Linear Equations
▶1. Solve for x: 3x - 7 = 11
Answer: 6
Explanation: 3x = 18 ⇒ x = 6.
▶2. If 4x + 5 = 29, what is x?
Answer: 6
Explanation: 4x = 24 ⇒ x = 6.
▶3. The sum of a number and 8 is 20. Find the number.
Answer: 12
Explanation: x + 8 = 20 ⇒ x = 12.
▶4. If 2x - 3 = 5x + 9, what is x?
Answer: -4
Explanation: 2x - 5x = 9 + 3 ⇒ -3x = 12 ⇒ x = -4.
▶5. If 7x = 2x + 25, find x.
Answer: 5
Explanation: 7x - 2x = 25 ⇒ 5x = 25 ⇒ x = 5.
▶6. If 5x - 2 = 3x + 10, what is x?
Answer: 6
Explanation: 5x - 3x = 10 + 2 ⇒ 2x = 12 ⇒ x = 6.
▶7. If 2x + 3 = 3x - 4, what is x?
Answer: 7
Explanation: 3x - 2x = 3 + 4 ⇒ x = 7.
▶8. If 4x - 5 = 3x + 2, what is x?
Answer: 7
Explanation: 4x - 3x = 2 + 5 ⇒ x = 7.
▶9. If 6x + 2 = 3x + 20, what is x?
Answer: 6
Explanation: 6x - 3x = 20 - 2 ⇒ 3x = 18 ⇒ x = 6.
▶10. If 8x - 3 = 5x + 12, what is x?
Answer: 5
Explanation: 8x - 5x = 12 + 3 ⇒ 3x = 15 ⇒ x = 5.
▶11. If 9x + 4 = 7x + 18, what is x?
Answer: 7
Explanation: 9x - 7x = 18 - 4 ⇒ 2x = 14 ⇒ x = 7.
▶12. If 10x - 5 = 3x + 16, what is x?
Answer: 3
Explanation: 10x - 3x = 16 + 5 ⇒ 7x = 21 ⇒ x = 3.
▶13. If 11x + 7 = 8x + 22, what is x?
Answer: 5
Explanation: 11x - 8x = 22 - 7 ⇒ 3x = 15 ⇒ x = 5.
▶14. If 12x - 8 = 4x + 24, what is x?
Answer: 4
Explanation: 12x - 4x = 24 + 8 ⇒ 8x = 32 ⇒ x = 4.
▶15. If 13x + 9 = 7x + 33, what is x?
Answer: 4
Explanation: 13x - 7x = 33 - 9 ⇒ 6x = 24 ⇒ x = 4.
Quadratic Equations
▶16. Find the roots of x² - 6x + 9 = 0
Answer: 3 (double root)
Explanation: (x-3)²=0 ⇒ x=3.
▶17. If x² - 8x + 15 = 0, what are the roots?
Answer: 5, 3
Explanation: (x-5)(x-3)=0 ⇒ x=5,3.
▶18. If x² + 2x + 1 = 0, what is x?
Answer: -1 (double root)
Explanation: (x+1)²=0 ⇒ x=-1.
▶19. If x² - 10x + 21 = 0, what are the roots?
Answer: 7, 3
Explanation: (x-7)(x-3)=0 ⇒ x=7,3.
▶20. If x² - 4x + 3 = 0, what are the roots?
Answer: 3, 1
Explanation: (x-3)(x-1)=0 ⇒ x=3,1.
▶21. If x² - 9x + 20 = 0, what are the roots?
Answer: 5, 4
Explanation: (x-5)(x-4)=0 ⇒ x=5,4.
▶22. If x² - 12x + 36 = 0, what is x?
Answer: 6 (double root)
Explanation: (x-6)²=0 ⇒ x=6.
▶23. If x² - 5x + 6 = 0, what are the roots?
Answer: 3, 2
Explanation: (x-3)(x-2)=0 ⇒ x=3,2.
▶24. If x² - 7x + 12 = 0, what are the roots?
Answer: 4, 3
Explanation: (x-4)(x-3)=0 ⇒ x=4,3.
▶25. If x² - 13x + 42 = 0, what are the roots?
Answer: 7, 6
Explanation: (x-7)(x-6)=0 ⇒ x=7,6.
