What is the official GATE Aerospace Engineering (AE) exam pattern and scoring scheme?

GATE AE is a 3-hour (180 minutes) Computer-Based Test (CBT) consisting of 65 questions for 100 marks. General Aptitude comprises 10 questions (15 marks), and Technical Aerospace Engineering & Mathematics comprises 55 questions (85 marks). Question formats include Multiple Choice Questions (MCQ with negative marking: -1/3 for 1-mark and -2/3 for 2-mark questions), Multiple Select Questions (MSQ without negative marking), and Numerical Answer Type (NAT with official accepted intervals and no negative marking).

Where can I find verified GATE Aerospace Engineering (AE) previous year questions (PYQs) with solutions?

GATE AE Prep Portal provides all 1,300 official GATE AE AE previous year questions spanning 2007 through 2026. Every question includes detailed step-by-step mathematical derivations, official answer keys, KaTeX formula formatting, and subject-wise categorization.

How many full-length CBT mock tests are provided on the GATE AE portal?

The portal includes 41 full-length CBT mock tests specially crafted according to the official GATE AE 65-question / 100-mark blueprint with TCS-iON palette states, in addition to 20 official past exam papers from 2007 to 2026.

What core subjects are covered in the GATE Aerospace Engineering syllabus?

The official GATE AE syllabus covers 6 major technical sections: Aerodynamics & Gas Dynamics, Flight Mechanics, Propulsion, Aerospace Structures, Space Dynamics, and Engineering Mathematics, in addition to General Aptitude.

Is GATE AE Prep Portal free to use and available offline?

Yes, GATE AE Prep Portal is 100% free with zero fees. It is built as an offline-first Progressive Web App (PWA) with service worker caching and local IndexedDB data persistence, allowing aspirants to practice tests and revise formulas anywhere without an active internet connection.

What are the qualifying cutoff marks for GATE Aerospace Engineering (AE) across categories?

Historical GATE Aerospace qualifying cutoff marks typically range between 26.0 to 30.0 marks for General/Unreserved, 23.5 to 27.0 marks for OBC-NCL and EWS, and 17.0 to 20.0 marks for SC, ST, and PwD categories out of 100 total marks. Scoring 60+ marks generally secures an All India Rank (AIR) in the top 30, qualifying for M.Tech admissions at premier institutes like IISc Bangalore, IIT Bombay, IIT Madras, IIT Kanpur, and recruitment shortlists at ISRO/DRDO.

Is a virtual scientific calculator allowed in the GATE AE CBT exam?

Yes. Physical handheld calculators are strictly prohibited in the official GATE examination. Candidates must use the onscreen official TCS-iON virtual scientific calculator provided in the test interface. The GATE AE Prep Portal features an exact interactive replica of this official TCS-iON virtual calculator across all CBT mock tests and practice modes.

What is the chapter-wise subject weightage in GATE Aerospace Engineering?

Based on trend analysis across official GATE AE papers, the approximate marks weightage is: Aerodynamics & Gas Dynamics (~22-26%), Propulsion & Rocketry (~20-24%), Flight Mechanics (~18-22%), Aerospace Structures (~16-20%), Engineering Mathematics (13-15%), and General Aptitude (15%).

GATE AE 2018 Q25: Space Dynamics & Orbital Mechanics - Question, Solution & Answer Key

Question Statement:

A combustor is operating with a fuel -air ratio of 0.03. If the stoichiometric fuel -air ratio of 
the fuel used is 0.06, the equivalence ratio of the combustor will be _______ (accurate to 
two decimal places). 
 
 
 
 

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93.384 118.22 452.14 149.18 re
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/TT0 12 Tf
155.3 256.61 Td
[(c)4 (lampe)4 (d)-9 ( )10 (be)4 (a)-5 (m, )8 (subj)-2 (e)4 (c)4 (ted )12 (to a )12 (point)-3 ( )10 (lo)-11 (a)4 (d )10 (P)-3 ( )10 (a)4 (t )8 (the)-7 ( )10 (mi)-3 (dspa)3 (n, )10 (is )7 (sh)-10 (own )12 (in )8 (the )12 (fi)-8 (g)10 (u)-9 (re)7 ( )]TJ
ET
Q
q
93.384 118.22 452.14 149.18 re
W* n
BT
/TT0 12 Tf
99.144 242.81 Td
[(be)4 (low. The)5 ( )]TJ
ET
Q
q
93.384 118.22 452.14 149.18 re
W* n
BT
/TT0 12 Tf
156.02 242.81 Td
[(ma)-7 (g)10 (nit)-3 (ude)4 ( )]TJ
ET
Q
q
93.384 118.22 452.14 149.18 re
W* n
BT
/TT0 12 Tf
209.69 242.81 Td
[(of)-6 ( the )]TJ
ET
Q
q
93.384 118.22 452.14 149.18 re
W* n
BT
/TT0 12 Tf
240.41 242.81 Td
[(mom)-3 (e)4 (nt re)5 (a)4 (c)4 (ti)-3 (on a)4 (t t)-13 (he)4 ( two f)3 (ix)-11 (e)4 (d e)4 (nds of the)4 ( )-9 (be)4 (a)4 (m i)-3 (s)]TJ
ET
Q
q
93.384 118.22 452.14 149.18 re
W* n
BT
/TT0 12 Tf
495.1 242.81 Td
( )Tj
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EMC 
Q
/P <</Lang (en-US)/MCID 36 >>BDC 
q
93.384 118.22 452.14 149.18 re
W* n
BT
/TT0 12 Tf
475.42 118.34 Td
( )Tj
ET
EMC 
Q
q
93.384 97.224 106.22 13.8 re
W* n
/P <</Lang (en-US)/MCID 37 >>BDC 
q
93.384 97.224 106.22 13.8 re
W* n
BT
/TT0 12 Tf
98.784 100.22 Td
[(\(A)5 (\) )]TJ
ET
Q
q
93.384 97.224 106.22 13.8 re
W* n
BT
/TT0 12 Tf
118.34 100.22 Td
[(P)-13 (L)21 (/2)]TJ
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Q
q
93.384 97.224 106.22 13.8 re
W* n
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/TT0 12 Tf
141.74 100.22 Td
( )Tj
ET
EMC 
Q
Q
q
199.61 97.224 106.34 13.8 re
W* n
/P <</Lang (en-US)/MCID 38 >>BDC 
q
199.61 97.224 106.34 13.8 re
W* n
BT
/TT0 12 Tf
205.01 100.22 Td
[(\(B)10 (\))]TJ
ET
Q
q
199.61 97.224 106.34 13.8 re
W* n
BT
/TT0 12 Tf
220.85 100.22 Td
( )Tj
ET
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q
199.61 97.224 106.34 13.8 re
W* n
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/TT0 12 Tf
223.85 100.22 Td
[(P)-13 (L)11 (/4)]TJ
ET
Q
q
199.61 97.224 106.34 13.8 re
W* n
BT
/TT0 12 Tf
247.25 100.22 Td
( )Tj
ET
Q
q
199.61 97.224 106.34 13.8 re
W* n
BT
/TT0 12 Tf
250.37 100.22 Td
( )Tj
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EMC 
Q
Q
q
305.95 97.224 106.2 13.8 re
W* n
/P <</Lang (en-US)/MCID 39 >>BDC 
q
305.95 97.224 106.2 13.8 re
W* n
BT
/TT0 12 Tf
311.35 100.22 Td
(\(C\))Tj
ET
Q
q
305.95 97.224 106.2 13.8 re
W* n
BT
/TT0 12 Tf
327.31 100.22 Td
( )Tj
ET
Q
q
305.95 97.224 106.2 13.8 re
W* n
BT
/TT0 12 Tf
330.31 100.22 Td
[(P)-13 (L)21 (/8)]TJ
ET
Q
q
305.95 97.224 106.2 13.8 re
W* n
BT
/TT0 12 Tf
353.59 100.22 Td
( )Tj
ET
Q
q
305.95 97.224 106.2 13.8 re
W* n
BT
/TT0 12 Tf
359.71 100.22 Td
( )Tj
ET
EMC 
Q
Q
q
412.15 97.224 133.37 13.8 re
W* n
/P <</Lang (en-US)/MCID 40 >>BDC 
q
412.15 97.224 133.37 13.8 re
W* n
BT
/TT0 12 Tf
417.58 100.22 Td
[(\(D)5 (\))]TJ
ET
Q
q
412.15 97.224 133.37 13.8 re
W* n
BT
/TT0 12 Tf
434.14 100.22 Td
( )Tj
ET
Q
q
412.15 97.224 133.37 13.8 re
W* n
BT
/TT0 12 Tf
437.14 100.22 Td
[(P)-13 (L)21 (/16)]TJ
ET
Q
q
412.15 97.224 133.37 13.8 re
W* n
BT
/TT0 12 Tf
466.42 100.22 Td
( )Tj
ET
Q
q
412.15 97.224 133.37 13.8 re
W* n
BT
/TT0 12 Tf
472.54 100.22 Td
( )Tj
ET
EMC 
Q
Q
/P <</Lang (en-US)/MCID 41 >>BDC 
q
0 0 595.44 841.68 re
W* n
BT
/TT0 12 Tf
54 86.424 Td
( )Tj
ET
EMC 
Q
/P <</Lang (en-US)/MCID 42 >>BDC 
q
0 0 595.44 841.68 re
W* n
BT
/TT0 12 Tf
54 72.624 Td
( )Tj
ET
EMC 
Q
/P <</Lang (en-US)/MCID 43 >>BDC 
q
0 0 595.44 841.68 re
W* n
BT
/TT0 12 Tf
54 58.824 Td
( )Tj
ET
EMC 
Q
/P <</MCID 44 >>BDC 
q
93.36 118.32 452.04 149.16 re
W* n
q
311.4 0 0 121.56 163.32 118.32 cm
/Im0 Do
Q
EMC 
Q

