FE Exam Flashcards: Engineering Equations, Units & Handbook Skills

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1. Κάρτα 1

   Ερώτηση

   SI base unit of length?

   Απάντηση

   The metre, symbol $m$.

2. Κάρτα 2

   Ερώτηση

   Slope between two points?

   Απάντηση

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

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

3. Κάρτα 3

   Ερώτηση

   Net force relationship for constant mass in an inertial frame?

   Απάντηση

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

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

4. Κάρτα 4

   Ερώτηση

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

   Απάντηση

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

5. Κάρτα 5

   Ερώτηση

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

   Απάντηση

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

6. Κάρτα 6

   Ερώτηση

   Probability of the complement of event $A$?

   Απάντηση

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

7. Κάρτα 7

   Ερώτηση

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

   Απάντηση

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

8. Κάρτα 8

   Ερώτηση

   SI base unit of mass?

   Απάντηση

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

9. Κάρτα 9

   Ερώτηση

   Linear equation from slope and intercept?

   Απάντηση

   $$y=mx+b$$

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

10. Κάρτα 10

    Ερώτηση

    Velocity after time $t$ under constant acceleration?

    Απάντηση

    $$v=v_0+at$$

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

11. Κάρτα 11

    Ερώτηση

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

    Απάντηση

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

12. Κάρτα 12

    Ερώτηση

    How is the FE reference handbook available during the exam?

    Απάντηση

    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. Κάρτα 13

    Ερώτηση

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

    Απάντηση

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

14. Κάρτα 14

    Ερώτηση

    SI base unit of time?

    Απάντηση

    The second, symbol $s$.

15. Κάρτα 15

    Ερώτηση

    Linear equation from one point and a slope?

    Απάντηση

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

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

16. Κάρτα 16

    Ερώτηση

    Position after time $t$ under constant acceleration?

    Απάντηση

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

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

17. Κάρτα 17

    Ερώτηση

    A project assignment lies outside your competence. Proper response?

    Απάντηση

    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. Κάρτα 18

    Ερώτηση

    SI base unit of electric current?

    Απάντηση

    The ampere, symbol $A$.

19. Κάρτα 19

    Ερώτηση

    Future amount under simple interest?

    Απάντηση

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

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

20. Κάρτα 20

    Ερώτηση

    Distance between two Cartesian points?

    Απάντηση

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

    The coordinate components must use the same length unit.

21. Κάρτα 21

    Ερώτηση

    Constant-acceleration relation without elapsed time?

    Απάντηση

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

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

22. Κάρτα 22

    Ερώτηση

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

    Απάντηση

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

23. Κάρτα 23

    Ερώτηση

    Conditional probability of $A$ given $B$?

    Απάντηση

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

    This requires $P(B)>0$.

24. Κάρτα 24

    Ερώτηση

    SI base unit of thermodynamic temperature?

    Απάντηση

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

25. Κάρτα 25

    Ερώτηση

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

    Απάντηση

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

26. Κάρτα 26

    Ερώτηση

    Midpoint of two Cartesian points?

    Απάντηση

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

    Average corresponding coordinates.

27. Κάρτα 27

    Ερώτηση

    Linear momentum of a particle?

    Απάντηση

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

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

28. Κάρτα 28

    Ερώτηση

    Future amount under compound interest?

    Απάντηση

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

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

29. Κάρτα 29

    Ερώτηση

    Should an FE search workflow depend on Ctrl+F?

    Απάντηση

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

30. Κάρτα 30

    Ερώτηση

    SI base unit of amount of substance?

    Απάντηση

    The mole, symbol $mol$.

31. Κάρτα 31

    Ερώτηση

    Joint probability expressed with a conditional probability?

    Απάντηση

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

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

32. Κάρτα 32

    Ερώτηση

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

    Απάντηση

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

    The coordinate units and radius unit must match.

33. Κάρτα 33

    Ερώτηση

    Impulse-momentum relationship?

    Απάντηση

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

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

34. Κάρτα 34

    Ερώτηση

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

    Απάντηση

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

35. Κάρτα 35

    Ερώτηση

    SI base unit of luminous intensity?

    Απάντηση

    The candela, symbol $cd$.

36. Κάρτα 36

    Ερώτηση

    Present worth of one future amount?

    Απάντηση

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

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

37. Κάρτα 37

    Ερώτηση

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

    Απάντηση

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

    This form requires $a\ne0$.