▶26. If x² - 14x + 49 = 0, what is x?
Answer: 7 (double root)
Explanation: (x-7)²=0 ⇒ x=7.
▶27. If x² - 15x + 56 = 0, what are the roots?
Answer: 8, 7
Explanation: (x-8)(x-7)=0 ⇒ x=8,7.
▶28. If x² - 16x + 64 = 0, what is x?
Answer: 8 (double root)
Explanation: (x-8)²=0 ⇒ x=8.
▶29. If x² - 17x + 72 = 0, what are the roots?
Answer: 9, 8
Explanation: (x-9)(x-8)=0 ⇒ x=9,8.
▶30. If x² - 18x + 81 = 0, what is x?
Answer: 9 (double root)
Explanation: (x-9)²=0 ⇒ x=9.
Simultaneous Equations
▶31. Solve: x + y = 12, x - y = 4
Answer: x=8, y=4
Explanation: Add: 2x=16 ⇒ x=8; then y=4.
▶32. If 2x + 3y = 13 and 3x + 4y = 18, find x and y.
Answer: x=2, y=3
Explanation: Multiply first by 4, second by 3: 8x+12y=52, 9x+12y=54. Subtract: x=2, then y=3.
▶33. Solve: 3x - 2y = 7, 2x + y = 4
Answer: x=2, y=0
Explanation: y=4-2x; substitute in first: 3x-2(4-2x)=7 ⇒ 3x-8+4x=7 ⇒ 7x=15 ⇒ x=15/7, y=4-2(15/7)=4-30/7= (28-30)/7= -2/7.
▶34. If x + 2y = 8 and 2x - y = 3, what are x and y?
Answer: x=3, y=2.5
Explanation: y=(8-x)/2; 2x-(8-x)/2=3 ⇒ 2x-4+x/2=3 ⇒ (4x+x)/2=7 ⇒ 5x=14 ⇒ x=2.8, y=2.6.
▶35. If 4x + 5y = 23 and 3x - 2y = 4, find x and y.
Answer: x=3, y=2
Explanation: Multiply second by 5: 15x-10y=20; first by 2: 8x+10y=46. Add: 23x=66 ⇒ x=66/23, y=(23-4x)/5.
▶36. If 2x + y = 7 and x - y = 1, what are x and y?
Answer: x=2.67, y=1.67
Explanation: x=1+y; 2(1+y)+y=7 ⇒ 2+2y+y=7 ⇒ 3y=5 ⇒ y=1.67, x=2.67.
▶37. If x + y + z = 6, x - y + z = 2, x + y - z = 4, find x, y, z.
Answer: x=2, y=2, z=2
Explanation: Add all: 3x=6+2+4=12 ⇒ x=4; then y=2, z=2.
▶38. If 2x + 3y - z = 7, x - y + 2z = 4, x + 2y + z = 9, find x, y, z.
Answer: x=1, y=2, z=3
Explanation: Solve by elimination or substitution.
▶39. If x + y = 5, x - y = 1, what are x and y?
Answer: x=3, y=2
Explanation: Add: 2x=6 ⇒ x=3; then y=2.
▶40. If 3x + 2y = 12, 2x - y = 3, find x and y.
Answer: x=2, y=3
Explanation: Substitute y=3 in first: 3x+6=12 ⇒ 3x=6 ⇒ x=2.
▶41. If x + 2y + 3z = 14, 2x + 3y + z = 13, 3x + y + 2z = 13, find x, y, z.
Answer: x=2, y=3, z=2
Explanation: Solve by elimination or substitution.
▶42. If x + y = 10, x - y = 2, what are x and y?
Answer: x=6, y=4
Explanation: Add: 2x=12 ⇒ x=6; then y=4.
▶43. If 2x + 3y = 13, 3x + 4y = 18, what are x and y?
Answer: x=2, y=3
Explanation: Multiply first by 4, second by 3: 8x+12y=52, 9x+12y=54. Subtract: x=2, then y=3.
▶44. If x + y = 7, x - y = 1, what are x and y?
Answer: x=4, y=3
Explanation: Add: 2x=8 ⇒ x=4; then y=3.
▶45. If 2x + 3y = 12, 3x + 2y = 13, what are x and y?
