1. Where has the सूत्रम् 2-3-19 सहयुक्तेऽप्रधाने been used for the first time in the गीता?
Answer: The सूत्रम् 2-3-19 सहयुक्तेऽप्रधाने has been used for the first time in the गीता in the form कैः (सर्वनाम-प्रातिपदिकम् ‘किम्’, पुंलिङ्गे तृतीया-बहुवचनम्) in the following verse –
यावदेतान्निरीक्षेऽहं योद्धुकामानवस्थितान्‌ |
कैर्मया सह योद्धव्यमस्मिन्‌ रणसमुद्यमे || 1-22||

2. In the सिद्धान्तकौमुदी the वृत्ति: of the सूत्रम् 2-3-19 सहयुक्तेऽप्रधाने reads सहार्थेन युक्ते अप्रधाने तृतीया स्‍यात् । Commenting on the word सहार्थेन used in the वृत्ति: the तत्त्वबोधिनी commentary says – ‘सहेनाप्रधाने’ इत्येव वाच्ये युक्तग्रहणादर्थग्रहणमित्याह– सहार्थेनेति। सहार्थकशब्देन सह–साकं–सार्धमित्यादिनेत्यर्थः। Please explain.
Answer: If पाणिनि: did not want to include the synonyms of सह he could have composed a more compact सूत्रम् as सहेनाप्रधाने। The fact that he has included युक्ते in the सूत्रम् 2-3-19 सहयुक्तेऽप्रधाने tells us that it applies not only when सह itself is used but also when a synonym of सह is used. The synonyms of सह are साकम्, सार्धम् etc.

3. Which सूत्रम् prescribes the दीर्घादेश: (elongation) in the form शान्त:?
Answer: : The सूत्रम् 6-4-15 अनुनासिकस्य क्विझलोः क्ङिति prescribes the दीर्घादेश: (elongation) in the form शान्तः (प्रातिपदिकम् ‘शान्त’, पुंलिङ्गे प्रथमा-एकवचनम्) – derived from the verbal root √शम् (शमुँ उपशमे ४. ९८).

The ending letter ‘उ’ of ‘शमुँ’ is a इत् by 1-3-2 उपदेशेऽजनुनासिक इत् and takes elision by 1-3-9 तस्य लोपः। Hence ‘शमुँ’ is a उदित्।

शम् + क्त । By 3-2-102 निष्ठा। Note: The affix ‘क्त’ has been used कर्तरि (to denote the agent) here as per 3-4-72 गत्यर्थाकर्मकश्लिषशीङ्स्थासवसजनरुहजीर्यतिभ्यश्च – When following one of the verbal roots listed below, the affix “क्त” may be used in the active voice (कर्तरि) as well as in the passive voice (भावकर्मणो:) –
(i) any verbal root used in the sense of motion (गत्यर्थ:)
(ii) any verbal root used intransitively (अकर्मक:)
(iii) the verbal roots √श्लिष् (श्लिषँ आलिङ्गने ४. ८३), √शी (शीङ् स्वप्ने २. २६), √स्था (ष्ठा गतिनिवृत्तौ १. १०७७), √आस् (आसँ उपवेशने २. ११), √वस् (वसँ निवासे १. ११६०), √जन् (जनीँ प्रादुर्भावे ४. ४४), √रुह् (रुहँ बीजजन्मनि प्रादुर्भावे च १. ९९५) and √जॄ (जॄष् वयोहानौ ४. २५)।
Note: The verbal roots listed in (iii) are intransitive. But when used with a उपसर्ग: they may become transitive. In that case they would not be covered by (ii). This is the reason for listing them separately in (iii).
= शम् + त । अनुबन्ध-लोप: by 1-3-8 लशक्वतद्धिते and 1-3-9 तस्य लोपः। The affix ‘त’ is prevented from taking the augment ‘इट्’ (which would have been done by 7-2-35 आर्धधातुकस्येड् वलादेः) by 7-2-15 यस्य विभाषा – If a verbal root optionally allows an augment इट् in some case, then following that verbal root a निष्ठा affix (ref. 1-1-26) is prohibited from taking the augment इट्। Note: 7-2-15 is applicable here because as per 7-2-56 उदितो वा – When following a verbal root which is उदित् (has the letter ‘उ’ as a इत्), the affix ‘क्त्वा’ optionally takes the augment इट्।
= शाम् + त । By 6-4-15 अनुनासिकस्य क्विझलोः क्ङिति – The penultimate vowel of a अङ्गम् which ends in a nasal consonant is elongated when followed by either i) the affix ‘क्वि’ or ii) an affix which begins with a झल् letter and is either कित् or ङित् – has a the letter ‘क्’ or the letter ‘ङ्’ as a इत्।
= शां + त । By 8-3-23 मोऽनुस्वारः।
= शान्त । By 8-4-58 अनुस्वारस्य ययि परसवर्णः।

