FE Exam Flashcards: Engineering Equations, Units & Handbook Skills

اس ڈیک کے کارڈز

1. کارڈ 1

   سوال

   SI base unit of length?

   جواب

   The metre, symbol $m$.

2. کارڈ 2

   سوال

   Slope between two points?

   جواب

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

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

3. کارڈ 3

   سوال

   Net force relationship for constant mass in an inertial frame?

   جواب

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

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

4. کارڈ 4

   سوال

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

   جواب

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

5. کارڈ 5

   سوال

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

   جواب

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

6. کارڈ 6

   سوال

   Probability of the complement of event $A$?

   جواب

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

7. کارڈ 7

   سوال

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

   جواب

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

8. کارڈ 8

   سوال

   SI base unit of mass?

   جواب

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

9. کارڈ 9

   سوال

   Linear equation from slope and intercept?

   جواب

   $$y=mx+b$$

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

10. کارڈ 10

    سوال

    Velocity after time $t$ under constant acceleration?

    جواب

    $$v=v_0+at$$

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

11. کارڈ 11

    سوال

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

    جواب

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

12. کارڈ 12

    سوال

    How is the FE reference handbook available during the exam?

    جواب

    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. کارڈ 13

    سوال

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

    جواب

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

14. کارڈ 14

    سوال

    SI base unit of time?

    جواب

    The second, symbol $s$.

15. کارڈ 15

    سوال

    Linear equation from one point and a slope?

    جواب

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

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

16. کارڈ 16

    سوال

    Position after time $t$ under constant acceleration?

    جواب

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

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

17. کارڈ 17

    سوال

    A project assignment lies outside your competence. Proper response?

    جواب

    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. کارڈ 18

    سوال

    SI base unit of electric current?

    جواب

    The ampere, symbol $A$.

19. کارڈ 19

    سوال

    Future amount under simple interest?

    جواب

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

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

20. کارڈ 20

    سوال

    Distance between two Cartesian points?

    جواب

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

    The coordinate components must use the same length unit.

21. کارڈ 21

    سوال

    Constant-acceleration relation without elapsed time?

    جواب

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

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

22. کارڈ 22

    سوال

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

    جواب

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

23. کارڈ 23

    سوال

    Conditional probability of $A$ given $B$?

    جواب

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

    This requires $P(B)>0$.

24. کارڈ 24

    سوال

    SI base unit of thermodynamic temperature?

    جواب

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

25. کارڈ 25

    سوال

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

    جواب

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

26. کارڈ 26

    سوال

    Midpoint of two Cartesian points?

    جواب

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

    Average corresponding coordinates.

27. کارڈ 27

    سوال

    Linear momentum of a particle?

    جواب

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

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

28. کارڈ 28

    سوال

    Future amount under compound interest?

    جواب

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

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

29. کارڈ 29

    سوال

    Should an FE search workflow depend on Ctrl+F?

    جواب

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

30. کارڈ 30

    سوال

    SI base unit of amount of substance?

    جواب

    The mole, symbol $mol$.

31. کارڈ 31

    سوال

    Joint probability expressed with a conditional probability?

    جواب

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

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

32. کارڈ 32

    سوال

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

    جواب

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

    The coordinate units and radius unit must match.

33. کارڈ 33

    سوال

    Impulse-momentum relationship?

    جواب

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

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

34. کارڈ 34

    سوال

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

    جواب

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

35. کارڈ 35

    سوال

    SI base unit of luminous intensity?

    جواب

    The candela, symbol $cd$.

36. کارڈ 36

    سوال

    Present worth of one future amount?

    جواب

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

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

37. کارڈ 37

    سوال

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

    جواب

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

    This form requires $a\ne0$.

38. کارڈ 38

    سوال

    When is total linear momentum conserved?

    جواب

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

39. کارڈ 39

    سوال

    Why avoid memorizing FE handbook page numbers?

    جواب

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

40. کارڈ 40

    سوال

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

    جواب

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

    Independence is not the same as mutual exclusivity.

41. کارڈ 41

    سوال

    SI unit of frequency in base-unit form?

    جواب

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

42. کارڈ 42

    سوال

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

    جواب

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

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

43. کارڈ 43

    سوال

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

    جواب

    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. کارڈ 44

    سوال

    Work done by a constant force through displacement?