38. Κάρτα 38

    Ερώτηση

    When is total linear momentum conserved?

    Απάντηση

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

39. Κάρτα 39

    Ερώτηση

    Why avoid memorizing FE handbook page numbers?

    Απάντηση

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

40. Κάρτα 40

    Ερώτηση

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

    Απάντηση

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

    Independence is not the same as mutual exclusivity.

41. Κάρτα 41

    Ερώτηση

    SI unit of frequency in base-unit form?

    Απάντηση

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

42. Κάρτα 42

    Ερώτηση

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

    Απάντηση

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

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

43. Κάρτα 43

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 44

    Ερώτηση

    Work done by a constant force through displacement?

    Απάντηση

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

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

45. Κάρτα 45

    Ερώτηση

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

    Απάντηση

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

46. Κάρτα 46

    Ερώτηση

    SI unit of force in base-unit form?

    Απάντηση

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

47. Κάρτα 47

    Ερώτηση

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

    Απάντηση

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

48. Κάρτα 48

    Ερώτηση

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

    Απάντηση

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

    This requires $P(B)>0$.

49. Κάρτα 49

    Ερώτηση

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

    Απάντηση

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

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

50. Κάρτα 50

    Ερώτηση

    Translational kinetic energy of a particle?

    Απάντηση

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

    Mass and speed must use a coherent unit system.

51. Κάρτα 51

    Ερώτηση

    SI unit of pressure in force-area form?

    Απάντηση

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

52. Κάρτα 52

    Ερώτηση

    Equivalent annual amount for present worth $P$?

    Απάντηση

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

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

53. Κάρτα 53

    Ερώτηση

    May confidential client information be shared for personal benefit?

    Απάντηση

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

54. Κάρτα 54

    Ερώτηση

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

    Απάντηση

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

55. Κάρτα 55

    Ερώτηση

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

    Απάντηση

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

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

56. Κάρτα 56

    Ερώτηση

    Net work-energy relationship for a particle?

    Απάντηση

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

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

57. Κάρτα 57

    Ερώτηση

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

    Απάντηση

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

    Use this when order matters and selection is without replacement.

58. Κάρτα 58

    Ερώτηση

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

    Απάντηση

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

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

59. Κάρτα 59

    Ερώτηση

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

    Απάντηση

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

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

60. Κάρτα 60

    Ερώτηση

    SI unit of energy in force-distance form?

    Απάντηση

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

61. Κάρτα 61

    Ερώτηση

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

    Απάντηση

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

62. Κάρτα 62

    Ερώτηση

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

    Απάντηση

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

63. Κάρτα 63

    Ερώτηση

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

    Απάντηση

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

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

64. Κάρτα 64

    Ερώτηση

    Change in gravitational potential energy near Earth's surface?

    Απάντηση

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

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

65. Κάρτα 65

    Ερώτηση

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

    Απάντηση

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

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

66. Κάρτα 66

    Ερώτηση

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

    Απάντηση

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

    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. Κάρτα 67

    Ερώτηση

    A handbook search for an acronym fails. Next move?

    Απάντηση

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

68. Κάρτα 68

    Ερώτηση

    SI unit of power in energy-time form?

    Απάντηση

    The watt, $W=J/s$.

69. Κάρτα 69

    Ερώτηση

    Logarithm of a product?

    Απάντηση

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

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

70. Κάρτα 70

    Ερώτηση

    Conditions for a binomial model?

    Απάντηση

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

71. Κάρτα 71

    Ερώτηση

    Using another engineer's analysis in your deliverable?

    Απάντηση

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

72. Κάρτα 72

    Ερώτηση

    SI unit of electric charge in current-time form?

    Απάντηση

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

73. Κάρτα 73

    Ερώτηση

    Average power from work over elapsed time?

    Απάντηση

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

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

74. Κάρτα 74

    Ερώτηση

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

    Απάντηση

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

    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. Κάρτα 75

    Ερώτηση

    Logarithm of a positive quantity raised to a power?

    Απάντηση

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

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

76. Κάρτα 76

    Ερώτηση

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

    Απάντηση

    The volt, $V=J/C$.

77. Κάρτα 77

    Ερώτηση

    Instantaneous mechanical power delivered by a force?

