the roots of the equation \(x^2 + -3x + 19 = 0\).

A) 6.11, -3.11
B) No real roots
C) N/A (1)
D) N/A (2)

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -3x + 19 = 0\), here \(a=1, b=-3, c=19\).

Discriminant \(D = b^2 - 4ac = -3^2 - 4(1)(19)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(No real roots\)

Q111. Find the roots of the equation \(x^2 + -4x + -6 = 0\).

A) 10.32, -2.32
B) 5.16, -1.16
C) No real roots
D) 1.16, -5.16

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -4x + -6 = 0\), here \(a=1, b=-4, c=-6\).

Discriminant \(D = b^2 - 4ac = -4^2 - 4(1)(-6)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(5.16, -1.16\)

Q112. Find the roots of the equation \(x^2 + 11x + 11 = 0\).

A) 0.92, -11.92
B) -1.11, -9.89
C) -2.23, -19.77
D) 9.89, 1.11

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 11x + 11 = 0\), here \(a=1, b=11, c=11\).

Discriminant \(D = b^2 - 4ac = 11^2 - 4(1)(11)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(-1.11, -9.89\)

Q113. Find the roots of the equation \(x^2 + 6x + -9 = 0\).

A) 1.24, -7.24
B) 2.49, -14.49
C) -3.0, -3.0
D) 7.24, -1.24

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 6x + -9 = 0\), here \(a=1, b=6, c=-9\).

Discriminant \(D = b^2 - 4ac = 6^2 - 4(1)(-9)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(1.24, -7.24\)

Q114. Find the roots of the equation \(x^2 + -7x + -4 = 0\).

A) 0.53, -7.53
B) 7.53, -0.53
C) 15.06, -1.06
D) 6.37, 0.63

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -7x + -4 = 0\), here \(a=1, b=-7, c=-4\).

Discriminant \(D = b^2 - 4ac = -7^2 - 4(1)(-4)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(7.53, -0.53\)

Q115. Find the roots of the equation \(x^2 + -4x + -16 = 0\).

A) No real roots
B) 12.94, -4.94
C) 6.47, -2.47
D) 2.47, -6.47

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -4x + -16 = 0\), here \(a=1, b=-4, c=-16\).

Discriminant \(D = b^2 - 4ac = -4^2 - 4(1)(-16)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(6.47, -2.47\)

Q116. Find the roots of the equation \(x^2 + 6x + -14 = 0\).

A) 1.8, -7.8
B) 3.59, -15.59
C) No real roots
D) 7.8, -1.8

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 6x + -14 = 0\), here \(a=1, b=6, c=-14\).

Discriminant \(D = b^2 - 4ac = 6^2 - 4(1)(-14)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(1.8, -7.8\)

Q117. Find the roots of the equation \(x^2 + -9x + 17 = 0\).

A) -2.7, -6.3
B) 12.61, 5.39
C) 10.6, -1.6
D) 6.3, 2.7

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -9x + 17 = 0\), here \(a=1, b=-9, c=17\).

Discriminant \(D = b^2 - 4ac = -9^2 - 4(1)(17)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(6.3, 2.7\)

Q118. Find the roots of the equation \(x^2 + 9x + -8 = 0\).

A) 9.82, -0.82
B) 1.63, -19.63
C) -1.0, -8.0
D) 0.82, -9.82

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 9x + -8 = 0\), here \(a=1, b=9, c=-8\).

Discriminant \(D = b^2 - 4ac = 9^2 - 4(1)(-8)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(0.82, -9.82\)

Q119. Find the roots of the equation \(x^2 + 1x + 13 = 0\).

A) 3.14, -4.14
B) No real roots
C) N/A (1)
D) N/A (2)

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 1x + 13 = 0\), here \(a=1, b=1, c=13\).

Discriminant \(D = b^2 - 4ac = 1^2 - 4(1)(13)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(No real roots\)

Q120. Find the roots of the equation \(x^2 + 4x + -10 = 0\).

A) 1.74, -5.74
B) 3.48, -11.48
C) 5.74, -1.74
D) No real roots

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 4x + -10 = 0\), here \(a=1, b=4, c=-10\).

Discriminant \(D = b^2 - 4ac = 4^2 - 4(1)(-10)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(1.74, -5.74\)

Q121. Find the roots of the equation \(x^2 + 8x + -5 = 0\).

A) -0.68, -7.32
B) 0.58, -8.58
C) 8.58, -0.58
D) 1.17, -17.17

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 8x + -5 = 0\), here \(a=1, b=8, c=-5\).

Discriminant \(D = b^2 - 4ac = 8^2 - 4(1)(-5)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(0.58, -8.58\)

Q122. Find the roots of the equation \(x^2 + 0x + 4 = 0\).