Answer: x=2, y=2.67
Explanation: Substitute x=2 in first: 4+3y=12 ⇒ 3y=8 ⇒ y=2.67.
Advanced & CAT-Style
▶46. If the sum of two numbers is 20 and their difference is 4, what are the numbers?
Answer: 12, 8
Explanation: x+y=20, x-y=4 ⇒ x=12, y=8.
▶47. If the product of two numbers is 35 and their sum is 12, what are the numbers?
Answer: 5, 7
Explanation: x+y=12, xy=35 ⇒ x=5, y=7.
▶48. If the sum of the squares of two numbers is 25 and their product is 12, what is the sum of the numbers?
Answer: 7
Explanation: (x+y)² = x² + 2xy + y² = 25 + 24 = 49 ⇒ x+y=7.
▶49. If the sum of the cubes of two numbers is 35 and their sum is 5, what is their product?
Answer: 5
Explanation: x³+y³=35, x+y=5 ⇒ x³+y³=(x+y)³-3xy(x+y) ⇒ 35=125-15xy ⇒ xy=6.
▶50. If the sum of three consecutive integers is 51, what are the integers?
Answer: 16, 17, 18
Explanation: x+(x+1)+(x+2)=51 ⇒ 3x=48 ⇒ x=16.
CAT-Level Challengers
▶51. Solve for x: (3x - 2)/5 + (2x + 3)/3 = (4x + 7)/15 + 2
Answer: 28/15
Solution: Multiply both sides by 15: 3(3x-2) + 5(2x+3) = (4x+7) + 30. 9x - 6 + 10x + 15 = 4x + 37. 19x + 9 = 4x + 37. 15x = 28, so x = 28/15.
▶52. If one root of x^2 - p x + 8 = 0 is twice the other, find p.
Answer: 6 or -6
Solution: Let roots be a and 2a. a + 2a = p, so 3a = p. a * 2a = 8, so 2a^2 = 8, a^2 = 4, a = 2 or -2. If a = 2, p = 6. If a = -2, p = -6.
▶53. Solve: 2x + 3y = 7, 6x + 9y = k. For what value of k will the system have infinitely many solutions?
Answer: 21
Solution: The second equation is a multiple of the first if k = 3*7 = 21.
▶54. Find k such that x^2 - 4x + k = 0 has real and distinct roots.
Answer: k < 4
Solution: Discriminant D > 0: 16 - 4k > 0, so k < 4.
▶55. Ten years ago, A was twice as old as B. Five years from now, A will be 1.5 times as old as B. Find their current ages.
Answer: A = 40, B = 25
Solution: Let current ages be A and B. A - 10 = 2(B - 10), A + 5 = 1.5(B + 5). First: A = 2B - 10. Second: 2B - 10 + 5 = 1.5B + 7.5, so 0.5B = 12.5, B = 25. Then A = 2*25 - 10 = 40.
▶56. Solve x^3 - 7x^2 + 14x - 8 = 0, given that the roots are in geometric progression.
Answer: 1, 2, 4
Solution: Let the roots be a/r, a, and ar. Product: (a/r) * a * (ar) = 8, so a^3 = 8, so a = 2. Sum: (2/r) + 2 + 2r = 7. (2/r) + 2r = 5, so 2 + 2r^2 = 5r, so 2r^2 - 5r + 2 = 0. Solving, r = 2 or r = 1/2. So the roots are 1, 2, and 4.
▶57. Solve |2x - 5| ≤ 3.
Answer: x ∈ [1, 4]
Solution: -3 ≤ 2x - 5 ≤ 3 Add 5: 2 ≤ 2x ≤ 8 ⇒ 1 ≤ x ≤ 4
▶58. Find the minimum value of 2x^2 - 8x + 7.
Answer: -1
Solution: Minimum at x = 2. Value: 2(2)^2 - 8(2) + 7 = -1
▶59. Solve:
x + y = 7, \quad x^2 + y^2 = 25
Answer: (x,y) = (3,4), (4,3)
Solution: (x+y)^2 = 49 ⇒ x^2 + 2xy + y^2 = 49 ⇒ 25 + 2xy = 49 ⇒ xy = 12
Quadratic: t^2 - 7t + 12 = 0 ⇒ t = 3, 4
▶60. A two-digit number is 4 times the sum of its digits. If 18 is added, the number reverses. Find the number.
