FE Exam Flashcards: Engineering Equations, Units & Handbook Skills

การ์ดในเด็คนี้

1. การ์ด 1

   คำถาม

   SI base unit of length?

   คำตอบ

   The metre, symbol $m$.

2. การ์ด 2

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   Slope between two points?

   คำตอบ

   $$m=\frac{y_2-y_1}{x_2-x_1}$$

   This requires $x_2\ne x_1$; slope is the change in the dependent variable per unit change in the independent variable.

3. การ์ด 3

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   Net force relationship for constant mass in an inertial frame?

   คำตอบ

   $$\sum\mathbf{F}=m\mathbf{a}$$

   Use a coherent unit system and include only external forces on the chosen body.

4. การ์ด 4

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   Cash-flow sign convention from the decision maker's viewpoint?

   คำตอบ

   Receipts are positive and disbursements are negative. Choose one viewpoint and keep it consistent across the cash-flow diagram and equations.

5. การ์ด 5

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   Where should an FE candidate get the current reference handbook for practice?

   คำตอบ

   From the candidate's MyNCEES account. Use the current supplied file rather than an old indexed copy or a third-party mirror.

6. การ์ด 6

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   Probability of the complement of event $A$?

   คำตอบ

   $$P(A^c)=1-P(A)$$

7. การ์ด 7

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   An engineering judgment conflicts with pressure to protect schedule or cost. First duty?

   คำตอบ

   Protect public health, safety, and welfare. Document the technical concern and escalate through appropriate channels.

8. การ์ด 8

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   SI base unit of mass?

   คำตอบ

   The kilogram, symbol $kg$. It is a mass unit, not a force unit.

9. การ์ด 9

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   Linear equation from slope and intercept?

   คำตอบ

   $$y=mx+b$$

   Here $m$ is slope and $b$ is the value of $y$ where $x=0$.

10. การ์ด 10

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    Velocity after time $t$ under constant acceleration?

    คำตอบ

    $$v=v_0+at$$

    This one-dimensional form requires constant $a$ and a consistent positive direction.

11. การ์ด 11

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    Where is an end-of-period cash flow placed?

    คำตอบ

    At the boundary marking the end of that period. Interest factors assume consistent period lengths and cash-flow timing.

12. การ์ด 12

    คำถาม

    How is the FE reference handbook available during the exam?

    คำตอบ

    It is supplied onscreen as a searchable PDF. Practice locating information in the same current reference rather than relying on memory of a printed layout.

13. การ์ด 13

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    Probability of $A$ or $B$ for possibly overlapping events?

    คำตอบ

    $$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$

14. การ์ด 14

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    SI base unit of time?

    คำตอบ

    The second, symbol $s$.

15. การ์ด 15

    คำถาม

    Linear equation from one point and a slope?

    คำตอบ

    $$y-y_1=m(x-x_1)$$

    Use it when the line passes through $(x_1,y_1)$ and has slope $m$.

16. การ์ด 16

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    Position after time $t$ under constant acceleration?

    คำตอบ

    $$x=x_0+v_0t+\frac{1}{2}at^2$$

    The acceleration is constant and all signed components use one coordinate direction.

17. การ์ด 17

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    A project assignment lies outside your competence. Proper response?

    คำตอบ

    Accept only the parts you are qualified to perform, or obtain qualified supervision or support. Do not present unfamiliar work as your own competent judgment.

18. การ์ด 18

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    SI base unit of electric current?

    คำตอบ

    The ampere, symbol $A$.

19. การ์ด 19

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    Future amount under simple interest?

    คำตอบ

    $$F=P(1+in)$$

    Here $P$ is principal, $i$ is interest per period, and $n$ is the number of periods; interest is earned only on principal.

20. การ์ด 20

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    Distance between two Cartesian points?

    คำตอบ

    $$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$

    The coordinate components must use the same length unit.

21. การ์ด 21

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    Constant-acceleration relation without elapsed time?

    คำตอบ

    $$v^2=v_0^2+2a(x-x_0)$$

    Use signed one-dimensional components and constant $a$.

22. การ์ด 22

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    Where do you search the supplied reference material in the exam interface?

    คำตอบ

    Use the exam software's search box beside the supplied reference material.

23. การ์ด 23

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    Conditional probability of $A$ given $B$?

    คำตอบ

    $$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$$

    This requires $P(B)>0$.

24. การ์ด 24

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    SI base unit of thermodynamic temperature?

    คำตอบ

    The kelvin, symbol $K$. Kelvin is written without a degree sign.

25. การ์ด 25

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    A client asks you to remove an unfavorable test result from a report. Proper response?

    คำตอบ

    Keep the material result and explain it accurately. Engineering statements should be objective, complete enough to avoid deception, and supported by evidence.

26. การ์ด 26

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    Midpoint of two Cartesian points?

    คำตอบ

    $$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$$

    Average corresponding coordinates.

27. การ์ด 27

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    Linear momentum of a particle?

    คำตอบ

    $$\mathbf{p}=m\mathbf{v}$$

    For constant mass, SI units are $kg\cdot m/s$.

28. การ์ด 28

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    Future amount under compound interest?

    คำตอบ

    $$F=P(1+i)^n$$

    The rate period must match the period used for $n$.

29. การ์ด 29

    คำถาม

    Should an FE search workflow depend on Ctrl+F?

    คำตอบ

    No. The May 2026 NCEES Examinee Guide says Ctrl+F isn't available; practice with the exam interface's own search box.

30. การ์ด 30

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    SI base unit of amount of substance?

    คำตอบ

    The mole, symbol $mol$.

31. การ์ด 31

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    Joint probability expressed with a conditional probability?

    คำตอบ

    $$P(A\cap B)=P(A\mid B)P(B)$$

    This elementary formula requires $P(B)>0$.

32. การ์ด 32

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    Standard equation of a circle centered at $(h,k)$?

    คำตอบ

    $$(x-h)^2+(y-k)^2=r^2$$

    The coordinate units and radius unit must match.

33. การ์ด 33

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    Impulse-momentum relationship?

    คำตอบ

    $$\int_{t_1}^{t_2}\sum\mathbf{F}\,dt=\mathbf{p}_2-\mathbf{p}_1$$

    The force is the net external force on the chosen system.

34. การ์ด 34

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    You discover a financial interest that could affect your recommendation. Proper response?

    คำตอบ

    Disclose the conflict promptly to affected parties and obtain appropriate consent or withdraw. Hidden personal interests undermine independent judgment.

35. การ์ด 35

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    SI base unit of luminous intensity?

    คำตอบ

    The candela, symbol $cd$.

36. การ์ด 36

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    Present worth of one future amount?

    คำตอบ

    $$P=\frac{F}{(1+i)^n}$$

    Use an effective rate $i$ per period consistent with $n$.

37. การ์ด 37

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    Roots of $ax^2+bx+c=0$?

    คำตอบ

    $$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

    This form requires $a\ne0$.

38. การ์ด 38

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    When is total linear momentum conserved?

    คำตอบ

    When net external impulse on the system is zero. Internal forces can redistribute momentum but do not change the system total.

39. การ์ด 39

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    Why avoid memorizing FE handbook page numbers?

    คำตอบ

    Pagination and layout can change. Remember section names, concepts, and strong search terms instead.

40. การ์ด 40

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    Probability condition that defines independence of $A$ and $B$?

    คำตอบ

    $$P(A\cap B)=P(A)P(B)$$

    Independence is not the same as mutual exclusivity.

