FE Exam Flashcards: Engineering Equations, Units & Handbook Skills

Karty w tej talii

1. Karta 1

   Pytanie

   SI base unit of length?

   Odpowiedź

   The metre, symbol $m$.

2. Karta 2

   Pytanie

   Slope between two points?

   Odpowiedź

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

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

3. Karta 3

   Pytanie

   Net force relationship for constant mass in an inertial frame?

   Odpowiedź

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

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

4. Karta 4

   Pytanie

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

   Odpowiedź

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

5. Karta 5

   Pytanie

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

   Odpowiedź

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

6. Karta 6

   Pytanie

   Probability of the complement of event $A$?

   Odpowiedź

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

7. Karta 7

   Pytanie

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

   Odpowiedź

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

8. Karta 8

   Pytanie

   SI base unit of mass?

   Odpowiedź

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

9. Karta 9

   Pytanie

   Linear equation from slope and intercept?

   Odpowiedź

   $$y=mx+b$$

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

10. Karta 10

    Pytanie

    Velocity after time $t$ under constant acceleration?

    Odpowiedź

    $$v=v_0+at$$

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

11. Karta 11

    Pytanie

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

    Odpowiedź

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

12. Karta 12

    Pytanie

    How is the FE reference handbook available during the exam?

    Odpowiedź

    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. Karta 13

    Pytanie

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

    Odpowiedź

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

14. Karta 14

    Pytanie

    SI base unit of time?

    Odpowiedź

    The second, symbol $s$.

15. Karta 15

    Pytanie

    Linear equation from one point and a slope?

    Odpowiedź

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

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

16. Karta 16

    Pytanie

    Position after time $t$ under constant acceleration?

    Odpowiedź

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

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

17. Karta 17

    Pytanie

    A project assignment lies outside your competence. Proper response?

    Odpowiedź

    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. Karta 18

    Pytanie

    SI base unit of electric current?

    Odpowiedź

    The ampere, symbol $A$.

19. Karta 19

    Pytanie

    Future amount under simple interest?

    Odpowiedź

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

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

20. Karta 20

    Pytanie

    Distance between two Cartesian points?

    Odpowiedź

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

    The coordinate components must use the same length unit.

21. Karta 21

    Pytanie

    Constant-acceleration relation without elapsed time?

    Odpowiedź

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

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

22. Karta 22

    Pytanie

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

    Odpowiedź

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

23. Karta 23

    Pytanie

    Conditional probability of $A$ given $B$?

    Odpowiedź

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

    This requires $P(B)>0$.

24. Karta 24

    Pytanie

    SI base unit of thermodynamic temperature?

    Odpowiedź

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

25. Karta 25

    Pytanie

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

    Odpowiedź

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

26. Karta 26

    Pytanie

    Midpoint of two Cartesian points?

    Odpowiedź

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

    Average corresponding coordinates.

27. Karta 27

    Pytanie

    Linear momentum of a particle?

    Odpowiedź

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

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

28. Karta 28

    Pytanie

    Future amount under compound interest?

    Odpowiedź

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

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

29. Karta 29

    Pytanie

    Should an FE search workflow depend on Ctrl+F?

    Odpowiedź

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

30. Karta 30

    Pytanie

    SI base unit of amount of substance?

    Odpowiedź

    The mole, symbol $mol$.

31. Karta 31

    Pytanie

    Joint probability expressed with a conditional probability?

    Odpowiedź

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

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

32. Karta 32

    Pytanie

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

    Odpowiedź

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

    The coordinate units and radius unit must match.

33. Karta 33

    Pytanie

    Impulse-momentum relationship?

    Odpowiedź

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

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

34. Karta 34

    Pytanie

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

    Odpowiedź

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

35. Karta 35

    Pytanie

    SI base unit of luminous intensity?

    Odpowiedź

    The candela, symbol $cd$.

36. Karta 36

    Pytanie

    Present worth of one future amount?

    Odpowiedź

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

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

37. Karta 37

    Pytanie

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

    Odpowiedź

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

    This form requires $a\ne0$.