Answer: 24
Solution: Let number be 10a + b. 10a + b = 4(a + b) ⇒ 6a = 3b ⇒ 2a = b Also, 10a + b + 18 = 10b + a ⇒ a - b = -2 Substitute b = 2a: a = 2, b = 4
▶61. If x^2 - 5x + 6 = 0 and 2x^2 - 9x + k = 0 have a common root, find k.
Answer: k = 9 or 10
Solution: Roots of first: 2, 3. If common root 2: k=10; if 3: k=9.
▶62. For what a and b will the system have no solution?
2x + 3y = 7, \quad ax + by = 9
Answer: 2b = 3a and a ≠ 18/7
Solution: No solution if 2/a = 3/b ≠ 7/9. Set 2b = 3a, a ≠ 18/7.
▶63. If a(b-c)x^2 + b(c-a)x + c(a-b) = 0 has equal roots, show that a, b, c are in harmonic progression.
Answer: a, b, c are in HP
Solution: Discriminant D = 0: [b(c-a)]^2 - 4[a(b-c)] [c(a-b)] = 0 After simplification: 2/b = 1/a + 1/c, so HP.
▶64. A chemist has 20% and 50% acid solutions. How many liters of each to make 30L of 40% acid?
Answer: 10L of 20%, 20L of 50%
Solution: Let x L of 20%, (30-x) L of 50%. 0.2x + 0.5(30-x) = 0.4 * 30 0.2x + 15 - 0.5x = 12 ⇒ -0.3x = -3 ⇒ x = 10
▶65. Solve |x-1| + |x-2| = 3.
Answer: x = 0, 3
Solution: Critical points: x=1, 2. Case 1: x < 1: -(x-1) - (x-2) = 3 ⇒ x = 0 Case 3: x ≥ 2: (x-1) + (x-2) = 3 ⇒ x = 3
▶66. If α, β are roots of x^2 - 5x + 6 = 0, find α^2 + β^2.
Answer: 13
Solution: α + β = 5, αβ = 6 α^2 + β^2 = (α+β)^2 - 2αβ = 25 - 12 = 13
▶67. Solve:
x + y + z = 6, \quad 2x + y + 3z = 14, \quad -x + 2y + z = 0
Answer: x = 16/5, y = 2/5, z = 12/5
Solution: Solve the system using substitution and elimination as shown in the detailed solution above.
▶68. A and B complete a work in 12 days. A alone takes 10 days more than B alone. Find time taken by B alone.
Answer: 20 days
Solution: Let B take x days, then A takes x+10 days. 1/x + 1/(x+10) = 1/12 x^2 - 14x - 120 = 0 gives x = 20.
▶69. The graph of y = x^2 - 6x + 8 intersects the x-axis at points P and Q. Find the length of PQ.
Answer: 2 units
Solution: Roots: x=2,4. Distance: |4-2|=2.
▶70. Solve x^2 - 5x + 6 > 0.
Answer: x ∈ (-∞, 2) ∪ (3, ∞)
Solution: Roots: 2, 3. Parabola opens upwards. Positive when x < 2 or x > 3.
▶71. Solve:
log_2 x + log_4 y = 4, \quad log_4 x + log_2 y = 5
Answer: x = 4, y = 16
Solution: Set a = log_2 x, b = log_2 y. Solve the system to get x=4, y=16.
▶72. An item is sold at 10% profit. If cost decreases by 20% and selling price increases by 10%, profit increases by ₹24. Find original cost.
Answer: 2400/31
Solution: Let cost = C. Profit increase: 0.31C = 24 ⇒ C = 2400/31.
▶73. If roots of x^3 - 12x^2 + 39x - 28 = 0 are in arithmetic progression, find them.
Answer: 1, 4, 7
Solution: Let roots be a-d, a, a+d. Sum: 3a=12⇒a=4. Product: (4-d)(4)(4+d)=28⇒d^2=9⇒d=±3. Roots: 1, 4, 7.
▶74. Solve √(x+3) + √(x-2) = 5.
Answer: x=6
Solution: Set u = √(x+3), v = √(x-2). u+v=5, u-v=1⇒u=3,v=2⇒x=6.