A) N/A (2)
B) N/A (1)
C) No real roots
D) 2.0, -2.0

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 0x + 4 = 0\), here \(a=1, b=0, c=4\).

Discriminant \(D = b^2 - 4ac = 0^2 - 4(1)(4)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(No real roots\)

Q123. Find the roots of the equation \(x^2 + -5x + -15 = 0\).

A) 2.11, -7.11
B) No real roots
C) 14.22, -4.22
D) 7.11, -2.11

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -5x + -15 = 0\), here \(a=1, b=-5, c=-15\).

Discriminant \(D = b^2 - 4ac = -5^2 - 4(1)(-15)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(7.11, -2.11\)

Q124. Find the roots of the equation \(x^2 + 11x + -17 = 0\).

A) 2.75, -24.75
B) 12.37, -1.37
C) -1.86, -9.14
D) 1.37, -12.37

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 11x + -17 = 0\), here \(a=1, b=11, c=-17\).

Discriminant \(D = b^2 - 4ac = 11^2 - 4(1)(-17)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(1.37, -12.37\)

Q125. Find the roots of the equation \(x^2 + 5x + 16 = 0\).

A) 2.22, -7.22
B) N/A (1)
C) No real roots
D) N/A (2)

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 5x + 16 = 0\), here \(a=1, b=5, c=16\).

Discriminant \(D = b^2 - 4ac = 5^2 - 4(1)(16)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(No real roots\)

Q126. Find the roots of the equation \(x^2 + -4x + 1 = 0\).

A) 3.73, 0.27
B) -0.27, -3.73
C) 4.24, -0.24
D) 7.46, 0.54

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -4x + 1 = 0\), here \(a=1, b=-4, c=1\).

Discriminant \(D = b^2 - 4ac = -4^2 - 4(1)(1)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(3.73, 0.27\)

Q127. Find the roots of the equation \(x^2 + -2x + 18 = 0\).

A) 5.36, -3.36
B) N/A (1)
C) N/A (2)
D) No real roots

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -2x + 18 = 0\), here \(a=1, b=-2, c=18\).

Discriminant \(D = b^2 - 4ac = -2^2 - 4(1)(18)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(No real roots\)

Q128. Find the roots of the equation \(x^2 + 6x + -10 = 0\).

A) No real roots
B) 7.36, -1.36
C) 2.72, -14.72
D) 1.36, -7.36

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 6x + -10 = 0\), here \(a=1, b=6, c=-10\).

Discriminant \(D = b^2 - 4ac = 6^2 - 4(1)(-10)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(1.36, -7.36\)

Q129. Find the roots of the equation \(x^2 + -6x + 11 = 0\).

A) N/A (1)
B) 7.47, -1.47
C) No real roots
D) N/A (2)

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -6x + 11 = 0\), here \(a=1, b=-6, c=11\).

Discriminant \(D = b^2 - 4ac = -6^2 - 4(1)(11)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(No real roots\)

Q130. Find the roots of the equation \(x^2 + 12x + 4 = 0\).

A) -0.69, -23.31
B) 0.32, -12.32
C) 11.66, 0.34
D) -0.34, -11.66

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 12x + 4 = 0\), here \(a=1, b=12, c=4\).

Discriminant \(D = b^2 - 4ac = 12^2 - 4(1)(4)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(-0.34, -11.66\)

Q131. Find the roots of the equation \(x^2 + -5x + 18 = 0\).

A) 7.42, -2.42
B) N/A (2)
C) No real roots
D) N/A (1)

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -5x + 18 = 0\), here \(a=1, b=-5, c=18\).

Discriminant \(D = b^2 - 4ac = -5^2 - 4(1)(18)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(No real roots\)

Q132. Find the roots of the equation \(x^2 + 7x + 17 = 0\).

A) N/A (2)
B) 1.91, -8.91
C) No real roots
D) N/A (1)

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 7x + 17 = 0\), here \(a=1, b=7, c=17\).

Discriminant \(D = b^2 - 4ac = 7^2 - 4(1)(17)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(No real roots\)

Q133. Find the roots of the equation \(x^2 + 3x + 10 = 0\).

A) No real roots
B) N/A (2)
C) 2.0, -5.0
D) N/A (1)

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 3x + 10 = 0\), here \(a=1, b=3, c=10\).

