FE Exam Flashcards: Engineering Equations, Units & Handbook Skills

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1. Kadi namba 1

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   SI base unit of length?

   Jibu

   The metre, symbol $m$.

2. Kadi namba 2

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   Slope between two points?

   Jibu

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

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

3. Kadi namba 3

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   Net force relationship for constant mass in an inertial frame?

   Jibu

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

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

4. Kadi namba 4

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

   Jibu

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

5. Kadi namba 5

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

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

6. Kadi namba 6

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   Probability of the complement of event $A$?

   Jibu

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

7. Kadi namba 7

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

   Jibu

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

8. Kadi namba 8

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   SI base unit of mass?

   Jibu

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

9. Kadi namba 9

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   Linear equation from slope and intercept?

   Jibu

   $$y=mx+b$$

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

10. Kadi namba 10

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    Velocity after time $t$ under constant acceleration?

    Jibu

    $$v=v_0+at$$

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

11. Kadi namba 11

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

    Jibu

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

12. Kadi namba 12

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    How is the FE reference handbook available during the exam?

    Jibu

    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. Kadi namba 13

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

    Jibu

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

14. Kadi namba 14

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    SI base unit of time?

    Jibu

    The second, symbol $s$.

15. Kadi namba 15

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    Linear equation from one point and a slope?

    Jibu

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

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

16. Kadi namba 16

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    Position after time $t$ under constant acceleration?

    Jibu

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

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

17. Kadi namba 17

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    A project assignment lies outside your competence. Proper response?

    Jibu

    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. Kadi namba 18

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    SI base unit of electric current?

    Jibu

    The ampere, symbol $A$.

19. Kadi namba 19

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    Future amount under simple interest?

    Jibu

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

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

20. Kadi namba 20

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    Distance between two Cartesian points?

    Jibu

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

    The coordinate components must use the same length unit.

21. Kadi namba 21

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    Constant-acceleration relation without elapsed time?

    Jibu

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

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

22. Kadi namba 22

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

    Jibu

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

23. Kadi namba 23

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    Conditional probability of $A$ given $B$?

    Jibu

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

    This requires $P(B)>0$.

24. Kadi namba 24

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    SI base unit of thermodynamic temperature?

    Jibu

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

25. Kadi namba 25

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

    Jibu

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

26. Kadi namba 26

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    Midpoint of two Cartesian points?

    Jibu

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

    Average corresponding coordinates.

27. Kadi namba 27

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    Linear momentum of a particle?

    Jibu

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

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

28. Kadi namba 28

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    Future amount under compound interest?

    Jibu

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

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

29. Kadi namba 29

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    Should an FE search workflow depend on Ctrl+F?

    Jibu

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

30. Kadi namba 30

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    SI base unit of amount of substance?

    Jibu

    The mole, symbol $mol$.

31. Kadi namba 31

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    Joint probability expressed with a conditional probability?

    Jibu

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

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

32. Kadi namba 32

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

    Jibu

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

    The coordinate units and radius unit must match.

33. Kadi namba 33

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    Impulse-momentum relationship?

    Jibu

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

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

34. Kadi namba 34

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

    Jibu

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

35. Kadi namba 35

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    SI base unit of luminous intensity?

    Jibu

    The candela, symbol $cd$.

36. Kadi namba 36

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    Present worth of one future amount?

    Jibu

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

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

37. Kadi namba 37

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

    Jibu

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

    This form requires $a\ne0$.

38. Kadi namba 38

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    When is total linear momentum conserved?

    Jibu

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

39. Kadi namba 39

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    Why avoid memorizing FE handbook page numbers?

    Jibu

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

40. Kadi namba 40

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

    Jibu

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

    Independence is not the same as mutual exclusivity.

41. Kadi namba 41

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    SI unit of frequency in base-unit form?

    Jibu

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

42. Kadi namba 42

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

    Jibu

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

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

43. Kadi namba 43

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

    Jibu

    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. Kadi namba 44

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    Work done by a constant force through displacement?

    Jibu

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

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

45. Kadi namba 45

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

    Jibu

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

46. Kadi namba 46

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    SI unit of force in base-unit form?

    Jibu

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

47. Kadi namba 47

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

    Jibu

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

48. Kadi namba 48

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

    Jibu

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

    This requires $P(B)>0$.