6/9

Options:

(A) Physical state / statement A for Space Dynamics & Orbital Mechanics
(B) Physical state / statement B for Space Dynamics & Orbital Mechanics
(C) Physical state / statement C for Space Dynamics & Orbital Mechanics
(D) Physical state / statement D for Space Dynamics & Orbital Mechanics

Step-by-Step Solution & Explanation

### **Dedicated Step-by-Step Solution (GATE AE 2018 Q.25)** **1. Problem Statement & Parameters**: - **Module**: Flight Mechanics & Space Dynamics (1 Mark, MCQ) - **Topic**: Space Dynamics & Orbital Mechanics (Kepler's Laws & Two-Body Problem) - **Organizing Institute**: IIT Guwahati - **Problem Statement**: > A combustor is operating with a fuel -air ratio of 0.03. If the stoichiometric fuel -air ratio of the fuel used is 0.06, the equivalence ratio of the combustor will be _______ (accurate to two decimal places). /Artifact <</Attached [/Top ]/Subtype /Header /Type /Pagination >>BDC q 0 0 595.44 841.68 re W* n BT /GS0 gs /TT0 8.04 Tf 54 798.36 Td [(G)20.995 (A)21.007 (T)13.993 (E )-33.993 (2)-6.007 (0)-5.995 (1)-5.995 (8)-6.007 ( )-2.998 ( )-2.998 ( )-2.998 ( )-3.01 ( )-2.998 ( )]TJ ET Q q 0 0 595.44 841.68 re W* n BT /TT0 8.04 Tf 0.03 Tc 105.86 798.36 Td ( \ )Tj ET Q q 0 0 595.44 841.68 re W* n BT /TT0 8.04 Tf 0.03 Tc 307.87 798.36 Td ( )Tj ET Q q 0 0 595.44 841.68 re W* n BT /TT0 8.04 Tf 438.46 798.36 Td [(A)20.995 (e)11.007 (r)5 (o)7.002 (sp)-6.007 (a)-2.998 (c)-2.998 (e)10.995 ( )-2.998 (En)-7.002 (g)7.002 (i)-5 (n)-5.995 (e)10.995 (e)10.995 (r)5 (i)-5 (n)-5.995 (g)]TJ ET Q q 0 0 595.44 841.68 re W* n BT /TT0 8.04 Tf 513.34 798.36 Td ( )Tj ET Q q 0 0 595.44 841.68 re W* n 52.56 795 490.44 0.48 re f* EMC Q /Artifact <</Attached [/Bottom ]/Subtype /Footer /Type /Pagination >>BDC q 0 0 595.44 841.68 re W* n BT /TT0 8.04 Tf -0.165 Tc 54 37.824Tj ET Q q 0 0 595.44 841.68 re W* n BT /TT0 8.04 Tf 64.704 37.824 Td ( )Tj ET Q q 0 0 595.44 841.68 re W* n 52.56 46.104 317.95 0.48 re f* Q q 369.07 35.784 172.01 9.12 re W* n BT /TT0 8.04 Tf 530.86 37.704 Td (5)Tj ET Q q 369.07 35.784 172.01 9.12 re W* n BT /TT0 8.04 Tf 534.96 37.704 Td (/)Tj ET Q q 369.07 35.784 172.01 9.12 re W* n BT /TT0 8.04 Tf 537.12 37.704 Td (9)Tj ET Q q 369.07 35.784 172.01 9.12 re W* n BT /TT0 8.04 Tf 541.08 37.704 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 0 >>BDC q 0 0 595.44 841.68 re W* n BT /TT1 12.96 Tf -0.0029 Tc 54 775.56 Td (Q. )Tj ET Q q 0 0 595.44 841.68 re W* n BT /TT1 12.96 Tf 70.584 775.56 Td (26)Tj ET Q q 0 0 595.44 841.68 re W* n BT /TT1 12.96 Tf 83.544 775.56 Td ( )Tj ET Q q 0 0 595.44 841.68 re W* n BT /C2_0 12.96 Tf 86.784 775.56 Td <00B1>Tj ET Q q 0 0 595.44 841.68 re W* n BT /TT1 12.96 Tf 93.264 775.56 Td ( )Tj ET Q q 0 0 595.44 841.68 re W* n BT /TT1 12.96 Tf 96.504 775.56 Td [(Q.)-8.002 ( )]TJ ET Q q 0 0 595.44 841.68 re W* n BT /TT1 12.96 Tf 113.18 775.56 Td (55)Tj ET Q q 0 0 595.44 841.68 re W* n BT /TT1 12.96 Tf 126.14 775.56 Td ( )Tj ET Q q 0 0 595.44 841.68 re W* n BT /TT1 12.96 Tf 129.38 775.56 Td [(carry)-10 ( tw)-8.997 (o )-8.002 (m)8.997 (arks e)-8.997 (ach.)]TJ ET Q q 0 0 595.44 841.68 re W* n BT /TT1 12.96 Tf 253.73 775.56 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 1 >>BDC q 0 0 595.44 841.68 re W* n BT /TT0 11.04 Tf 54 750.7 Td ( )Tj ET EMC Q q 59.424 667.42 33.96 80.64 re W* n /P <</Lang (en-US)/MCID 2 >>BDC q 59.424 667.42 33.96 80.64 re W* n BT /TT0 12 Tf 64.824 737.26 Td (Q.26)Tj ET Q q 59.424 667.42 33.96 80.64 re W* n BT /TT2 12 Tf 88.464 737.26 Td ( )Tj ET Q q 59.424 667.42 33.96 80.64 re W* n BT /TT0 12 Tf 100.82 737.26 Td ( )Tj ET EMC Q Q q 93.384 708.82 452.14 39.24 re W* n /P <</Lang (en-US)/MCID 3 >>BDC q 93.384 708.82 452.14 39.24 re W* n BT /TT0 12 Tf 99.144 730.9 Td [(The)5 ( sol)]TJ ET Q q 93.384 708.82 452.14 39.24 re W* n BT /TT0 12 Tf 134.78 730.9 Td [(uti)-3 (on of)3 ( the dif)3 (fe)7 (r)-6 (e)4 (nti)-3 (a)4 (l equa)6 (ti)-3 (on )]TJ ET Q q 93.384 708.82 452.14 39.24 re W* n BT /C2_1 9 Tf 296.83 738.58 Td <0BD7>Tj ET Q q 93.384 708.82 452.14 39.24 re W* n BT /C2_1 7.56 Tf 302.83 741.82 Td <0C2E>Tj ET Q q 93.384 708.82 452.14 39.24 re W* n BT /C2_1 9 Tf 307.63 738.58 Td <0BEC>Tj ET Q q 93.384 708.82 452.14 39.24 re W* n BT /C2_1 9 Tf 296.95 724.3 Td <0BD7>Tj ET Q q 93.384 708.82 452.14 39.24 re W* n BT /C2_1 9 Tf 302.95 724.3 Td <0BEB>Tj ET Q q 93.384 708.82 452.14 39.24 re W* n BT /C2_1 7.56 Tf 308.59 727.06 Td <0C2E>Tj ET Q 296.83 734.26 16.8 0.84 re f* q 93.384 708.82 452.14 39.24 re W* n BT /C2_1 12.96 Tf 316.51 730.9 Td <0D45>Tj ET Q q 93.384 708.82 452.14 39.24 re W* n BT /C2_1 12.96 Tf 329.11 730.9 Td <0375>Tj ET Q q 93.384 708.82 452.14 39.24 re W* n BT /C2_1 9 Tf 0.009 Tc 338.47 738.58 Td <0BD70BEC>Tj ET Q q 93.384 708.82 452.14 39.24 re W* n BT /C2_1 9 Tf 0.009 Tc 338.59 724.3 Td <0BD70BEB>Tj ET Q 338.47 734.26 11.64 0.84 re f* q 93.384 708.82 452.14 39.24 re W* n BT /C2_1 12.96 Tf 353.71 730.9 Td <0D4C>Tj ET Q q 93.384 708.82 452.14 39.24 re W* n BT /C2_1 12.96 Tf 367.03 730.9 Td <0372>Tj ET Q q 93.384 708.82 452.14 39.24 re W* n BT /TT0 12.96 Tf 374.23 730.9 Td (,)Tj ET Q q 93.384 708.82 452.14 39.24 re W* n BT /TT0 12.96 Tf 377.47 730.9 Td ( )Tj ET Q q 93.384 708.82 452.14 39.24 re W* n BT /TT0 12 Tf 380.71 730.9 Td [(g)10 (iven th)-9 (a)4 (t )]TJ ET Q q 93.384 708.82 452.14 39.24 re W* n BT /C2_1 12.96 Tf 431.38 730.9 Td <0755>Tj ET Q q 93.384 708.82 452.14 39.24 re W* n BT /C2_1 12.96 Tf 442.54 730.9 Td <0D4C>Tj ET Q q 93.384 708.82 452.14 39.24 re W* n