Discriminant \(D = b^2 - 4ac = 3^2 - 4(1)(10)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(No real roots\)

Q134. Solve the pair of linear equations: \(2x + 5y = 24\) \(4x + 1y = 14\)

A) 2.56, 3.78
B) -2.09, -3.09
C) -5.22, -6.89
D) 12.0, 3.5

Difficulty: Medium · Topic: Solving simultaneous linear equations

Solution

Using elimination/substitution method:

\(2x + 5y = 24\) ... (1)

\(4x + 1y = 14\) ... (2)

Solution: \(x, y = 2.56, 3.78\)

Q135. Solve the pair of linear equations: \(1x + 2y = 8\) \(3x + 1y = 9\)

A) -1.43, -2.14
B) 8.0, 3.0
C) 2.0, 3.0
D) -5.2, -6.6

Difficulty: Medium · Topic: Solving simultaneous linear equations

Solution

Using elimination/substitution method:

\(1x + 2y = 8\) ... (1)

\(3x + 1y = 9\) ... (2)

Solution: \(x, y = 2.0, 3.0\)

Q136. Solve the pair of linear equations: \(3x + 2y = 19\) \(1x + 4y = 17\)

A) 11.0, 7.0
B) 4.2, 3.2
C) 3.0, 2.29
D) 6.33, 17.0

Difficulty: Medium · Topic: Solving simultaneous linear equations

Solution

Using elimination/substitution method:

\(3x + 2y = 19\) ... (1)

\(1x + 4y = 17\) ... (2)

Solution: \(x, y = 4.2, 3.2\)

Q137. Solve the pair of linear equations: \(3x + 4y = 25\) \(2x + 5y = 26\)

A) 32.71, 18.29
B) 0.91, 1.22
C) 3.0, 4.0
D) 8.33, 13.0

Difficulty: Medium · Topic: Solving simultaneous linear equations

Solution

Using elimination/substitution method:

\(3x + 4y = 25\) ... (1)

\(2x + 5y = 26\) ... (2)

Solution: \(x, y = 3.0, 4.0\)

Q138. Solve the pair of linear equations: \(4x + 1y = 14\) \(1x + 3y = 13\)

A) 2.23, 2.92
B) 2.64, 3.45
C) 3.5, 13.0
D) 5.0, 6.0

Difficulty: Medium · Topic: Solving simultaneous linear equations

Solution

Using elimination/substitution method:

\(4x + 1y = 14\) ... (1)

\(1x + 3y = 13\) ... (2)

Solution: \(x, y = 2.64, 3.45\)

Q139. Solve the pair of linear equations: \(5x + 1y = 22\) \(2x + 3y = 17\)

A) 6.38, 9.92
B) 2.88, 2.41
C) 3.77, 3.15
D) 4.4, 8.5

Difficulty: Medium · Topic: Solving simultaneous linear equations

Solution

Using elimination/substitution method:

\(5x + 1y = 22\) ... (1)

\(2x + 3y = 17\) ... (2)

Solution: \(x, y = 3.77, 3.15\)

Q140. Solve the pair of linear equations: \(1x + 3y = 10\) \(2x + 1y = 7\)

A) -6.2, -5.4
B) 10.0, 3.5
C) 2.2, 2.6
D) -1.57, -1.86

Difficulty: Medium · Topic: Solving simultaneous linear equations

Solution

Using elimination/substitution method:

\(1x + 3y = 10\) ... (1)

\(2x + 1y = 7\) ... (2)

Solution: \(x, y = 2.2, 2.6\)

Q141. Solve the pair of linear equations: \(1x + 1y = 7\) \(2x + 3y = 16\)

A) 5.0, 2.0
B) 7.0, 8.0
C) 37.0, 30.0
D) 1.0, 0.4

Difficulty: Medium · Topic: Solving simultaneous linear equations

Solution

Using elimination/substitution method:

\(1x + 1y = 7\) ... (1)

\(2x + 3y = 16\) ... (2)

Solution: \(x, y = 5.0, 2.0\)

Q142. Solve the pair of linear equations: \(5x + 2y = 19\) \(3x + 4y = 21\)

A) 1.31, 1.85
B) 2.43, 3.43
C) 8.43, 11.57
D) 3.8, 7.0

Difficulty: Medium · Topic: Solving simultaneous linear equations

Solution

Using elimination/substitution method:

\(5x + 2y = 19\) ... (1)

\(3x + 4y = 21\) ... (2)

Solution: \(x, y = 2.43, 3.43\)

Q143. Solve the pair of linear equations: \(2x + 3y = 13\) \(3x + 2y = 12\)

A) -12.4, -12.6
B) -0.77, -1.15
C) 6.5, 4.0
D) 2.0, 3.0

Difficulty: Medium · Topic: Solving simultaneous linear equations

Solution

Using elimination/substitution method:

\(2x + 3y = 13\) ... (1)

\(3x + 2y = 12\) ... (2)

Solution: \(x, y = 2.0, 3.0\)

Q144. Find the roots of the equation \(x^2 + 0x + -11 = 0\).

A) 3.32, -3.32
B) No real roots
C) N/A (1)
D) 6.63, -6.63

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 0x + -11 = 0\), here \(a=1, b=0, c=-11\).

Discriminant \(D = b^2 - 4ac = 0^2 - 4(1)(-11)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(3.32, -3.32\)

Q145. Find the roots of the equation \(x^2 + -8x + 1 = 0\).