41. การ์ด 41

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    SI unit of frequency in base-unit form?

    คำตอบ

    The hertz, $Hz=s^{-1}$, applies to periodic phenomena. One hertz is one cycle per second.

42. การ์ด 42

    คำถาม

    Present worth of an end-of-period uniform series $A$?

    คำตอบ

    $$P=A\frac{(1+i)^n-1}{i(1+i)^n}$$

    This requires $i\ne0$; at $i=0$, $P=An$.

43. การ์ด 43

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    What does $b^2-4ac$ reveal for a real-coefficient quadratic?

    คำตอบ

    It determines the root type: positive gives two distinct real roots, zero gives one repeated real root, and negative gives a complex-conjugate pair.

44. การ์ด 44

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    Work done by a constant force through displacement?

    คำตอบ

    $$W=\mathbf{F}\cdot\mathbf{s}=Fs\cos\theta$$

    Use the angle between force and displacement; SI units are joules.

45. การ์ด 45

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    A vendor offers a valuable gift while you evaluate bids. Proper response?

    คำตอบ

    Decline it and follow the organization's disclosure policy. Even an intended courtesy can create an actual or apparent conflict.

46. การ์ด 46

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    SI unit of force in base-unit form?

    คำตอบ

    The newton, $N=kg\cdot m/s^2$.

47. การ์ด 47

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    First search cue for a named quantity such as coefficient of thermal expansion?

    คำตอบ

    Search the exact technical noun phrase. Distinctive nouns usually narrow results better than a whole question sentence.

48. การ์ด 48

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    Bayes' rule for $P(A\mid B)$?

    คำตอบ

    $$P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}$$

    This requires $P(B)>0$.

49. การ์ด 49

    คำถาม

    Product of powers with the same positive base and real exponents?

    คำตอบ

    $$a^m a^n=a^{m+n}$$

    This form assumes $a>0$ and real exponents $m$ and $n$.

50. การ์ด 50

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    Translational kinetic energy of a particle?

    คำตอบ

    $$K=\frac{1}{2}mv^2$$

    Mass and speed must use a coherent unit system.

51. การ์ด 51

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    SI unit of pressure in force-area form?

    คำตอบ

    The pascal, $Pa=N/m^2$. It is the SI derived unit for pressure and stress.

52. การ์ด 52

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    Equivalent annual amount for present worth $P$?

    คำตอบ

    $$A=P\frac{i(1+i)^n}{(1+i)^n-1}$$

    For the displayed formula, $i\ne0$; at $i=0$, $A=P/n$. Payments occur at period ends and $i$ matches that period.

53. การ์ด 53

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    May confidential client information be shared for personal benefit?

    คำตอบ

    No. Protect it unless the client authorizes disclosure or law, safety, or professional duty requires it.

54. การ์ด 54

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    A handbook search term returns too many results. Best refinement?

    คำตอบ

    Add one meaningful qualifier, such as the phenomenon, geometry, distribution, or material property you need.

55. การ์ด 55

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    Quotient of powers with the same positive base and real exponents?

    คำตอบ

    $$\frac{a^m}{a^n}=a^{m-n}$$

    This form assumes $a>0$ and real exponents $m$ and $n$.

56. การ์ด 56

    คำถาม

    Net work-energy relationship for a particle?

    คำตอบ

    $$W_{net}=K_2-K_1$$

    The net work includes the work of all forces represented in the particle model.

57. การ์ด 57

    คำถาม

    Ordered selections of $r$ objects from $n$ distinct objects?

    คำตอบ

    $$P(n,r)=\frac{n!}{(n-r)!}$$

    Use this when order matters and selection is without replacement.

58. การ์ด 58

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    Future worth of an end-of-period uniform series $A$?

    คำตอบ

    $$F=A\frac{(1+i)^n-1}{i}$$

    This requires $i\ne0$; at $i=0$, $F=An$.

59. การ์ด 59

    คำถาม

    Power of a power for a positive base and real exponents?

    คำตอบ

    $$(a^m)^n=a^{mn}$$

    This form assumes $a>0$ and real exponents $m$ and $n$.

60. การ์ด 60

    คำถาม

    SI unit of energy in force-distance form?

    คำตอบ

    The joule, $J=N\cdot m$.

61. การ์ด 61

    คำถาม

    A problem gives only an unfamiliar symbol. Best search preparation?

    คำตอบ

    Translate the symbol into its likely quantity name and discipline context before searching; symbol-only searches are often ambiguous.

62. การ์ด 62

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    Confidential information reveals an imminent public-safety risk. What comes first?

    คำตอบ

    Take steps required to protect people, using appropriate legal and organizational channels. Share only what is necessary and document the basis for escalation.

63. การ์ด 63

    คำถาม

    Unordered selections of $r$ objects from $n$ distinct objects?

    คำตอบ

    $$C(n,r)=\frac{n!}{r!(n-r)!}$$

    Use this when order does not matter and selection is without replacement.

64. การ์ด 64

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    Change in gravitational potential energy near Earth's surface?

    คำตอบ

    $$\Delta U_g=mg\Delta h$$

    Use approximately constant $g$ and a clearly chosen height datum.

65. การ์ด 65

    คำถาม

    Negative-exponent rule for a positive base and real exponent?

    คำตอบ

    $$a^{-n}=\frac{1}{a^n}$$

    This form assumes $a>0$ and real exponent $n$.

66. การ์ด 66

    คำถาม

    Uniform end-of-period deposit needed for future amount $F$?

    คำตอบ

    $$A=F\frac{i}{(1+i)^n-1}$$

    For the displayed formula, $i\ne0$; at $i=0$, $A=F/n$. The first deposit is one period from now and the last occurs at period $n$.

67. การ์ด 67

    คำถาม

    A handbook search for an acronym fails. Next move?

    คำตอบ

    Try the expanded term and a common alternate name. Acronyms vary by section and discipline.

68. การ์ด 68

    คำถาม

    SI unit of power in energy-time form?

    คำตอบ

    The watt, $W=J/s$.

69. การ์ด 69

    คำถาม

    Logarithm of a product?

    คำตอบ

    $$\log_b(xy)=\log_b x+\log_b y$$

    For real logarithms, $b>0$, $b\ne1$, and both arguments are positive.

70. การ์ด 70

    คำถาม

    Conditions for a binomial model?

    คำตอบ

    A fixed number of independent trials, two outcomes per trial, and the same success probability on every trial.

71. การ์ด 71

    คำถาม

    Using another engineer's analysis in your deliverable?

    คำตอบ

    Credit the source and verify the analysis before relying on it. Do not present someone else's work as your own.

72. การ์ด 72

    คำถาม

    SI unit of electric charge in current-time form?

    คำตอบ

    The coulomb, $C=A\cdot s$.

73. การ์ด 73

    คำถาม

    Average power from work over elapsed time?

    คำตอบ

    $$P_{avg}=\frac{W}{\Delta t}$$

    Use consistent energy and time units; watts are joules per second.

74. การ์ด 74

    คำถาม

    Annual equivalent of cost $P$ with salvage $S$ after $n$ periods?

    คำตอบ

    $$A=P\frac{i(1+i)^n}{(1+i)^n-1}-S\frac{i}{(1+i)^n-1}$$

    The first factor converts present cost to an annual amount; the second converts future salvage. Use the same effective rate and period basis. At $i=0$, $A=(P-S)/n$.

75. การ์ด 75

    คำถาม

    Logarithm of a positive quantity raised to a power?

