Pre-Calculus II
Introduction
Pre-Calculus II bridges elementary algebra and calculus through the study of functions, limits, derivatives, integrals and coordinate geometry.
Recommended Background
Pre-Calculus I is helpful.
Topic-Wise Course Outline
Functions and Graphs
Polynomial, rational, exponential, logarithmic and trigonometric functions; inverse and composite functions; even and odd functions; parametric descriptions.
Limits and Continuity
Meaning of a limit; algebraic limit laws; one-sided limits; continuity and discontinuities.
Differentiation
Rates of change and slope; derivatives from first principles; differentiation rules; introduction to Taylor’s theorem.
Integration: Foundations
Antiderivatives, standard integral formulas and simple definite integrals.
Techniques and Applications of Integration
Substitution; integration by parts; integrals involving limits; areas bounded by curves.
Analytic Geometry
Coordinates and quadrants; distance and slope; forms of a straight-line equation; translation and rotation of axes.
Conic Sections
Circles and their equations; center, radius and diameter; equations from given geometric data.
How to Study This Subject
Start with the definitions and core concepts in each topic. Work through examples, check every step of a solution, and then practise questions to test understanding.
Begin with the MCQs and worked examples below, then review each topic in the outline. More practice material will be added progressively.
Foundational MCQs and Step-by-Step Practice
Ten introductory topic-wise questions with explanations, followed by two fully worked examples. This is a starter set; further coverage is planned.
Functions
Q1. If f(x)=x² and g(x)=x+1, find (f∘g)(2).
- A. 5
- B. 9
- C. 6
- D. 3
Show correct answer and step-by-step explanation
Correct answer: B. 9
g(2)=3, then f(3)=9.
Graphs
Q2. The graph y=(x−3)² is y=x² shifted in which direction?
- A. 3 units left
- B. 3 units down
- C. 3 units right
- D. 3 units up
Show correct answer and step-by-step explanation
Correct answer: C. 3 units right
Replacing x with x−3 shifts the graph right by 3.
Limits
Q3. Evaluate lim(x→2) (x²−4)/(x−2).
- A. 0
- B. 2
- C. 4
- D. No limit
Show correct answer and step-by-step explanation
Correct answer: C. 4
Factor as (x−2)(x+2) and take the limit of x+2.
Continuity
Q4. Which is required for continuity at x=a?
- A. Only f(a) exists
- B. lim(x→a) f(x)=f(a)
- C. f(a)=0
- D. f'(a)>0
Show correct answer and step-by-step explanation
Correct answer: B. lim(x→a) f(x)=f(a)
The function value and two-sided limit must exist and coincide.
Differentiation
Q5. What is d(x³)/dx?
- A. x²
- B. 3x
- C. 3x²
- D. x⁴/4
Show correct answer and step-by-step explanation
Correct answer: C. 3x²
Apply the power rule.
Product Rule
Q6. Differentiate x² sin x.
- A. 2x sin x+x² cos x
- B. 2x cos x
- C. x² cos x
- D. sin x+2x
Show correct answer and step-by-step explanation
Correct answer: A. 2x sin x+x² cos x
Product rule: (uv)’=u’v+uv’.
Integration
Q7. Find ∫2x dx.
- A. 2x²+C
- B. x²+C
- C. x+C
- D. ln|x|+C
Show correct answer and step-by-step explanation
Correct answer: B. x²+C
An antiderivative of 2x is x².
Definite Integrals
Q8. Evaluate ∫₀¹ 2x dx.
- A. 0
- B. 2
- C. 1
- D. 1/2
Show correct answer and step-by-step explanation
Correct answer: C. 1
Evaluate [x²] from 0 to 1.
Analytic Geometry
Q9. Find the slope through (1,2) and (3,6).
- A. 1
- B. 2
- C. 3
- D. 4
Show correct answer and step-by-step explanation
Correct answer: B. 2
Slope=(6−2)/(3−1)=2.
Conic Sections
Q10. Find the radius of (x−2)²+(y+1)²=9.
- A. 9
- B. 6
- C. 2
- D. 3
Show correct answer and step-by-step explanation
Correct answer: D. 3
The right side equals r²=9, so r=3.
Solved Examples with Step-by-Step Answers
Example 1: Algebraic limit
Problem: Find lim(x→3) (x²−9)/(x−3).
- Factor numerator: (x−3)(x+3).
- Cancel x−3 for x≠3.
- Evaluate x+3 as x approaches 3.
Final answer: The limit is 6.
Example 2: Definite integral
Problem: Evaluate ∫₀² 3x² dx.
- The antiderivative of 3x² is x³.
- Compute [x³] from 0 to 2.
Final answer: The integral is 8.
Additional Topic-Wise MCQs — Questions 11–20
Additional concept and calculation questions. Select a question’s expandable answer to review the reasoning.
Exponential Functions
Q11. Solve 2ˣ=8.