38. Karta 38

    Pytanie

    When is total linear momentum conserved?

    Odpowiedź

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

39. Karta 39

    Pytanie

    Why avoid memorizing FE handbook page numbers?

    Odpowiedź

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

40. Karta 40

    Pytanie

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

    Odpowiedź

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

    Independence is not the same as mutual exclusivity.

41. Karta 41

    Pytanie

    SI unit of frequency in base-unit form?

    Odpowiedź

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

42. Karta 42

    Pytanie

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

    Odpowiedź

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

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

43. Karta 43

    Pytanie

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

    Odpowiedź

    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. Karta 44

    Pytanie

    Work done by a constant force through displacement?

    Odpowiedź

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

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

45. Karta 45

    Pytanie

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

    Odpowiedź

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

46. Karta 46

    Pytanie

    SI unit of force in base-unit form?

    Odpowiedź

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

47. Karta 47

    Pytanie

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

    Odpowiedź

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

48. Karta 48

    Pytanie

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

    Odpowiedź

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

    This requires $P(B)>0$.

49. Karta 49

    Pytanie

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

    Odpowiedź

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

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

50. Karta 50

    Pytanie

    Translational kinetic energy of a particle?

    Odpowiedź

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

    Mass and speed must use a coherent unit system.

51. Karta 51

    Pytanie

    SI unit of pressure in force-area form?

    Odpowiedź

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

52. Karta 52

    Pytanie

    Equivalent annual amount for present worth $P$?

    Odpowiedź

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

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

53. Karta 53

    Pytanie

    May confidential client information be shared for personal benefit?

    Odpowiedź

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

54. Karta 54

    Pytanie

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

    Odpowiedź

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

55. Karta 55

    Pytanie

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

    Odpowiedź

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

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

56. Karta 56

    Pytanie

    Net work-energy relationship for a particle?

    Odpowiedź

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

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

57. Karta 57

    Pytanie

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

    Odpowiedź

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

    Use this when order matters and selection is without replacement.

58. Karta 58

    Pytanie

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

    Odpowiedź

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

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

59. Karta 59

    Pytanie

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

    Odpowiedź

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

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

60. Karta 60

    Pytanie

    SI unit of energy in force-distance form?

    Odpowiedź

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

61. Karta 61

    Pytanie

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

    Odpowiedź

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

62. Karta 62

    Pytanie

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

    Odpowiedź

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

63. Karta 63

    Pytanie

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

    Odpowiedź

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

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

64. Karta 64

    Pytanie

    Change in gravitational potential energy near Earth's surface?

    Odpowiedź

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

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

65. Karta 65

    Pytanie

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

    Odpowiedź

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

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

66. Karta 66

    Pytanie

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

    Odpowiedź

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

    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. Karta 67

    Pytanie

    A handbook search for an acronym fails. Next move?

    Odpowiedź

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

68. Karta 68

    Pytanie

    SI unit of power in energy-time form?

    Odpowiedź

    The watt, $W=J/s$.

69. Karta 69

    Pytanie

    Logarithm of a product?

    Odpowiedź

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

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

70. Karta 70

    Pytanie

    Conditions for a binomial model?

    Odpowiedź

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

71. Karta 71

    Pytanie

    Using another engineer's analysis in your deliverable?

    Odpowiedź

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

72. Karta 72

    Pytanie

    SI unit of electric charge in current-time form?

    Odpowiedź

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

73. Karta 73

    Pytanie

    Average power from work over elapsed time?

    Odpowiedź

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

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

74. Karta 74

    Pytanie

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

    Odpowiedź

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

    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. Karta 75

    Pytanie

    Logarithm of a positive quantity raised to a power?

    Odpowiedź

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

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

76. Karta 76

    Pytanie

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

    Odpowiedź

    The volt, $V=J/C$.

77. Karta 77

    Pytanie

    Instantaneous mechanical power delivered by a force?

    Odpowiedź

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

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

78. Karta 78

    Pytanie

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

    Odpowiedź

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

79. Karta 79

    Pytanie

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

    Odpowiedź

    $$E[X]=np$$

80. Karta 80

    Pytanie

    SI unit of electrical resistance in voltage-current form?