    คำตอบ

    $$\log_b(x^p)=p\log_b x$$

    For real logarithms, $x>0$, $b>0$, and $b\ne1$.

76. การ์ด 76

    คำถาม

    SI unit of electric potential difference in energy-charge form?

    คำตอบ

    The volt, $V=J/C$.

77. การ์ด 77

    คำถาม

    Instantaneous mechanical power delivered by a force?

    คำตอบ

    $$P=\mathbf{F}\cdot\mathbf{v}$$

    Use the velocity of the force's point of application.

78. การ์ด 78

    คำถาม

    A precise handbook term gives no useful match. Next move?

    คำตอบ

    Try a standard synonym or the broader parent concept, then narrow within the relevant section.

79. การ์ด 79

    คำถาม

    Mean of $X\sim\operatorname{Binomial}(n,p)$?

    คำตอบ

    $$E[X]=np$$

80. การ์ด 80

    คำถาม

    SI unit of electrical resistance in voltage-current form?

    คำตอบ

    The ohm, $\Omega=V/A$.

81. การ์ด 81

    คำถาม

    Moment of a force about a point?

    คำตอบ

    $$\mathbf{M}=\mathbf{r}\times\mathbf{F}$$

    Vector $\mathbf{r}$ runs from the moment point to a point on the force's line of action.

82. การ์ด 82

    คำถาม

    Cash-flow pattern described by an arithmetic gradient $G$?

    คำตอบ

    A series that changes by the same amount $G$ each period. In the usual convention, the first gradient increment appears after the first base payment.

83. การ์ด 83

    คำถาม

    Inverse relationship between the natural log and exponential?

    คำตอบ

    $$\ln(e^x)=x,\qquad e^{\ln x}=x$$

    The second identity requires $x>0$.

84. การ์ด 84

    คำถาม

    SI prefix giga?

    คำตอบ

    $G=10^9$.

85. การ์ด 85

    คำถาม

    Angular velocity after time $t$ under constant angular acceleration?

    คำตอบ

    $$\omega=\omega_0+\alpha t$$

    Use consistent angular sign convention and radians for angular measures.

86. การ์ด 86

    คำถาม

    When should an engineer sign or seal technical work?

    คำตอบ

    Only when authorized and when the work was prepared by the engineer or under the level of responsible charge required by the applicable jurisdiction.

87. การ์ด 87

    คำถาม

    Best memory target for returning to a handbook formula?

    คำตอบ

    The section or concept name plus a distinctive search phrase. That survives layout changes better than a page number.

88. การ์ด 88

    คำถาม

    SI prefix mega?

    คำตอบ

    $M=10^6$. Uppercase matters: lowercase $m$ means milli.

89. การ์ด 89

    คำถาม

    Convert degrees to radians?

    คำตอบ

    $$\theta_{rad}=\theta_{deg}\frac{\pi}{180}$$

    Use radians in calculus formulas unless a formula explicitly says otherwise.

90. การ์ด 90

    คำถาม

    Variance of $X\sim\operatorname{Binomial}(n,p)$?

    คำตอบ

    $$\operatorname{Var}(X)=np(1-p)$$

91. การ์ด 91

    คำถาม

    Cash-flow pattern described by a geometric gradient rate $g$?

    คำตอบ

    A series in which each payment is the previous payment multiplied by $1+g$. Keep $g$ on the same period basis as the interest rate.

92. การ์ด 92

    คำถาม

    Where do licensure and sealing requirements come from?

    คำตอบ

    The applicable jurisdiction's current laws and board rules. Confirm local requirements rather than assuming one state's rules apply everywhere.

93. การ์ด 93

    คำถาม

    Rotational kinetic energy about a fixed axis?

    คำตอบ

    $$K_{rot}=\frac{1}{2}I\omega^2$$

    $I$ is the mass moment of inertia about that axis.

94. การ์ด 94

    คำถาม

    SI prefix kilo?

    คำตอบ

    $k=10^3$.

95. การ์ด 95

    คำถาม

    Right-triangle definition of sine?

    คำตอบ

    Sine is opposite over hypotenuse. This definition applies to an acute angle in a right triangle.

96. การ์ด 96

    คำถาม

    After search lands near a result, what should you scan first?

    คำตอบ

    The section heading and nearby subheading. They establish whether the result belongs to the right model and subject.

97. การ์ด 97

    คำถาม

    Rotational analogue of Newton's second law about a fixed principal axis?

    คำตอบ

    $$\sum M=I\alpha$$

    This scalar form assumes the stated fixed-axis model and a consistent sign convention.

98. การ์ด 98

    คำถาม

    SI prefix milli?

    คำตอบ

    $m=10^{-3}$. Context distinguishes the prefix from the metre symbol.

99. การ์ด 99

    คำถาม

    Meaning of a nominal annual rate compounded $m$ times per year?

    คำตอบ

    It is a quoted annual rate equal to $m$ times the periodic rate. It is not itself the effective annual rate unless compounding is annual.

100. การ์ด 100

     คำถาม

     Standardize an observation $x$ from a distribution with mean $\mu$ and standard deviation $\sigma$?

     คำตอบ

     $$z=\frac{x-\mu}{\sigma}$$

     This requires $\sigma>0$; using a normal probability table also assumes an appropriate normal model.

101. การ์ด 101

     คำถาม

     Right-triangle definition of cosine?

     คำตอบ

     Cosine is adjacent over hypotenuse. This definition applies to an acute angle in a right triangle.

102. การ์ด 102

     คำถาม

     Force-displacement relation for an ideal linear spring?

     คำตอบ

     $$F=-kx$$

     The minus sign shows restoring direction; the model applies within the spring's linear range.

103. การ์ด 103

     คำถาม

     May qualifications or project responsibility be exaggerated in a proposal?

     คำตอบ

     No. State education, licensure, experience, and responsibility accurately, including the actual roles of collaborators.

104. การ์ด 104

     คำถาม

     SI prefix micro?

     คำตอบ

     $\mu=10^{-6}$.

105. การ์ด 105

     คำถาม

     A handbook symbol conflicts with your usual notation. What controls?

     คำตอบ

     The definition beside the selected relationship. Symbols can represent different quantities in different sections.

106. การ์ด 106

     คำถาม

     Periodic rate from nominal annual rate $r$ compounded $m$ times per year?

     คำตอบ

     $$i=\frac{r}{m}$$

     Use consistent decimal units, and pair this rate with the number of compounding periods.

107. การ์ด 107

     คำถาม

     Right-triangle definition of tangent?

     คำตอบ

     Tangent is opposite over adjacent, or $\sin\theta/\cos\theta$ where $\cos\theta\ne0$.

108. การ์ด 108

     คำถาม

     Average normal stress under centered axial load?

     คำตอบ

     $$\sigma=\frac{F}{A}$$

     $F$ is normal to area $A$; the average ignores local stress concentrations.

109. การ์ด 109

     คำถาม

     Mean, median, and mode of a symmetric normal distribution?

     คำตอบ

     They are equal and located at $\mu$. The distribution is symmetric about that center.

110. การ์ด 110

     คำถาม

     SI prefix nano?

     คำตอบ

     $n=10^{-9}$.

111. การ์ด 111

     คำถาม

     Fundamental Pythagorean trigonometric identity?

     คำตอบ

     $$\sin^2\theta+\cos^2\theta=1$$

112. การ์ด 112

     คำถาม

     Engineering normal strain?