- A. 2
- B. 3
- C. 4
- D. 8
Show correct answer and explanation
Correct answer: B. 3
Because 8=2³, x=3.
Logarithms
Q12. For x>0, what is log₁₀(1000)?
- A. 2
- B. 3
- C. 10
- D. 100
Show correct answer and explanation
Correct answer: B. 3
10³=1000.
Inverse Functions
Q13. If f(x)=x+5, what is f⁻¹(x)?
- A. x−5
- B. x+5
- C. 5x
- D. 1/(x+5)
Show correct answer and explanation
Correct answer: A. x−5
Undo addition of 5 by subtracting 5.
Limits
Q14. Find lim(x→0) (sin 3x)/x, where angles use radians.
- A. 0
- B. 1
- C. 3
- D. ∞
Show correct answer and explanation
Correct answer: C. 3
Write sin(3x)/x=3 sin(3x)/(3x), whose limit is 3.
Derivative Definition
Q15. Find the derivative of f(x)=x² at x=1.
- A. 0
- B. 1
- C. 2
- D. 4
Show correct answer and explanation
Correct answer: C. 2
The derivative is 2x, giving 2 at x=1.
Chain Rule
Q16. Differentiate sin(2x).
- A. cos(2x)
- B. 2cos(2x)
- C. −sin(2x)
- D. 2sin(2x)
Show correct answer and explanation
Correct answer: B. 2cos(2x)
The chain rule multiplies cos(2x) by the inner derivative 2.
Antiderivatives
Q17. Evaluate ∫(1/x)dx on x>0.
- A. x²/2+C
- B. ln x+C
- C. 1/x²+C
- D. eˣ+C
Show correct answer and explanation
Correct answer: B. ln x+C
The derivative of ln x equals 1/x for positive x.
Integration by Parts
Q18. Which formula is integration by parts?
- A. ∫u dv=uv−∫v du
- B. ∫u dv=uv+∫v du
- C. ∫u dv=u+v
- D. ∫u dv=u/v
Show correct answer and explanation
Correct answer: A. ∫u dv=uv−∫v du
Integration by parts follows from the product rule.
Coordinate Geometry
Q19. What is the midpoint of (2,4) and (6,8)?
- A. (2,2)
- B. (3,6)
- C. (4,6)
- D. (8,12)
Show correct answer and explanation
Correct answer: C. (4,6)
Average corresponding coordinates: ((2+6)/2,(4+8)/2)=(4,6).
Circles
Q20. A circle has centre (0,0) and radius 5. Which is its equation?
- A. x²+y²=5
- B. x+y=25
- C. x²−y²=25
- D. x²+y²=25
Show correct answer and explanation
Correct answer: D. x²+y²=25
Standard circle equation centered at the origin is x²+y²=r²=25.
Additional Practice MCQs — Questions 21–30
Further foundational and intermediate practice, with four answer options and explanations.
Functions
Q21. What is the range of f(x)=x² on the real numbers?
- A. All real numbers
- B. [0,∞)
- C. (−∞,0]
- D. (0,∞)
Show correct answer and explanation
Correct answer: B. [0,∞)
Squares are nonnegative, and every nonnegative value is attainable.
Exponentials
Q22. Solve eˣ=1.
- A. 0
- B. 1
- C. e
- D. No solution
Show correct answer and explanation
Correct answer: A. 0
e⁰=1.
Limits
Q23. Evaluate lim(x→1) (x³−1)/(x−1).
- A. 1
- B. 2
- C. 3
- D. Does not exist
Show correct answer and explanation
Correct answer: C. 3
Simplify to x²+x+1 before taking the limit.
Continuity
Q24. If f(x)=(x²−1)/(x−1) for x≠1, with f(1) undefined, what discontinuity occurs at 1?
- A. Jump
- B. Infinite
- C. Oscillatory
- D. Removable
Show correct answer and explanation
Correct answer: D. Removable
The function approaches the finite limit 2.
Derivatives
Q25. Differentiate e^(3x).
- A. e^(3x)
- B. 3e^(3x)
- C. eˣ/3
- D. 3xe^(3x)
Show correct answer and explanation
Correct answer: B. 3e^(3x)
Use the chain rule.
Q26. For f(x)=x³−3x, find f'(1).
- A. 0
- B. 1
- C. 3
- D. −3
Show correct answer and explanation
Correct answer: A. 0
f'(x)=3x²−3, which is 0 at x=1.
Integration
Q27. Evaluate ∫₀² x dx.
- A. 1
- B. 4
- C. 2
- D. 8
Show correct answer and explanation
Correct answer: C. 2
[x²/2] evaluated from 0 to 2 gives 2.
Integration by Parts
Q28. Evaluate ∫x eˣ dx.
- A. xeˣ+C
- B. eˣ(x−1)+C
- C. eˣ(x+1)+C
- D. x²eˣ+C
Show correct answer and explanation
Correct answer: B. eˣ(x−1)+C
Integrate by parts: xeˣ−eˣ+C.