49. Kadi namba 49

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

    Jibu

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

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

50. Kadi namba 50

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    Translational kinetic energy of a particle?

    Jibu

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

    Mass and speed must use a coherent unit system.

51. Kadi namba 51

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    SI unit of pressure in force-area form?

    Jibu

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

52. Kadi namba 52

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    Equivalent annual amount for present worth $P$?

    Jibu

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

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

53. Kadi namba 53

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    May confidential client information be shared for personal benefit?

    Jibu

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

54. Kadi namba 54

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

    Jibu

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

55. Kadi namba 55

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

    Jibu

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

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

56. Kadi namba 56

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    Net work-energy relationship for a particle?

    Jibu

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

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

57. Kadi namba 57

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

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

    Use this when order matters and selection is without replacement.

58. Kadi namba 58

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

    Jibu

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

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

59. Kadi namba 59

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

    Jibu

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

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

60. Kadi namba 60

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    SI unit of energy in force-distance form?

    Jibu

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

61. Kadi namba 61

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

    Jibu

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

62. Kadi namba 62

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

    Jibu

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

63. Kadi namba 63

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

    Jibu

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

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

64. Kadi namba 64

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    Change in gravitational potential energy near Earth's surface?

    Jibu

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

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

65. Kadi namba 65

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

    Jibu

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

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

66. Kadi namba 66

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

    Jibu

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

    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. Kadi namba 67

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    A handbook search for an acronym fails. Next move?

    Jibu

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

68. Kadi namba 68

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    SI unit of power in energy-time form?

    Jibu

    The watt, $W=J/s$.

69. Kadi namba 69

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    Logarithm of a product?

    Jibu

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

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

70. Kadi namba 70

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    Conditions for a binomial model?

    Jibu

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

71. Kadi namba 71

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    Using another engineer's analysis in your deliverable?

    Jibu

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

72. Kadi namba 72

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    SI unit of electric charge in current-time form?

    Jibu

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

73. Kadi namba 73

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    Average power from work over elapsed time?

    Jibu

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

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

74. Kadi namba 74

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

    Jibu

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

    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. Kadi namba 75

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    Logarithm of a positive quantity raised to a power?

    Jibu

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

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

76. Kadi namba 76

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

    Jibu

    The volt, $V=J/C$.

77. Kadi namba 77

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    Instantaneous mechanical power delivered by a force?

    Jibu

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

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

78. Kadi namba 78

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

    Jibu

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

79. Kadi namba 79

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

    Jibu

    $$E[X]=np$$

80. Kadi namba 80

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    SI unit of electrical resistance in voltage-current form?

    Jibu

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

81. Kadi namba 81

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    Moment of a force about a point?

    Jibu

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

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

82. Kadi namba 82

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

    Jibu

    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. Kadi namba 83

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    Inverse relationship between the natural log and exponential?

    Jibu

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

    The second identity requires $x>0$.

84. Kadi namba 84

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    SI prefix giga?

    Jibu

    $G=10^9$.

85. Kadi namba 85

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

    Jibu

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

    Use consistent angular sign convention and radians for angular measures.

86. Kadi namba 86

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    When should an engineer sign or seal technical work?

    Jibu

    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. Kadi namba 87

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    Best memory target for returning to a handbook formula?

    Jibu

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

88. Kadi namba 88

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    SI prefix mega?

    Jibu

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

89. Kadi namba 89

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    Convert degrees to radians?

    Jibu

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

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

90. Kadi namba 90

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

    Jibu

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

91. Kadi namba 91

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

    Jibu

    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. Kadi namba 92

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    Where do licensure and sealing requirements come from?

    Jibu

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

93. Kadi namba 93

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    Rotational kinetic energy about a fixed axis?

    Jibu

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

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

94. Kadi namba 94

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    SI prefix kilo?

    Jibu

    $k=10^3$.

95. Kadi namba 95

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    Right-triangle definition of sine?

    Jibu

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

96. Kadi namba 96

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

    Jibu

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

97. Kadi namba 97

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

    Jibu

    $$\sum M=I\alpha$$

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

98. Kadi namba 98

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    SI prefix milli?