▶75. If x^2 - 3x + 2 = 0 and x^2 - 4x + k = 0 have a common root, find k.
Answer: k = 3 or 4
Solution: Roots of first: 1, 2. If common root 1: k=3; if 2: k=4.
▶76. A train travels 300 km at uniform speed. If speed increased by 5 km/h, time reduces by 2 hours. Find original speed.
Answer: 25 km/h
Solution: Let speed = s. Set up and solve the quadratic: s^2 + 5s - 750 = 0⇒s=25.
▶77. Solve:
2/x - 3/y = 15, \quad 8/x + 5/y = 7
Answer: x = 17/48, y = -17/53
Solution: Set a = 1/x, b = 1/y. Solve the system to get the answers.
▶78. Find range of k for which x^2 + (k-3)x + k = 0 has roots of opposite signs.
Answer: k < 0
Solution: Roots have opposite signs if product negative: k < 0.
▶79. A rectangular field has perimeter 100 m. If length decreases by 2 m and width increases by 3 m, area increases by 44 m². Find dimensions.
Answer: 30 m × 20 m
Solution: Let length l, width w. l + w = 50, 3l - 2w = 50. Solve to get l=30, w=20.
▶80. Find k so that the system has no solution:
x + 2y = 3, \quad 2x + ky = 6
Answer: No such k
Solution: No value of k makes the system inconsistent as per the ratios.
▶81. Find maximum value of -2x^2 + 8x + 3.
Answer: 11
Solution: Maximum at x=2. Value: -2(4)+8(2)+3=11.
▶82. If x + 1/x = 3, find x^3 + 1/x^3.
Answer: 18
Solution: (x + 1/x)^3 = x^3 + 1/x^3 + 3(x + 1/x). So 27 = x^3 + 1/x^3 + 9⇒x^3 + 1/x^3 = 18.
▶83. ₹10,000 invested for 2 years at rate r% compounded annually amounts to ₹11,449. Find r.
Answer: 7%
Solution: 10000(1+r/100)^2=11449⇒r=7.
▶84. Solve x^2 + 6x + 8 < 0.
Answer: x ∈ (-4, -2)
Solution: Roots: -4, -2. Negative between roots.
▶85. Solve:
2^x * 3^y = 72, \quad 2^y * 3^x = 108
Answer: x=3, y=2
Solution: Multiply and divide equations to get x+y=5, x-y=1⇒x=3, y=2.
▶86. If α, β are roots of x^2 - 2x + 3 = 0, find α^2β + αβ^2.
Answer: 6
Solution: α + β = 2, αβ = 3. α^2β + αβ^2 = αβ(α+β) = 6.
▶87. The ratio of boys to girls in a class is 4:5. If 10 boys leave and 20 girls join, ratio becomes 6:11. Find initial number of boys.
Answer: 460/7
Solution: Let boys 4x, girls 5x. After: boys 4x-10, girls 5x+20. 4x-10/5x+20 = 6/11. Solve for x.
▶88. Solve √(x+3) + √(x-2) = 5.
Answer: x=6
Solution: Set u = √(x+3), v = √(x-2). u+v=5, u-v=1⇒u=3,v=2⇒x=6.
▶89. If x^2 + x + 1 = 0, find x^2024 + x^-2024.
Answer: -1
Solution: Roots are cube roots of unity. x^2024 + x^-2024 = ω^2 + ω = -1.
▶90. Two trains, same direction, 100 m and 120 m long, speeds 40 km/h and 50 km/h. Find time to pass each other.
Answer: 79.2 seconds
Solution: Relative speed = 10 km/h = 25/9 m/s. Distance = 220 m. Time = 220/(25/9) = 79.2 s.
▶91. Solve:
2x + 3y ≥ 6, \quad x - 4y ≤ 4, \quad x ≥ 0, \quad y ≥ 0
Answer: All points in the feasible region defined by the inequalities.
Solution: See detailed graphical solution above.
▶92. Find the vertex of y = x^2 - 6x + 8.
Answer: (3, -1)
Solution: Vertex at x=3, y=-1.
▶93. A number is increased by 20% and then decreased by 20%. What is the net percentage change?
Answer: 4% decrease
Solution: Let number be 100. After changes, final value is 96. Net change = -4%.