    جواب

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

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

45. کارڈ 45

    سوال

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

    جواب

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

46. کارڈ 46

    سوال

    SI unit of force in base-unit form?

    جواب

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

47. کارڈ 47

    سوال

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

    جواب

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

48. کارڈ 48

    سوال

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

    جواب

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

    This requires $P(B)>0$.

49. کارڈ 49

    سوال

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

    جواب

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

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

50. کارڈ 50

    سوال

    Translational kinetic energy of a particle?

    جواب

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

    Mass and speed must use a coherent unit system.

51. کارڈ 51

    سوال

    SI unit of pressure in force-area form?

    جواب

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

52. کارڈ 52

    سوال

    Equivalent annual amount for present worth $P$?

    جواب

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

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

53. کارڈ 53

    سوال

    May confidential client information be shared for personal benefit?

    جواب

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

54. کارڈ 54

    سوال

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

    جواب

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

55. کارڈ 55

    سوال

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

    جواب

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

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

56. کارڈ 56

    سوال

    Net work-energy relationship for a particle?

    جواب

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

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

57. کارڈ 57

    سوال

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

    جواب

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

    Use this when order matters and selection is without replacement.

58. کارڈ 58

    سوال

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

    جواب

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

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

59. کارڈ 59

    سوال

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

    جواب

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

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

60. کارڈ 60

    سوال

    SI unit of energy in force-distance form?

    جواب

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

61. کارڈ 61

    سوال

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

    جواب

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

62. کارڈ 62

    سوال

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

    جواب

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

63. کارڈ 63

    سوال

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

    جواب

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

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

64. کارڈ 64

    سوال

    Change in gravitational potential energy near Earth's surface?

    جواب

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

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

65. کارڈ 65

    سوال

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

    جواب

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

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

66. کارڈ 66

    سوال

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

    جواب

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

    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. کارڈ 67

    سوال

    A handbook search for an acronym fails. Next move?

    جواب

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

68. کارڈ 68

    سوال

    SI unit of power in energy-time form?

    جواب

    The watt, $W=J/s$.

69. کارڈ 69

    سوال

    Logarithm of a product?

    جواب

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

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

70. کارڈ 70

    سوال

    Conditions for a binomial model?

    جواب

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

71. کارڈ 71

    سوال

    Using another engineer's analysis in your deliverable?

    جواب

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

72. کارڈ 72

    سوال

    SI unit of electric charge in current-time form?

    جواب

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

73. کارڈ 73

    سوال

    Average power from work over elapsed time?

    جواب

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

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

74. کارڈ 74

    سوال

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

    جواب

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

    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. کارڈ 75

    سوال

    Logarithm of a positive quantity raised to a power?

    جواب

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

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

76. کارڈ 76

    سوال

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

    جواب

    The volt, $V=J/C$.

77. کارڈ 77

    سوال

    Instantaneous mechanical power delivered by a force?

    جواب

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

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

78. کارڈ 78

    سوال

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

    جواب

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

79. کارڈ 79

    سوال

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

    جواب

    $$E[X]=np$$

80. کارڈ 80

    سوال

    SI unit of electrical resistance in voltage-current form?

    جواب

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

81. کارڈ 81

    سوال

    Moment of a force about a point?

    جواب

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

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

82. کارڈ 82

    سوال

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

    جواب

    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. کارڈ 83

    سوال

    Inverse relationship between the natural log and exponential?

    جواب

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

    The second identity requires $x>0$.

84. کارڈ 84

    سوال

    SI prefix giga?

    جواب

    $G=10^9$.

85. کارڈ 85

    سوال

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

    جواب

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

    Use consistent angular sign convention and radians for angular measures.

86. کارڈ 86

    سوال

    When should an engineer sign or seal technical work?

    جواب

    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. کارڈ 87

    سوال

    Best memory target for returning to a handbook formula?

    جواب

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

88. کارڈ 88

    سوال

    SI prefix mega?

    جواب

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

89. کارڈ 89

    سوال

    Convert degrees to radians?

    جواب

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

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

90. کارڈ 90

    سوال

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

    جواب

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

91. کارڈ 91

    سوال

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

    جواب

    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. کارڈ 92

    سوال

    Where do licensure and sealing requirements come from?