     คำตอบ

     $$\varepsilon=\frac{\Delta L}{L_0}$$

     It is dimensionless and uses original gauge length $L_0$.

113. การ์ด 113

     คำถาม

     A candidate retrieves a formula from the handbook. What must be checked with it?

     คำตอบ

     Its applicability assumptions and variable definitions. A familiar equation used under the wrong model is still wrong.

114. การ์ด 114

     คำถาม

     Technical duty to an employer or client?

     คำตอบ

     Act as a faithful professional within lawful, ethical limits, communicate material risks, and preserve independent technical judgment.

115. การ์ด 115

     คำถาม

     How does a length conversion factor apply to area?

     คำตอบ

     Square it. If $1\,u=a\,v$, then $1\,u^2=a^2\,v^2$.

116. การ์ด 116

     คำถาม

     Effective annual rate from nominal rate $r$ compounded $m$ times per year?

     คำตอบ

     $$i_{eff}=\left(1+\frac{r}{m}\right)^m-1$$

117. การ์ด 117

     คำถาม

     When is a Poisson count model reasonable?

     คำตอบ

     For counts in a fixed interval when events occur independently at an approximately constant average rate and simultaneous events are negligible.

118. การ์ด 118

     คำถาม

     Relationship for the third side of a non-right triangle?

     คำตอบ

     $$c^2=a^2+b^2-2ab\cos C$$

     Angle $C$ lies between sides $a$ and $b$, opposite side $c$.

119. การ์ด 119

     คำถาม

     Linear elastic stress-strain relation in uniaxial loading?

     คำตอบ

     $$\sigma=E\varepsilon$$

     Use it in the material's linear elastic range; $E$ has stress units.

120. การ์ด 120

     คำถาม

     How does a length conversion factor apply to volume?

     คำตอบ

     Cube it. If $1\,u=a\,v$, then $1\,u^3=a^3\,v^3$.

121. การ์ด 121

     คำถาม

     What should you record for every variable in a retrieved equation?

     คำตอบ

     Its meaning and expected unit or dimension. This catches swapped symbols and incompatible substitutions.

122. การ์ด 122

     คำถาม

     Effective annual rate for continuous compounding at nominal rate $r$?

     คำตอบ

     $$i_{eff}=e^r-1$$

     Here $r$ is expressed as a decimal per year.

123. การ์ด 123

     คำถาม

     A supervisor directs an illegal engineering action. Proper response?

     คำตอบ

     Refuse the illegal action, document the direction, and escalate through appropriate legal or organizational channels.

124. การ์ด 124

     คำถาม

     Magnitude of a three-component vector $\mathbf{v}$?

     คำตอบ

     $$\lVert\mathbf{v}\rVert=\sqrt{v_x^2+v_y^2+v_z^2}$$

     All components must represent the same physical quantity in compatible units.

125. การ์ด 125

     คำถาม

     Mass density?

     คำตอบ

     $$\rho=\frac{m}{V}$$

     Use mass, not weight, in the numerator.

126. การ์ด 126

     คำถาม

     Inches in one foot?

     คำตอบ

     $12\,in=1\,ft$.

127. การ์ด 127

     คำถาม

     Mean and variance of $X\sim\operatorname{Poisson}(\lambda)$?

     คำตอบ

     Both equal $\lambda$. The standard deviation is $\sqrt{\lambda}$.

128. การ์ด 128

     คำถาม

     How should subscripts be handled in a retrieved relationship?

     คำตอบ

     Read what each subscript identifies—state, direction, phase, location, or reference—before substituting values.

129. การ์ด 129

     คำถาม

     Dot product from vector magnitudes and included angle?

     คำตอบ

     $$\mathbf{a}\cdot\mathbf{b}=\lVert\mathbf{a}\rVert\lVert\mathbf{b}\rVert\cos\theta$$

     The result is a scalar.

130. การ์ด 130

     คำถาม

     Average normal pressure on an area?

     คำตอบ

     $$p=\frac{F_n}{A}$$

     $F_n$ is the normal force; pressure is scalar in a fluid at rest.

131. การ์ด 131

     คำถาม

     Feet in one yard?

     คำตอบ

     $3\,ft=1\,yd$.

132. การ์ด 132

     คำถาม

     When are two interest rates equivalent?

     คำตอบ

     When they produce the same accumulation over the same time interval. Convert both rates to a common effective period before comparing them.

133. การ์ด 133

     คำถาม

     You discover a material error in issued work. Proper response?

     คำตอบ

     Notify responsible parties promptly, assess the safety impact, correct the record, and document the response. Do not hide the error.

134. การ์ด 134

     คำถาม

     Expected value of a discrete random variable?

     คำตอบ

     $$E[X]=\sum_x xP(X=x)$$

     Sum over all possible values with a valid probability mass function.

135. การ์ด 135

     คำถาม

     Dot-product test for perpendicular nonzero vectors?

     คำตอบ

     Their dot product is zero. The zero vector also has zero dot product with every vector, so require both vectors to be nonzero before concluding perpendicularity.

136. การ์ด 136

     คำถาม

     Hydrostatic pressure change with vertical depth in a constant-density fluid?

     คำตอบ

     $$\Delta p=\rho g\Delta h$$

     Depth increases downward and the fluid is at rest.

137. การ์ด 137

     คำถาม

     Feet in one statute mile?

     คำตอบ

     $5280\,ft=1\,mi$.

138. การ์ด 138

     คำถาม

     Present-worth acceptance rule for an independent investment at the minimum attractive rate of return?

     คำตอบ

     Accept it when net present worth is nonnegative. All cash flows must be discounted to the same time using the chosen comparison rate.

     

     276 การ์ด

     FE Exam Flashcards: Engineering Equations, Units & Handbook Skills

     เรียนเด็คนี้ฟรี

     Nibomo จะเปิดขึ้นเพื่อให้คุณเริ่มเรียนได้ทันที

139. การ์ด 139

     คำถาม

     A retrieved formula contains signed quantities. What should you confirm?

     คำตอบ

     The stated sign convention and positive direction. Do not mix magnitudes with signed components silently.

140. การ์ด 140

     คำถาม

     Magnitude of a vector cross product?

     คำตอบ

     $$\lVert\mathbf{a}\times\mathbf{b}\rVert=\lVert\mathbf{a}\rVert\lVert\mathbf{b}\rVert\sin\theta$$

     The direction follows the right-hand rule.

141. การ์ด 141

     คำถาม

     Exact inch-to-metre relationship?

     คำตอบ

     $1\,in=0.0254\,m$ exactly.

142. การ์ด 142

     คำถาม

     Volumetric flow rate through a cross-section?

     คำตอบ

     $$Q=Av_{avg}$$

     Velocity is the area-average component normal to the section.

143. การ์ด 143

     คำถาม

     Variance from the mean $\mu$?

     คำตอบ

     $$\operatorname{Var}(X)=E[(X-\mu)^2]$$

     Variance has squared units; standard deviation has the original units.

144. การ์ด 144

     คำถาม

     Future-worth acceptance rule for an independent investment?

     คำตอบ

     Accept it when net future worth at the study horizon is nonnegative, using the minimum attractive rate of return consistently.

145. การ์ด 145

     คำถาม

     What coordinate detail should be checked before using a vector or mechanics formula?

     คำตอบ

     Axis directions and the reference point or origin. Components and moments depend on those choices.

146. การ์ด 146

     คำถาม

     Exact foot-to-metre relationship?

     คำตอบ

     $1\,ft=0.3048\,m$ exactly.