Lines
Q29. Which line has slope 2 and y-intercept −3?
- A. y=−3x+2
- B. y=2x+3
- C. y=x−3
- D. y=2x−3
Show correct answer and explanation
Correct answer: D. y=2x−3
In y=mx+b, m=2 and b=−3.
Circles
Q30. What is the centre of x²+y²−6x+4y=12?
- A. (3,−2)
- B. (−3,2)
- C. (6,−4)
- D. (0,0)
Show correct answer and explanation
Correct answer: A. (3,−2)
Complete squares to obtain (x−3)²+(y+2)²=25.
Additional Step-by-Step Solved Example
Integration by substitution
Problem: Evaluate ∫₀¹ 2x(x²+1)³ dx.
- Set u=x²+1, so du=2x dx.
- The limits become u=1 and u=2.
- Integrate ∫₁² u³du=[u⁴/4]₁².
Final answer: The value is 15/4.
Additional Practice MCQs — Questions 31–150
Q31. Evaluate lim(x→1) (2x+3).
- A. 6
- B. 4
- C. 8
- D. 5
Show correct answer and explanation
Correct answer: D. 5
Polynomials are continuous, so substitute x=1 to obtain 5.
Q32. Evaluate lim(x→2) (3x+4).
- A. 10
- B. 11
- C. 9
- D. 13
Show correct answer and explanation
Correct answer: A. 10
Polynomials are continuous, so substitute x=2 to obtain 10.
Q33. Evaluate lim(x→3) (4x+5).
- A. 20
- B. 17
- C. 18
- D. 16
Show correct answer and explanation
Correct answer: B. 17
Polynomials are continuous, so substitute x=3 to obtain 17.
Q34. Evaluate lim(x→4) (5x+6).
- A. 25
- B. 29
- C. 26
- D. 27
Show correct answer and explanation
Correct answer: C. 26
Polynomials are continuous, so substitute x=4 to obtain 26.
Q35. Evaluate lim(x→5) (6x+7).
- A. 38
- B. 36
- C. 40
- D. 37
Show correct answer and explanation
Correct answer: D. 37
Polynomials are continuous, so substitute x=5 to obtain 37.
Q36. Evaluate lim(x→6) (7x+8).
- A. 50
- B. 51
- C. 49
- D. 53
Show correct answer and explanation
Correct answer: A. 50
Polynomials are continuous, so substitute x=6 to obtain 50.
Q37. Evaluate lim(x→7) (8x+9).
- A. 68
- B. 65
- C. 66
- D. 64
Show correct answer and explanation
Correct answer: B. 65
Polynomials are continuous, so substitute x=7 to obtain 65.
Q38. Evaluate lim(x→8) (9x+10).
- A. 81
- B. 85
- C. 82
- D. 83
Show correct answer and explanation
Correct answer: C. 82
Polynomials are continuous, so substitute x=8 to obtain 82.
Q39. Evaluate lim(x→9) (10x+11).
- A. 102
- B. 100
- C. 104
- D. 101
Show correct answer and explanation
Correct answer: D. 101
Polynomials are continuous, so substitute x=9 to obtain 101.
Q40. Evaluate lim(x→10) (11x+12).
- A. 122
- B. 123
- C. 121
- D. 125
Show correct answer and explanation
Correct answer: A. 122
Polynomials are continuous, so substitute x=10 to obtain 122.
Q41. Find the derivative at x=1 of f(x)=2x².
- A. 7
- B. 4
- C. 5
- D. 3
Show correct answer and explanation
Correct answer: B. 4
f'(x)=4x, so f'(1)=4.
Q42. Find the derivative at x=2 of f(x)=3x².
- A. 11
- B. 15
- C. 12
- D. 13
Show correct answer and explanation
Correct answer: C. 12
f'(x)=6x, so f'(2)=12.
Q43. Find the derivative at x=3 of f(x)=4x².
- A. 25
- B. 23
- C. 27
- D. 24
Show correct answer and explanation
Correct answer: D. 24
f'(x)=8x, so f'(3)=24.
Q44. Find the derivative at x=4 of f(x)=5x².
- A. 40
- B. 41
- C. 39
- D. 43
Show correct answer and explanation
Correct answer: A. 40
f'(x)=10x, so f'(4)=40.
Q45. Find the derivative at x=5 of f(x)=6x².
- A. 63
- B. 60
- C. 61
- D. 59
Show correct answer and explanation
Correct answer: B. 60
f'(x)=12x, so f'(5)=60.
Q46. Find the derivative at x=6 of f(x)=7x².
- A. 83
- B. 87
- C. 84
- D. 85
Show correct answer and explanation
Correct answer: C. 84
f'(x)=14x, so f'(6)=84.
Q47. Find the derivative at x=7 of f(x)=8x².
- A. 113
- B. 111
- C. 115
- D. 112
Show correct answer and explanation
Correct answer: D. 112
f'(x)=16x, so f'(7)=112.