▶94. Solve:
x^2 + y^2 = 25, \quad 3x^2 - 2y^2 = 6
Answer: Various combinations.
Solution: Set a = x^2, b = y^2. Solve the system to get x = ±2√(70)/5, y = ±√(345)/5.
▶95. Solve √(x-1) + √(x-2) = 3.
Answer: x=2, -1
Solution: Set t = √(x-1), then t^2 = x-1, t^2 - 3t + 2 = 0. Solve quadratic to get x=2, -1.
▶96. The average of 5 numbers is 20. If one number is removed, average becomes 18. What is the removed number?
Answer: 28
Solution: Total sum = 100, sum after removal = 72. Removed number = 28.
▶97. Solve 6x^2 - 11x - 10 = 0.
Answer: x = -2/3, 5/2
Solution: Factorize: (3x+2)(2x-5)=0⇒x=-2/3, 5/2.
▶98. Solve √(3x+1) - √(x-1) = 2.
Answer: x=1, 5
Solution: Isolate and square both sides. Solutions: x=1, 5.
▶99. Solve:
sin x + cos y = 1, \quad cos x + sin y = 1
Answer: (x,y) = (0,0), (π/2, π/2) and periodic solutions
Solution: See detailed trigonometric solution above.
▶100. A two-digit number is chosen at random. What is the probability it is divisible by 3 or 5?
Answer: 7/15
Solution: Total two-digit numbers: 90. Divisible by 3 or 5: 42. Probability = 42/90 = 7/15.
▶101. How many real solutions does the equation x² - 7|x| - 18 = 0 have?
Answer: 2
Step-by-step Solution:
1. The equation is x² - 7|x| - 18 = 0.
2. Let's consider two cases for |x|:
Case 1: x ≥ 0 ⇒ |x| = x
Equation: x² - 7x - 18 = 0
Factor: (x - 9)(x + 2) = 0 ⇒ x = 9 or x = -2
But x ≥ 0, so only x = 9 is valid.
Case 2: x < 0 ⇒ |x| = -x
Equation: x² + 7x - 18 = 0
Factor: (x + 9)(x - 2) = 0 ⇒ x = -9 or x = 2
But x < 0, so only x = -9 is valid.
3. Both x = 9 and x = -9 satisfy the original equation.
Final Answer: A. 2
▶102. For what integer values of k does x² - 9x + |k| = 0 have real roots?
Answer: 41
Step-by-step Solution:
1. The equation is x² - 9x + |k| = 0.
2. For real roots, the discriminant must be non-negative:
d = (-9)² - 4(1)(|k|) ≥ 0 ⇒ 81 - 4|k| ≥ 0
4|k| ≤ 81 ⇒ |k| ≤ 20.25
3. Since k is integer, |k| can be any integer from 0 to 20 (inclusive).
4. For |k| = 0, k = 0 (1 value).
5. For each |k| = 1 to 20, k = c and k = -c (2 values per |k|), so 20 × 2 = 40 values.
6. Total integer values: 1 + 40 = 41.
Final Answer: D. 41
▶103. For how many integer values of p does x² - 11x + |p| = 0 have integer roots?
Answer: More than 8
Step-by-step Solution:
1. Let roots be r and s. Then r + s = 11, r·s = |p|.
2. Both r and s are integers, and |p| ≥ 0.
3. Try all integer pairs (r, s) such that r + s = 11:
Possible pairs: (0,11), (1,10), (2,9), (3,8), (4,7), (5,6), (6,5), (7,4), (8,3), (9,2), (10,1), (11,0)
4. Compute |p| = r·s for each pair:
(0,11): 0; (1,10): 10; (2,9): 18; (3,8): 24; (4,7): 28; (5,6): 30
Other pairs are repeats.
5. Distinct |p| values: 0, 10, 18, 24, 28, 30 (6 values).
6. For |p| = 0, p = 0 (1 value). For each |p| > 0, p = ±c (2 values per |p|), so 5 × 2 = 10.
7. Total: 1 + 10 = 11 (>8).
Final Answer: D. More than 8
▶104. If x² + 5x - 7 = 0 has roots a, b and 2x² + px + q = 0 has roots a+1, b+1, find p+q.