147. การ์ด 147

     คำถาม

     How should important uncertainty be communicated?

     คำตอบ

     State the assumptions, limits, and plausible consequences clearly enough for the decision maker to understand the risk.

148. การ์ด 148

     คำถาม

     When is the matrix product $AB$ defined?

     คำตอบ

     It is defined when the number of columns of $A$ equals the number of rows of $B$. If $A$ is $m\times n$ and $B$ is $n\times p$, then $AB$ is $m\times p$.

149. การ์ด 149

     คำถาม

     Steady one-inlet, one-outlet mass continuity?

     คำตอบ

     $$\rho_1A_1v_1=\rho_2A_2v_2$$

     Here $v_1$ and $v_2$ are cross-sectional average velocity components normal to their sections, with density treated as uniform over each section. For incompressible flow with constant density, this reduces to $A_1v_1=A_2v_2$.

150. การ์ด 150

     คำถาม

     Sample mean of observations $x_1,\ldots,x_n$?

     คำตอบ

     $$\bar{x}=\frac{1}{n}\sum_{i=1}^n x_i$$

151. การ์ด 151

     คำถาม

     Square feet in one acre?

     คำตอบ

     $1\,acre=43{,}560\,ft^2$.

152. การ์ด 152

     คำถาม

     Determinant of $\begin{bmatrix}a&b\\c&d\end{bmatrix}$?

     คำตอบ

     $$ad-bc$$

153. การ์ด 153

     คำถาม

     Bernoulli relationship along a streamline for ideal steady flow?

     คำตอบ

     $$\frac{p}{\rho g}+\frac{v^2}{2g}+z=\text{constant}$$

     This basic form assumes steady, incompressible, negligible-viscosity flow with no shaft work or heat effects between points.

154. การ์ด 154

     คำถาม

     May inconvenient measurements be discarded without a technical basis?

     คำตอบ

     No. Use a documented, technically justified rule for excluding data and preserve the original record.

155. การ์ด 155

     คำถาม

     Annual-worth acceptance rule for an independent investment?

     คำตอบ

     Accept it when equivalent annual worth is nonnegative. Compare alternatives over compatible service assumptions.

156. การ์ด 156

     คำถาม

     Before using a handbook equation, what unit-system check matters?

     คำตอบ

     Confirm that every input and any embedded constant use one compatible unit system. Convert first when necessary.

157. การ์ด 157

     คำถาม

     Cubic metres in one litre?

     คำตอบ

     $1\,L=10^{-3}\,m^3$.

158. การ์ด 158

     คำถาม

     What does a zero determinant mean for a square matrix?

     คำตอบ

     The matrix is singular and has no inverse. A corresponding linear system cannot have one unique solution.

159. การ์ด 159

     คำถาม

     Reynolds number for internal flow using characteristic length $L$?

     คำตอบ

     $$Re=\frac{\rho vL}{\mu}=\frac{vL}{\nu}$$

     It is dimensionless; use consistent properties and the geometry-appropriate characteristic length.

160. การ์ด 160

     คำถาม

     Unbiased sample variance for an independent sample from one population?

     คำตอบ

     $$s^2=\frac{1}{n-1}\sum_{i=1}^n(x_i-\bar{x})^2$$

     This requires at least two observations, so $n\ge2$. The $n-1$ denominator estimates population variance after fitting the sample mean.

161. การ์ด 161

     คำถาม

     How should mutually exclusive alternatives be compared?

     คำตอบ

     Use one consistent economic measure over a common study period and select the economically preferred feasible alternative; do not accept every alternative with a positive individual worth.

162. การ์ด 162

     คำถาม

     Cubic centimetres in one millilitre?

     คำตอบ

     $1\,mL=1\,cm^3$.

163. การ์ด 163

     คำถาม

     Why keep contemporaneous engineering records?

     คำตอบ

     They preserve assumptions, decisions, calculations, approvals, and changes so the work can be checked and responsibilities understood.

164. การ์ด 164

     คำถาม

     Fast check after substituting into a handbook formula?

     คำตอบ

     Verify that the resulting dimensions match the requested quantity and that additive terms share dimensions.

165. การ์ด 165

     คำถาม

     Matrix solution of $A\mathbf{x}=\mathbf{b}$ when $A$ is invertible?

     คำตอบ

     $$\mathbf{x}=A^{-1}\mathbf{b}$$

     In numerical work, direct factorization is usually preferable to explicitly forming the inverse.

166. การ์ด 166

     คำถาม

     Sensible heat for constant specific heat and no phase change?

     คำตอบ

     $$Q=mc\Delta T$$

     $c$ must match the material, temperature range, mass unit, and temperature-difference unit.

167. การ์ด 167

     คำถาม

     Standard error of the mean for an i.i.d. sample with population standard deviation $\sigma$?

     คำตอบ

     $$SE(\bar{x})=\frac{\sigma}{\sqrt{n}}$$

     The observations are independent and identically distributed with common variance $\sigma^2$. Use $s$ for unknown $\sigma$ only when the selected inference method allows it.

168. การ์ด 168

     คำถาม

     Safe setup for converting $25\,ft^2$ to square metres?

     คำตอบ

     $$25\,ft^2\left(\frac{0.3048\,m}{1\,ft}\right)^2$$

     The squared length factor cancels $ft^2$ and leaves $m^2$.

169. การ์ด 169

     คำถาม

     Purpose of incremental analysis for mutually exclusive alternatives?

     คำตอบ

     It tests whether the additional investment in the higher-cost option earns the required return compared with the lower-cost option.

170. การ์ด 170

     คำถาม

     Physical meaning of $dy/dx$?

     คำตอบ

     It is the instantaneous rate of change of $y$ with respect to $x$, with units of the $y$ unit divided by the $x$ unit.

171. การ์ด 171

     คำถาม

     One-dimensional steady conduction through a plane wall?

     คำตอบ

     $$\dot Q=-kA\frac{dT}{dx}$$

     The minus sign indicates heat flows toward lower temperature; $k$ is thermal conductivity.

172. การ์ด 172

     คำถาม

     When should you algebraically rearrange a handbook relationship?

     คำตอบ

     After confirming the correct equation, definitions, assumptions, and sign convention. Then isolate the requested variable without changing its domain restrictions.

173. การ์ด 173

     คำถาม

     Safe setup for converting $2\,in^3$ to cubic metres?

     คำตอบ

     $$2\,in^3\left(\frac{0.0254\,m}{1\,in}\right)^3$$

     Cube the exact length conversion because the original quantity is volume.

174. การ์ด 174

     คำถาม

     A client changes a safety-critical design requirement verbally. Proper response?

     คำตอบ

     Confirm the change in writing, analyze its consequences, and obtain the required approvals before relying on it.

175. การ์ด 175

     คำถาม

     Derivative of $x^n$ for constant $n$?

     คำตอบ

     $$\frac{d}{dx}x^n=nx^{n-1}$$

     Apply domain restrictions when $n$ is non-integer or negative.

176. การ์ด 176

     คำถาม

     Convective heat-transfer rate at a surface?

     คำตอบ

     $$\dot Q=hA(T_s-T_\infty)$$

     $h$ depends on the flow and geometry; the sign follows the chosen temperature difference.

177. การ์ด 177

     คำถาม

     What does the central limit theorem say about a sample mean?

     คำตอบ

     For sufficiently large independent, identically distributed samples with finite variance, the sampling distribution of the properly standardized mean approaches normality. It does not make the raw data normal.