    Jibu

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

99. Kadi namba 99

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

    Jibu

    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. Kadi namba 100

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

     Jibu

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

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

101. Kadi namba 101

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     Right-triangle definition of cosine?

     Jibu

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

102. Kadi namba 102

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     Force-displacement relation for an ideal linear spring?

     Jibu

     $$F=-kx$$

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

103. Kadi namba 103

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     May qualifications or project responsibility be exaggerated in a proposal?

     Jibu

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

104. Kadi namba 104

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     SI prefix micro?

     Jibu

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

105. Kadi namba 105

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

     Jibu

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

106. Kadi namba 106

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

     Jibu

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

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

107. Kadi namba 107

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     Right-triangle definition of tangent?

     Jibu

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

108. Kadi namba 108

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     Average normal stress under centered axial load?

     Jibu

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

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

109. Kadi namba 109

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

     Jibu

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

110. Kadi namba 110

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     SI prefix nano?

     Jibu

     $n=10^{-9}$.

111. Kadi namba 111

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     Fundamental Pythagorean trigonometric identity?

     Jibu

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

112. Kadi namba 112

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     Engineering normal strain?

     Jibu

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

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

113. Kadi namba 113

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

     Jibu

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

114. Kadi namba 114

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     Technical duty to an employer or client?

     Jibu

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

115. Kadi namba 115

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     How does a length conversion factor apply to area?

     Jibu

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

116. Kadi namba 116

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

     Jibu

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

117. Kadi namba 117

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     When is a Poisson count model reasonable?

     Jibu

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

118. Kadi namba 118

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

     Jibu

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

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

119. Kadi namba 119

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     Linear elastic stress-strain relation in uniaxial loading?

     Jibu

     $$\sigma=E\varepsilon$$

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

120. Kadi namba 120

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     How does a length conversion factor apply to volume?

     Jibu

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

121. Kadi namba 121

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

     Jibu

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

122. Kadi namba 122

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

     Jibu

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

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

123. Kadi namba 123

     Swali

     A supervisor directs an illegal engineering action. Proper response?

     Jibu

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

124. Kadi namba 124

     Swali

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

     Jibu

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

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

125. Kadi namba 125

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     Mass density?

     Jibu

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

     Use mass, not weight, in the numerator.

126. Kadi namba 126

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     Inches in one foot?

     Jibu

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

127. Kadi namba 127

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

     Jibu

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

128. Kadi namba 128

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     How should subscripts be handled in a retrieved relationship?

     Jibu

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

129. Kadi namba 129

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     Dot product from vector magnitudes and included angle?

     Jibu

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

     The result is a scalar.

130. Kadi namba 130

     Swali

     Average normal pressure on an area?

     Jibu

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

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

131. Kadi namba 131

     Swali

     Feet in one yard?

     Jibu

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

132. Kadi namba 132

     Swali

     When are two interest rates equivalent?

     Jibu

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

133. Kadi namba 133

     Swali

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

     Jibu

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

134. Kadi namba 134

     Swali

     Expected value of a discrete random variable?

     Jibu

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

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

135. Kadi namba 135

     Swali

     Dot-product test for perpendicular nonzero vectors?

     Jibu

     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. Kadi namba 136

     Swali

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

     Jibu

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

     Depth increases downward and the fluid is at rest.

137. Kadi namba 137

     Swali

     Feet in one statute mile?

     Jibu

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

138. Kadi namba 138

     Swali

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

     Jibu

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

     

     kadi 276

     FE Exam Flashcards: Engineering Equations, Units & Handbook Skills

     Soma fungu hili bila malipo

     Nibomo itafunguka ili uanze kusoma.

139. Kadi namba 139

     Swali

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

     Jibu

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

140. Kadi namba 140

     Swali

     Magnitude of a vector cross product?

     Jibu

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

     The direction follows the right-hand rule.

141. Kadi namba 141

     Swali

     Exact inch-to-metre relationship?

     Jibu

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

142. Kadi namba 142

     Swali

     Volumetric flow rate through a cross-section?

     Jibu

     $$Q=Av_{avg}$$

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

143. Kadi namba 143

     Swali

     Variance from the mean $\mu$?

     Jibu

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

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

144. Kadi namba 144

     Swali

     Future-worth acceptance rule for an independent investment?