    Odpowiedź

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

81. Karta 81

    Pytanie

    Moment of a force about a point?

    Odpowiedź

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

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

82. Karta 82

    Pytanie

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

    Odpowiedź

    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. Karta 83

    Pytanie

    Inverse relationship between the natural log and exponential?

    Odpowiedź

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

    The second identity requires $x>0$.

84. Karta 84

    Pytanie

    SI prefix giga?

    Odpowiedź

    $G=10^9$.

85. Karta 85

    Pytanie

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

    Odpowiedź

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

    Use consistent angular sign convention and radians for angular measures.

86. Karta 86

    Pytanie

    When should an engineer sign or seal technical work?

    Odpowiedź

    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. Karta 87

    Pytanie

    Best memory target for returning to a handbook formula?

    Odpowiedź

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

88. Karta 88

    Pytanie

    SI prefix mega?

    Odpowiedź

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

89. Karta 89

    Pytanie

    Convert degrees to radians?

    Odpowiedź

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

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

90. Karta 90

    Pytanie

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

    Odpowiedź

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

91. Karta 91

    Pytanie

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

    Odpowiedź

    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. Karta 92

    Pytanie

    Where do licensure and sealing requirements come from?

    Odpowiedź

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

93. Karta 93

    Pytanie

    Rotational kinetic energy about a fixed axis?

    Odpowiedź

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

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

94. Karta 94

    Pytanie

    SI prefix kilo?

    Odpowiedź

    $k=10^3$.

95. Karta 95

    Pytanie

    Right-triangle definition of sine?

    Odpowiedź

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

96. Karta 96

    Pytanie

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

    Odpowiedź

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

97. Karta 97

    Pytanie

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

    Odpowiedź

    $$\sum M=I\alpha$$

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

98. Karta 98

    Pytanie

    SI prefix milli?

    Odpowiedź

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

99. Karta 99

    Pytanie

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

    Odpowiedź

    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. Karta 100

     Pytanie

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

     Odpowiedź

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

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

101. Karta 101

     Pytanie

     Right-triangle definition of cosine?

     Odpowiedź

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

102. Karta 102

     Pytanie

     Force-displacement relation for an ideal linear spring?

     Odpowiedź

     $$F=-kx$$

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

103. Karta 103

     Pytanie

     May qualifications or project responsibility be exaggerated in a proposal?

     Odpowiedź

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

104. Karta 104

     Pytanie

     SI prefix micro?

     Odpowiedź

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

105. Karta 105

     Pytanie

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

     Odpowiedź

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

106. Karta 106

     Pytanie

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

     Odpowiedź

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

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

107. Karta 107

     Pytanie

     Right-triangle definition of tangent?

     Odpowiedź

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

108. Karta 108

     Pytanie

     Average normal stress under centered axial load?

     Odpowiedź

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

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

109. Karta 109

     Pytanie

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

     Odpowiedź

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

110. Karta 110

     Pytanie

     SI prefix nano?

     Odpowiedź

     $n=10^{-9}$.

111. Karta 111

     Pytanie

     Fundamental Pythagorean trigonometric identity?

     Odpowiedź

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

112. Karta 112

     Pytanie

     Engineering normal strain?

     Odpowiedź

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

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

113. Karta 113

     Pytanie

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

     Odpowiedź

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

114. Karta 114

     Pytanie

     Technical duty to an employer or client?

     Odpowiedź

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

115. Karta 115

     Pytanie

     How does a length conversion factor apply to area?

     Odpowiedź

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

116. Karta 116

     Pytanie

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

     Odpowiedź

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

117. Karta 117

     Pytanie

     When is a Poisson count model reasonable?

     Odpowiedź

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

118. Karta 118

     Pytanie

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

     Odpowiedź

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

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

119. Karta 119

     Pytanie

     Linear elastic stress-strain relation in uniaxial loading?