Q48. Find the derivative at x=8 of f(x)=9x².
- A. 144
- B. 145
- C. 143
- D. 147
Show correct answer and explanation
Correct answer: A. 144
f'(x)=18x, so f'(8)=144.
Q49. Find the derivative at x=9 of f(x)=10x².
- A. 183
- B. 180
- C. 181
- D. 179
Show correct answer and explanation
Correct answer: B. 180
f'(x)=20x, so f'(9)=180.
Q50. Find the derivative at x=10 of f(x)=11x².
- A. 219
- B. 223
- C. 220
- D. 221
Show correct answer and explanation
Correct answer: C. 220
f'(x)=22x, so f'(10)=220.
Q51. Compute ∫₀¹ 2x^1 dx.
- A. 2
- B. 0
- C. 4
- D. 1
Show correct answer and explanation
Correct answer: D. 1
Using the power rule, the definite integral equals 2/(2)=1.
Q52. Compute ∫₀¹ 3x^2 dx.
- A. 1
- B. 2
- C. 0
- D. 4
Show correct answer and explanation
Correct answer: A. 1
Using the power rule, the definite integral equals 3/(3)=1.
Q53. Compute ∫₀¹ 4x^3 dx.
- A. 4
- B. 1
- C. 2
- D. 0
Show correct answer and explanation
Correct answer: B. 1
Using the power rule, the definite integral equals 4/(4)=1.
Q54. Compute ∫₀¹ 5x^4 dx.
- A. 0
- B. 4
- C. 1
- D. 2
Show correct answer and explanation
Correct answer: C. 1
Using the power rule, the definite integral equals 5/(5)=1.
Q55. Compute ∫₀¹ 6x^5 dx.
- A. 2
- B. 0
- C. 4
- D. 1
Show correct answer and explanation
Correct answer: D. 1
Using the power rule, the definite integral equals 6/(6)=1.
Q56. Compute ∫₀¹ 7x^6 dx.
- A. 1
- B. 2
- C. 0
- D. 4
Show correct answer and explanation
Correct answer: A. 1
Using the power rule, the definite integral equals 7/(7)=1.
Q57. Compute ∫₀¹ 8x^7 dx.
- A. 4
- B. 1
- C. 2
- D. 0
Show correct answer and explanation
Correct answer: B. 1
Using the power rule, the definite integral equals 8/(8)=1.
Q58. Compute ∫₀¹ 9x^8 dx.
- A. 0
- B. 4
- C. 1
- D. 2
Show correct answer and explanation
Correct answer: C. 1
Using the power rule, the definite integral equals 9/(9)=1.
Q59. Compute ∫₀¹ 10x^9 dx.
- A. 2
- B. 0
- C. 4
- D. 1
Show correct answer and explanation
Correct answer: D. 1
Using the power rule, the definite integral equals 10/(10)=1.
Q60. Compute ∫₀¹ 11x^10 dx.
- A. 1
- B. 2
- C. 0
- D. 4
Show correct answer and explanation
Correct answer: A. 1
Using the power rule, the definite integral equals 11/(11)=1.
Q61. For f(x)=2x+3, what is the slope of its graph?
- A. 5
- B. 2
- C. 3
- D. 1
Show correct answer and explanation
Correct answer: B. 2
In y=mx+b, the coefficient of x is the slope m=2.
Q62. For f(x)=3x+4, what is the slope of its graph?
- A. 2
- B. 6
- C. 3
- D. 4
Show correct answer and explanation
Correct answer: C. 3
In y=mx+b, the coefficient of x is the slope m=3.
Q63. For f(x)=4x+5, what is the slope of its graph?
- A. 5
- B. 3
- C. 7
- D. 4
Show correct answer and explanation
Correct answer: D. 4
In y=mx+b, the coefficient of x is the slope m=4.
Q64. For f(x)=5x+6, what is the slope of its graph?
- A. 5
- B. 6
- C. 4
- D. 8
Show correct answer and explanation
Correct answer: A. 5
In y=mx+b, the coefficient of x is the slope m=5.
Q65. For f(x)=6x+7, what is the slope of its graph?
- A. 9
- B. 6
- C. 7
- D. 5
Show correct answer and explanation
Correct answer: B. 6
In y=mx+b, the coefficient of x is the slope m=6.
Q66. For f(x)=7x+8, what is the slope of its graph?
- A. 6
- B. 10
- C. 7
- D. 8
Show correct answer and explanation
Correct answer: C. 7
In y=mx+b, the coefficient of x is the slope m=7.
Q67. For f(x)=8x+9, what is the slope of its graph?
- A. 9
- B. 7
- C. 11
- D. 8
Show correct answer and explanation
Correct answer: D. 8
In y=mx+b, the coefficient of x is the slope m=8.
Q68. For f(x)=9x+10, what is the slope of its graph?