178. การ์ด 178

     คำถาม

     Internal rate of return?

     คำตอบ

     The discount rate that makes net present worth zero. Multiple sign changes in cash flow can produce multiple or no meaningful real rates.

179. การ์ด 179

     คำถาม

     Several similar handbook equations appear together. How do you choose?

     คำตอบ

     Match each equation's stated geometry, boundary conditions, material model, and variable definitions to the problem.

180. การ์ด 180

     คำถาม

     Convert Celsius temperature to kelvin?

     คำตอบ

     $$T_K=T_{^\circ C}+273.15$$

     This conversion is for absolute temperature, not temperature difference.

181. การ์ด 181

     คำถาม

     Derivative of a product $u(x)v(x)$?

     คำตอบ

     $$(uv)'=u'v+uv'$$

182. การ์ด 182

     คำถาม

     Internal escalation does not resolve a serious public-safety threat. Next step?

     คำตอบ

     Use the next appropriate authority or reporting channel, consistent with law and the urgency of the hazard. Keep a factual record.

183. การ์ด 183

     คำถาม

     Two-sided confidence interval for a population mean with known $\sigma$?

     คำตอบ

     $$\bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$$

     Use the normal-model conditions and the critical value for confidence level $1-\alpha$.

184. การ์ด 184

     คำถาม

     Net thermal radiation from a small gray surface to large surroundings?

     คำตอบ

     $$\dot Q=\varepsilon\sigma A(T_s^4-T_{sur}^4)$$

     Use absolute temperatures; the model assumes the stated gray-surface surroundings.

185. การ์ด 185

     คำถาม

     Basic benefit-cost acceptance rule for an independent public project?

     คำตอบ

     Accept when the consistently defined benefit-cost ratio is at least one. For mutually exclusive choices, use an incremental comparison rather than ranking by individual ratios alone.

186. การ์ด 186

     คำถาม

     Derivative of a composite function $f(g(x))$?

     คำตอบ

     $$\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$$

187. การ์ด 187

     คำถาม

     Why inspect boundary or initial conditions near a formula?

     คำตอบ

     They determine which solution form applies and can eliminate constants or terms. A formula without its conditions may be unusable.

188. การ์ด 188

     คำถาม

     Convert Celsius temperature to Fahrenheit?

     คำตอบ

     $$T_{^\circ F}=\frac{9}{5}T_{^\circ C}+32$$

189. การ์ด 189

     คำถาม

     Ethical basis for expert technical testimony?

     คำตอบ

     Offer opinions only within your competence, disclose important assumptions and limitations, and distinguish facts from professional judgment.

190. การ์ด 190

     คำถาม

     Meaning of a breakeven value?

     คำตอบ

     The value of a selected variable that makes two alternatives economically equivalent or makes net worth zero. Other assumptions are held fixed.

191. การ์ด 191

     คำถาม

     Derivative of $e^x$?

     คำตอบ

     $$\frac{d}{dx}e^x=e^x$$

192. การ์ด 192

     คำถาม

     First-law energy balance for a closed system with work positive out?

     คำตอบ

     $$\Delta E=Q-W$$

     $Q$ is heat into the system and $W$ is work done by the system; state a different sign convention before using it.

193. การ์ด 193

     คำถาม

     Frequentist meaning of a 95% confidence procedure?

     คำตอบ

     Across repeated samples, about 95% of intervals built by the procedure would contain the fixed population parameter. It is not a 95% probability assigned to this already-computed interval.

194. การ์ด 194

     คำถาม

     A needed value falls between handbook table entries. What is the cue?

     คำตอบ

     Check whether interpolation is appropriate, keep both axes and units straight, and avoid extrapolating unless the source supports it.

195. การ์ด 195

     คำถาม

     Derivative of $\ln x$ over the real domain?

     คำตอบ

     $$\frac{d}{dx}\ln x=\frac{1}{x}$$

     This real-valued form requires $x>0$.

196. การ์ด 196

     คำถาม

     Relationship between Celsius and kelvin temperature differences?

     คำตอบ

     $1\,K$ of temperature difference equals $1\,^{\circ}C$ of temperature difference. Do not add $273.15$ to a difference.

197. การ์ด 197

     คำถาม

     How should credit for team engineering work be assigned?

     คำตอบ

     Accurately recognize each contributor's work and responsibility. Do not claim sole credit for a collaborative result.

198. การ์ด 198

     คำถาม

     Main limitation of simple payback period?

     คำตอบ

     It ignores the time value of money and usually ignores cash flows after payback. Discounted payback fixes the first issue, not the second.

199. การ์ด 199

     คำถาม

     Roles of the null and alternative hypotheses?

     คำตอบ

     The null states the baseline parameter claim tested; the alternative states the competing direction or difference supported when evidence is strong enough.

200. การ์ด 200

     คำถาม

     Ideal-gas equation of state?

     คำตอบ

     $$pV=nRT$$

     Use absolute temperature and a gas constant consistent with pressure, volume, amount, and energy units.

201. การ์ด 201

     คำถาม

     First check when reading a handbook table?

     คำตอบ

     Read the title, column headings, units, notes, and basis conditions before selecting a value.

202. การ์ด 202

     คำถาม

     Candidates for absolute extrema of a continuous function on a closed interval?

     คำตอบ

     The endpoints and every interior point where the first derivative is zero or does not exist. Evaluate the function at each candidate; the largest and smallest values are the absolute extrema.

203. การ์ด 203

     คำถาม

     How should a sunk cost affect a current engineering-economy decision?

     คำตอบ

     It should not affect the choice because it cannot be changed by the decision. Use future incremental cash flows instead.

204. การ์ด 204

     คำถาม

     Convert Fahrenheit absolute temperature to Rankine?

     คำตอบ

     $$T_{^\circ R}=T_{^\circ F}+459.67$$

     Rankine uses the Fahrenheit-sized degree from absolute zero.

205. การ์ด 205

     คำถาม

     Ohm's law for an ideal resistor?

     คำตอบ

     $$V=IR$$

     Use a consistent passive sign convention; $R$ is constant only over the assumed operating range.

206. การ์ด 206

     คำถาม

     You have confidential details about a competitor's bid. Proper response?

     คำตอบ

     Do not use or disclose improperly obtained information. Notify the appropriate procurement or ethics contact.

207. การ์ด 207

     คำถาม

     Physical meaning of a definite integral $\int_a^b f(x)\,dx$?

     คำตอบ

     It is signed accumulation of $f$ from $a$ to $b$. Its units are the units of $f$ multiplied by the units of $x$.

208. การ์ด 208

     คำถาม

     Is a radian dimensionless?

     คำตอบ

     Yes. It is the ratio of arc length to radius, so the length units cancel; keep $rad$ when it clarifies angular meaning.

209. การ์ด 209

     คำถาม

     Electrical power absorbed under the passive sign convention?

     คำตอบ

     $$P=VI$$

     For a resistor, this also gives $P=I^2R=V^2/R$ when Ohm's law applies.

210. การ์ด 210

     คำถาม

     Meaning of a p-value?

     คำตอบ

     Assuming the null hypothesis and model are correct, it is the probability of a result at least as extreme as the observed result. It is not the probability that the null is true.

211. การ์ด 211

     คำถาม

     First check when reading a handbook graph?

     คำตอบ

     Identify both axes, scales, units, curve labels, and any log scale before estimating a value.

212. การ์ด 212

     คำถาม

     Difference between mass and weight?