    Απάντηση

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

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

78. Κάρτα 78

    Ερώτηση

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

    Απάντηση

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

79. Κάρτα 79

    Ερώτηση

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

    Απάντηση

    $$E[X]=np$$

80. Κάρτα 80

    Ερώτηση

    SI unit of electrical resistance in voltage-current form?

    Απάντηση

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

81. Κάρτα 81

    Ερώτηση

    Moment of a force about a point?

    Απάντηση

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

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

82. Κάρτα 82

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 83

    Ερώτηση

    Inverse relationship between the natural log and exponential?

    Απάντηση

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

    The second identity requires $x>0$.

84. Κάρτα 84

    Ερώτηση

    SI prefix giga?

    Απάντηση

    $G=10^9$.

85. Κάρτα 85

    Ερώτηση

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

    Απάντηση

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

    Use consistent angular sign convention and radians for angular measures.

86. Κάρτα 86

    Ερώτηση

    When should an engineer sign or seal technical work?

    Απάντηση

    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. Κάρτα 87

    Ερώτηση

    Best memory target for returning to a handbook formula?

    Απάντηση

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

88. Κάρτα 88

    Ερώτηση

    SI prefix mega?

    Απάντηση

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

89. Κάρτα 89

    Ερώτηση

    Convert degrees to radians?

    Απάντηση

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

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

90. Κάρτα 90

    Ερώτηση

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

    Απάντηση

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

91. Κάρτα 91

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 92

    Ερώτηση

    Where do licensure and sealing requirements come from?

    Απάντηση

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

93. Κάρτα 93

    Ερώτηση

    Rotational kinetic energy about a fixed axis?

    Απάντηση

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

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

94. Κάρτα 94

    Ερώτηση

    SI prefix kilo?

    Απάντηση

    $k=10^3$.

95. Κάρτα 95

    Ερώτηση

    Right-triangle definition of sine?

    Απάντηση

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

96. Κάρτα 96

    Ερώτηση

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

    Απάντηση

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

97. Κάρτα 97

    Ερώτηση

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

    Απάντηση

    $$\sum M=I\alpha$$

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

98. Κάρτα 98

    Ερώτηση

    SI prefix milli?

    Απάντηση

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

99. Κάρτα 99

    Ερώτηση

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

    Απάντηση

    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. Κάρτα 100

     Ερώτηση

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

     Απάντηση

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

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

101. Κάρτα 101

     Ερώτηση

     Right-triangle definition of cosine?

     Απάντηση

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

102. Κάρτα 102

     Ερώτηση

     Force-displacement relation for an ideal linear spring?

     Απάντηση

     $$F=-kx$$

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

103. Κάρτα 103

     Ερώτηση

     May qualifications or project responsibility be exaggerated in a proposal?

     Απάντηση

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

104. Κάρτα 104

     Ερώτηση

     SI prefix micro?

     Απάντηση

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

105. Κάρτα 105

     Ερώτηση

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

     Απάντηση

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

106. Κάρτα 106

     Ερώτηση

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

     Απάντηση

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

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

107. Κάρτα 107

     Ερώτηση

     Right-triangle definition of tangent?

     Απάντηση

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

108. Κάρτα 108

     Ερώτηση

     Average normal stress under centered axial load?

     Απάντηση

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

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

109. Κάρτα 109

     Ερώτηση

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

     Απάντηση

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

110. Κάρτα 110

     Ερώτηση

     SI prefix nano?

     Απάντηση

     $n=10^{-9}$.

111. Κάρτα 111

     Ερώτηση

     Fundamental Pythagorean trigonometric identity?

     Απάντηση

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

112. Κάρτα 112

     Ερώτηση

     Engineering normal strain?

     Απάντηση

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

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

113. Κάρτα 113

     Ερώτηση

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

     Απάντηση

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

114. Κάρτα 114

     Ερώτηση

     Technical duty to an employer or client?

     Απάντηση

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

115. Κάρτα 115

     Ερώτηση

     How does a length conversion factor apply to area?

     Απάντηση

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

116. Κάρτα 116

     Ερώτηση

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

     Απάντηση

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

117. Κάρτα 117

     Ερώτηση

     When is a Poisson count model reasonable?

     Απάντηση

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

118. Κάρτα 118

     Ερώτηση

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

     Απάντηση

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

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

119. Κάρτα 119

     Ερώτηση

     Linear elastic stress-strain relation in uniaxial loading?

     Απάντηση

     $$\sigma=E\varepsilon$$

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

120. Κάρτα 120

     Ερώτηση

     How does a length conversion factor apply to volume?