    جواب

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

93. کارڈ 93

    سوال

    Rotational kinetic energy about a fixed axis?

    جواب

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

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

94. کارڈ 94

    سوال

    SI prefix kilo?

    جواب

    $k=10^3$.

95. کارڈ 95

    سوال

    Right-triangle definition of sine?

    جواب

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

96. کارڈ 96

    سوال

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

    جواب

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

97. کارڈ 97

    سوال

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

    جواب

    $$\sum M=I\alpha$$

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

98. کارڈ 98

    سوال

    SI prefix milli?

    جواب

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

99. کارڈ 99

    سوال

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

    جواب

    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. کارڈ 100

     سوال

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

     جواب

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

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

101. کارڈ 101

     سوال

     Right-triangle definition of cosine?

     جواب

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

102. کارڈ 102

     سوال

     Force-displacement relation for an ideal linear spring?

     جواب

     $$F=-kx$$

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

103. کارڈ 103

     سوال

     May qualifications or project responsibility be exaggerated in a proposal?

     جواب

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

104. کارڈ 104

     سوال

     SI prefix micro?

     جواب

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

105. کارڈ 105

     سوال

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

     جواب

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

106. کارڈ 106

     سوال

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

     جواب

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

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

107. کارڈ 107

     سوال

     Right-triangle definition of tangent?

     جواب

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

108. کارڈ 108

     سوال

     Average normal stress under centered axial load?

     جواب

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

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

109. کارڈ 109

     سوال

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

     جواب

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

110. کارڈ 110

     سوال

     SI prefix nano?

     جواب

     $n=10^{-9}$.

111. کارڈ 111

     سوال

     Fundamental Pythagorean trigonometric identity?

     جواب

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

112. کارڈ 112

     سوال

     Engineering normal strain?

     جواب

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

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

113. کارڈ 113

     سوال

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

     جواب

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

114. کارڈ 114

     سوال

     Technical duty to an employer or client?

     جواب

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

115. کارڈ 115

     سوال

     How does a length conversion factor apply to area?

     جواب

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

116. کارڈ 116

     سوال

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

     جواب

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

117. کارڈ 117

     سوال

     When is a Poisson count model reasonable?

     جواب

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

118. کارڈ 118

     سوال

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

     جواب

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

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

119. کارڈ 119

     سوال

     Linear elastic stress-strain relation in uniaxial loading?

     جواب

     $$\sigma=E\varepsilon$$

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

120. کارڈ 120

     سوال

     How does a length conversion factor apply to volume?

     جواب

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

121. کارڈ 121

     سوال

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

     جواب

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

122. کارڈ 122

     سوال

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

     جواب

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

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

123. کارڈ 123

     سوال

     A supervisor directs an illegal engineering action. Proper response?

     جواب

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

124. کارڈ 124

     سوال

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

     جواب

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

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

125. کارڈ 125

     سوال

     Mass density?

     جواب

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

     Use mass, not weight, in the numerator.

126. کارڈ 126

     سوال

     Inches in one foot?

     جواب

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

127. کارڈ 127

     سوال

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

     جواب

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

128. کارڈ 128

     سوال

     How should subscripts be handled in a retrieved relationship?

     جواب

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

129. کارڈ 129

     سوال

     Dot product from vector magnitudes and included angle?

     جواب

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

     The result is a scalar.

130. کارڈ 130

     سوال

     Average normal pressure on an area?

     جواب

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

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

131. کارڈ 131

     سوال

     Feet in one yard?

     جواب

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

132. کارڈ 132

     سوال

     When are two interest rates equivalent?

     جواب

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

133. کارڈ 133

     سوال

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

     جواب

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

134. کارڈ 134

     سوال

     Expected value of a discrete random variable?

     جواب

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

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

135. کارڈ 135

     سوال

     Dot-product test for perpendicular nonzero vectors?

     جواب

     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. کارڈ 136

     سوال

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

     جواب

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

     Depth increases downward and the fluid is at rest.

137. کارڈ 137

     سوال

     Feet in one statute mile?