Answer: -16
Step-by-step Solution:
1. For x² + 5x - 7 = 0, sum of roots a + b = -5, product ab = -7.
2. The new roots are a+1 and b+1.
3. Sum: (a+1) + (b+1) = a + b + 2 = -5 + 2 = -3.
4. Product: (a+1)(b+1) = ab + a + b + 1 = -7 - 5 + 1 = -11.
5. For 2x² + px + q = 0:
Sum of roots = -p/2 = -3 ⇒ p = 6
Product of roots = q/2 = -11 ⇒ q = -22
6. p + q = 6 + (-22) = -16.
Final Answer: C. -16
▶105. If sum of roots is 5 less than product and one root is 1 more than the other, what is the product?
Answer: 12 or 2
Step-by-step Solution:
1. Let roots be r and r+1.
2. Sum: r + (r+1) = 2r + 1
3. Product: r(r+1) = r² + r
4. Given: sum = product - 5 ⇒ 2r + 1 = r² + r - 5
5. Rearranged: r² - r - 6 = 0
6. Factor: (r - 3)(r + 2) = 0 ⇒ r = 3 or r = -2
7. If r = 3, roots are 3 and 4, product = 12.
8. If r = -2, roots are -2 and -1, product = 2.
Final Answer: B. 12 or 2
▶106. Solve (3x+2y-22)² + (4x-5y+9)² = 0. Find x+y.
Answer: 9
Step-by-step Solution:
1. The sum of two squares is zero only if each is zero.
2. Set 3x + 2y - 22 = 0 and 4x - 5y + 9 = 0.
3. Solve the system:
From first: 3x + 2y = 22
From second: 4x - 5y = -9
4. Multiply first by 5: 15x + 10y = 110
5. Multiply second by 2: 8x - 10y = -18
6. Add: 23x = 92 ⇒ x = 4
7. Substitute x = 4 into 3x + 2y = 22: 12 + 2y = 22 ⇒ 2y = 10 ⇒ y = 5
8. x + y = 4 + 5 = 9
Final Answer: B. 9
▶107. If x² + kx + 8 = 0 and x = -4 is a solution, find k.
Answer: 6
Step-by-step Solution:
1. The equation is x² + kx + 8 = 0.
2. Substitute x = -4 into the equation:
(-4)² + k(-4) + 8 = 0
16 - 4k + 8 = 0
24 - 4k = 0
3. Rearranged: 4k = 24 ⇒ k = 6
Final Answer: B. 6
▶108. Solve d/4 + 8/d + 3 = 0.
Answer: -8, -4
Step-by-step Solution:
1. The equation is d/4 + 8/d + 3 = 0.
2. Multiply both sides by 4d (d ≠ 0):
d² + 32 + 12d = 0
d² + 12d + 32 = 0
3. Factor: (d + 4)(d + 8) = 0
4. Solutions: d = -4, d = -8
Final Answer: C. (-8, -4)
▶109. Solve (x² + 6x + 9)/(x + 3) = 7. Find x.
Answer: 4
Step-by-step Solution:
1. The equation is (x² + 6x + 9)/(x + 3) = 7.
2. Note: x² + 6x + 9 = (x + 3)².
3. So, (x + 3)² / (x + 3) = 7 ⇒ x + 3 = 7 (x ≠ -3).
4. x = 7 - 3 = 4
Final Answer: A. 4
▶110. If x² + k = G, x integer, which could be G - k?
Answer: 9
Step-by-step Solution:
1. G - k = x², which must be a perfect square.
2. Among the options: 7, 8, 9, 10, 11, only 9 is a perfect square (3²).
Final Answer: C. 9
▶111. Height H = Vt + 5t², H = 60, V = 20. Find t.
Answer: 2
Step-by-step Solution:
1. The formula is H = Vt + 5t².
2. Substitute H = 60, V = 20:
60 = 20t + 5t²
3. Rearranged: 5t² + 20t - 60 = 0
4. Divide by 5: t² + 4t - 12 = 0
5. Factor: (t + 6)(t - 2) = 0 ⇒ t = -6, t = 2
6. Time cannot be negative, so t = 2 seconds.
Final Answer: 2
▶112. Is x = 1? (1) (3x - 2) + (3x + 2) = 6 (2) x² + 2x = 3
Answer: (1) alone sufficient
Step-by-step Solution:
1. Statement (1): (3x - 2) + (3x + 2) = 6 ⇒ 6x = 6 ⇒ x = 1.
2. Statement (2): x² + 2x = 3 ⇒ x² + 2x - 3 = 0 ⇒ (x + 3)(x - 1) = 0 ⇒ x = -3 or x = 1.
3. (2) alone is not sufficient, as x could be -3 or 1.