A) 15.75, 0.25
B) 7.87, 0.13
C) 8.12, -0.12
D) -0.13, -7.87

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -8x + 1 = 0\), here \(a=1, b=-8, c=1\).

Discriminant \(D = b^2 - 4ac = -8^2 - 4(1)(1)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(7.87, 0.13\)

Q146. Find the roots of the equation \(x^2 + 0x + 1 = 0\).

A) N/A (1)
B) 1.0, -1.0
C) N/A (2)
D) No real roots

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 0x + 1 = 0\), here \(a=1, b=0, c=1\).

Discriminant \(D = b^2 - 4ac = 0^2 - 4(1)(1)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(No real roots\)

Q147. Find the roots of the equation \(x^2 + -11x + 5 = 0\).

A) 11.44, -0.44
B) 10.52, 0.48
C) -0.48, -10.52
D) 21.05, 0.95

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -11x + 5 = 0\), here \(a=1, b=-11, c=5\).

Discriminant \(D = b^2 - 4ac = -11^2 - 4(1)(5)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(10.52, 0.48\)

Q148. Find the roots of the equation \(x^2 + 1x + 2 = 0\).

A) 1.0, -2.0
B) N/A (2)
C) N/A (1)
D) No real roots

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 1x + 2 = 0\), here \(a=1, b=1, c=2\).

Discriminant \(D = b^2 - 4ac = 1^2 - 4(1)(2)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(No real roots\)

Q149. Find the roots of the equation \(x^2 + -8x + -17 = 0\).

A) 19.49, -3.49
B) 9.74, -1.74
C) 1.74, -9.74
D) No real roots

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -8x + -17 = 0\), here \(a=1, b=-8, c=-17\).

Discriminant \(D = b^2 - 4ac = -8^2 - 4(1)(-17)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(9.74, -1.74\)

Q150. Find the roots of the equation \(x^2 + 6x + 14 = 0\).

A) N/A (2)
B) N/A (1)
C) 1.8, -7.8
D) No real roots

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 6x + 14 = 0\), here \(a=1, b=6, c=14\).

Discriminant \(D = b^2 - 4ac = 6^2 - 4(1)(14)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(No real roots\)

Q151. Find the roots of the equation \(x^2 + 10x + -3 = 0\).

A) 0.58, -20.58
B) 10.29, -0.29
C) 0.29, -10.29
D) -0.31, -9.69

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 10x + -3 = 0\), here \(a=1, b=10, c=-3\).

Discriminant \(D = b^2 - 4ac = 10^2 - 4(1)(-3)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(0.29, -10.29\)

Q152. Find the roots of the equation \(x^2 + 5x + 7 = 0\).

A) N/A (2)
B) No real roots
C) N/A (1)
D) 1.14, -6.14

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 5x + 7 = 0\), here \(a=1, b=5, c=7\).

Discriminant \(D = b^2 - 4ac = 5^2 - 4(1)(7)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(No real roots\)

Q153. Find the roots of the equation \(x^2 + 6x + 0 = 0\).

A) 6.0, 0.0
B) 0.0, -6.0
C) 0.0, -12.0
D) N/A (1)

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 6x + 0 = 0\), here \(a=1, b=6, c=0\).

Discriminant \(D = b^2 - 4ac = 6^2 - 4(1)(0)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(0.0, -6.0\)

Q154. Find the roots of the equation \(x^2 + 1x + 19 = 0\).

A) N/A (2)
B) 3.89, -4.89
C) N/A (1)
D) No real roots

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 1x + 19 = 0\), here \(a=1, b=1, c=19\).

Discriminant \(D = b^2 - 4ac = 1^2 - 4(1)(19)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(No real roots\)

Q155. Find the roots of the equation \(x^2 + -10x + 10 = 0\).

A) 17.75, 2.25
B) 8.87, 1.13
C) 10.92, -0.92
D) -1.13, -8.87

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -10x + 10 = 0\), here \(a=1, b=-10, c=10\).

Discriminant \(D = b^2 - 4ac = -10^2 - 4(1)(10)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(8.87, 1.13\)

Q156. Find the roots of the equation \(x^2 + -3x + 14 = 0\).

A) N/A (2)
B) No real roots
C) N/A (1)
D) 5.53, -2.53

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -3x + 14 = 0\), here \(a=1, b=-3, c=14\).

Discriminant \(D = b^2 - 4ac = -3^2 - 4(1)(14)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(No real roots\)

Q157. Find the roots of the equation \(x^2 + 10x + 13 = 0\).

A) 8.46, 1.54
B) 1.16, -11.16
C) -3.07, -16.93
D) -1.54, -8.46

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 10x + 13 = 0\), here \(a=1, b=10, c=13\).