     คำตอบ

     Mass measures inertia; weight is gravitational force on that mass. Their dimensions and units are different.

213. การ์ด 213

     คำถาม

     Kirchhoff's current law at a node?

     คำตอบ

     The algebraic sum of currents at a node is zero. Choose one sign convention, such as currents entering positive, and apply it consistently.

214. การ์ด 214

     คำถาม

     Evaluate $\int_a^b f(x)\,dx$ from an antiderivative $F$?

     คำตอบ

     $$\int_a^b f(x)\,dx=F(b)-F(a)$$

     This applies when $F'=f$ on the interval under the usual continuity conditions.

215. การ์ด 215

     คำถาม

     Opportunity cost?

     คำตอบ

     The value of the best forgone alternative. Include it when choosing one use of a scarce resource prevents another valuable use.

216. การ์ด 216

     คำถาม

     Weight from mass in a gravitational field?

     คำตอบ

     $$W=mg$$

     Use compatible units; $g$ is local gravitational acceleration, not a universal constant for every location.

217. การ์ด 217

     คำถาม

     Kirchhoff's voltage law around a closed loop?

     คำตอบ

     The algebraic sum of voltage rises and drops around the loop is zero. Traverse the loop once with a consistent polarity convention.

218. การ์ด 218

     คำถาม

     Decision rule using significance level $\alpha$?

     คำตอบ

     Reject the null when the p-value is at most $\alpha$; otherwise fail to reject it. Failing to reject does not prove the null.

219. การ์ด 219

     คำถาม

     Can proprietary methods from a former employer be reused freely?

     คำตอบ

     Only if you have the legal right or permission to use them. General skill and experience may transfer; confidential proprietary material does not.

220. การ์ด 220

     คำถาม

     Pound-force to newtons?

     คำตอบ

     $1\,lbf\approx4.44822\,N$. Use more digits only when the required precision justifies them.

221. การ์ด 221

     คำถาม

     When does a definite integral equal ordinary geometric area?

     คำตอบ

     When the integrand is nonnegative over the interval. If it changes sign, the integral is signed area; integrate the absolute value or split the interval for total area.

222. การ์ด 222

     คำถาม

     Charge-voltage relation for an ideal capacitor?

     คำตอบ

     $$q=CV$$

     $C$ is capacitance in farads; $V$ is the voltage across the capacitor under the chosen polarity.

223. การ์ด 223

     คำถาม

     Searching for a material or fluid property: what qualifiers often matter?

     คำตอบ

     Material identity, phase, temperature, pressure, and composition. Use only the qualifiers relevant to that property source.

224. การ์ด 224

     คำถาม

     Force produced when one slug accelerates at $1\,ft/s^2$?

     คำตอบ

     $1\,lbf$. Equivalently, $1\,lbf=1\,slug\cdot ft/s^2$.

225. การ์ด 225

     คำถาม

     Relationship among nominal rate $i$, real rate $r$, and inflation $f$?

     คำตอบ

     $$1+i=(1+r)(1+f)$$

     Keep cash-flow dollars and discount rates on the same real-or-nominal basis.

226. การ์ด 226

     คำถาม

     Antiderivative of $x^n$ for $n\ne-1$?

     คำตอบ

     $$\int x^n\,dx=\frac{x^{n+1}}{n+1}+C$$

227. การ์ด 227

     คำถาม

     Time constant of a first-order resistor-capacitor circuit?

     คำตอบ

     $$\tau=RC$$

     Use the resistance seen by the capacitor in the applicable first-order linear circuit.

228. การ์ด 228

     คำถาม

     What physical quantity does pound-mass measure?

     คำตอบ

     Mass. Do not substitute $lbm$ directly where a force unit such as $lbf$ is required.

229. การ์ด 229

     คำถาม

     Type I error?

     คำตอบ

     Rejecting a true null hypothesis. Its probability is controlled by the chosen significance level under the test assumptions.

230. การ์ด 230

     คำถาม

     A search hit contains the right words. Is that enough?

     คำตอบ

     No. Read the surrounding definition, equation label, units, and notes to confirm the hit answers the actual question.

231. การ์ด 231

     คำถาม

     Annual straight-line depreciation from basis $B$ to salvage $S$ over $n$ years?

     คำตอบ

     $$D=\frac{B-S}{n}$$

     This is an accounting allocation, not a market-value forecast.

232. การ์ด 232

     คำถาม

     Purpose of $g_c$ in some US customary force-mass equations?

     คำตอบ

     It reconciles chosen $lbm$, $lbf$, foot, and second units. Include it only when the equation and unit convention require it; coherent SI equations do not use it.

233. การ์ด 233

     คำถาม

     Recognizable form of a separable first-order differential equation?

     คำตอบ

     $$\frac{dy}{dx}=g(x)h(y)$$

     Rearrange to place the $y$ terms with $dy$ and the $x$ terms with $dx$, while checking for solutions lost by division.

234. การ์ด 234

     คำถาม

     Using unverified software output in a safety-relevant calculation?

     คำตอบ

     Check inputs, assumptions, units, and results with an independent method appropriate to the risk. The engineer remains responsible for professional judgment.

235. การ์ด 235

     คำถาม

     Equivalent resistance of ideal resistors in series?

     คำตอบ

     $$R_{eq}=\sum_i R_i$$

     Series elements carry the same current with no branching node between them.

236. การ์ด 236

     คำถาม

     Pressure conversion from psi to pascals?

     คำตอบ

     $1\,psi\approx6894.76\,Pa$. Keep enough significant figures for the problem's data.

237. การ์ด 237

     คำถาม

     Type II error?

     คำตอบ

     Failing to reject a false null hypothesis. Its probability is $\beta$ for a specified alternative.

238. การ์ด 238

     คำถาม

     Useful flashcard to make after a failed handbook search?

     คำตอบ

     A short scenario-to-search-cue card recording the section name and terms that eventually worked—not a copied problem or page number.

239. การ์ด 239

     คำถาม

     Standard form of a first-order linear differential equation?

     คำตอบ

     $$\frac{dy}{dx}+P(x)y=Q(x)$$

     An integrating factor is $e^{\int P(x)\,dx}$.

240. การ์ด 240

     คำถาม

     Equivalent resistance of ideal resistors in parallel?

     คำตอบ

     $$\frac{1}{R_{eq}}=\sum_i\frac{1}{R_i}$$

     Parallel elements share the same two nodes and therefore the same voltage.

241. การ์ด 241

     คำถาม

     Book value after $k$ years of straight-line depreciation?

     คำตอบ

     $$BV_k=B-kD$$

     Apply it within the depreciable life and do not depreciate below the assumed salvage value.

242. การ์ด 242

     คำถาม

     Standard atmosphere in pascals?

     คำตอบ

     $1\,atm=101{,}325\,Pa$ exactly.

243. การ์ด 243

     คำถาม

     Using AI-generated technical content in engineering work?

     คำตอบ

     Treat it as unverified input: protect confidential data, check sources and calculations, disclose use when required, and keep accountable human review.

244. การ์ด 244

     คำถาม

     Trial form for a homogeneous linear ODE with constant coefficients?

     คำตอบ

     Use $y=e^{rx}$ to obtain the characteristic polynomial in $r$. Repeated or complex roots require the corresponding standard solution forms.

245. การ์ด 245

     คำถาม

     Wave-speed relationship among frequency and wavelength?

     คำตอบ

     $$v=f\lambda$$

     Use frequency in cycles per second and wavelength in compatible length units.