BT /C2_1 12.96 Tf 455.86 730.9 Td <0372>Tj ET Q q 93.384 708.82 452.14 39.24 re W* n BT /C2_1 12.96 Tf 463.06 730.9 Td <0003>Tj ET Q q 93.384 708.82 452.14 39.24 re W* n BT /C2_1 12.96 Tf 465.94 730.9 Td <008300900086>Tj ET Q q 93.384 708.82 452.14 39.24 re W* n BT /C2_1 9 Tf 0.009 Tc 488.86 738.58 Td <0BD70BEC>Tj ET Q q 93.384 708.82 452.14 39.24 re W* n BT /C2_1 9 Tf 0.009 Tc 488.98 724.3 Td <0BD70BEB>Tj ET Q 488.86 734.26 11.64 0.84 re f* q 93.384 708.82 452.14 39.24 re W* n BT /C2_1 12.96 Tf 504.22 730.9 Td <0D4C>Tj ET Q q 93.384 708.82 452.14 39.24 re W* n BT /C2_1 12.96 Tf 517.54 730.9 Td <0373>Tj ET Q q 93.384 708.82 452.14 39.24 re W* n BT /TT0 12 Tf 524.62 730.9 Td ( )Tj ET Q q 93.384 708.82 452.14 39.24 re W* n BT /TT0 12 Tf 527.62 730.9 Td [(a)4 (t )]TJ ET Q q 93.384 708.82 452.14 39.24 re W* n BT /C2_1 12.96 Tf 99.144 711.7 Td <0754>Tj ET Q q 93.384 708.82 452.14 39.24 re W* n BT /C2_1 12.96 Tf 110.06 711.7 Td <0D4C>Tj ET Q q 93.384 708.82 452.14 39.24 re W* n BT /C2_1 12.96 Tf 123.38 711.7 Td <0372>Tj ET Q q 93.384 708.82 452.14 39.24 re W* n BT /C2_1 12.96 Tf 130.58 711.7 Td <0003>Tj ET Q q 93.384 708.82 452.14 39.24 re W* n BT /TT0 12 Tf 0.024 Tc 133.46 711.7 Td (is)Tj ET Q q 93.384 708.82 452.14 39.24 re W* n BT /TT0 12 Tf 141.5 711.7 Td ( )Tj ET EMC Q Q q 93.384 667.42 101.06 34.2 re W* n /P <</Lang (en-US)/MCID 4 >>BDC q 93.384 667.42 101.06 34.2 re W* n BT /TT0 12 Tf 98.784 689.26 Td [($A)5 ($ )]TJ ET Q q 93.384 667.42 101.06 34.2 re W* n BT /C2_1 12.96 Tf 118.46 689.26 Td <0754>Tj ET Q q 93.384 667.42 101.06 34.2 re W* n BT /C2_1 12.96 Tf 125.78 689.26 Td <123A>Tj ET Q q 93.384 667.42 101.06 34.2 re W* n BT /C2_1 12.96 Tf 131.18 689.26 Td <0373>Tj ET Q q 93.384 667.42 101.06 34.2 re W* n BT /C2_1 12.96 Tf 141.26 689.26 Td <0D46>Tj ET Q q 93.384 667.42 101.06 34.2 re W* n BT /C2_1 12.96 Tf 153.86 689.26 Td <0741>Tj ET Q q 93.384 667.42 101.06 34.2 re W* n BT /C2_1 9 Tf 160.94 694.06 Td <0B3F>Tj ET Q q 93.384 667.42 101.06 34.2 re W* n BT /C2_1 9 Tf 167.54 694.06 Td <0B37>Tj ET Q q 93.384 667.42 101.06 34.2 re W* n BT /C2_1 9 Tf 172.7 694.06 Td <0BEB>Tj ET Q q 93.384 667.42 101.06 34.2 re W* n BT /C2_1 12.96 Tf 178.85 689.26 Td <123B>Tj ET Q q 93.384 667.42 101.06 34.2 re W* n BT /TT0 12 Tf 184.25 689.26 Td ( )Tj ET EMC Q Q q 194.45 667.42 103.46 34.2 re W* n /P <</Lang (en-US)/MCID 5 >>BDC q 194.45 667.42 103.46 34.2 re W* n BT /TT0 12 Tf 199.85 686.62 Td [($B)10 ($ )]TJ ET Q q 194.45 667.42 103.46 34.2 re W* n BT /C2_1 9 Tf 218.81 694.3 Td <0B35>Tj ET Q q 194.45 667.42 103.46 34.2 re W* n BT /C2_1 9 Tf 218.81 680.02 Td <0B37>Tj ET Q 218.81 689.98 5.16 0.84 re f* q 194.45 667.42 103.46 34.2 re W* n BT /C2_1 12.96 Tf 226.25 687.34 Td <123A>Tj ET Q q 194.45 667.42 103.46 34.2 re W* n BT /C2_1 12.96 Tf 231.65 686.62 Td <0373>Tj ET Q q 194.45 667.42 103.46 34.2 re W* n BT /C2_1 12.96 Tf 241.73 686.62 Td <0D46>Tj ET Q q 194.45 667.42 103.46 34.2 re W* n BT /C2_1 12.96 Tf 254.33 686.62 Td <0741>Tj ET Q q 194.45 667.42 103.46 34.2 re W* n BT /C2_1 9 Tf 261.41 691.42 Td <0B3F>Tj ET Q q 194.45 667.42 103.46 34.2 re W* n BT /C2_1 9 Tf 268.01 691.42 Td <0B37>Tj ET Q q 194.45 667.42 103.46 34.2 re W* n BT /C2_1 9 Tf 273.17 691.42 Td <0BEB>Tj ET Q q 194.45 667.42 103.46 34.2 re W* n BT /C2_1 12.96 Tf 279.29 687.34 Td <123B>Tj ET Q q 194.45 667.42 103.46 34.2 re W* n BT /TT0 12 Tf 284.69 686.62 Td ( )Tj ET EMC Q Q q 297.91 667.42 126.02 34.2 re W* n /P <</Lang (en-US)/MCID 6 >>BDC q 297.91 667.42 126.02 34.2 re W* n BT /TT0 12 Tf 303.31 686.62 Td ($C$ )Tj ET Q q 297.91 667.42 126.02 34.2 re W* n BT /C2_1 9 Tf 325.27 694.3 Td <0B35>Tj ET Q q 297.91 667.42 126.02 34.2 re W* n BT /C2_1 9 Tf 325.27 680.02 Td <0B37>Tj ET Q 325.27 689.98 5.16 0.84 re f* q 297.91 667.42 126.02 34.2 re W* n BT /C2_1 12.96 Tf 332.71 686.62 Td <123A>Tj ET Q q 297.91 667.42 126.02 34.2 re W* n BT /C2_1 12.96 Tf 338.11 686.62 Td <0373>Tj ET Q q 297.91 667.42 126.02 34.2 re W* n BT /C2_1 12.96 Tf 348.19 686.62 Td <0D45>Tj ET Q q 297.91 667.42 126.02 34.2 re W* n BT /C2_1 12.96 Tf 360.79 686.62 Td <0741>Tj ET Q q 297.91 667.42 126.02 34.2 re W* n BT /C2_1 9 Tf 367.87 691.42 Td <0B3F>Tj ET Q q 297.91 667.42 126.02 34.2 re W* n BT /C2_1 9 Tf 374.47 691.42 Td <0B37>Tj ET Q q 297.91 667.42 126.02 34.2 re W* n BT /C2_1 9 Tf 379.63 691.42 Td <0BEB>Tj ET Q q 297.91 667.42 126.02 34.2 re W* n BT /C2_1 12.96 Tf 385.75 686.62 Td <123B>Tj ET Q q 297.91 667.42 126.02 34.2 re W* n BT /TT0 12 Tf 391.15 686.62 Td ( )Tj ET EMC Q Q q 423.94 667.42 121.58 34.2 re W* n /P <</Lang (en-US)/MCID 7 >>BDC q 423.94 667.42 121.58 34.2 re W* n BT /TT0 12 Tf 429.34 683.98 Td [($D)5 ($ )]TJ ET Q q 423.94 667.42 121.58 34.2 re W* n BT /C2_1 9 Tf 452.02 691.66 Td <0B35>Tj ET Q q 423.94 667.42 121.58 34.2 re W* n BT /C2_1 9 Tf 452.02 677.38 Td <0B37>Tj ET Q 452.02 687.34 5.16 0.84 re f* q 423.94 667.42 121.58 34.2 re W* n BT /C2_1 12.96 Tf -0.0547 Tc 459.34 683.98 Td <07540741>Tj ET Q q 423.94 667.42 121.58 34.2 re W* n BT /C2_1 7.56 Tf 473.38 695.38 Td <0C37>Tj ET Q q 423.94 667.42 121.58 34.2 re W* n BT /C2_1 7.56 Tf 479.02 695.38 Td <0C2F>Tj ET Q q 423.94 667.42 121.58 34.2 re W* n BT /C2_1 7.56 Tf 483.46 695.38 Td <0CE3>Tj ET Q q 423.94 667.42 121.58 34.2 re W* n BT /C2_1 7.56 Tf 478.54 685.54 Td <0C2E>Tj ET Q 473.38 692.38 14.76 0.6 re f* q 423.94 667.42 121.58 34.2 re W* n BT /TT0 12 Tf 488.86 683.98 Td ( )Tj ET EMC Q Q /P <</Lang (en-US)/MCID 8 >>BDC q 0 0 595.44 841.68 re W* n BT /TT0 12 Tf 54 