     جواب

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

138. کارڈ 138

     سوال

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

     جواب

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

     

     276 کارڈز

     FE Exam Flashcards: Engineering Equations, Units & Handbook Skills

     یہ ڈیک مفت پڑھیں

     مطالعہ شروع کرنے کے لیے Nibomo کھلے گا۔

139. کارڈ 139

     سوال

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

     جواب

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

140. کارڈ 140

     سوال

     Magnitude of a vector cross product?

     جواب

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

     The direction follows the right-hand rule.

141. کارڈ 141

     سوال

     Exact inch-to-metre relationship?

     جواب

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

142. کارڈ 142

     سوال

     Volumetric flow rate through a cross-section?

     جواب

     $$Q=Av_{avg}$$

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

143. کارڈ 143

     سوال

     Variance from the mean $\mu$?

     جواب

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

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

144. کارڈ 144

     سوال

     Future-worth acceptance rule for an independent investment?

     جواب

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

145. کارڈ 145

     سوال

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

     جواب

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

146. کارڈ 146

     سوال

     Exact foot-to-metre relationship?

     جواب

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

147. کارڈ 147

     سوال

     How should important uncertainty be communicated?

     جواب

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

148. کارڈ 148

     سوال

     When is the matrix product $AB$ defined?

     جواب

     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. کارڈ 149

     سوال

     Steady one-inlet, one-outlet mass continuity?

     جواب

     $$\rho_1A_1v_1=\rho_2A_2v_2$$

     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. کارڈ 150

     سوال

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

     جواب

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

151. کارڈ 151

     سوال

     Square feet in one acre?

     جواب

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

152. کارڈ 152

     سوال

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

     جواب

     $$ad-bc$$

153. کارڈ 153

     سوال

     Bernoulli relationship along a streamline for ideal steady flow?

     جواب

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

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

154. کارڈ 154

     سوال

     May inconvenient measurements be discarded without a technical basis?

     جواب

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

155. کارڈ 155

     سوال

     Annual-worth acceptance rule for an independent investment?

     جواب

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

156. کارڈ 156

     سوال

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

     جواب

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

157. کارڈ 157

     سوال

     Cubic metres in one litre?

     جواب

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

158. کارڈ 158

     سوال

     What does a zero determinant mean for a square matrix?

     جواب

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

159. کارڈ 159

     سوال

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

     جواب

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

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

160. کارڈ 160

     سوال

     Unbiased sample variance for an independent sample from one population?

     جواب

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

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

161. کارڈ 161

     سوال

     How should mutually exclusive alternatives be compared?

     جواب

     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. کارڈ 162

     سوال

     Cubic centimetres in one millilitre?

     جواب

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

163. کارڈ 163

     سوال

     Why keep contemporaneous engineering records?

     جواب

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

164. کارڈ 164

     سوال

     Fast check after substituting into a handbook formula?

     جواب

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

165. کارڈ 165

     سوال

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

     جواب

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

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

166. کارڈ 166

     سوال

     Sensible heat for constant specific heat and no phase change?

     جواب

     $$Q=mc\Delta T$$

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

167. کارڈ 167

     سوال

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

     جواب

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

     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. کارڈ 168

     سوال

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

     جواب

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

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

169. کارڈ 169

     سوال

     Purpose of incremental analysis for mutually exclusive alternatives?

     جواب

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

170. کارڈ 170

     سوال

     Physical meaning of $dy/dx$?

     جواب

     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. کارڈ 171

     سوال

     One-dimensional steady conduction through a plane wall?

     جواب

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

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

172. کارڈ 172

     سوال

     When should you algebraically rearrange a handbook relationship?

     جواب

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

173. کارڈ 173

     سوال

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

     جواب

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

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

174. کارڈ 174

     سوال

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

     جواب

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

175. کارڈ 175

     سوال

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

     جواب

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

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

176. کارڈ 176

     سوال

     Convective heat-transfer rate at a surface?

     جواب

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

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

177. کارڈ 177

     سوال

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

     جواب

     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. کارڈ 178

     سوال

     Internal rate of return?

     جواب

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

179. کارڈ 179

     سوال

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

     جواب

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

180. کارڈ 180

     سوال

     Convert Celsius temperature to kelvin?

     جواب

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

     This conversion is for absolute temperature, not temperature difference.