- A. 9
- B. 10
- C. 8
- D. 12
Show correct answer and explanation
Correct answer: A. 9
In y=mx+b, the coefficient of x is the slope m=9.
Q69. For f(x)=10x+11, what is the slope of its graph?
- A. 13
- B. 10
- C. 11
- D. 9
Show correct answer and explanation
Correct answer: B. 10
In y=mx+b, the coefficient of x is the slope m=10.
Q70. For f(x)=11x+12, what is the slope of its graph?
- A. 10
- B. 14
- C. 11
- D. 12
Show correct answer and explanation
Correct answer: C. 11
In y=mx+b, the coefficient of x is the slope m=11.
Q71. Find the radius of the circle (x−1)²+(y+2)²=4.
- A. 3
- B. 1
- C. 5
- D. 2
Show correct answer and explanation
Correct answer: D. 2
The right-hand side is r²=4; hence radius r=2.
Q72. Find the radius of the circle (x−2)²+(y+3)²=9.
- A. 3
- B. 4
- C. 2
- D. 6
Show correct answer and explanation
Correct answer: A. 3
The right-hand side is r²=9; hence radius r=3.
Q73. Find the radius of the circle (x−3)²+(y+4)²=16.
- A. 7
- B. 4
- C. 5
- D. 3
Show correct answer and explanation
Correct answer: B. 4
The right-hand side is r²=16; hence radius r=4.
Q74. Find the radius of the circle (x−4)²+(y+5)²=25.
- A. 4
- B. 8
- C. 5
- D. 6
Show correct answer and explanation
Correct answer: C. 5
The right-hand side is r²=25; hence radius r=5.
Q75. Find the radius of the circle (x−5)²+(y+6)²=36.
- A. 7
- B. 5
- C. 9
- D. 6
Show correct answer and explanation
Correct answer: D. 6
The right-hand side is r²=36; hence radius r=6.
Q76. Find the radius of the circle (x−6)²+(y+7)²=49.
- A. 7
- B. 8
- C. 6
- D. 10
Show correct answer and explanation
Correct answer: A. 7
The right-hand side is r²=49; hence radius r=7.
Q77. Find the radius of the circle (x−7)²+(y+8)²=64.
- A. 11
- B. 8
- C. 9
- D. 7
Show correct answer and explanation
Correct answer: B. 8
The right-hand side is r²=64; hence radius r=8.
Q78. Find the radius of the circle (x−8)²+(y+9)²=81.
- A. 8
- B. 12
- C. 9
- D. 10
Show correct answer and explanation
Correct answer: C. 9
The right-hand side is r²=81; hence radius r=9.
Q79. Find the radius of the circle (x−9)²+(y+10)²=100.
- A. 11
- B. 9
- C. 13
- D. 10
Show correct answer and explanation
Correct answer: D. 10
The right-hand side is r²=100; hence radius r=10.
Q80. Find the radius of the circle (x−10)²+(y+11)²=121.
- A. 11
- B. 12
- C. 10
- D. 14
Show correct answer and explanation
Correct answer: A. 11
The right-hand side is r²=121; hence radius r=11.
Q81. Evaluate the second derivative of f(x)=2x³ at x=1.
- A. 15
- B. 12
- C. 13
- D. 11
Show correct answer and explanation
Correct answer: B. 12
Differentiate twice: f”(x)=12x; at 1 this is 12.
Q82. Evaluate the second derivative of f(x)=3x³ at x=2.
- A. 35
- B. 39
- C. 36
- D. 37
Show correct answer and explanation
Correct answer: C. 36
Differentiate twice: f”(x)=18x; at 2 this is 36.
Q83. Evaluate the second derivative of f(x)=4x³ at x=3.
- A. 73
- B. 71
- C. 75
- D. 72
Show correct answer and explanation
Correct answer: D. 72
Differentiate twice: f”(x)=24x; at 3 this is 72.
Q84. Evaluate the second derivative of f(x)=5x³ at x=4.
- A. 120
- B. 121
- C. 119
- D. 123
Show correct answer and explanation
Correct answer: A. 120
Differentiate twice: f”(x)=30x; at 4 this is 120.
Q85. Evaluate the second derivative of f(x)=6x³ at x=5.
- A. 183
- B. 180
- C. 181
- D. 179
Show correct answer and explanation
Correct answer: B. 180
Differentiate twice: f”(x)=36x; at 5 this is 180.
Q86. Evaluate the second derivative of f(x)=7x³ at x=6.
- A. 251
- B. 255
- C. 252
- D. 253
Show correct answer and explanation
Correct answer: C. 252
Differentiate twice: f”(x)=42x; at 6 this is 252.
Q87. Evaluate the second derivative of f(x)=8x³ at x=7.
- A. 337
- B. 335
- C. 339
- D. 336
Show correct answer and explanation
Correct answer: D. 336
Differentiate twice: f”(x)=48x; at 7 this is 336.