Discriminant \(D = b^2 - 4ac = 10^2 - 4(1)(13)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(-1.54, -8.46\)

Q158. Find the roots of the equation \(x^2 + 12x + -1 = 0\).

A) 0.08, -12.08
B) 0.17, -24.17
C) -0.08, -11.92
D) 12.08, -0.08

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 12x + -1 = 0\), here \(a=1, b=12, c=-1\).

Discriminant \(D = b^2 - 4ac = 12^2 - 4(1)(-1)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(0.08, -12.08\)

Q159. Find the roots of the equation \(x^2 + -8x + -11 = 0\).

A) 1.2, -9.2
B) 6.24, 1.76
C) 18.39, -2.39
D) 9.2, -1.2

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -8x + -11 = 0\), here \(a=1, b=-8, c=-11\).

Discriminant \(D = b^2 - 4ac = -8^2 - 4(1)(-11)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(9.2, -1.2\)

Q160. Find the roots of the equation \(x^2 + 2x + -4 = 0\).

A) 2.47, -6.47
B) 3.24, -1.24
C) No real roots
D) 1.24, -3.24

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 2x + -4 = 0\), here \(a=1, b=2, c=-4\).

Discriminant \(D = b^2 - 4ac = 2^2 - 4(1)(-4)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(1.24, -3.24\)

Q161. Find the roots of the equation \(x^2 + -10x + 1 = 0\).

A) 19.8, 0.2
B) 10.1, -0.1
C) 9.9, 0.1
D) -0.1, -9.9

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -10x + 1 = 0\), here \(a=1, b=-10, c=1\).

Discriminant \(D = b^2 - 4ac = -10^2 - 4(1)(1)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(9.9, 0.1\)

Q162. Find the roots of the equation \(x^2 + 10x + -19 = 0\).

A) -2.55, -7.45
B) 3.27, -23.27
C) 1.63, -11.63
D) 11.63, -1.63

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 10x + -19 = 0\), here \(a=1, b=10, c=-19\).

Discriminant \(D = b^2 - 4ac = 10^2 - 4(1)(-19)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(1.63, -11.63\)

Q163. Find the roots of the equation \(x^2 + -11x + 14 = 0\).

A) -1.47, -9.53
B) 12.15, -1.15
C) 19.06, 2.94
D) 9.53, 1.47

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -11x + 14 = 0\), here \(a=1, b=-11, c=14\).

Discriminant \(D = b^2 - 4ac = -11^2 - 4(1)(14)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(9.53, 1.47\)

Q164. Find the roots of the equation \(x^2 + -10x + -17 = 0\).

A) 1.48, -11.48
B) 7.83, 2.17
C) 22.96, -2.96
D) 11.48, -1.48

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -10x + -17 = 0\), here \(a=1, b=-10, c=-17\).

Discriminant \(D = b^2 - 4ac = -10^2 - 4(1)(-17)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(11.48, -1.48\)

Q165. Find the roots of the equation \(x^2 + -4x + 16 = 0\).

A) No real roots
B) 6.47, -2.47
C) N/A (1)
D) N/A (2)

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -4x + 16 = 0\), here \(a=1, b=-4, c=16\).

Discriminant \(D = b^2 - 4ac = -4^2 - 4(1)(16)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(No real roots\)

Q166. Find the roots of the equation \(x^2 + -10x + -11 = 0\).

A) 11.0, -1.0
B) 8.74, 1.26
C) 1.0, -11.0
D) 22.0, -2.0

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -10x + -11 = 0\), here \(a=1, b=-10, c=-11\).

Discriminant \(D = b^2 - 4ac = -10^2 - 4(1)(-11)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(11.0, -1.0\)

Q167. Find the roots of the equation \(x^2 + 5x + -3 = 0\).

A) 1.08, -11.08
B) -0.7, -4.3
C) 0.54, -5.54
D) 5.54, -0.54

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 5x + -3 = 0\), here \(a=1, b=5, c=-3\).

Discriminant \(D = b^2 - 4ac = 5^2 - 4(1)(-3)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(0.54, -5.54\)

Q168. Find the roots of the equation \(x^2 + -9x + -5 = 0\).

A) 8.41, 0.59
B) 0.52, -9.52
C) 9.52, -0.52
D) 19.05, -1.05

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -9x + -5 = 0\), here \(a=1, b=-9, c=-5\).

Discriminant \(D = b^2 - 4ac = -9^2 - 4(1)(-5)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(9.52, -0.52\)

Q169. Find the roots of the equation \(x^2 + -6x + 10 = 0\).

A) N/A (2)
B) N/A (1)
C) No real roots
D) 7.36, -1.36

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -6x + 10 = 0\), here \(a=1, b=-6, c=10\).

Discriminant \(D = b^2 - 4ac = -6^2 - 4(1)(10)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(No real roots\)

Q170. Find the roots of the equation \(x^2 + 4x + -8 = 0\).