     Odpowiedź

     $$\sigma=E\varepsilon$$

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

120. Karta 120

     Pytanie

     How does a length conversion factor apply to volume?

     Odpowiedź

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

121. Karta 121

     Pytanie

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

     Odpowiedź

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

122. Karta 122

     Pytanie

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

     Odpowiedź

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

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

123. Karta 123

     Pytanie

     A supervisor directs an illegal engineering action. Proper response?

     Odpowiedź

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

124. Karta 124

     Pytanie

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

     Odpowiedź

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

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

125. Karta 125

     Pytanie

     Mass density?

     Odpowiedź

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

     Use mass, not weight, in the numerator.

126. Karta 126

     Pytanie

     Inches in one foot?

     Odpowiedź

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

127. Karta 127

     Pytanie

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

     Odpowiedź

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

128. Karta 128

     Pytanie

     How should subscripts be handled in a retrieved relationship?

     Odpowiedź

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

129. Karta 129

     Pytanie

     Dot product from vector magnitudes and included angle?

     Odpowiedź

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

     The result is a scalar.

130. Karta 130

     Pytanie

     Average normal pressure on an area?

     Odpowiedź

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

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

131. Karta 131

     Pytanie

     Feet in one yard?

     Odpowiedź

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

132. Karta 132

     Pytanie

     When are two interest rates equivalent?

     Odpowiedź

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

133. Karta 133

     Pytanie

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

     Odpowiedź

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

134. Karta 134

     Pytanie

     Expected value of a discrete random variable?

     Odpowiedź

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

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

135. Karta 135

     Pytanie

     Dot-product test for perpendicular nonzero vectors?

     Odpowiedź

     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. Karta 136

     Pytanie

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

     Odpowiedź

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

     Depth increases downward and the fluid is at rest.

137. Karta 137

     Pytanie

     Feet in one statute mile?

     Odpowiedź

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

138. Karta 138

     Pytanie

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

     Odpowiedź

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

     

     276 kart

     FE Exam Flashcards: Engineering Equations, Units & Handbook Skills

     Ucz się z tej talii za darmo

     Otworzy się Nibomo, żeby od razu zacząć naukę.

139. Karta 139

     Pytanie

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

     Odpowiedź

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

140. Karta 140

     Pytanie

     Magnitude of a vector cross product?

     Odpowiedź

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

     The direction follows the right-hand rule.

141. Karta 141

     Pytanie

     Exact inch-to-metre relationship?

     Odpowiedź

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

142. Karta 142

     Pytanie

     Volumetric flow rate through a cross-section?

     Odpowiedź

     $$Q=Av_{avg}$$

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

143. Karta 143

     Pytanie

     Variance from the mean $\mu$?

     Odpowiedź

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

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

144. Karta 144

     Pytanie

     Future-worth acceptance rule for an independent investment?

     Odpowiedź

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

145. Karta 145

     Pytanie

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

     Odpowiedź

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

146. Karta 146

     Pytanie

     Exact foot-to-metre relationship?

     Odpowiedź

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

147. Karta 147

     Pytanie

     How should important uncertainty be communicated?

     Odpowiedź

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

148. Karta 148

     Pytanie

     When is the matrix product $AB$ defined?

     Odpowiedź

     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. Karta 149

     Pytanie

     Steady one-inlet, one-outlet mass continuity?

     Odpowiedź

     $$\rho_1A_1v_1=\rho_2A_2v_2$$

     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. Karta 150

     Pytanie

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

     Odpowiedź

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

151. Karta 151

     Pytanie

     Square feet in one acre?

     Odpowiedź

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

152. Karta 152

     Pytanie

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

     Odpowiedź

     $$ad-bc$$

153. Karta 153

     Pytanie

     Bernoulli relationship along a streamline for ideal steady flow?

     Odpowiedź

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

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

154. Karta 154

     Pytanie

     May inconvenient measurements be discarded without a technical basis?

     Odpowiedź

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

155. Karta 155

     Pytanie

     Annual-worth acceptance rule for an independent investment?