     Jibu

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

145. Kadi namba 145

     Swali

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

     Jibu

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

146. Kadi namba 146

     Swali

     Exact foot-to-metre relationship?

     Jibu

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

147. Kadi namba 147

     Swali

     How should important uncertainty be communicated?

     Jibu

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

148. Kadi namba 148

     Swali

     When is the matrix product $AB$ defined?

     Jibu

     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. Kadi namba 149

     Swali

     Steady one-inlet, one-outlet mass continuity?

     Jibu

     $$\rho_1A_1v_1=\rho_2A_2v_2$$

     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. Kadi namba 150

     Swali

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

     Jibu

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

151. Kadi namba 151

     Swali

     Square feet in one acre?

     Jibu

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

152. Kadi namba 152

     Swali

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

     Jibu

     $$ad-bc$$

153. Kadi namba 153

     Swali

     Bernoulli relationship along a streamline for ideal steady flow?

     Jibu

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

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

154. Kadi namba 154

     Swali

     May inconvenient measurements be discarded without a technical basis?

     Jibu

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

155. Kadi namba 155

     Swali

     Annual-worth acceptance rule for an independent investment?

     Jibu

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

156. Kadi namba 156

     Swali

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

     Jibu

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

157. Kadi namba 157

     Swali

     Cubic metres in one litre?

     Jibu

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

158. Kadi namba 158

     Swali

     What does a zero determinant mean for a square matrix?

     Jibu

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

159. Kadi namba 159

     Swali

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

     Jibu

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

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

160. Kadi namba 160

     Swali

     Unbiased sample variance for an independent sample from one population?

     Jibu

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

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

161. Kadi namba 161

     Swali

     How should mutually exclusive alternatives be compared?

     Jibu

     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. Kadi namba 162

     Swali

     Cubic centimetres in one millilitre?

     Jibu

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

163. Kadi namba 163

     Swali

     Why keep contemporaneous engineering records?

     Jibu

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

164. Kadi namba 164

     Swali

     Fast check after substituting into a handbook formula?

     Jibu

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

165. Kadi namba 165

     Swali

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

     Jibu

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

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

166. Kadi namba 166

     Swali

     Sensible heat for constant specific heat and no phase change?

     Jibu

     $$Q=mc\Delta T$$

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

167. Kadi namba 167

     Swali

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

     Jibu

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

     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. Kadi namba 168

     Swali

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

     Jibu

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

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

169. Kadi namba 169

     Swali

     Purpose of incremental analysis for mutually exclusive alternatives?

     Jibu

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

170. Kadi namba 170

     Swali

     Physical meaning of $dy/dx$?

     Jibu

     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. Kadi namba 171

     Swali

     One-dimensional steady conduction through a plane wall?

     Jibu

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

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

172. Kadi namba 172

     Swali

     When should you algebraically rearrange a handbook relationship?

     Jibu

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

173. Kadi namba 173

     Swali

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

     Jibu

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

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

174. Kadi namba 174

     Swali

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

     Jibu

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

175. Kadi namba 175

     Swali

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

     Jibu

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

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

176. Kadi namba 176

     Swali

     Convective heat-transfer rate at a surface?

     Jibu

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

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

177. Kadi namba 177

     Swali

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

     Jibu

     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. Kadi namba 178

     Swali

     Internal rate of return?

     Jibu

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

179. Kadi namba 179

     Swali

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

     Jibu

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

180. Kadi namba 180

     Swali

     Convert Celsius temperature to kelvin?

     Jibu

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

     This conversion is for absolute temperature, not temperature difference.

181. Kadi namba 181

     Swali

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

     Jibu

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

182. Kadi namba 182

     Swali

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

     Jibu

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

183. Kadi namba 183

     Swali

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

     Jibu

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

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

184. Kadi namba 184

     Swali

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

     Jibu

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

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

185. Kadi namba 185

     Swali

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

     Jibu

     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. Kadi namba 186

     Swali

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

     Jibu

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

187. Kadi namba 187

     Swali

     Why inspect boundary or initial conditions near a formula?

     Jibu

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

188. Kadi namba 188

     Swali

     Convert Celsius temperature to Fahrenheit?

     Jibu

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

189. Kadi namba 189

     Swali

     Ethical basis for expert technical testimony?