181. کارڈ 181

     سوال

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

     جواب

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

182. کارڈ 182

     سوال

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

     جواب

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

183. کارڈ 183

     سوال

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

     جواب

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

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

184. کارڈ 184

     سوال

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

     جواب

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

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

185. کارڈ 185

     سوال

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

     جواب

     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. کارڈ 186

     سوال

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

     جواب

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

187. کارڈ 187

     سوال

     Why inspect boundary or initial conditions near a formula?

     جواب

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

188. کارڈ 188

     سوال

     Convert Celsius temperature to Fahrenheit?

     جواب

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

189. کارڈ 189

     سوال

     Ethical basis for expert technical testimony?

     جواب

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

190. کارڈ 190

     سوال

     Meaning of a breakeven value?

     جواب

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

191. کارڈ 191

     سوال

     Derivative of $e^x$?

     جواب

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

192. کارڈ 192

     سوال

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

     جواب

     $$\Delta E=Q-W$$

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

193. کارڈ 193

     سوال

     Frequentist meaning of a 95% confidence procedure?

     جواب

     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. کارڈ 194

     سوال

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

     جواب

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

195. کارڈ 195

     سوال

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

     جواب

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

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

196. کارڈ 196

     سوال

     Relationship between Celsius and kelvin temperature differences?

     جواب

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

197. کارڈ 197

     سوال

     How should credit for team engineering work be assigned?

     جواب

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

198. کارڈ 198

     سوال

     Main limitation of simple payback period?

     جواب

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

199. کارڈ 199

     سوال

     Roles of the null and alternative hypotheses?

     جواب

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

200. کارڈ 200

     سوال

     Ideal-gas equation of state?

     جواب

     $$pV=nRT$$

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

201. کارڈ 201

     سوال

     First check when reading a handbook table?

     جواب

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

202. کارڈ 202

     سوال

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

     جواب

     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. کارڈ 203

     سوال

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

     جواب

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

204. کارڈ 204

     سوال

     Convert Fahrenheit absolute temperature to Rankine?

     جواب

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

     Rankine uses the Fahrenheit-sized degree from absolute zero.

205. کارڈ 205

     سوال

     Ohm's law for an ideal resistor?

     جواب

     $$V=IR$$

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

206. کارڈ 206

     سوال

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

     جواب

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

207. کارڈ 207

     سوال

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

     جواب

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

208. کارڈ 208

     سوال

     Is a radian dimensionless?

     جواب

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

209. کارڈ 209

     سوال

     Electrical power absorbed under the passive sign convention?

     جواب

     $$P=VI$$

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

210. کارڈ 210

     سوال

     Meaning of a p-value?

     جواب

     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. کارڈ 211

     سوال

     First check when reading a handbook graph?

     جواب

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

212. کارڈ 212

     سوال

     Difference between mass and weight?

     جواب

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

213. کارڈ 213

     سوال

     Kirchhoff's current law at a node?

     جواب

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

214. کارڈ 214

     سوال

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

     جواب

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

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

215. کارڈ 215

     سوال

     Opportunity cost?

     جواب

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

216. کارڈ 216

     سوال

     Weight from mass in a gravitational field?

     جواب

     $$W=mg$$

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

217. کارڈ 217

     سوال

     Kirchhoff's voltage law around a closed loop?

     جواب

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

218. کارڈ 218

     سوال

     Decision rule using significance level $\alpha$?

     جواب

     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. کارڈ 219

     سوال

     Can proprietary methods from a former employer be reused freely?

     جواب

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

220. کارڈ 220

     سوال

     Pound-force to newtons?

     جواب

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

221. کارڈ 221

     سوال

     When does a definite integral equal ordinary geometric area?

     جواب

     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. کارڈ 222

     سوال

     Charge-voltage relation for an ideal capacitor?

     جواب

     $$q=CV$$

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

223. کارڈ 223

     سوال

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

     جواب

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

224. کارڈ 224

     سوال

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

     جواب

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

225. کارڈ 225

     سوال

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

     جواب

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

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

226. کارڈ 226

     سوال

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

     جواب

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

227. کارڈ 227

     سوال

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

     جواب

     $$\tau=RC$$

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

228. کارڈ 228

     سوال

     What physical quantity does pound-mass measure?

     جواب

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

229. کارڈ 229

     سوال

     Type I error?