     Odpowiedź

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

156. Karta 156

     Pytanie

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

     Odpowiedź

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

157. Karta 157

     Pytanie

     Cubic metres in one litre?

     Odpowiedź

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

158. Karta 158

     Pytanie

     What does a zero determinant mean for a square matrix?

     Odpowiedź

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

159. Karta 159

     Pytanie

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

     Odpowiedź

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

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

160. Karta 160

     Pytanie

     Unbiased sample variance for an independent sample from one population?

     Odpowiedź

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

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

161. Karta 161

     Pytanie

     How should mutually exclusive alternatives be compared?

     Odpowiedź

     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. Karta 162

     Pytanie

     Cubic centimetres in one millilitre?

     Odpowiedź

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

163. Karta 163

     Pytanie

     Why keep contemporaneous engineering records?

     Odpowiedź

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

164. Karta 164

     Pytanie

     Fast check after substituting into a handbook formula?

     Odpowiedź

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

165. Karta 165

     Pytanie

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

     Odpowiedź

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

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

166. Karta 166

     Pytanie

     Sensible heat for constant specific heat and no phase change?

     Odpowiedź

     $$Q=mc\Delta T$$

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

167. Karta 167

     Pytanie

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

     Odpowiedź

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

     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. Karta 168

     Pytanie

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

     Odpowiedź

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

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

169. Karta 169

     Pytanie

     Purpose of incremental analysis for mutually exclusive alternatives?

     Odpowiedź

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

170. Karta 170

     Pytanie

     Physical meaning of $dy/dx$?

     Odpowiedź

     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. Karta 171

     Pytanie

     One-dimensional steady conduction through a plane wall?

     Odpowiedź

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

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

172. Karta 172

     Pytanie

     When should you algebraically rearrange a handbook relationship?

     Odpowiedź

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

173. Karta 173

     Pytanie

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

     Odpowiedź

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

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

174. Karta 174

     Pytanie

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

     Odpowiedź

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

175. Karta 175

     Pytanie

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

     Odpowiedź

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

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

176. Karta 176

     Pytanie

     Convective heat-transfer rate at a surface?

     Odpowiedź

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

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

177. Karta 177

     Pytanie

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

     Odpowiedź

     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. Karta 178

     Pytanie

     Internal rate of return?

     Odpowiedź

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

179. Karta 179

     Pytanie

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

     Odpowiedź

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

180. Karta 180

     Pytanie

     Convert Celsius temperature to kelvin?

     Odpowiedź

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

     This conversion is for absolute temperature, not temperature difference.

181. Karta 181

     Pytanie

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

     Odpowiedź

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

182. Karta 182

     Pytanie

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

     Odpowiedź

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

183. Karta 183

     Pytanie

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

     Odpowiedź

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

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

184. Karta 184

     Pytanie

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

     Odpowiedź

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

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

185. Karta 185

     Pytanie

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

     Odpowiedź

     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. Karta 186

     Pytanie

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

     Odpowiedź

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

187. Karta 187

     Pytanie

     Why inspect boundary or initial conditions near a formula?

     Odpowiedź

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

188. Karta 188

     Pytanie

     Convert Celsius temperature to Fahrenheit?

     Odpowiedź

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

189. Karta 189

     Pytanie

     Ethical basis for expert technical testimony?

     Odpowiedź

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

190. Karta 190

     Pytanie

     Meaning of a breakeven value?

     Odpowiedź

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

191. Karta 191

     Pytanie

     Derivative of $e^x$?

     Odpowiedź

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

192. Karta 192

     Pytanie

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

     Odpowiedź

     $$\Delta E=Q-W$$

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

193. Karta 193

     Pytanie

     Frequentist meaning of a 95% confidence procedure?

     Odpowiedź

     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. Karta 194

     Pytanie

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

     Odpowiedź

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

195. Karta 195

     Pytanie

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

     Odpowiedź

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

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

196. Karta 196

     Pytanie

     Relationship between Celsius and kelvin temperature differences?

     Odpowiedź

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

197. Karta 197

     Pytanie

     How should credit for team engineering work be assigned?