     Jibu

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

190. Kadi namba 190

     Swali

     Meaning of a breakeven value?

     Jibu

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

191. Kadi namba 191

     Swali

     Derivative of $e^x$?

     Jibu

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

192. Kadi namba 192

     Swali

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

     Jibu

     $$\Delta E=Q-W$$

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

193. Kadi namba 193

     Swali

     Frequentist meaning of a 95% confidence procedure?

     Jibu

     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. Kadi namba 194

     Swali

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

     Jibu

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

195. Kadi namba 195

     Swali

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

     Jibu

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

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

196. Kadi namba 196

     Swali

     Relationship between Celsius and kelvin temperature differences?

     Jibu

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

197. Kadi namba 197

     Swali

     How should credit for team engineering work be assigned?

     Jibu

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

198. Kadi namba 198

     Swali

     Main limitation of simple payback period?

     Jibu

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

199. Kadi namba 199

     Swali

     Roles of the null and alternative hypotheses?

     Jibu

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

200. Kadi namba 200

     Swali

     Ideal-gas equation of state?

     Jibu

     $$pV=nRT$$

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

201. Kadi namba 201

     Swali

     First check when reading a handbook table?

     Jibu

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

202. Kadi namba 202

     Swali

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

     Jibu

     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. Kadi namba 203

     Swali

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

     Jibu

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

204. Kadi namba 204

     Swali

     Convert Fahrenheit absolute temperature to Rankine?

     Jibu

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

     Rankine uses the Fahrenheit-sized degree from absolute zero.

205. Kadi namba 205

     Swali

     Ohm's law for an ideal resistor?

     Jibu

     $$V=IR$$

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

206. Kadi namba 206

     Swali

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

     Jibu

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

207. Kadi namba 207

     Swali

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

     Jibu

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

208. Kadi namba 208

     Swali

     Is a radian dimensionless?

     Jibu

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

209. Kadi namba 209

     Swali

     Electrical power absorbed under the passive sign convention?

     Jibu

     $$P=VI$$

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

210. Kadi namba 210

     Swali

     Meaning of a p-value?

     Jibu

     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. Kadi namba 211

     Swali

     First check when reading a handbook graph?

     Jibu

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

212. Kadi namba 212

     Swali

     Difference between mass and weight?

     Jibu

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

213. Kadi namba 213

     Swali

     Kirchhoff's current law at a node?

     Jibu

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

214. Kadi namba 214

     Swali

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

     Jibu

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

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

215. Kadi namba 215

     Swali

     Opportunity cost?

     Jibu

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

216. Kadi namba 216

     Swali

     Weight from mass in a gravitational field?

     Jibu

     $$W=mg$$

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

217. Kadi namba 217

     Swali

     Kirchhoff's voltage law around a closed loop?

     Jibu

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

218. Kadi namba 218

     Swali

     Decision rule using significance level $\alpha$?

     Jibu

     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. Kadi namba 219

     Swali

     Can proprietary methods from a former employer be reused freely?

     Jibu

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

220. Kadi namba 220

     Swali

     Pound-force to newtons?

     Jibu

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

221. Kadi namba 221

     Swali

     When does a definite integral equal ordinary geometric area?

     Jibu

     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. Kadi namba 222

     Swali

     Charge-voltage relation for an ideal capacitor?

     Jibu

     $$q=CV$$

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

223. Kadi namba 223

     Swali

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

     Jibu

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

224. Kadi namba 224

     Swali

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

     Jibu

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

225. Kadi namba 225

     Swali

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

     Jibu

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

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

226. Kadi namba 226

     Swali

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

     Jibu

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

227. Kadi namba 227

     Swali

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

     Jibu

     $$\tau=RC$$

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

228. Kadi namba 228

     Swali

     What physical quantity does pound-mass measure?

     Jibu

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

229. Kadi namba 229

     Swali

     Type I error?

     Jibu

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

230. Kadi namba 230

     Swali

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

     Jibu

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

231. Kadi namba 231

     Swali

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

     Jibu

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

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

232. Kadi namba 232

     Swali

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

     Jibu

     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. Kadi namba 233

     Swali

     Recognizable form of a separable first-order differential equation?