     Απάντηση

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

121. Κάρτα 121

     Ερώτηση

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

     Απάντηση

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

122. Κάρτα 122

     Ερώτηση

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

     Απάντηση

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

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

123. Κάρτα 123

     Ερώτηση

     A supervisor directs an illegal engineering action. Proper response?

     Απάντηση

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

124. Κάρτα 124

     Ερώτηση

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

     Απάντηση

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

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

125. Κάρτα 125

     Ερώτηση

     Mass density?

     Απάντηση

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

     Use mass, not weight, in the numerator.

126. Κάρτα 126

     Ερώτηση

     Inches in one foot?

     Απάντηση

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

127. Κάρτα 127

     Ερώτηση

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

     Απάντηση

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

128. Κάρτα 128

     Ερώτηση

     How should subscripts be handled in a retrieved relationship?

     Απάντηση

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

129. Κάρτα 129

     Ερώτηση

     Dot product from vector magnitudes and included angle?

     Απάντηση

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

     The result is a scalar.

130. Κάρτα 130

     Ερώτηση

     Average normal pressure on an area?

     Απάντηση

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

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

131. Κάρτα 131

     Ερώτηση

     Feet in one yard?

     Απάντηση

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

132. Κάρτα 132

     Ερώτηση

     When are two interest rates equivalent?

     Απάντηση

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

133. Κάρτα 133

     Ερώτηση

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

     Απάντηση

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

134. Κάρτα 134

     Ερώτηση

     Expected value of a discrete random variable?

     Απάντηση

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

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

135. Κάρτα 135

     Ερώτηση

     Dot-product test for perpendicular nonzero vectors?

     Απάντηση

     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. Κάρτα 136

     Ερώτηση

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

     Απάντηση

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

     Depth increases downward and the fluid is at rest.

137. Κάρτα 137

     Ερώτηση

     Feet in one statute mile?

     Απάντηση

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

138. Κάρτα 138

     Ερώτηση

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

     Απάντηση

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

     

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139. Κάρτα 139

     Ερώτηση

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

     Απάντηση

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

140. Κάρτα 140

     Ερώτηση

     Magnitude of a vector cross product?

     Απάντηση

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

     The direction follows the right-hand rule.

141. Κάρτα 141

     Ερώτηση

     Exact inch-to-metre relationship?

     Απάντηση

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

142. Κάρτα 142

     Ερώτηση

     Volumetric flow rate through a cross-section?

     Απάντηση

     $$Q=Av_{avg}$$

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

143. Κάρτα 143

     Ερώτηση

     Variance from the mean $\mu$?

     Απάντηση

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

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

144. Κάρτα 144

     Ερώτηση

     Future-worth acceptance rule for an independent investment?

     Απάντηση

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

145. Κάρτα 145

     Ερώτηση

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

     Απάντηση

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

146. Κάρτα 146

     Ερώτηση

     Exact foot-to-metre relationship?

     Απάντηση

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

147. Κάρτα 147

     Ερώτηση

     How should important uncertainty be communicated?

     Απάντηση

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

148. Κάρτα 148

     Ερώτηση

     When is the matrix product $AB$ defined?

     Απάντηση

     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. Κάρτα 149

     Ερώτηση

     Steady one-inlet, one-outlet mass continuity?

     Απάντηση

     $$\rho_1A_1v_1=\rho_2A_2v_2$$

     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. Κάρτα 150

     Ερώτηση

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

     Απάντηση

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

151. Κάρτα 151

     Ερώτηση

     Square feet in one acre?

     Απάντηση

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

152. Κάρτα 152

     Ερώτηση

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

     Απάντηση

     $$ad-bc$$

153. Κάρτα 153

     Ερώτηση

     Bernoulli relationship along a streamline for ideal steady flow?

     Απάντηση

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

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

154. Κάρτα 154

     Ερώτηση

     May inconvenient measurements be discarded without a technical basis?

     Απάντηση

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

155. Κάρτα 155

     Ερώτηση

     Annual-worth acceptance rule for an independent investment?

     Απάντηση

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

156. Κάρτα 156

     Ερώτηση

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

     Απάντηση

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

157. Κάρτα 157

     Ερώτηση

     Cubic metres in one litre?

     Απάντηση

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

158. Κάρτα 158

     Ερώτηση

     What does a zero determinant mean for a square matrix?