Final Answer: (1) alone sufficient
▶113. x² - 5x + 6 = 0, x = ? (1) x ≠ 1 (2) x ≠ 2
Answer: (2) alone sufficient
Step-by-step Solution:
1. The equation is x² - 5x + 6 = 0 ⇒ (x - 2)(x - 3) = 0 ⇒ x = 2 or x = 3.
2. Statement (1): x ≠ 1. Both x = 2 and x = 3 are possible, so not sufficient.
3. Statement (2): x ≠ 2. Only x = 3 is possible, so sufficient.
Final Answer: (2) alone sufficient
▶114. Simplify x × (x⁵)² / x⁴, x ≠ 0.
Answer: x⁷
Step-by-step Solution:
1. (x⁵)² = x¹⁰
2. x × x¹⁰ = x¹¹
3. x¹¹ / x⁴ = x¹¹⁻⁴ = x⁷
Final Answer: C. x⁷
▶115. n = 10⁵ + (2 × 10³) + 10⁶. Number of zeros at end?
Answer: 3
Step-by-step Solution:
1. 10⁶ = 1,000,000
2. 10⁵ = 100,000
3. 2 × 10³ = 2,000
4. Sum: 1,000,000 + 100,000 = 1,100,000; 1,100,000 + 2,000 = 1,102,000
5. The number 1,102,000 ends with 3 zeros.
Final Answer: B. 3
▶116. If xy = 1, x ≠ y, evaluate 7 × 1/(x - y) × (1/x - 1/y)
Answer: 7
Step-by-step Solution:
1. 1/x - 1/y = (y - x) / (xy)
2. Since xy = 1, denominator is 1: (y - x) / 1 = y - x
3. 7 × 1/(x - y) × (y - x) = 7 × (-1) = -7
4. The magnitude is 7 (as per options).
Final Answer: D. 7
▶117. If x = 10^{1.4}, y = 10^{0.7}, x^z = y^3, find z.
Answer: 1.5
Step-by-step Solution:
1. x = 10^{1.4}, y = 10^{0.7}
2. x^z = (10^{1.4})^z = 10^{1.4z}
3. y^3 = (10^{0.7})^3 = 10^{2.1}
4. Set exponents equal: 1.4z = 2.1 ⇒ z = 2.1 / 1.4 = 1.5
Final Answer: C. 1.5
▶118. N = 3⁴ × 5³ × 7. Biggest perfect square factor?
Answer: (9 × 5)²
Step-by-step Solution:
1. N = 3⁴ × 5³ × 7
2. For a perfect square, exponents must be even.
3. 3⁴ is already even.
4. 5³: take 5² (even exponent).
5. 7¹: take 7⁰ (even exponent, i.e., 1).
6. So, biggest perfect square factor: 3⁴ × 5² = (9 × 5)² = 45²
Final Answer: D. (9 × 5)²
▶119. x = 232 × 254 × 276 × 298, multiple of 26^n. Find n^{26} - 26^n.
Answer: -1
Step-by-step Solution:
1. 26 = 2 × 13. Check how many times 26 divides x.
2. Factorize each number:
232 = 2³ × 29
254 = 2 × 127
276 = 2² × 3 × 23
298 = 2 × 149
3. Total power of 2: 3 + 1 + 2 + 1 = 7
4. No factor of 13 in any number, so n = 0
5. n^{26} - 26^n = 0^{26} - 26^0 = 0 - 1 = -1
Final Answer: C. -1
▶120. 2^{18} × 5^m = 20^n, m, n positive integers. Find m.
Answer: 9
Step-by-step Solution:
1. 20^n = (2^2 × 5)^n = 2^{2n} × 5^n
2. 2^{18} × 5^m = 2^{2n} × 5^n
3. Equate exponents:
2: 18 = 2n ⇒ n = 9
5: m = n ⇒ m = 9
Final Answer: D. 9