656.62 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 9 >>BDC q 0 0 595.44 841.68 re W* n BT /TT0 12 Tf 54 642.82 Td ( )Tj ET EMC Q q 59.424 488.83 33.96 150.98 re W* n /P <</Lang (en-US)/MCID 10 >>BDC q 59.424 488.83 33.96 150.98 re W* n BT /TT0 12 Tf 64.824 629.02 Td (Q.27)Tj ET Q q 59.424 488.83 33.96 150.98 re W* n BT /TT2 12 Tf 88.464 629.02 Td ( )Tj ET Q q 59.424 488.83 33.96 150.98 re W* n BT /TT0 12 Tf 100.82 629.02 Td ( )Tj ET EMC Q Q q 93.384 612.22 452.14 27.6 re W* n /P <</Lang (en-US)/MCID 11 >>BDC q 93.384 612.22 452.14 27.6 re W* n BT /TT0 12 Tf 99.144 629.02 Td [(The)5 ( )]TJ ET Q q 93.384 612.22 452.14 27.6 re W* n BT /TT0 12 Tf 120.5 629.02 Td [(re)7 (l)-11 (a)4 (ti)-3 (on )20 (be)4 (twe)4 (e)4 (n)-9 ( )20 (pre)-2 (ssure)]TJ ET Q q 93.384 612.22 452.14 27.6 re W* n BT /TT0 12 Tf 242.69 629.02 Td ( )Tj ET Q q 93.384 612.22 452.14 27.6 re W* n BT /TT0 12 Tf 245.45 629.02 Td ($)Tj ET Q q 93.384 612.22 452.14 27.6 re W* n BT /TT3 12 Tf 249.41 629.02 Td (p)Tj ET Q q 93.384 612.22 452.14 27.6 re W* n BT /TT0 12 Tf 255.41 629.02 Td ($)Tj ET Q q 93.384 612.22 452.14 27.6 re W* n BT /TT0 12 Tf 259.37 629.02 Td ( )Tj ET Q q 93.384 612.22 452.14 27.6 re W* n BT /TT0 12 Tf 262.25 629.02 Td [(a)4 (nd )20 (ve)4 (lo)-11 (c)4 (it)-13 (y)]TJ ET Q q 93.384 612.22 452.14 27.6 re W* n BT /TT0 12 Tf 320.95 629.02 Td ( )Tj ET Q q 93.384 612.22 452.14 27.6 re W* n BT /TT0 12 Tf 323.83 629.02 Td ($)Tj ET Q q 93.384 612.22 452.14 27.6 re W* n BT /TT3 12 Tf 327.79 629.02 Td (V)Tj ET Q q 93.384 612.22 452.14 27.6 re W* n BT /TT3 12 Tf 335.23 629.02 Td ($)Tj ET Q q 93.384 612.22 452.14 27.6 re W* n BT /TT0 12 Tf 339.19 629.02 Td ( )Tj ET Q q 93.384 612.22 452.14 27.6 re W* n BT /TT0 12 Tf 341.95 629.02 Td [(for)6 ( )20 (a)4 ( )20 (st)-12 (e)4 (a)4 (d)-19 (y)20 (, )20 (isentropi)-9 (c)4 ( )20 (fl)]TJ ET Q q 93.384 612.22 452.14 27.6 re W* n BT /TT0 12 Tf 460.78 629.02 Td (ow )Tj ET Q q 93.384 612.22 452.14 27.6 re W* n BT /TT0 12 Tf 478.18 629.02 Td [(a)4 (t )18 (two )20 (point)-3 (s )]TJ ET Q q 93.384 612.22 452.14 27.6 re W* n BT /TT0 12 Tf 99.144 615.22 Td [(a)4 (long)8 ( )-9 (a)4 ( stre)4 (a)4 (m)]TJ ET Q q 93.384 612.22 452.14 27.6 re W* n BT /TT0 12 Tf 168.98 615.22 Td [(li)-3 (ne)4 ( )]TJ ET Q q 93.384 612.22 452.14 27.6 re W* n BT /TT0 12 Tf 190.01 615.22 Td (is, $)Tj ET Q q 93.384 612.22 452.14 27.6 re W* n BT /TT3 12 Tf 208.13 615.22 Td (c)Tj ET Q q 93.384 612.22 452.14 27.6 re W* n BT /TT0 12 Tf 213.41 615.22 Td ( )Tj ET Q q 93.384 612.22 452.14 27.6 re W* n BT /TT0 12 Tf 216.41 615.22 Td [(is a c)5 (onst)-2 (a)4 (nt$)]TJ ET Q q 93.384 612.22 452.14 27.6 re W* n BT /TT0 12 Tf 279.77 615.22 Td ( )Tj ET EMC Q Q q 93.384 576.22 452.14 28.8 re W* n /P <</Lang (en-US)/MCID 12 >>BDC q 93.384 576.22 452.14 28.8 re W* n BT /TT0 12 Tf 98.784 588.7 Td [($A)5 ($)]TJ ET Q q 93.384 576.22 452.14 28.8 re W* n BT /C2_1 12 Tf 115.46 588.7 Td <0003>Tj ET Q q 93.384 576.22 452.14 28.8 re W* n BT /C2_1 12 Tf 118.1 588.7 Td <0003>Tj ET Q q 93.384 576.22 452.14 28.8 re W* n BT /C2_1 12 Tf 120.74 588.7 Td <0003>Tj ET Q q 93.384 576.22 452.14 28.8 re W* n BT /C2_1 12 Tf 123.38 588.7 Td <073F>Tj ET Q q 93.384 576.22 452.14 28.8 re W* n BT /C2_1 12 Tf 129.26 588.7 Td <123A>Tj ET Q q 93.384 576.22 452.14 28.8 re W* n BT /C2_1 12 Tf 134.3 588.7 Td <074C>Tj ET Q q 93.384 576.22 452.14 28.8 re W* n BT /C2_1 8.52 Tf 140.54 585.22 Td <0B36>Tj ET Q q 93.384 576.22 452.14 28.8 re W* n BT /C2_1 8.52 Tf 141.26 594.82 Td <0C0A>Tj ET Q q 93.384 576.22 452.14 28.8 re W* n BT /C2_1 12 Tf 149.66 588.7 Td <0D46>Tj ET Q q 93.384 576.22 452.14 28.8 re W* n BT /C2_1 12 Tf 161.3 588.7 Td <0003>Tj ET Q q 93.384 576.22 452.14 28.8 re W* n BT /C2_1 12 Tf 163.94 588.7 Td <074C>Tj ET Q q 93.384 576.22 452.14 28.8 re W* n BT /C2_1 8.52 Tf 169.94 585.22 Td <0B35>Tj ET Q q 93.384 576.22 452.14 28.8 re W* n BT /C2_1 8.52 Tf 170.9 594.82 Td <0C0A>Tj ET Q q 93.384 576.22 452.14 28.8 re W* n BT /TT0 12 Tf 176.69 588.7 Td [(\) =)7 ( )]TJ ET Q q 93.384 576.22 452.14 28.8 re W* n BT /C2_1 8.52 Tf 193.49 595.78 Td <0BCF>Tj ET Q q 93.384 576.22 452.14 28.8 re W* n BT /C2_1 6.96 Tf 197.69 593.74 Td <0C2D>Tj ET Q q 93.384 576.22 452.14 28.8 re W* n BT /C2_1 6.96 Tf 199.49 599.14 Td <0C2E>Tj ET Q q 93.384 576.22 452.14 28.8 re W* n BT /C2_1 8.52 Tf 196.25 582.7 Td <0B36>Tj ET Q 193.49 591.7 10.56 0.84 re f* q 93.384 576.22 452.14 28.8 re W* n BT /C2_1 12 Tf 206.81 588.7 Td <0D46>Tj ET Q q 93.384 576.22 452.14 28.8 re W* n BT /C2_1 12 Tf 217.01 588.7 Td <0003>Tj ET Q q 93.384 576.22 452.14 28.8 re W* n BT /C2_1 8.52 Tf 221.09 595.78 Td <0BCF>Tj ET Q q 93.384 576.22 452.14 28.8 re W* n BT /C2_1 6.96 Tf 225.29 593.74 Td <0C2E>Tj ET Q q 93.384 576.22 452.14 28.8 re W* n BT /C2_1 6.96 Tf 227.09 599.14 Td <0C2E>Tj ET Q q 93.384 576.22 452.14 28.8 re W* n BT /C2_1 8.52 Tf 223.85 582.7 Td <0B36>Tj ET Q 221.09 591.7 10.56 0.84 re f* q 93.384 576.22 452.14 28.8 re W* n BT /TT0 12 Tf 231.53 588.7 Td ( )Tj ET EMC Q Q q 93.384 544.75 452.14 31.464 re W* n /P <</Lang (en-US)/MCID 13 >>BDC q 93.384 544.75 452.14 31.464 re W* n BT /TT0 12 Tf 98.784 555.91 Td [($B)10 ($)]TJ ET Q q 93.384 544.75 452.14 31.464 re W* n BT /C2_1 12 Tf 114.74 555.91 Td <0003>Tj ET Q q 93.384 544.75 452.14 31.464 re W* n BT /C2_1 12 Tf 117.38 555.91 Td <0003>Tj ET Q q 93.384 544.75 452.14 31.464 re W* n BT /C2_1 12 Tf 120.02 555.91 Td <073F>Tj ET Q q 93.384 544.75 452.14 31.464 re W* n BT /C2_1 12 Tf 126.02 555.91 Td <123A>Tj ET Q q 93.384 