     Απάντηση

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

159. Κάρτα 159

     Ερώτηση

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

     Απάντηση

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

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

160. Κάρτα 160

     Ερώτηση

     Unbiased sample variance for an independent sample from one population?

     Απάντηση

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

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

161. Κάρτα 161

     Ερώτηση

     How should mutually exclusive alternatives be compared?

     Απάντηση

     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. Κάρτα 162

     Ερώτηση

     Cubic centimetres in one millilitre?

     Απάντηση

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

163. Κάρτα 163

     Ερώτηση

     Why keep contemporaneous engineering records?

     Απάντηση

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

164. Κάρτα 164

     Ερώτηση

     Fast check after substituting into a handbook formula?

     Απάντηση

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

165. Κάρτα 165

     Ερώτηση

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

     Απάντηση

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

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

166. Κάρτα 166

     Ερώτηση

     Sensible heat for constant specific heat and no phase change?

     Απάντηση

     $$Q=mc\Delta T$$

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

167. Κάρτα 167

     Ερώτηση

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

     Απάντηση

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

     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. Κάρτα 168

     Ερώτηση

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

     Απάντηση

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

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

169. Κάρτα 169

     Ερώτηση

     Purpose of incremental analysis for mutually exclusive alternatives?

     Απάντηση

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

170. Κάρτα 170

     Ερώτηση

     Physical meaning of $dy/dx$?

     Απάντηση

     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. Κάρτα 171

     Ερώτηση

     One-dimensional steady conduction through a plane wall?

     Απάντηση

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

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

172. Κάρτα 172

     Ερώτηση

     When should you algebraically rearrange a handbook relationship?

     Απάντηση

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

173. Κάρτα 173

     Ερώτηση

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

     Απάντηση

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

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

174. Κάρτα 174

     Ερώτηση

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

     Απάντηση

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

175. Κάρτα 175

     Ερώτηση

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

     Απάντηση

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

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

176. Κάρτα 176

     Ερώτηση

     Convective heat-transfer rate at a surface?

     Απάντηση

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

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

177. Κάρτα 177

     Ερώτηση

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

     Απάντηση

     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. Κάρτα 178

     Ερώτηση

     Internal rate of return?

     Απάντηση

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

179. Κάρτα 179

     Ερώτηση

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

     Απάντηση

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

180. Κάρτα 180

     Ερώτηση

     Convert Celsius temperature to kelvin?

     Απάντηση

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

     This conversion is for absolute temperature, not temperature difference.

181. Κάρτα 181

     Ερώτηση

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

     Απάντηση

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

182. Κάρτα 182

     Ερώτηση

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

     Απάντηση

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

183. Κάρτα 183

     Ερώτηση

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

     Απάντηση

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

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

184. Κάρτα 184

     Ερώτηση

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

     Απάντηση

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

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

185. Κάρτα 185

     Ερώτηση

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

     Απάντηση

     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. Κάρτα 186

     Ερώτηση

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

     Απάντηση

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

187. Κάρτα 187

     Ερώτηση

     Why inspect boundary or initial conditions near a formula?

     Απάντηση

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

188. Κάρτα 188

     Ερώτηση

     Convert Celsius temperature to Fahrenheit?

     Απάντηση

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

189. Κάρτα 189

     Ερώτηση

     Ethical basis for expert technical testimony?

     Απάντηση

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

190. Κάρτα 190

     Ερώτηση

     Meaning of a breakeven value?

     Απάντηση

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

191. Κάρτα 191

     Ερώτηση

     Derivative of $e^x$?

     Απάντηση

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

192. Κάρτα 192

     Ερώτηση

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

     Απάντηση

     $$\Delta E=Q-W$$

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

193. Κάρτα 193

     Ερώτηση

     Frequentist meaning of a 95% confidence procedure?

     Απάντηση

     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. Κάρτα 194

     Ερώτηση

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

     Απάντηση

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

195. Κάρτα 195

     Ερώτηση

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

     Απάντηση

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

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

196. Κάρτα 196

     Ερώτηση

     Relationship between Celsius and kelvin temperature differences?

     Απάντηση

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

197. Κάρτα 197

     Ερώτηση

     How should credit for team engineering work be assigned?

     Απάντηση

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

198. Κάρτα 198

     Ερώτηση

     Main limitation of simple payback period?