Q88. Evaluate the second derivative of f(x)=9x³ at x=8.
- A. 432
- B. 433
- C. 431
- D. 435
Show correct answer and explanation
Correct answer: A. 432
Differentiate twice: f”(x)=54x; at 8 this is 432.
Q89. Evaluate the second derivative of f(x)=10x³ at x=9.
- A. 543
- B. 540
- C. 541
- D. 539
Show correct answer and explanation
Correct answer: B. 540
Differentiate twice: f”(x)=60x; at 9 this is 540.
Q90. Evaluate the second derivative of f(x)=11x³ at x=10.
- A. 659
- B. 663
- C. 660
- D. 661
Show correct answer and explanation
Correct answer: C. 660
Differentiate twice: f”(x)=66x; at 10 this is 660.
Q91. Find ∫₀^2 2x dx.
- A. 5
- B. 3
- C. 7
- D. 4
Show correct answer and explanation
Correct answer: D. 4
The antiderivative is x²; evaluating at 2 and 0 gives 4.
Q92. Find ∫₀^3 2x dx.
- A. 9
- B. 10
- C. 8
- D. 12
Show correct answer and explanation
Correct answer: A. 9
The antiderivative is x²; evaluating at 3 and 0 gives 9.
Q93. Find ∫₀^4 2x dx.
- A. 19
- B. 16
- C. 17
- D. 15
Show correct answer and explanation
Correct answer: B. 16
The antiderivative is x²; evaluating at 4 and 0 gives 16.
Q94. Find ∫₀^5 2x dx.
- A. 24
- B. 28
- C. 25
- D. 26
Show correct answer and explanation
Correct answer: C. 25
The antiderivative is x²; evaluating at 5 and 0 gives 25.
Q95. Find ∫₀^6 2x dx.
- A. 37
- B. 35
- C. 39
- D. 36
Show correct answer and explanation
Correct answer: D. 36
The antiderivative is x²; evaluating at 6 and 0 gives 36.
Q96. Find ∫₀^7 2x dx.
- A. 49
- B. 50
- C. 48
- D. 52
Show correct answer and explanation
Correct answer: A. 49
The antiderivative is x²; evaluating at 7 and 0 gives 49.
Q97. Find ∫₀^8 2x dx.
- A. 67
- B. 64
- C. 65
- D. 63
Show correct answer and explanation
Correct answer: B. 64
The antiderivative is x²; evaluating at 8 and 0 gives 64.
Q98. Find ∫₀^9 2x dx.
- A. 80
- B. 84
- C. 81
- D. 82
Show correct answer and explanation
Correct answer: C. 81
The antiderivative is x²; evaluating at 9 and 0 gives 81.
Q99. Find ∫₀^10 2x dx.
- A. 101
- B. 99
- C. 103
- D. 100
Show correct answer and explanation
Correct answer: D. 100
The antiderivative is x²; evaluating at 10 and 0 gives 100.
Q100. Find ∫₀^11 2x dx.
- A. 121
- B. 122
- C. 120
- D. 124
Show correct answer and explanation
Correct answer: A. 121
The antiderivative is x²; evaluating at 11 and 0 gives 121.
Q101. Solve 2x=6 for x.
- A. 6
- B. 3
- C. 4
- D. 2
Show correct answer and explanation
Correct answer: B. 3
Divide both sides by 2 to obtain x=3.
Q102. Solve 3x=12 for x.
- A. 3
- B. 7
- C. 4
- D. 5
Show correct answer and explanation
Correct answer: C. 4
Divide both sides by 3 to obtain x=4.
Q103. Solve 4x=20 for x.
- A. 6
- B. 4
- C. 8
- D. 5
Show correct answer and explanation
Correct answer: D. 5
Divide both sides by 4 to obtain x=5.
Q104. Solve 5x=30 for x.
- A. 6
- B. 7
- C. 5
- D. 9
Show correct answer and explanation
Correct answer: A. 6
Divide both sides by 5 to obtain x=6.
Q105. Solve 6x=42 for x.
- A. 10
- B. 7
- C. 8
- D. 6
Show correct answer and explanation
Correct answer: B. 7
Divide both sides by 6 to obtain x=7.
Q106. Solve 7x=56 for x.
- A. 7
- B. 11
- C. 8
- D. 9
Show correct answer and explanation
Correct answer: C. 8
Divide both sides by 7 to obtain x=8.
Q107. Solve 8x=72 for x.
- A. 10
- B. 8
- C. 12
- D. 9
Show correct answer and explanation
Correct answer: D. 9
Divide both sides by 8 to obtain x=9.
Q108. Solve 9x=90 for x.
- A. 10
- B. 11
- C. 9
- D. 13
Show correct answer and explanation
Correct answer: A. 10
Divide both sides by 9 to obtain x=10.
Q109. Solve 10x=110 for x.
- A. 14
- B. 11
- C. 12
- D. 10
Show correct answer and explanation
Correct answer: B. 11
Divide both sides by 10 to obtain x=11.