     جواب

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

230. کارڈ 230

     سوال

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

     جواب

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

231. کارڈ 231

     سوال

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

     جواب

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

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

232. کارڈ 232

     سوال

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

     جواب

     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. کارڈ 233

     سوال

     Recognizable form of a separable first-order differential equation?

     جواب

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

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

234. کارڈ 234

     سوال

     Using unverified software output in a safety-relevant calculation?

     جواب

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

235. کارڈ 235

     سوال

     Equivalent resistance of ideal resistors in series?

     جواب

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

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

236. کارڈ 236

     سوال

     Pressure conversion from psi to pascals?

     جواب

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

237. کارڈ 237

     سوال

     Type II error?

     جواب

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

238. کارڈ 238

     سوال

     Useful flashcard to make after a failed handbook search?

     جواب

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

239. کارڈ 239

     سوال

     Standard form of a first-order linear differential equation?

     جواب

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

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

240. کارڈ 240

     سوال

     Equivalent resistance of ideal resistors in parallel?

     جواب

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

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

241. کارڈ 241

     سوال

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

     جواب

     $$BV_k=B-kD$$

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

242. کارڈ 242

     سوال

     Standard atmosphere in pascals?

     جواب

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

243. کارڈ 243

     سوال

     Using AI-generated technical content in engineering work?

     جواب

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

244. کارڈ 244

     سوال

     Trial form for a homogeneous linear ODE with constant coefficients?

     جواب

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

245. کارڈ 245

     سوال

     Wave-speed relationship among frequency and wavelength?

     جواب

     $$v=f\lambda$$

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

246. کارڈ 246

     سوال

     Statistical power?

     جواب

     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. کارڈ 247

     سوال

     Bar in pascals?

     جواب

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

248. کارڈ 248

     سوال

     How should handbook navigation be practiced before the FE exam?

     جواب

     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. کارڈ 249

     سوال

     Is depreciation itself a cash outflow?

     جواب

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

250. کارڈ 250

     سوال

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

     جواب

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

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

251. کارڈ 251

     سوال

     Relationship between period and frequency?

     جواب

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

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

252. کارڈ 252

     سوال

     How should foreseeable environmental harm enter an engineering decision?

     جواب

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

253. کارڈ 253

     سوال

     Foot-pound-force in joules?

     جواب

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

254. کارڈ 254

     سوال

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

     جواب

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

255. کارڈ 255

     سوال

     When should you search instead of relying on recall?

     جواب

     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. کارڈ 256

     سوال

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

     جواب

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

257. کارڈ 257

     سوال

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

     جواب

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

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

258. کارڈ 258

     سوال

     General material balance on a control volume?

     جواب

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

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

259. کارڈ 259

     سوال

     Approximate joules in one International Table Btu?

     جواب

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

260. کارڈ 260

     سوال

     What is the defender in a replacement analysis?

     جواب

     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. کارڈ 261

     سوال

     Speaking publicly about a technical matter outside your expertise?

     جواب

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

262. کارڈ 262

     سوال

     Single-interval trapezoidal approximation to an integral?

     جواب

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

     It is exact for linear $f$.

263. کارڈ 263

     سوال

     Energy efficiency of a device?

     جواب

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

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

264. کارڈ 264

     سوال

     Purpose of sensitivity analysis in engineering economics?

     جواب

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

265. کارڈ 265

     سوال

     Can a strong correlation establish causation?

     جواب

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

266. کارڈ 266

     سوال

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

     جواب

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

267. کارڈ 267

     سوال

     Mechanical horsepower in watts?

     جواب

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

268. کارڈ 268

     سوال

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

     جواب

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

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

269. کارڈ 269

     سوال

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

     جواب

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

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

270. کارڈ 270

     سوال

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

     جواب

     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. کارڈ 271

     سوال

     Kilowatt-hour in joules?

     جواب

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

272. کارڈ 272

     سوال

     Percent error relative to an accepted nonzero value?

     جواب

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

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

273. کارڈ 273

     سوال

     Does legal compliance alone settle an engineering ethics question?

     جواب

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

274. کارڈ 274

     سوال

     Does handbook fluency replace discipline-specific problem practice?

     جواب

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

275. کارڈ 275

     سوال

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

     جواب

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

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

276. کارڈ 276

     سوال

     Dimensional requirement for a physically meaningful additive equation?

     جواب

     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.