246. การ์ด 246

     คำถาม

     Statistical power?

     คำตอบ

     The probability of rejecting the null when a specified alternative is true: $1-\beta$. Larger effects, larger samples, or lower variability commonly increase power.

247. การ์ด 247

     คำถาม

     Bar in pascals?

     คำตอบ

     $1\,bar=100{,}000\,Pa$ exactly.

248. การ์ด 248

     คำถาม

     How should handbook navigation be practiced before the FE exam?

     คำตอบ

     Use the current handbook from MyNCEES while solving original or properly licensed practice problems, and rehearse the exact search terms that find definitions and assumptions.

249. การ์ด 249

     คำถาม

     Is depreciation itself a cash outflow?

     คำตอบ

     No. It is a noncash accounting expense, though it can affect taxes and therefore after-tax cash flow.

250. การ์ด 250

     คำถาม

     First-order linear approximation of $f(x)$ near $x=a$?

     คำตอบ

     $$f(x)\approx f(a)+f'(a)(x-a)$$

     It is most accurate for $x$ close to $a$ when higher derivatives stay bounded.

251. การ์ด 251

     คำถาม

     Relationship between period and frequency?

     คำตอบ

     $$T=\frac{1}{f}$$

     This requires $f>0$; $T$ has time units.

252. การ์ด 252

     คำถาม

     How should foreseeable environmental harm enter an engineering decision?

     คำตอบ

     Identify and communicate it, follow applicable requirements, and consider practicable ways to reduce harm without compromising public safety.

253. การ์ด 253

     คำถาม

     Foot-pound-force in joules?

     คำตอบ

     $1\,ft\cdot lbf\approx1.35582\,J$. This is a work or energy conversion; keep torque labeled as force times distance even though its dimensions match energy.

254. การ์ด 254

     คำถาม

     Allowed range and sign meaning of Pearson correlation $r$?

     คำตอบ

     $-1\le r\le1$. The sign gives the direction of linear association, and magnitude measures its linear strength.

255. การ์ด 255

     คำถาม

     When should you search instead of relying on recall?

     คำตอบ

     Search when a constant, coefficient, convention, specialized relationship, or applicability detail could change the result. Recall can start the route; the reference should confirm details.

256. การ์ด 256

     คำถาม

     After-tax effect of a deductible expense $E$ at marginal tax rate $t$?

     คำตอบ

     The net cost is $E(1-t)$ under the simplified assumption that the deduction can be used fully in that period.

257. การ์ด 257

     คำถาม

     Centered finite-difference approximation to $f'(x)$?

     คำตอบ

     $$f'(x)\approx\frac{f(x+h)-f(x-h)}{2h}$$

     For smooth $f$, truncation error is proportional to $h^2$ before roundoff dominates.

258. การ์ด 258

     คำถาม

     General material balance on a control volume?

     คำตอบ

     $$\text{accumulation}=\text{in}-\text{out}+\text{generation}-\text{consumption}$$

     At steady state, accumulation is zero; generation and consumption depend on reactions or sources.

259. การ์ด 259

     คำถาม

     Approximate joules in one International Table Btu?

     คำตอบ

     $1\,Btu_{IT}\approx1055.06\,J$. Identify the Btu convention when precision matters.

260. การ์ด 260

     คำถาม

     What is the defender in a replacement analysis?

     คำตอบ

     The currently owned asset being compared with a proposed replacement. Its current market value is an opportunity cost; its original purchase price is sunk.

261. การ์ด 261

     คำถาม

     Speaking publicly about a technical matter outside your expertise?

     คำตอบ

     Limit claims to what you can support, identify the boundary of your expertise, and avoid implying qualifications you do not have.

262. การ์ด 262

     คำถาม

     Single-interval trapezoidal approximation to an integral?

     คำตอบ

     $$\int_a^b f(x)\,dx\approx\frac{b-a}{2}[f(a)+f(b)]$$

     It is exact for linear $f$.

263. การ์ด 263

     คำถาม

     Energy efficiency of a device?

     คำตอบ

     $$\eta=\frac{\text{useful output energy}}{\text{input energy}}$$

     Use the same energy basis in numerator and denominator; a real passive device has $0\le\eta\le1$.

264. การ์ด 264

     คำถาม

     Purpose of sensitivity analysis in engineering economics?

     คำตอบ

     It shows how the decision changes when an uncertain input changes. Vary inputs over plausible ranges while preserving units and model relationships.

265. การ์ด 265

     คำถาม

     Can a strong correlation establish causation?

     คำตอบ

     No. Correlation alone cannot rule out confounding, reverse direction, selection effects, or coincidence.

266. การ์ด 266

     คำถาม

     Where should you confirm the tested topics for your FE discipline?

     คำตอบ

     Use the discipline specification linked from the official NCEES FE exam hub. This common deck does not replace that discipline-specific boundary.

267. การ์ด 267

     คำถาม

     Mechanical horsepower in watts?

     คำตอบ

     $1\,hp\approx745.7\,W$, based on $550\,ft\cdot lbf/s$. Distinguish it from metric horsepower.

268. การ์ด 268

     คำถาม

     Combine independent standard uncertainties $u_1,\ldots,u_n$ in a sum or difference?

     คำตอบ

     $$u_c=\sqrt{u_1^2+u_2^2+\cdots+u_n^2}$$

     This root-sum-square form assumes uncorrelated inputs and coefficients of magnitude one; otherwise include sensitivities and covariance.

269. การ์ด 269

     คำถาม

     Newton iteration for a root of $f(x)=0$?

     คำตอบ

     $$x_{k+1}=x_k-\frac{f(x_k)}{f'(x_k)}$$

     It requires $f'(x_k)\ne0$ and a suitable starting value; convergence is not guaranteed.

270. การ์ด 270

     คำถาม

     Meaning of slope in simple linear regression of $y$ on $x$?

     คำตอบ

     It is the estimated change in mean response $y$ for a one-unit increase in $x$, under the fitted linear model. Its units are $y$ units per $x$ unit.

271. การ์ด 271

     คำถาม

     Kilowatt-hour in joules?

     คำตอบ

     $1\,kW\cdot h=3.6\times10^6\,J$. It is energy, not power.

272. การ์ด 272

     คำถาม

     Percent error relative to an accepted nonzero value?

     คำตอบ

     $$\%\,error=\left|\frac{x_{measured}-x_{accepted}}{x_{accepted}}\right|100\%$$

     The accepted value must be nonzero and both values must use compatible units.

273. การ์ด 273

     คำถาม

     Does legal compliance alone settle an engineering ethics question?

     คำตอบ

     No. Law sets mandatory boundaries, while professional duties can require stronger action, especially when safety, honesty, or conflicts are involved.

274. การ์ด 274

     คำถาม

     Does handbook fluency replace discipline-specific problem practice?

     คำตอบ

     No. Pair navigation and common recall with the current discipline specification and timed, rights-safe problems that match your selected FE discipline.

275. การ์ด 275

     คำถาม

     Linear interpolation between $(x_1,y_1)$ and $(x_2,y_2)$?

     คำตอบ

     $$y\approx y_1+(y_2-y_1)\frac{x-x_1}{x_2-x_1}$$

     Use it between distinct $x$ values when local linear behavior is reasonable.

276. การ์ด 276

     คำถาม

     Dimensional requirement for a physically meaningful additive equation?

     คำตอบ

     Every term being added or equated must have the same dimensions. A dimensionally consistent equation can still be physically wrong, but an inconsistent one is wrong.