     Απάντηση

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

199. Κάρτα 199

     Ερώτηση

     Roles of the null and alternative hypotheses?

     Απάντηση

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

200. Κάρτα 200

     Ερώτηση

     Ideal-gas equation of state?

     Απάντηση

     $$pV=nRT$$

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

201. Κάρτα 201

     Ερώτηση

     First check when reading a handbook table?

     Απάντηση

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

202. Κάρτα 202

     Ερώτηση

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

     Απάντηση

     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. Κάρτα 203

     Ερώτηση

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

     Απάντηση

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

204. Κάρτα 204

     Ερώτηση

     Convert Fahrenheit absolute temperature to Rankine?

     Απάντηση

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

     Rankine uses the Fahrenheit-sized degree from absolute zero.

205. Κάρτα 205

     Ερώτηση

     Ohm's law for an ideal resistor?

     Απάντηση

     $$V=IR$$

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

206. Κάρτα 206

     Ερώτηση

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

     Απάντηση

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

207. Κάρτα 207

     Ερώτηση

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

     Απάντηση

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

208. Κάρτα 208

     Ερώτηση

     Is a radian dimensionless?

     Απάντηση

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

209. Κάρτα 209

     Ερώτηση

     Electrical power absorbed under the passive sign convention?

     Απάντηση

     $$P=VI$$

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

210. Κάρτα 210

     Ερώτηση

     Meaning of a p-value?

     Απάντηση

     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. Κάρτα 211

     Ερώτηση

     First check when reading a handbook graph?

     Απάντηση

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

212. Κάρτα 212

     Ερώτηση

     Difference between mass and weight?

     Απάντηση

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

213. Κάρτα 213

     Ερώτηση

     Kirchhoff's current law at a node?

     Απάντηση

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

214. Κάρτα 214

     Ερώτηση

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

     Απάντηση

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

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

215. Κάρτα 215

     Ερώτηση

     Opportunity cost?

     Απάντηση

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

216. Κάρτα 216

     Ερώτηση

     Weight from mass in a gravitational field?

     Απάντηση

     $$W=mg$$

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

217. Κάρτα 217

     Ερώτηση

     Kirchhoff's voltage law around a closed loop?

     Απάντηση

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

218. Κάρτα 218

     Ερώτηση

     Decision rule using significance level $\alpha$?

     Απάντηση

     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. Κάρτα 219

     Ερώτηση

     Can proprietary methods from a former employer be reused freely?

     Απάντηση

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

220. Κάρτα 220

     Ερώτηση

     Pound-force to newtons?

     Απάντηση

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

221. Κάρτα 221

     Ερώτηση

     When does a definite integral equal ordinary geometric area?

     Απάντηση

     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. Κάρτα 222

     Ερώτηση

     Charge-voltage relation for an ideal capacitor?

     Απάντηση

     $$q=CV$$

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

223. Κάρτα 223

     Ερώτηση

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

     Απάντηση

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

224. Κάρτα 224

     Ερώτηση

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

     Απάντηση

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

225. Κάρτα 225

     Ερώτηση

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

     Απάντηση

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

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

226. Κάρτα 226

     Ερώτηση

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

     Απάντηση

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

227. Κάρτα 227

     Ερώτηση

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

     Απάντηση

     $$\tau=RC$$

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

228. Κάρτα 228

     Ερώτηση

     What physical quantity does pound-mass measure?

     Απάντηση

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

229. Κάρτα 229

     Ερώτηση

     Type I error?

     Απάντηση

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

230. Κάρτα 230

     Ερώτηση

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

     Απάντηση

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

231. Κάρτα 231

     Ερώτηση

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

     Απάντηση

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

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

232. Κάρτα 232

     Ερώτηση

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

     Απάντηση

     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. Κάρτα 233

     Ερώτηση

     Recognizable form of a separable first-order differential equation?

     Απάντηση

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

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

234. Κάρτα 234

     Ερώτηση

     Using unverified software output in a safety-relevant calculation?

     Απάντηση

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

235. Κάρτα 235

     Ερώτηση

     Equivalent resistance of ideal resistors in series?

     Απάντηση

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

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

236. Κάρτα 236

     Ερώτηση

     Pressure conversion from psi to pascals?

     Απάντηση

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

237. Κάρτα 237

     Ερώτηση

     Type II error?

     Απάντηση

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

238. Κάρτα 238

     Ερώτηση

     Useful flashcard to make after a failed handbook search?