     Odpowiedź

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

198. Karta 198

     Pytanie

     Main limitation of simple payback period?

     Odpowiedź

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

199. Karta 199

     Pytanie

     Roles of the null and alternative hypotheses?

     Odpowiedź

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

200. Karta 200

     Pytanie

     Ideal-gas equation of state?

     Odpowiedź

     $$pV=nRT$$

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

201. Karta 201

     Pytanie

     First check when reading a handbook table?

     Odpowiedź

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

202. Karta 202

     Pytanie

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

     Odpowiedź

     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. Karta 203

     Pytanie

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

     Odpowiedź

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

204. Karta 204

     Pytanie

     Convert Fahrenheit absolute temperature to Rankine?

     Odpowiedź

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

     Rankine uses the Fahrenheit-sized degree from absolute zero.

205. Karta 205

     Pytanie

     Ohm's law for an ideal resistor?

     Odpowiedź

     $$V=IR$$

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

206. Karta 206

     Pytanie

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

     Odpowiedź

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

207. Karta 207

     Pytanie

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

     Odpowiedź

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

208. Karta 208

     Pytanie

     Is a radian dimensionless?

     Odpowiedź

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

209. Karta 209

     Pytanie

     Electrical power absorbed under the passive sign convention?

     Odpowiedź

     $$P=VI$$

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

210. Karta 210

     Pytanie

     Meaning of a p-value?

     Odpowiedź

     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. Karta 211

     Pytanie

     First check when reading a handbook graph?

     Odpowiedź

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

212. Karta 212

     Pytanie

     Difference between mass and weight?

     Odpowiedź

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

213. Karta 213

     Pytanie

     Kirchhoff's current law at a node?

     Odpowiedź

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

214. Karta 214

     Pytanie

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

     Odpowiedź

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

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

215. Karta 215

     Pytanie

     Opportunity cost?

     Odpowiedź

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

216. Karta 216

     Pytanie

     Weight from mass in a gravitational field?

     Odpowiedź

     $$W=mg$$

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

217. Karta 217

     Pytanie

     Kirchhoff's voltage law around a closed loop?

     Odpowiedź

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

218. Karta 218

     Pytanie

     Decision rule using significance level $\alpha$?

     Odpowiedź

     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. Karta 219

     Pytanie

     Can proprietary methods from a former employer be reused freely?

     Odpowiedź

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

220. Karta 220

     Pytanie

     Pound-force to newtons?

     Odpowiedź

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

221. Karta 221

     Pytanie

     When does a definite integral equal ordinary geometric area?

     Odpowiedź

     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. Karta 222

     Pytanie

     Charge-voltage relation for an ideal capacitor?

     Odpowiedź

     $$q=CV$$

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

223. Karta 223

     Pytanie

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

     Odpowiedź

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

224. Karta 224

     Pytanie

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

     Odpowiedź

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

225. Karta 225

     Pytanie

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

     Odpowiedź

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

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

226. Karta 226

     Pytanie

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

     Odpowiedź

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

227. Karta 227

     Pytanie

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

     Odpowiedź

     $$\tau=RC$$

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

228. Karta 228

     Pytanie

     What physical quantity does pound-mass measure?

     Odpowiedź

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

229. Karta 229

     Pytanie

     Type I error?

     Odpowiedź

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

230. Karta 230

     Pytanie

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

     Odpowiedź

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

231. Karta 231

     Pytanie

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

     Odpowiedź

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

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

232. Karta 232

     Pytanie

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

     Odpowiedź

     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. Karta 233

     Pytanie

     Recognizable form of a separable first-order differential equation?

     Odpowiedź

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

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

234. Karta 234

     Pytanie

     Using unverified software output in a safety-relevant calculation?

     Odpowiedź

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

235. Karta 235

     Pytanie

     Equivalent resistance of ideal resistors in series?

     Odpowiedź

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

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

236. Karta 236

     Pytanie

     Pressure conversion from psi to pascals?

     Odpowiedź

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

237. Karta 237

     Pytanie

     Type II error?