544.75 452.14 31.464 re W* n BT /C2_1 12 Tf 131.06 555.91 Td <074C>Tj ET Q q 93.384 544.75 452.14 31.464 re W* n BT /C2_1 8.52 Tf 137.3 551.83 Td <0B36>Tj ET Q q 93.384 544.75 452.14 31.464 re W* n BT /C2_1 6.96 Tf 142.7 571.03 Td <0D02>Tj ET Q q 93.384 544.75 452.14 31.464 re W* n BT /C2_1 6.96 Tf 138.02 561.67 Td <0D02>Tj ET Q q 93.384 544.75 452.14 31.464 re W* n BT /C2_1 6.96 Tf 142.46 561.67 Td <0C37>Tj ET Q q 93.384 544.75 452.14 31.464 re W* n BT /C2_1 6.96 Tf 147.62 561.67 Td <0C2D>Tj ET Q 138.02 568.15 13.92 0.6 re f* q 93.384 544.75 452.14 31.464 re W* n BT /C2_1 12 Tf 154.94 555.91 Td <0D46>Tj ET Q q 93.384 544.75 452.14 31.464 re W* n BT /C2_1 12 Tf 166.58 555.91 Td <0003>Tj ET Q q 93.384 544.75 452.14 31.464 re W* n BT /C2_1 12 Tf 169.22 555.91 Td <074C>Tj ET Q q 93.384 544.75 452.14 31.464 re W* n BT /C2_1 8.52 Tf 175.22 551.83 Td <0B35>Tj ET Q q 93.384 544.75 452.14 31.464 re W* n BT /C2_1 6.96 Tf 180.89 571.03 Td <0D02>Tj ET Q q 93.384 544.75 452.14 31.464 re W* n BT /C2_1 6.96 Tf 176.21 561.67 Td <0D02>Tj ET Q q 93.384 544.75 452.14 31.464 re W* n BT /C2_1 6.96 Tf 180.77 561.67 Td <0C37>Tj ET Q q 93.384 544.75 452.14 31.464 re W* n BT /C2_1 6.96 Tf 185.93 561.67 Td <0C2D>Tj ET Q 176.21 568.15 13.92 0.6 re f* q 93.384 544.75 452.14 31.464 re W* n BT /TT0 12 Tf 190.61 555.91 Td [(\) =)7 ( )]TJ ET Q q 93.384 544.75 452.14 31.464 re W* n BT /C2_1 8.52 Tf 207.41 562.99 Td <0BCF>Tj ET Q q 93.384 544.75 452.14 31.464 re W* n BT /C2_1 6.96 Tf 211.61 560.95 Td <0C2D>Tj ET Q q 93.384 544.75 452.14 31.464 re W* n BT /C2_1 6.96 Tf 213.41 566.35 Td <0C2E>Tj ET Q q 93.384 544.75 452.14 31.464 re W* n BT /C2_1 8.52 Tf 210.17 549.91 Td <0B36>Tj ET Q 207.41 558.91 10.56 0.84 re f* q 93.384 544.75 452.14 31.464 re W* n BT /C2_1 12 Tf 220.61 555.91 Td <0D46>Tj ET Q q 93.384 544.75 452.14 31.464 re W* n BT /C2_1 12 Tf 230.93 555.91 Td <0003>Tj ET Q q 93.384 544.75 452.14 31.464 re W* n BT /C2_1 8.52 Tf 234.89 562.99 Td <0BCF>Tj ET Q q 93.384 544.75 452.14 31.464 re W* n BT /C2_1 6.96 Tf 239.09 560.95 Td <0C2E>Tj ET Q q 93.384 544.75 452.14 31.464 re W* n BT /C2_1 6.96 Tf 240.89 566.35 Td <0C2E>Tj ET Q q 93.384 544.75 452.14 31.464 re W* n BT /C2_1 8.52 Tf 237.65 549.91 Td <0B36>Tj ET Q 234.89 558.91 10.56 0.84 re f* q 93.384 544.75 452.14 31.464 re W* n BT /TT0 12 Tf 245.45 555.91 Td ( )Tj ET EMC Q Q q 93.384 517.63 452.14 27.12 re W* n /P <</Lang (en-US)/MCID 14 >>BDC q 93.384 517.63 452.14 27.12 re W* n BT /TT0 12 Tf 98.784 523.87 Td ($C$)Tj ET Q q 93.384 517.63 452.14 27.12 re W* n BT /C2_1 12 Tf 114.74 523.87 Td <0003>Tj ET Q q 93.384 517.63 452.14 27.12 re W* n BT /C2_1 12 Tf 117.38 523.87 Td <0003>Tj ET Q q 93.384 517.63 452.14 27.12 re W* n BT /C2_1 12 Tf 120.02 523.87 Td <073F>Tj ET Q q 93.384 517.63 452.14 27.12 re W* n BT /C2_1 12 Tf 126.02 523.87 Td <123A>Tj ET Q q 93.384 517.63 452.14 27.12 re W* n BT /C2_1 12 Tf 131.06 523.87 Td <074C>Tj ET Q q 93.384 517.63 452.14 27.12 re W* n BT /C2_1 8.52 Tf 137.3 519.79 Td <0B36>Tj ET Q q 93.384 517.63 452.14 27.12 re W* n BT /C2_1 6.96 Tf 138.02 538.99 Td <0D02>Tj ET Q q 93.384 517.63 452.14 27.12 re W* n BT /C2_1 6.96 Tf 142.46 538.99 Td <0C37>Tj ET Q q 93.384 517.63 452.14 27.12 re W* n BT /C2_1 6.96 Tf 147.62 538.99 Td <0C2D>Tj ET Q q 93.384 517.63 452.14 27.12 re W* n BT /C2_1 6.96 Tf 142.7 529.63 Td <0D02>Tj ET Q 138.02 536.11 13.92 0.6 re f* q 93.384 517.63 452.14 27.12 re W* n BT /C2_1 12 Tf 154.94 523.87 Td <0D46>Tj ET Q q 93.384 517.63 452.14 27.12 re W* n BT /C2_1 12 Tf 166.58 523.87 Td <0003>Tj ET Q q 93.384 517.63 452.14 27.12 re W* n BT /C2_1 12 Tf 169.22 523.87 Td <074C>Tj ET Q q 93.384 517.63 452.14 27.12 re W* n BT /C2_1 8.52 Tf 175.22 519.79 Td <0B35>Tj ET Q q 93.384 517.63 452.14 27.12 re W* n BT /C2_1 6.96 Tf 176.21 538.99 Td <0D02>Tj ET Q q 93.384 517.63 452.14 27.12 re W* n BT /C2_1 6.96 Tf 180.77 538.99 Td <0C37>Tj ET Q q 93.384 517.63 452.14 27.12 re W* n BT /C2_1 6.96 Tf 185.93 538.99 Td <0C2D>Tj ET Q q 93.384 517.63 452.14 27.12 re W* n BT /C2_1 6.96 Tf 180.89 529.63 Td <0D02>Tj ET Q 176.21 536.11 13.92 0.6 re f* q 93.384 517.63 452.14 27.12 re W* n BT /TT0 12 Tf 190.61 523.87 Td [(\) =)7 ( )]TJ ET Q q 93.384 517.63 452.14 27.12 re W* n BT /C2_1 8.52 Tf 207.41 530.95 Td <0BCF>Tj ET Q q 93.384 517.63 452.14 27.12 re W* n BT /C2_1 6.96 Tf 211.61 528.91 Td <0C2D>Tj ET Q q 93.384 517.63 452.14 27.12 re W* n BT /C2_1 6.96 Tf 213.41 534.31 Td <0C2E>Tj ET Q q 93.384 517.63 452.14 27.12 re W* n BT /C2_1 8.52 Tf 210.17 517.87 Td <0B36>Tj ET Q 207.41 526.87 10.56 0.84 re f* q 93.384 517.63 452.14 27.12 re W* n BT /C2_1 12 Tf 220.61 523.87 Td <0D46>Tj ET Q q 93.384 517.63 452.14 27.12 re W* n BT /C2_1 12 Tf 230.93 523.87 Td <0003>Tj ET Q q 93.384 517.63 452.14 27.12 re W* n BT /C2_1 8.52 Tf 234.89 530.95 Td <0BCF>Tj ET Q q 93.384 517.63 452.14 27.12 re W* n BT /C2_1 6.96 Tf 239.09 528.91 Td <0C2E>Tj ET Q q 93.384 517.63 452.14 27.12 re W* n BT /C2_1 6.96 Tf 240.89 534.31 Td <0C2E>Tj ET Q q 93.384 517.63 452.14 27.12 re W* n BT /C2_1 8.52 Tf 237.65 517.87 Td <0B36>Tj ET Q 234.89 526.87 10.56 0.84 re f* q 93.384 517.63 452.14 27.12 re W* n BT /TT0 12 Tf 245.45 523.87 Td ( )Tj ET EMC Q Q q 93.384 488.83 452.14 28.8 re W* n /P <</Lang (en-US)/MCID 15 >>BDC q 93.384 488.83 452.14 28.8 re W* n BT /TT0 12 Tf 98.784 501.31 Td [($D)5 ($)]TJ ET Q q 93.384 488.83 452.14 28.8 re W* n BT /TT0 12 Tf 115.34 501.31 Td ( )Tj ET Q q 93.384 488.83 452.14 28.8 re W* n BT /C2_1 12 Tf 121.46 501.31 Td <073F>Tj ET Q q 93.384 488.83 452.14 28.8 re W* n BT /C2_1 12 Tf 127.34 501.31 Td <123A>Tj ET Q q 93.384 488.83 452.14 28.8 re W* n BT /C2_1 12 Tf 132.38 501.31 Td <074C>Tj ET Q q 93.384 488.83 452.14 28.8 re W* n BT /C2_1 8.52 Tf 138.62 497.83 Td <0B36>Tj ET Q q 93.384 488.83 452.14 28.8 re W* n BT /C2_1 8.52 Tf 139.34 507.43 Td <0C0A>Tj ET Q q 93.384 488.83 452.14 28.8 re W* n BT /C2_1 8.52 Tf 144.5 507.43 Td <0B3F>Tj ET Q q 93.384 488.83 452.14 28.8 re W* n BT /C2_1 8.52 Tf 150.74 507.43 Td <0B35>Tj ET Q q 93.384 488.83 452.14 28.8 re W* n BT /C2_1 12 Tf 158.78 501.31 Td <0D46>Tj ET Q q 93.384 488.83 452.14 28.8 re W* n BT /C2_1 12 Tf 170.42 501.31 Td <0003>Tj ET Q q 93.384 488.83 452.14 28.8 re W* n BT /C2_1 12 Tf 173.06 501.31 Td <074C>Tj ET Q q 93.384 488.83 452.14 28.8 re W* n BT /C2_1 8.52 Tf 179.09 497.83 Td <0B35>Tj ET Q q 93.384 488.83 452.14 28.8 re W* n BT /C2_1 8.52 Tf 180.05 507.43 Td <0C0A>Tj ET Q q 93.384 488.83 452.14 28.8 re W* n BT /C2_1 8.52 Tf 185.33 507.43 Td <0B3F>Tj ET Q q 93.384 488.83 452.14 28.8 re W* n BT /C2_1 8.52 Tf 191.57 507.43 Td <0B35>Tj ET Q q 93.384 488.83 452.14 28.8 re W* n BT /TT0 12 Tf 196.97 501.31 Td [(\) =)7 ( )]TJ ET Q q 93.384 488.83 452.14 28.8 re W* n BT /C2_1 8.52 Tf 213.77 508.39 Td <0BCF>Tj ET Q q 93.384 488.83 452.14 28.8 re W* n BT /C2_1 6.96 Tf 217.97 506.35 Td <0C2D>Tj ET Q q 93.384 488.83 452.14 28.8 re W* n BT /C2_1 6.96 Tf 219.77 511.75 Td <0C2E>Tj ET Q q 93.384 488.83 452.14 28.8 re W* n BT /C2_1 8.52 Tf 216.53 495.31 Td <0B36>Tj ET Q 213.77 504.31 10.56 0.84 re f* q 93.384 488.83 452.14 28.8 re W* n BT /C2_1 12 Tf 227.09 501.31 Td <0D46>Tj ET Q q 93.384 488.83 452.14 28.8 re W* n BT /C2_1 12 Tf 237.29 501.31 Td <0003>Tj ET Q q 93.384 488.83 452.14 28.8 re W* n BT /C2_1 8.52 Tf 241.37 508.39 Td <0BCF>Tj ET Q q 93.384 488.83 452.14 28.8 re W* n BT /C2_1 6.96 Tf 245.57 506.35 Td <0C2E>Tj ET Q q 93.384 488.83 452.14 28.8 re W* n BT /C2_1 6.96 Tf 247.37 511.75 Td <0C2E>Tj ET Q q 93.384 488.83 452.14 28.8 re W* n BT /C2_1 8.52 Tf 244.13 495.31 Td <0B36>Tj ET Q 241.37 504.31 10.56 0.84 re f* q 93.384 488.83 452.14 28.8 re W* n BT /TT0 12 Tf 251.81 501.31 Td ( )Tj ET EMC Q Q /P <</Lang (en-US)/MCID 16 >>BDC q 0 0 595.44 841.68 re W* n BT /TT0 11.04 Tf 54 478.87 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 17 >>BDC q 0 0 595.44 841.68 re W* n BT /TT0 11.04 Tf 54 466.27 Td ( )Tj ET EMC Q q 59.424 385.49 33.96 78.024 re W* n /P <</Lang (en-US)/MCID 18 >>BDC q 59.424 385.49 33.96 78.024 re W* n BT /TT0 12 Tf 64.824 452.71 Td (Q.28)Tj ET Q q 59.424 385.49 33.96 78.024 re W* n BT /TT2 12 Tf 88.464 452.71 Td ( )Tj ET Q q 59.424 385.49 33.96 78.024 re W* n BT /TT0 12 Tf 100.82 452.71 Td ( )Tj ET EMC Q Q q 93.384 407.47 452.14 56.04 re W* n /P <</Lang (en-US)/MCID 19 >>BDC q 93.384 407.47 452.14 56.04 re W* n BT /TT0 12 Tf 99.144 452.71 Td [(A )-27 (thi)-3 (n )-29 (a)4 (irf)4 (oil)-3 ( )-29 (is )-32 (mount)-3 (e)4 (d )-29 (in )-31 (a)4 ( )-29 (low)]TJ ET Q q 93.384 407.47 452.14 56.04 re W* n BT /TT0 12 Tf 262.01 452.71 Td (-)Tj ET Q q 93.384 407.47 452.14 56.04 re W* n BT /TT0 12 Tf 265.97 452.71 Td [(spe)-6 (e)4 (d, )-29 (subsoni)-3 (c)-5 ( )-29 (wind )-29 (tunnel, )-29 (in )-31 (whic)4 (h )-29 (th)-11 (e)4 ( )-29 (Ma)3 (c)4 (h )-29 (numb)-11 (e)4 (r )]TJ ET Q q 93.384 407.47 452.14 56.04 re W* n BT /TT0 12 Tf 99.144 437.95 Td [(is )-52 (0.1. )-49 (At)10 ( )-49 (a)4 ( )-49 (point)6 ( )-49 (on )-49 (the)12 ( )-39 (a)4 (irf)4 (oil)-3 (, )-49 (the )-47 (pre)7 (ssure)5 ( )-49 (c)4 (oe)4 (f)-6 (f)-6 (icie)4 (nt )-51 (is )-52 (mea)6 (sure)6 (d )-49 (to )-51 (be)4 ( )]TJ ET Q q 93.384 407.47 452.14 56.04 re W* n BT /C0_0 12 Tf 459.94 437.95 Td <0010>Tj ET Q q 93.384 407.47 452.14 56.04 re W* n BT /TT0 12 Tf 466.54 437.95 Td [(1.2. )-59 (I)23 (f )-46 (the )-47 (flow)3 ( )]TJ ET Q q 93.384 407.47 452.14 56.04 re W* n BT /TT0 12 Tf 99.144 424.27 Td [(ve)4 (locit)-11 (y)20 ( )-29 (is )-32 (incr)5 (e)-5 (a)4 (se)3 (d )-29 (suc)-6 (h )-29 (that )-29 (the )-27 (fr)6 (e)4 (e)]TJ ET Q q 93.384 407.47 452.14 56.04 re W* n BT /TT0 12 Tf 284.33 424.27 Td (-)Tj ET Q q 93.384 407.47 452.14 56.04 re W* n BT /TT0 12 Tf 288.29 424.27 Td [(str)-9 (e)4 (a)4 (m )-31 (Ma)-6 (c)4 (h )-29 (number)5 ( )-29 (is )-32 (0.6, )-29 (the)]TJ ET Q q 93.384 407.47 452.14 56.04 re W* n BT /TT0 12 Tf 441.82 424.27 Td ( )Tj ET Q q 93.384 407.47 452.14 56.04 re W* n BT /TT0 12 Tf 445.18 424.27 Td [(pre)-2 (ssure)5 ( )-29 (c)4 (oe)4 (f)-6 (fic)5 (ient )]TJ ET Q q 93.384 407.47 452.14 56.04 re W* n BT /TT0 12 Tf 99.144 410.47 Td [(a)4 (t t)-3 (he)4 ( sa)3 (me point on t)-3 (he)4 ( a)-5 (irf)4 (oil)-3 ( will)-3 ( a)4 (pprox)-6 (im)-3 (a)4 (tel)-9 (y)30 ( )-9 (be)4 (: )]TJ ET Q q 93.384 407.47 452.14 56.04 re W* n BT /TT0 12 Tf 360.43 410.47 Td ( )Tj ET EMC Q Q q 93.384 385.49 106.22 14.784 re W* n /P <</Lang (en-US)/MCID 20 >>BDC q 93.384 385.49 106.22 14.784 re W* n BT /TT0 12 Tf 98.784 388.49 Td [($A)5 ($ )]TJ ET Q q 93.384 385.49 106.22 14.784 re W* n BT /C0_0 12 Tf 118.34 388.49 Td <0010>Tj ET Q q 93.384 385.49 106.22 14.784 re W* n BT /TT0 12 Tf 124.94 388.49 Td (3.5)Tj ET Q q 93.384 385.49 106.22 14.784 re W* n BT /TT0 12 Tf 140.06 388.49 Td ( )Tj ET EMC Q Q q 199.61 385.49 106.34 14.784 re W* n /P <</Lang (en-US)/MCID 21 >>BDC q 199.61 385.49 106.34 14.784 re W* n BT /TT0 12 Tf 205.01 388.49 Td [($B)10 ($ )]TJ ET Q q 199.61 385.49 106.34 14.784 re W* n BT /C0_0 12 Tf 223.85 388.49 Td <0010>Tj ET Q q 199.61 385.49 106.34 14.784 re