A) 5.46, -1.46
B) 2.93, -10.93
C) 1.46, -5.46
D) No real roots

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 4x + -8 = 0\), here \(a=1, b=4, c=-8\).

Discriminant \(D = b^2 - 4ac = 4^2 - 4(1)(-8)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(1.46, -5.46\)

Q171. Find the roots of the equation \(x^2 + 0x + 13 = 0\).

A) N/A (1)
B) N/A (2)
C) No real roots
D) 3.61, -3.61

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 0x + 13 = 0\), here \(a=1, b=0, c=13\).

Discriminant \(D = b^2 - 4ac = 0^2 - 4(1)(13)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(No real roots\)

Q172. Find the roots of the equation \(x^2 + -7x + 9 = 0\).

A) 5.3, 1.7
B) -1.7, -5.3
C) 8.11, -1.11
D) 10.61, 3.39

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -7x + 9 = 0\), here \(a=1, b=-7, c=9\).

Discriminant \(D = b^2 - 4ac = -7^2 - 4(1)(9)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(5.3, 1.7\)

Q173. Find the roots of the equation \(x^2 + 4x + 20 = 0\).

A) N/A (1)
B) No real roots
C) 2.9, -6.9
D) N/A (2)

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 4x + 20 = 0\), here \(a=1, b=4, c=20\).

Discriminant \(D = b^2 - 4ac = 4^2 - 4(1)(20)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(No real roots\)

Q174. Find the roots of the equation \(x^2 + 4x + 19 = 0\).

A) N/A (1)
B) N/A (2)
C) 2.8, -6.8
D) No real roots

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 4x + 19 = 0\), here \(a=1, b=4, c=19\).

Discriminant \(D = b^2 - 4ac = 4^2 - 4(1)(19)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(No real roots\)

Q175. Find the roots of the equation \(x^2 + 4x + 2 = 0\).

A) -1.17, -6.83
B) -0.59, -3.41
C) 3.41, 0.59
D) 0.45, -4.45

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 4x + 2 = 0\), here \(a=1, b=4, c=2\).

Discriminant \(D = b^2 - 4ac = 4^2 - 4(1)(2)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(-0.59, -3.41\)

Q176. Find the roots of the equation \(x^2 + -2x + -17 = 0\).

A) 3.24, -5.24
B) 5.24, -3.24
C) 10.49, -6.49
D) No real roots

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -2x + -17 = 0\), here \(a=1, b=-2, c=-17\).

Discriminant \(D = b^2 - 4ac = -2^2 - 4(1)(-17)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(5.24, -3.24\)

Q177. Find the roots of the equation \(x^2 + 7x + -3 = 0\).

A) 0.41, -7.41
B) 0.81, -14.81
C) 7.41, -0.41
D) -0.46, -6.54

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 7x + -3 = 0\), here \(a=1, b=7, c=-3\).

Discriminant \(D = b^2 - 4ac = 7^2 - 4(1)(-3)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(0.41, -7.41\)

Q178. Find the roots of the equation \(x^2 + 8x + 20 = 0\).

A) 2.0, -10.0
B) No real roots
C) N/A (1)
D) N/A (2)

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 8x + 20 = 0\), here \(a=1, b=8, c=20\).

Discriminant \(D = b^2 - 4ac = 8^2 - 4(1)(20)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(No real roots\)

Q179. Find the roots of the equation \(x^2 + 3x + -3 = 0\).

A) No real roots
B) 1.58, -7.58
C) 3.79, -0.79
D) 0.79, -3.79

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 3x + -3 = 0\), here \(a=1, b=3, c=-3\).

Discriminant \(D = b^2 - 4ac = 3^2 - 4(1)(-3)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(0.79, -3.79\)

Q180. Find the roots of the equation \(x^2 + 1x + 14 = 0\).

A) N/A (1)
B) N/A (2)
C) 3.27, -4.27
D) No real roots

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 1x + 14 = 0\), here \(a=1, b=1, c=14\).

Discriminant \(D = b^2 - 4ac = 1^2 - 4(1)(14)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(No real roots\)

Q181. Find the roots of the equation \(x^2 + 1x + -11 = 0\).

A) 5.71, -7.71
B) No real roots
C) 3.85, -2.85
D) 2.85, -3.85

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 1x + -11 = 0\), here \(a=1, b=1, c=-11\).

Discriminant \(D = b^2 - 4ac = 1^2 - 4(1)(-11)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(2.85, -3.85\)

Q182. Find the roots of the equation \(x^2 + 4x + -2 = 0\).

A) -0.59, -3.41
B) 0.45, -4.45
C) 0.9, -8.9
D) 4.45, -0.45

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 4x + -2 = 0\), here \(a=1, b=4, c=-2\).