Q110. Solve 11x=132 for x.
- A. 11
- B. 15
- C. 12
- D. 13
Show correct answer and explanation
Correct answer: C. 12
Divide both sides by 11 to obtain x=12.
Q111. Find the x-intercept of y=2x−6.
- A. 4
- B. 2
- C. 6
- D. 3
Show correct answer and explanation
Correct answer: D. 3
Set y=0 and solve 2x=6, giving x=3.
Q112. Find the x-intercept of y=3x−12.
- A. 4
- B. 5
- C. 3
- D. 7
Show correct answer and explanation
Correct answer: A. 4
Set y=0 and solve 3x=12, giving x=4.
Q113. Find the x-intercept of y=4x−20.
- A. 8
- B. 5
- C. 6
- D. 4
Show correct answer and explanation
Correct answer: B. 5
Set y=0 and solve 4x=20, giving x=5.
Q114. Find the x-intercept of y=5x−30.
- A. 5
- B. 9
- C. 6
- D. 7
Show correct answer and explanation
Correct answer: C. 6
Set y=0 and solve 5x=30, giving x=6.
Q115. Find the x-intercept of y=6x−42.
- A. 8
- B. 6
- C. 10
- D. 7
Show correct answer and explanation
Correct answer: D. 7
Set y=0 and solve 6x=42, giving x=7.
Q116. Find the x-intercept of y=7x−56.
- A. 8
- B. 9
- C. 7
- D. 11
Show correct answer and explanation
Correct answer: A. 8
Set y=0 and solve 7x=56, giving x=8.
Q117. Find the x-intercept of y=8x−72.
- A. 12
- B. 9
- C. 10
- D. 8
Show correct answer and explanation
Correct answer: B. 9
Set y=0 and solve 8x=72, giving x=9.
Q118. Find the x-intercept of y=9x−90.
- A. 9
- B. 13
- C. 10
- D. 11
Show correct answer and explanation
Correct answer: C. 10
Set y=0 and solve 9x=90, giving x=10.
Q119. Find the x-intercept of y=10x−110.
- A. 12
- B. 10
- C. 14
- D. 11
Show correct answer and explanation
Correct answer: D. 11
Set y=0 and solve 10x=110, giving x=11.
Q120. Find the x-intercept of y=11x−132.
- A. 12
- B. 13
- C. 11
- D. 15
Show correct answer and explanation
Correct answer: A. 12
Set y=0 and solve 11x=132, giving x=12.
Q121. Find the area of a triangle with base 2 and height 3.
- A. 6
- B. 3
- C. 4
- D. 2
Show correct answer and explanation
Correct answer: B. 3
Area=(1/2)×base×height=3.
Q122. Find the area of a triangle with base 3 and height 4.
- A. 5
- B. 9
- C. 6
- D. 7
Show correct answer and explanation
Correct answer: C. 6
Area=(1/2)×base×height=6.
Q123. Find the area of a triangle with base 4 and height 5.
- A. 11
- B. 9
- C. 13
- D. 10
Show correct answer and explanation
Correct answer: D. 10
Area=(1/2)×base×height=10.
Q124. Find the area of a triangle with base 5 and height 6.
- A. 15
- B. 16
- C. 14
- D. 18
Show correct answer and explanation
Correct answer: A. 15
Area=(1/2)×base×height=15.
Q125. Find the area of a triangle with base 6 and height 7.
- A. 24
- B. 21
- C. 22
- D. 20
Show correct answer and explanation
Correct answer: B. 21
Area=(1/2)×base×height=21.
Q126. Find the area of a triangle with base 7 and height 8.
- A. 27
- B. 31
- C. 28
- D. 29
Show correct answer and explanation
Correct answer: C. 28
Area=(1/2)×base×height=28.
Q127. Find the area of a triangle with base 8 and height 9.
- A. 37
- B. 35
- C. 39
- D. 36
Show correct answer and explanation
Correct answer: D. 36
Area=(1/2)×base×height=36.
Q128. Find the area of a triangle with base 9 and height 10.
- A. 45
- B. 46
- C. 44
- D. 48
Show correct answer and explanation
Correct answer: A. 45
Area=(1/2)×base×height=45.
Q129. Find the area of a triangle with base 10 and height 11.
- A. 58
- B. 55
- C. 56
- D. 54
Show correct answer and explanation
Correct answer: B. 55
Area=(1/2)×base×height=55.
Q130. Find the area of a triangle with base 11 and height 12.
- A. 65
- B. 69
- C. 66
- D. 67
Show correct answer and explanation
Correct answer: C. 66
Area=(1/2)×base×height=66.
Q131. Evaluate the derivative of f(x)=x^2 at x=1.
- A. 3
- B. 1
- C. 5
- D. 2
Show correct answer and explanation
Correct answer: D. 2
By the power rule, f'(x)=2x^1, hence f'(1)=2.