     Απάντηση

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

239. Κάρτα 239

     Ερώτηση

     Standard form of a first-order linear differential equation?

     Απάντηση

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

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

240. Κάρτα 240

     Ερώτηση

     Equivalent resistance of ideal resistors in parallel?

     Απάντηση

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

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

241. Κάρτα 241

     Ερώτηση

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

     Απάντηση

     $$BV_k=B-kD$$

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

242. Κάρτα 242

     Ερώτηση

     Standard atmosphere in pascals?

     Απάντηση

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

243. Κάρτα 243

     Ερώτηση

     Using AI-generated technical content in engineering work?

     Απάντηση

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

244. Κάρτα 244

     Ερώτηση

     Trial form for a homogeneous linear ODE with constant coefficients?

     Απάντηση

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

245. Κάρτα 245

     Ερώτηση

     Wave-speed relationship among frequency and wavelength?

     Απάντηση

     $$v=f\lambda$$

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

246. Κάρτα 246

     Ερώτηση

     Statistical power?

     Απάντηση

     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. Κάρτα 247

     Ερώτηση

     Bar in pascals?

     Απάντηση

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

248. Κάρτα 248

     Ερώτηση

     How should handbook navigation be practiced before the FE exam?

     Απάντηση

     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. Κάρτα 249

     Ερώτηση

     Is depreciation itself a cash outflow?

     Απάντηση

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

250. Κάρτα 250

     Ερώτηση

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

     Απάντηση

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

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

251. Κάρτα 251

     Ερώτηση

     Relationship between period and frequency?

     Απάντηση

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

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

252. Κάρτα 252

     Ερώτηση

     How should foreseeable environmental harm enter an engineering decision?

     Απάντηση

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

253. Κάρτα 253

     Ερώτηση

     Foot-pound-force in joules?

     Απάντηση

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

254. Κάρτα 254

     Ερώτηση

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

     Απάντηση

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

255. Κάρτα 255

     Ερώτηση

     When should you search instead of relying on recall?

     Απάντηση

     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. Κάρτα 256

     Ερώτηση

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

     Απάντηση

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

257. Κάρτα 257

     Ερώτηση

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

     Απάντηση

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

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

258. Κάρτα 258

     Ερώτηση

     General material balance on a control volume?

     Απάντηση

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

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

259. Κάρτα 259

     Ερώτηση

     Approximate joules in one International Table Btu?

     Απάντηση

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

260. Κάρτα 260

     Ερώτηση

     What is the defender in a replacement analysis?

     Απάντηση

     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. Κάρτα 261

     Ερώτηση

     Speaking publicly about a technical matter outside your expertise?

     Απάντηση

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

262. Κάρτα 262

     Ερώτηση

     Single-interval trapezoidal approximation to an integral?

     Απάντηση

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

     It is exact for linear $f$.

263. Κάρτα 263

     Ερώτηση

     Energy efficiency of a device?

     Απάντηση

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

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

264. Κάρτα 264

     Ερώτηση

     Purpose of sensitivity analysis in engineering economics?

     Απάντηση

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

265. Κάρτα 265

     Ερώτηση

     Can a strong correlation establish causation?

     Απάντηση

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

266. Κάρτα 266

     Ερώτηση

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

     Απάντηση

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

267. Κάρτα 267

     Ερώτηση

     Mechanical horsepower in watts?

     Απάντηση

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

268. Κάρτα 268

     Ερώτηση

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

     Απάντηση

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

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

269. Κάρτα 269

     Ερώτηση

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

     Απάντηση

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

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

270. Κάρτα 270

     Ερώτηση

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

     Απάντηση

     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. Κάρτα 271

     Ερώτηση

     Kilowatt-hour in joules?

     Απάντηση

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

272. Κάρτα 272

     Ερώτηση

     Percent error relative to an accepted nonzero value?

     Απάντηση

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

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

273. Κάρτα 273

     Ερώτηση

     Does legal compliance alone settle an engineering ethics question?

     Απάντηση

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

274. Κάρτα 274

     Ερώτηση

     Does handbook fluency replace discipline-specific problem practice?

     Απάντηση

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

275. Κάρτα 275

     Ερώτηση

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

     Απάντηση

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

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

276. Κάρτα 276

     Ερώτηση

     Dimensional requirement for a physically meaningful additive equation?

     Απάντηση

     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.