Discriminant \(D = b^2 - 4ac = 4^2 - 4(1)(-2)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(0.45, -4.45\)

Q183. Find the roots of the equation \(x^2 + 1x + -20 = 0\).

A) 4.0, -5.0
B) No real roots
C) 5.0, -4.0
D) 8.0, -10.0

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 1x + -20 = 0\), here \(a=1, b=1, c=-20\).

Discriminant \(D = b^2 - 4ac = 1^2 - 4(1)(-20)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(4.0, -5.0\)

Q184. Find the roots of the equation \(x^2 + -12x + 0 = 0\).

A) 12.0, 0.0
B) N/A (1)
C) 24.0, 0.0
D) 0.0, -12.0

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -12x + 0 = 0\), here \(a=1, b=-12, c=0\).

Discriminant \(D = b^2 - 4ac = -12^2 - 4(1)(0)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(12.0, 0.0\)

Q185. Find the roots of the equation \(x^2 + -12x + 15 = 0\).

A) 10.58, 1.42
B) 13.14, -1.14
C) 21.17, 2.83
D) -1.42, -10.58

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -12x + 15 = 0\), here \(a=1, b=-12, c=15\).

Discriminant \(D = b^2 - 4ac = -12^2 - 4(1)(15)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(10.58, 1.42\)

Q186. Find the roots of the equation \(x^2 + 9x + -3 = 0\).

A) 0.64, -18.64
B) 9.32, -0.32
C) 0.32, -9.32
D) -0.35, -8.65

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 9x + -3 = 0\), here \(a=1, b=9, c=-3\).

Discriminant \(D = b^2 - 4ac = 9^2 - 4(1)(-3)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(0.32, -9.32\)

Q187. Find the roots of the equation \(x^2 + 12x + 16 = 0\).

A) 1.21, -13.21
B) 10.47, 1.53
C) -3.06, -20.94
D) -1.53, -10.47

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 12x + 16 = 0\), here \(a=1, b=12, c=16\).

Discriminant \(D = b^2 - 4ac = 12^2 - 4(1)(16)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(-1.53, -10.47\)

Q188. Find the roots of the equation \(x^2 + -12x + 2 = 0\).

A) 12.16, -0.16
B) -0.17, -11.83
C) 11.83, 0.17
D) 23.66, 0.34

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -12x + 2 = 0\), here \(a=1, b=-12, c=2\).

Discriminant \(D = b^2 - 4ac = -12^2 - 4(1)(2)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(11.83, 0.17\)

Q189. Find the roots of the equation \(x^2 + -1x + 10 = 0\).

A) N/A (2)
B) No real roots
C) N/A (1)
D) 3.7, -2.7

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -1x + 10 = 0\), here \(a=1, b=-1, c=10\).

Discriminant \(D = b^2 - 4ac = -1^2 - 4(1)(10)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(No real roots\)

Q190. Find the roots of the equation \(x^2 + 1x + -18 = 0\).

A) 4.77, -3.77
B) No real roots
C) 3.77, -4.77
D) 7.54, -9.54

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 1x + -18 = 0\), here \(a=1, b=1, c=-18\).

Discriminant \(D = b^2 - 4ac = 1^2 - 4(1)(-18)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(3.77, -4.77\)

Q191. Find the roots of the equation \(x^2 + 7x + 13 = 0\).

A) 1.52, -8.52
B) N/A (1)
C) No real roots
D) N/A (2)

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 7x + 13 = 0\), here \(a=1, b=7, c=13\).

Discriminant \(D = b^2 - 4ac = 7^2 - 4(1)(13)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(No real roots\)

Q192. Find the roots of the equation \(x^2 + -5x + 6 = 0\).

A) 6.0, 4.0
B) 3.0, 2.0
C) 6.0, -1.0
D) -2.0, -3.0

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -5x + 6 = 0\), here \(a=1, b=-5, c=6\).

Discriminant \(D = b^2 - 4ac = -5^2 - 4(1)(6)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(3.0, 2.0\)

Q193. Find the roots of the equation \(x^2 + -8x + 3 = 0\).

A) 7.61, 0.39
B) -0.39, -7.61
C) 15.21, 0.79
D) 8.36, -0.36

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -8x + 3 = 0\), here \(a=1, b=-8, c=3\).

Discriminant \(D = b^2 - 4ac = -8^2 - 4(1)(3)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(7.61, 0.39\)

Q194. Find the roots of the equation \(x^2 + 6x + 5 = 0\).

A) -2.0, -10.0
B) 0.74, -6.74
C) -1.0, -5.0
D) 5.0, 1.0

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 6x + 5 = 0\), here \(a=1, b=6, c=5\).

Discriminant \(D = b^2 - 4ac = 6^2 - 4(1)(5)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(-1.0, -5.0\)

Q195. Find the roots of the equation \(x^2 + 9x + -11 = 0\).