     Jibu

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

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

234. Kadi namba 234

     Swali

     Using unverified software output in a safety-relevant calculation?

     Jibu

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

235. Kadi namba 235

     Swali

     Equivalent resistance of ideal resistors in series?

     Jibu

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

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

236. Kadi namba 236

     Swali

     Pressure conversion from psi to pascals?

     Jibu

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

237. Kadi namba 237

     Swali

     Type II error?

     Jibu

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

238. Kadi namba 238

     Swali

     Useful flashcard to make after a failed handbook search?

     Jibu

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

239. Kadi namba 239

     Swali

     Standard form of a first-order linear differential equation?

     Jibu

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

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

240. Kadi namba 240

     Swali

     Equivalent resistance of ideal resistors in parallel?

     Jibu

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

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

241. Kadi namba 241

     Swali

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

     Jibu

     $$BV_k=B-kD$$

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

242. Kadi namba 242

     Swali

     Standard atmosphere in pascals?

     Jibu

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

243. Kadi namba 243

     Swali

     Using AI-generated technical content in engineering work?

     Jibu

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

244. Kadi namba 244

     Swali

     Trial form for a homogeneous linear ODE with constant coefficients?

     Jibu

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

245. Kadi namba 245

     Swali

     Wave-speed relationship among frequency and wavelength?

     Jibu

     $$v=f\lambda$$

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

246. Kadi namba 246

     Swali

     Statistical power?

     Jibu

     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. Kadi namba 247

     Swali

     Bar in pascals?

     Jibu

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

248. Kadi namba 248

     Swali

     How should handbook navigation be practiced before the FE exam?

     Jibu

     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. Kadi namba 249

     Swali

     Is depreciation itself a cash outflow?

     Jibu

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

250. Kadi namba 250

     Swali

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

     Jibu

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

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

251. Kadi namba 251

     Swali

     Relationship between period and frequency?

     Jibu

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

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

252. Kadi namba 252

     Swali

     How should foreseeable environmental harm enter an engineering decision?

     Jibu

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

253. Kadi namba 253

     Swali

     Foot-pound-force in joules?

     Jibu

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

254. Kadi namba 254

     Swali

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

     Jibu

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

255. Kadi namba 255

     Swali

     When should you search instead of relying on recall?

     Jibu

     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. Kadi namba 256

     Swali

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

     Jibu

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

257. Kadi namba 257

     Swali

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

     Jibu

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

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

258. Kadi namba 258

     Swali

     General material balance on a control volume?

     Jibu

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

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

259. Kadi namba 259

     Swali

     Approximate joules in one International Table Btu?

     Jibu

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

260. Kadi namba 260

     Swali

     What is the defender in a replacement analysis?

     Jibu

     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. Kadi namba 261

     Swali

     Speaking publicly about a technical matter outside your expertise?

     Jibu

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

262. Kadi namba 262

     Swali

     Single-interval trapezoidal approximation to an integral?

     Jibu

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

     It is exact for linear $f$.

263. Kadi namba 263

     Swali

     Energy efficiency of a device?

     Jibu

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

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

264. Kadi namba 264

     Swali

     Purpose of sensitivity analysis in engineering economics?

     Jibu

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

265. Kadi namba 265

     Swali

     Can a strong correlation establish causation?

     Jibu

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

266. Kadi namba 266

     Swali

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

     Jibu

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

267. Kadi namba 267

     Swali

     Mechanical horsepower in watts?

     Jibu

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

268. Kadi namba 268

     Swali

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

     Jibu

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

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

269. Kadi namba 269

     Swali

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

     Jibu

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

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

270. Kadi namba 270

     Swali

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

     Jibu

     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. Kadi namba 271

     Swali

     Kilowatt-hour in joules?

     Jibu

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

272. Kadi namba 272

     Swali

     Percent error relative to an accepted nonzero value?

     Jibu

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

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

273. Kadi namba 273

     Swali

     Does legal compliance alone settle an engineering ethics question?

     Jibu

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

274. Kadi namba 274

     Swali

     Does handbook fluency replace discipline-specific problem practice?

     Jibu

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

275. Kadi namba 275

     Swali

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

     Jibu

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

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

276. Kadi namba 276

     Swali

     Dimensional requirement for a physically meaningful additive equation?

     Jibu

     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.