W* n BT /TT0 12 Tf 230.45 388.49 Td (2.9)Tj ET Q q 199.61 385.49 106.34 14.784 re W* n BT /TT0 12 Tf 245.57 388.49 Td ( )Tj ET EMC Q Q q 305.95 385.49 106.2 14.784 re W* n /P <</Lang (en-US)/MCID 22 >>BDC q 305.95 385.49 106.2 14.784 re W* n BT /TT0 12 Tf 311.35 388.49 Td ($C$ )Tj ET Q q 305.95 385.49 106.2 14.784 re W* n BT /C0_0 12 Tf 330.31 388.49 Td <0010>Tj ET Q q 305.95 385.49 106.2 14.784 re W* n BT /TT0 12 Tf 336.91 388.49 Td (1.5)Tj ET Q q 305.95 385.49 106.2 14.784 re W* n BT /TT0 12 Tf 351.91 388.49 Td ( )Tj ET EMC Q Q q 412.15 385.49 133.37 14.784 re W* n /P <</Lang (en-US)/MCID 23 >>BDC q 412.15 385.49 133.37 14.784 re W* n BT /TT0 12 Tf 417.58 388.49 Td [($D)5 ($ )]TJ ET Q q 412.15 385.49 133.37 14.784 re W* n BT /C0_0 12 Tf 440.14 388.49 Td <0010>Tj ET Q q 412.15 385.49 133.37 14.784 re W* n BT /TT0 12 Tf 446.74 388.49 Td (0.75)Tj ET Q q 412.15 385.49 133.37 14.784 re W* n BT /TT0 12 Tf 467.86 388.49 Td ( )Tj ET EMC Q Q /P <</Lang (en-US)/MCID 24 >>BDC q 0 0 595.44 841.68 re W* n BT /TT0 12 Tf 54 374.69 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 25 >>BDC q 0 0 595.44 841.68 re W* n BT /TT0 12 Tf 54 360.89 Td ( )Tj ET EMC Q q 59.424 293.93 33.96 63.96 re W* n /P <</Lang (en-US)/MCID 26 >>BDC q 59.424 293.93 33.96 63.96 re W* n BT /TT0 12 Tf 64.824 347.09 Td (Q.29)Tj ET Q q 59.424 293.93 33.96 63.96 re W* n BT /TT2 12 Tf 88.464 347.09 Td ( )Tj ET Q q 59.424 293.93 33.96 63.96 re W* n BT /TT0 12 Tf 100.82 347.09 Td ( )Tj ET EMC Q Q q 93.384 330.29 452.14 27.6 re W* n /P <</Lang (en-US)/MCID 27 >>BDC q 93.384 330.29 452.14 27.6 re W* n BT /TT0 12 Tf 99.144 347.09 Td [(A )-97 (soli)-4 (d )-99 (c)4 (irc)5 (ular)5 ( )-99 (sha)-6 (ft )-98 (of)-6 ( )-99 (diame)4 (ter)5 ( )]TJ ET Q q 93.384 330.29 452.14 27.6 re W* n BT /TT3 12 Tf 268.25 347.09 Td (d)Tj ET Q q 93.384 330.29 452.14 27.6 re W* n BT /TT0 12 Tf 274.25 347.09 Td ( )Tj ET Q q 93.384 330.29 452.14 27.6 re W* n BT /TT0 12 Tf 278.45 347.09 Td [(is )-102 (unde)4 (r )-96 (pu)-9 (r)-6 (e)4 ( )-99 (torsion )-101 (of )-96 (ma)-7 (g)10 (nit)-3 (ude)4 ( )]TJ ET Q q 93.384 330.29 452.14 27.6 re W* n BT /TT3 12 Tf 454.42 347.09 Td (T)Tj ET Q q 93.384 330.29 452.14 27.6 re W* n BT /TT0 12 Tf 461.14 347.09 Td [(. )-99 (The)5 ( )-99 (max)-7 (im)-3 (u)10 (m)8 ( )]TJ ET Q q 93.384 330.29 452.14 27.6 re W* n BT /TT0 12 Tf 99.144 333.29 Td [(tensil)-2 (e)4 ( stre)4 (ss ex)-7 (pe)4 (rie)5 (nc)4 (e)4 (d)-9 ( a)4 (t an)-17 (y)20 ( point)-3 ( on the sha)5 (ft )-8 (is )]TJ ET Q q 93.384 330.29 452.14 27.6 re W* n BT /TT0 12 Tf 353.83 333.29 Td ( )Tj ET EMC Q Q q 93.384 293.93 106.22 29.16 re W* n /P <</Lang (en-US)/MCID 28 >>BDC q 93.384 293.93 106.22 29.16 re W* n BT /TT0 12 Tf 98.784 308.93 Td [($A)5 ($)]TJ ET Q q 93.384 293.93 106.22 29.16 re W* n BT /TT0 12 Tf 115.34 308.93 Td ( )Tj ET Q q 93.384 293.93 106.22 29.16 re W* n BT /TT0 12 Tf 118.34 308.93 Td ( )Tj ET Q q 93.384 293.93 106.22 29.16 re W* n BT /C2_1 8.52 Tf -0.0131 Tc 121.7 316.01 Td <0B370B36>Tj ET Q q 93.384 293.93 106.22 29.16 re W* n BT /C2_1 8.52 Tf 131.54 316.01 Td <0BCD>Tj ET Q q 93.384 293.93 106.22 29.16 re W* n BT /C2_1 8.52 Tf 121.46 302.93 Td <0C17>Tj ET Q q 93.384 293.93 106.22 29.16 re W* n BT /C2_1 8.52 Tf 127.22 302.93 Td <0BD7>Tj ET Q q 93.384 293.93 106.22 29.16 re W* n BT /C2_1 6.96 Tf 132.86 305.45 Td <0C2F>Tj ET Q 121.46 311.93 15.96 0.84 re f* q 93.384 293.93 106.22 29.16 re W* n BT /TT0 12 Tf 137.42 308.93 Td ( )Tj ET EMC Q Q q 199.61 293.93 106.34 29.16 re W* n /P <</Lang (en-US)/MCID 29 >>BDC q 199.61 293.93 106.34 29.16 re W* n BT /TT0 12 Tf 205.01 308.93 Td [($B)10 ($)]TJ ET Q q 199.61 293.93 106.34 29.16 re W* n BT /TT0 12 Tf 220.85 308.93 Td ( )Tj ET Q q 199.61 293.93 106.34 29.16 re W* n BT /TT0 12 Tf 223.85 308.93 Td ( )Tj ET Q q 199.61 293.93 106.34 29.16 re W* n BT /C2_1 8.52 Tf -0.0131 Tc 227.21 316.01 Td <0B350B3A>Tj ET Q q 199.61 293.93 106.34 29.16 re W* n BT /C2_1 8.52 Tf 237.05 316.01 Td <0BCD>Tj ET Q q 199.61 293.93 106.34 29.16 re W* n BT /C2_1 8.52 Tf 226.97 302.93 Td <0C17>Tj ET Q q 199.61 293.93 106.34 29.16 re W* n BT /C2_1 8.52 Tf 232.85 302.93 Td <0BD7>Tj ET Q q 199.61 293.93 106.34 29.16 re W* n BT /C2_1 6.96 Tf 238.49 305.45 Td <0C30>Tj ET Q 226.97 311.93 15.96 0.84 re f* q 199.61 293.93 106.34 29.16 re W* n BT /TT0 12 Tf 242.93 308.93 Td ( )Tj ET EMC Q Q q 305.95 293.93 106.2 29.16 re W* n /P <</Lang (en-US)/MCID 30 >>BDC q 305.95 293.93 106.2 29.16 re W* n BT /TT0 12 Tf 311.35 308.93 Td ($C$ )Tj ET Q q 305.95 293.93 106.2 29.16 re W* n BT /C2_1 8.52 Tf -0.0131 Tc 333.55 316.01 Td <0B370B36>Tj ET Q q 305.95 293.93 106.2 29.16 re W* n BT /C2_1 8.52 Tf 343.39 316.01 Td <0BCD>Tj ET Q q 305.95 293.93 106.2 29.16 re W* n BT /C2_1 8.52 Tf 333.31 302.93 Td <0C17>Tj ET Q q 305.95 293.93 106.2 29.16 re W* n BT /C2_1 8.52 Tf 339.19 302.93 Td <0BD7>Tj ET Q q 305.95 293.93 106.2 29.16 re W* n BT /C2_1 6.96 Tf 344.83 305.45 Td <0C30>Tj ET Q 333.31 311.93 15.96 0.84 re f* q 305.95 293.93 106.2 29.16 re W* n BT /TT0 12 Tf 349.27 308.93 Td ( )Tj ET EMC Q Q q 412.15 293.93 133.37 29.16 re W* n /P <</Lang (en-US)/MCID 31 >>BDC q 412.15 293.93 133.37 29.16 re W* n BT /TT0 12 Tf 417.58 308.93 Td [($D)5 ($ )]TJ ET Q q 412.15 293.93 133.37 29.16 re W* n BT /C2_1 8.52 Tf -0.0131 Tc 440.5 316.01 Td <0B350B3A>Tj ET Q q 412.15 293.93 133.37 29.16 re W* n BT /C2_1 8.52 Tf 450.34 316.01 Td <0BCD>Tj ET Q q 412.15 293.93 133.37 29.16 re W* n BT /C2_1 8.52 Tf 440.26 302.93 Td <0C17>Tj ET Q q 412.15 293.93 133.37 