A) -1.46, -7.54
B) 1.09, -10.09
C) 2.18, -20.18
D) 10.09, -1.09

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 9x + -11 = 0\), here \(a=1, b=9, c=-11\).

Discriminant \(D = b^2 - 4ac = 9^2 - 4(1)(-11)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(1.09, -10.09\)

Q196. Find the roots of the equation \(x^2 + 2x + -6 = 0\).

A) 3.29, -7.29
B) 1.65, -3.65
C) No real roots
D) 3.65, -1.65

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 2x + -6 = 0\), here \(a=1, b=2, c=-6\).

Discriminant \(D = b^2 - 4ac = 2^2 - 4(1)(-6)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(1.65, -3.65\)

Q197. Find the roots of the equation \(x^2 + -6x + -9 = 0\).

A) 1.24, -7.24
B) 3.0, 3.0
C) 14.49, -2.49
D) 7.24, -1.24

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -6x + -9 = 0\), here \(a=1, b=-6, c=-9\).

Discriminant \(D = b^2 - 4ac = -6^2 - 4(1)(-9)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(7.24, -1.24\)

Q198. Find the roots of the equation \(x^2 + 10x + 10 = 0\).

A) -2.25, -17.75
B) 8.87, 1.13
C) -1.13, -8.87
D) 0.92, -10.92

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 10x + 10 = 0\), here \(a=1, b=10, c=10\).

Discriminant \(D = b^2 - 4ac = 10^2 - 4(1)(10)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(-1.13, -8.87\)

Q199. Find the roots of the equation \(x^2 + 9x + -19 = 0\).

A) 1.76, -10.76
B) 10.76, -1.76
C) 3.53, -21.53
D) -3.38, -5.62

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + 9x + -19 = 0\), here \(a=1, b=9, c=-19\).

Discriminant \(D = b^2 - 4ac = 9^2 - 4(1)(-19)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(1.76, -10.76\)

Q200. Find the roots of the equation \(x^2 + -6x + -12 = 0\).

A) 1.58, -7.58
B) 15.17, -3.17
C) 7.58, -1.58
D) No real roots

Difficulty: Medium · Topic: Solving using quadratic formula

Solution

For \(x^2 + -6x + -12 = 0\), here \(a=1, b=-6, c=-12\).

Discriminant \(D = b^2 - 4ac = -6^2 - 4(1)(-12)\)

\(x = \frac{-b \pm \sqrt{D}}{2a}\)

Roots: \(7.58, -1.58\)

Q201. Solve: 6x+3y=6xy, 2x+4y=5xy (x≠0, y≠0).

Difficulty: Medium-Hard · Topic: Equations Reducible to Linear Equations

Solution

Divide by xy: 6/y+3/x=6, 2/y+4/x=5. Let u=1/x, v=1/y.

3u+6v=6→u+2v=2, 4u+2v=5. Subtract: 3u=3→u=1→x=1, v=1/2→y=2.

Q202. Find a, b so that 2x+3y=7, (a−b)x+(a+b)y=3a+b−2 has infinitely many solutions.

Difficulty: Medium-Hard · Topic: Consistency of System

Solution

2/(a−b)=3/(a+b)→a=5b. 2/(a−b)=7/(3a+b−2)→a=9b−4. Solving: b=1, a=5.

Q203. A and B are 100 km apart. Cars start simultaneously. Same direction: meet in 5 h. Opposite: meet in 1 h. Find speeds.

Difficulty: Medium-Hard · Topic: Substitution Method

Solution

x−y=20 (same dir), x+y=100 (opposite). x=60, y=40.

Q204. Solve: 99x+101y=499, 101x+99y=501.

Difficulty: Hard · Topic: Elimination Method

Solution

Add: 200x+200y=1000→x+y=5. Subtract: 2x−2y=2→x−y=1. x=3, y=2.

Q205. Solve: (a−b)x+(a+b)y = a²−2ab−b², (a+b)(x+y) = a²+b².

Difficulty: Hard · Topic: Equations Reducible to Linear Equations

Solution

Eq.2: (a+b)x+(a+b)y = a²+b². Subtract eq.1: 2bx = 2b(a+b) → x = a+b.

(a+b)y = a²+b²−(a+b)² = −2ab → y = −2ab/(a+b).

Other Chapters in Mathematics

Ch 1: Real Numbers Ch 2: Polynomials Ch 4: Quadratic Equations Ch 5: Arithmetic Progressions Ch 6: Triangles Ch 7: Coordinate Geometry Ch 8: Introduction to Trigonometry Ch 9: Some Applications of Trigonometry Ch 10: Circles Ch 11: Areas Related to Circles Ch 12: Surface Areas and Volumes Ch 13: Statistics Ch 14: Probability

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