     Odpowiedź

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

238. Karta 238

     Pytanie

     Useful flashcard to make after a failed handbook search?

     Odpowiedź

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

239. Karta 239

     Pytanie

     Standard form of a first-order linear differential equation?

     Odpowiedź

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

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

240. Karta 240

     Pytanie

     Equivalent resistance of ideal resistors in parallel?

     Odpowiedź

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

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

241. Karta 241

     Pytanie

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

     Odpowiedź

     $$BV_k=B-kD$$

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

242. Karta 242

     Pytanie

     Standard atmosphere in pascals?

     Odpowiedź

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

243. Karta 243

     Pytanie

     Using AI-generated technical content in engineering work?

     Odpowiedź

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

244. Karta 244

     Pytanie

     Trial form for a homogeneous linear ODE with constant coefficients?

     Odpowiedź

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

245. Karta 245

     Pytanie

     Wave-speed relationship among frequency and wavelength?

     Odpowiedź

     $$v=f\lambda$$

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

246. Karta 246

     Pytanie

     Statistical power?

     Odpowiedź

     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. Karta 247

     Pytanie

     Bar in pascals?

     Odpowiedź

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

248. Karta 248

     Pytanie

     How should handbook navigation be practiced before the FE exam?

     Odpowiedź

     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. Karta 249

     Pytanie

     Is depreciation itself a cash outflow?

     Odpowiedź

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

250. Karta 250

     Pytanie

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

     Odpowiedź

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

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

251. Karta 251

     Pytanie

     Relationship between period and frequency?

     Odpowiedź

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

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

252. Karta 252

     Pytanie

     How should foreseeable environmental harm enter an engineering decision?

     Odpowiedź

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

253. Karta 253

     Pytanie

     Foot-pound-force in joules?

     Odpowiedź

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

254. Karta 254

     Pytanie

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

     Odpowiedź

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

255. Karta 255

     Pytanie

     When should you search instead of relying on recall?

     Odpowiedź

     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. Karta 256

     Pytanie

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

     Odpowiedź

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

257. Karta 257

     Pytanie

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

     Odpowiedź

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

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

258. Karta 258

     Pytanie

     General material balance on a control volume?

     Odpowiedź

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

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

259. Karta 259

     Pytanie

     Approximate joules in one International Table Btu?

     Odpowiedź

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

260. Karta 260

     Pytanie

     What is the defender in a replacement analysis?

     Odpowiedź

     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. Karta 261

     Pytanie

     Speaking publicly about a technical matter outside your expertise?

     Odpowiedź

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

262. Karta 262

     Pytanie

     Single-interval trapezoidal approximation to an integral?

     Odpowiedź

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

     It is exact for linear $f$.

263. Karta 263

     Pytanie

     Energy efficiency of a device?

     Odpowiedź

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

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

264. Karta 264

     Pytanie

     Purpose of sensitivity analysis in engineering economics?

     Odpowiedź

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

265. Karta 265

     Pytanie

     Can a strong correlation establish causation?

     Odpowiedź

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

266. Karta 266

     Pytanie

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

     Odpowiedź

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

267. Karta 267

     Pytanie

     Mechanical horsepower in watts?

     Odpowiedź

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

268. Karta 268

     Pytanie

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

     Odpowiedź

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

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

269. Karta 269

     Pytanie

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

     Odpowiedź

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

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

270. Karta 270

     Pytanie

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

     Odpowiedź

     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. Karta 271

     Pytanie

     Kilowatt-hour in joules?

     Odpowiedź

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

272. Karta 272

     Pytanie

     Percent error relative to an accepted nonzero value?

     Odpowiedź

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

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

273. Karta 273

     Pytanie

     Does legal compliance alone settle an engineering ethics question?

     Odpowiedź

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

274. Karta 274

     Pytanie

     Does handbook fluency replace discipline-specific problem practice?

     Odpowiedź

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

275. Karta 275

     Pytanie

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

     Odpowiedź

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

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

276. Karta 276

     Pytanie

     Dimensional requirement for a physically meaningful additive equation?

     Odpowiedź

     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.