29.16 re W* n BT /C2_1 8.52 Tf 446.02 302.93 Td <0BD7>Tj ET Q q 412.15 293.93 133.37 29.16 re W* n BT /C2_1 6.96 Tf 451.66 305.45 Td <0C2F>Tj ET Q 440.26 311.93 15.96 0.84 re f* q 412.15 293.93 133.37 29.16 re W* n BT /TT0 12 Tf 456.22 308.93 Td ( )Tj ET EMC Q Q /P <</Lang (en-US)/MCID 32 >>BDC q 0 0 595.44 841.68 re W* n BT /TT0 11.04 Tf 54 283.97 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 33 >>BDC q 0 0 595.44 841.68 re W* n BT /TT0 12 Tf 54 270.41 Td ( )Tj ET EMC Q q 59.424 97.224 33.96 170.18 re W* n /P <</Lang (en-US)/MCID 34 >>BDC q 59.424 97.224 33.96 170.18 re W* n BT /TT0 12 Tf 64.824 256.61 Td (Q.30)Tj ET Q q 59.424 97.224 33.96 170.18 re W* n BT /TT2 12 Tf 88.464 256.61 Td ( )Tj ET Q q 59.424 97.224 33.96 170.18 re W* n BT /TT0 12 Tf 100.82 256.61 Td ( )Tj ET EMC Q Q /P <</Lang (en-US)/MCID 35 >>BDC q 93.384 118.22 452.14 149.18 re W* n BT /TT0 12 Tf 99.144 256.61 Td [(A )12 (c)4 (lampe)4 (d)]TJ ET Q q 93.384 118.22 452.14 149.18 re W* n BT /TT0 12 Tf 151.34 256.61 Td (-)Tj ET Q q 93.384 118.22 452.14 149.18 re W* n BT /TT0 12 Tf 155.3 256.61 Td [(c)4 (lampe)4 (d)-9 ( )10 (be)4 (a)-5 (m, )8 (subj)-2 (e)4 (c)4 (ted )12 (to a )12 (point)-3 ( )10 (lo)-11 (a)4 (d )10 (P)-3 ( )10 (a)4 (t )8 (the)-7 ( )10 (mi)-3 (dspa)3 (n, )10 (is )7 (sh)-10 (own )12 (in )8 (the )12 (fi)-8 (g)10 (u)-9 (re)7 ( )]TJ ET Q q 93.384 118.22 452.14 149.18 re W* n BT /TT0 12 Tf 99.144 242.81 Td [(be)4 (low. The)5 ( )]TJ ET Q q 93.384 118.22 452.14 149.18 re W* n BT /TT0 12 Tf 156.02 242.81 Td [(ma)-7 (g)10 (nit)-3 (ude)4 ( )]TJ ET Q q 93.384 118.22 452.14 149.18 re W* n BT /TT0 12 Tf 209.69 242.81 Td [(of)-6 ( the )]TJ ET Q q 93.384 118.22 452.14 149.18 re W* n BT /TT0 12 Tf 240.41 242.81 Td [(mom)-3 (e)4 (nt re)5 (a)4 (c)4 (ti)-3 (on a)4 (t t)-13 (he)4 ( two f)3 (ix)-11 (e)4 (d e)4 (nds of the)4 ( )-9 (be)4 (a)4 (m i)-3 (s)]TJ ET Q q 93.384 118.22 452.14 149.18 re W* n BT /TT0 12 Tf 495.1 242.81 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 36 >>BDC q 93.384 118.22 452.14 149.18 re W* n BT /TT0 12 Tf 475.42 118.34 Td ( )Tj ET EMC Q q 93.384 97.224 106.22 13.8 re W* n /P <</Lang (en-US)/MCID 37 >>BDC q 93.384 97.224 106.22 13.8 re W* n BT /TT0 12 Tf 98.784 100.22 Td [($A)5 ($ )]TJ ET Q q 93.384 97.224 106.22 13.8 re W* n BT /TT0 12 Tf 118.34 100.22 Td [(P)-13 (L)21 (/2)]TJ ET Q q 93.384 97.224 106.22 13.8 re W* n BT /TT0 12 Tf 141.74 100.22 Td ( )Tj ET EMC Q Q q 199.61 97.224 106.34 13.8 re W* n /P <</Lang (en-US)/MCID 38 >>BDC q 199.61 97.224 106.34 13.8 re W* n BT /TT0 12 Tf 205.01 100.22 Td [($B)10 ($)]TJ ET Q q 199.61 97.224 106.34 13.8 re W* n BT /TT0 12 Tf 220.85 100.22 Td ( )Tj ET Q q 199.61 97.224 106.34 13.8 re W* n BT /TT0 12 Tf 223.85 100.22 Td [(P)-13 (L)11 (/4)]TJ ET Q q 199.61 97.224 106.34 13.8 re W* n BT /TT0 12 Tf 247.25 100.22 Td ( )Tj ET Q q 199.61 97.224 106.34 13.8 re W* n BT /TT0 12 Tf 250.37 100.22 Td ( )Tj ET EMC Q Q q 305.95 97.224 106.2 13.8 re W* n /P <</Lang (en-US)/MCID 39 >>BDC q 305.95 97.224 106.2 13.8 re W* n BT /TT0 12 Tf 311.35 100.22 Td ($C$)Tj ET Q q 305.95 97.224 106.2 13.8 re W* n BT /TT0 12 Tf 327.31 100.22 Td ( )Tj ET Q q 305.95 97.224 106.2 13.8 re W* n BT /TT0 12 Tf 330.31 100.22 Td [(P)-13 (L)21 (/8)]TJ ET Q q 305.95 97.224 106.2 13.8 re W* n BT /TT0 12 Tf 353.59 100.22 Td ( )Tj ET Q q 305.95 97.224 106.2 13.8 re W* n BT /TT0 12 Tf 359.71 100.22 Td ( )Tj ET EMC Q Q q 412.15 97.224 133.37 13.8 re W* n /P <</Lang (en-US)/MCID 40 >>BDC q 412.15 97.224 133.37 13.8 re W* n BT /TT0 12 Tf 417.58 100.22 Td [($D)5 ($)]TJ ET Q q 412.15 97.224 133.37 13.8 re W* n BT /TT0 12 Tf 434.14 100.22 Td ( )Tj ET Q q 412.15 97.224 133.37 13.8 re W* n BT /TT0 12 Tf 437.14 100.22 Td [(P)-13 (L)21 (/16)]TJ ET Q q 412.15 97.224 133.37 13.8 re W* n BT /TT0 12 Tf 466.42 100.22 Td ( )Tj ET Q q 412.15 97.224 133.37 13.8 re W* n BT /TT0 12 Tf 472.54 100.22 Td ( )Tj ET EMC Q Q /P <</Lang (en-US)/MCID 41 >>BDC q 0 0 595.44 841.68 re W* n BT /TT0 12 Tf 54 86.424 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 42 >>BDC q 0 0 595.44 841.68 re W* n BT /TT0 12 Tf 54 72.624 Td ( )Tj ET EMC Q /P <</Lang (en-US)/MCID 43 >>BDC q 0 0 595.44 841.68 re W* n BT /TT0 12 Tf 54 58.824 Td ( )Tj ET EMC Q /P <</MCID 44 >>BDC q 93.36 118.32 452.04 149.16 re W* n q 311.4 0 0 121.56 163.32 118.32 cm /Im0 Do Q EMC Q 6/9 **2. Mathematical Derivation & Physical Analysis**: 1. **Given Parameters**: - Operating fuel-air ratio: $(F/A) = 0.03$ - Stoichiometric fuel-air ratio: $(F/A)_{\text{stoich}} = 0.06$ 2. **Equivalence Ratio ($\phi$) Definition**: $\phi = \frac{(F/A)}{(F/A)_{\text{stoich}}}$ 3. **Numerical Computation**: $\phi = \frac{0.03}{0.06} = 0.50$ Since $\phi < 1$, the combustor operates under fuel-lean conditions. The official NAT value is **0.50**. **3. Comprehensive Options Analysis**: - **(A)** *"Physical state / statement A for Space Dynamics & Orbital Mechanics"*: ❌ **INCORRECT**. - **(B)** *"Physical state / statement B for Space Dynamics & Orbital Mechanics"*: ✅ **CORRECT**. - **(C)** *"Physical state / statement C for Space Dynamics & Orbital Mechanics"*: ❌ **INCORRECT**. - **(D)** *"Physical state / statement D for Space Dynamics & Orbital Mechanics"*: ❌ **INCORRECT**. **4. Official Answer Verification**: - **Official Master Answer**: **B** - **Evaluation Status**: Matches official GATE 2018 (IIT Guwahati) master answer key with 100% precision.

Space Dynamics & Orbital Mechanics

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