Q132. Evaluate the derivative of f(x)=x^3 at x=1.
- A. 3
- B. 4
- C. 2
- D. 6
Show correct answer and explanation
Correct answer: A. 3
By the power rule, f'(x)=3x^2, hence f'(1)=3.
Q133. Evaluate the derivative of f(x)=x^4 at x=1.
- A. 7
- B. 4
- C. 5
- D. 3
Show correct answer and explanation
Correct answer: B. 4
By the power rule, f'(x)=4x^3, hence f'(1)=4.
Q134. Evaluate the derivative of f(x)=x^5 at x=1.
- A. 4
- B. 8
- C. 5
- D. 6
Show correct answer and explanation
Correct answer: C. 5
By the power rule, f'(x)=5x^4, hence f'(1)=5.
Q135. Evaluate the derivative of f(x)=x^6 at x=1.
- A. 7
- B. 5
- C. 9
- D. 6
Show correct answer and explanation
Correct answer: D. 6
By the power rule, f'(x)=6x^5, hence f'(1)=6.
Q136. Evaluate the derivative of f(x)=x^7 at x=1.
- A. 7
- B. 8
- C. 6
- D. 10
Show correct answer and explanation
Correct answer: A. 7
By the power rule, f'(x)=7x^6, hence f'(1)=7.
Q137. Evaluate the derivative of f(x)=x^8 at x=1.
- A. 11
- B. 8
- C. 9
- D. 7
Show correct answer and explanation
Correct answer: B. 8
By the power rule, f'(x)=8x^7, hence f'(1)=8.
Q138. Evaluate the derivative of f(x)=x^9 at x=1.
- A. 8
- B. 12
- C. 9
- D. 10
Show correct answer and explanation
Correct answer: C. 9
By the power rule, f'(x)=9x^8, hence f'(1)=9.
Q139. Evaluate the derivative of f(x)=x^10 at x=1.
- A. 11
- B. 9
- C. 13
- D. 10
Show correct answer and explanation
Correct answer: D. 10
By the power rule, f'(x)=10x^9, hence f'(1)=10.
Q140. Evaluate the derivative of f(x)=x^11 at x=1.
- A. 11
- B. 12
- C. 10
- D. 14
Show correct answer and explanation
Correct answer: A. 11
By the power rule, f'(x)=11x^10, hence f'(1)=11.
Q141. For f(x)=(x−2)²+3, find the minimum value on R.
- A. 6
- B. 3
- C. 4
- D. 2
Show correct answer and explanation
Correct answer: B. 3
A square is nonnegative and vanishes at x=2; the minimum is 3.
Q142. For f(x)=(x−3)²+4, find the minimum value on R.
- A. 3
- B. 7
- C. 4
- D. 5
Show correct answer and explanation
Correct answer: C. 4
A square is nonnegative and vanishes at x=3; the minimum is 4.
Q143. For f(x)=(x−4)²+5, find the minimum value on R.
- A. 6
- B. 4
- C. 8
- D. 5
Show correct answer and explanation
Correct answer: D. 5
A square is nonnegative and vanishes at x=4; the minimum is 5.
Q144. For f(x)=(x−5)²+6, find the minimum value on R.
- A. 6
- B. 7
- C. 5
- D. 9
Show correct answer and explanation
Correct answer: A. 6
A square is nonnegative and vanishes at x=5; the minimum is 6.
Q145. For f(x)=(x−6)²+7, find the minimum value on R.
- A. 10
- B. 7
- C. 8
- D. 6
Show correct answer and explanation
Correct answer: B. 7
A square is nonnegative and vanishes at x=6; the minimum is 7.
Q146. For f(x)=(x−7)²+8, find the minimum value on R.
- A. 7
- B. 11
- C. 8
- D. 9
Show correct answer and explanation
Correct answer: C. 8
A square is nonnegative and vanishes at x=7; the minimum is 8.
Q147. For f(x)=(x−8)²+9, find the minimum value on R.
- A. 10
- B. 8
- C. 12
- D. 9
Show correct answer and explanation
Correct answer: D. 9
A square is nonnegative and vanishes at x=8; the minimum is 9.
Q148. For f(x)=(x−9)²+10, find the minimum value on R.
- A. 10
- B. 11
- C. 9
- D. 13
Show correct answer and explanation
Correct answer: A. 10
A square is nonnegative and vanishes at x=9; the minimum is 10.
Q149. For f(x)=(x−10)²+11, find the minimum value on R.
- A. 14
- B. 11
- C. 12
- D. 10
Show correct answer and explanation
Correct answer: B. 11
A square is nonnegative and vanishes at x=10; the minimum is 11.
Q150. For f(x)=(x−11)²+12, find the minimum value on R.
- A. 11
- B. 15
- C. 12
- D. 13
Show correct answer and explanation
Correct answer: C. 12
A square is nonnegative and vanishes at x=11; the minimum is 12.
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