‘शान्त’ gets प्रातिपदिक-सञ्ज्ञा by 1-2-46 कृत्तद्धितसमासाश्च।

4. Can you spot the affix ‘क’ in the commentary?
Answer: The affix ‘क’ occurs in the form यौवनस्थः (प्रातिपदिकम् ‘यौवनस्थ’, पुंलिङ्गे प्रथमा-एकवचनम्)।

यौवने तिष्ठतीति यौवनस्थः।
The derivation of the प्रातिपदिकम् ‘यौवनस्थ’ is similar to the derivation of the प्रातिपदिकम् ‘गजस्थ’ as shown in the following post – http://avg-sanskrit.org/2012/10/12/गजस्थस्य-mgs/

5. Which कृत् affix is used to form the प्रातिपदिकम् ‘अनुज’ (used as part of the compound सानुज: in the verses)?
Answer: The कृत् affix ‘ड’ is used to form the प्रातिपदिकम् ‘अनुज’।

अनुजातः = अनुजः।

अनु + जन् + ड । By 3-2-101 अन्येष्वपि दृश्यते – Even when in composition with a पदम् which is other than the ones mentioned in the preceding rules (from 3-2-97 to 3-2-100), the verbal root √जन् (जनीँ प्रादुर्भावे ४. ४४) may be seen taking the affix ‘ड’ provided the verbal root is used to denote an action in the past tense.
= अनु + जन् + अ । अनुबन्ध-लोप: by 1-3-7 चुटू and 1-3-9 तस्य लोपः।
= अनु + ज् + अ । By 6-4-143 टेः। Note: The अङ्गम् does not have the भ-सञ्ज्ञा here. But still टि-लोप: is done because otherwise no purpose would be served by having the letter ‘ड्’ as a इत् in ‘ड’। डित्वसामर्थ्यादभस्यापि टेर्लोपः।
= अनु + ज ।

Now we form a compound between ‘अनु’ (which is the उपपदम्) and ‘ज’ by using the सूत्रम् 2-2-19 उपपदमतिङ्।

In the compound, ‘अनु’ is placed in the prior position as per 2-2-30 उपसर्जनं पूर्वम्‌। Note: In the सूत्रम् 3-2-101, the term अन्येषु ends in the seventh (locative) case. Hence ‘अनु’ gets the उपपद-सञ्ज्ञा here by 3-1-92 तत्रोपपदं सप्तमीस्थम्‌। Note: Here ‘अनु’ is the उपपदम् and hence it gets the उपसर्जन-सञ्ज्ञा by 1-2-43 प्रथमानिर्दिष्टं समास उपसर्जनम् – A term denoted by a word (in this case ‘उपपदम्’) ending in a nominative case in a सूत्रम् (in this case 2-2-19 उपपदमतिङ्) which prescribes a compound gets the designation ‘उपसर्जन’।

अनु + ज = अनुज। ‘अनुज’ gets प्रातिपदिक-सञ्ज्ञा by 1-2-46 कृत्तद्धितसमासाश्च।

6. How would you say this in Sanskrit?
“Śrī Kṛṣṇa went to Mathurā with Balarāma.”
Answer: श्रीकृष्णः बलरामेण सह मथुराम् ययौ = श्रीकृष्णो बलरामेण सह मथुरां ययौ।

Easy questions:

1. Where has the सूत्रम् 6-4-8 सर्वनामस्थाने चासम्बुद्धौ been used in the verses?
Answer: The सूत्रम् 6-4-8 सर्वनामस्थाने चासम्बुद्धौ is used in the form युवा (पुंलिङ्ग-प्रातिपदिकम् ‘युवन्’, प्रथमा-एकवचनम्)।

युवन् + सुँ । By 4-1-2 स्वौजसमौट्छस्टा…। The affix ‘सुँ’ gets the सर्वनामस्थान-सञ्ज्ञा here by 1-1-43 सुडनपुंसकस्य। This allows the सूत्रम् 6-4-8 to apply below.
युवन् + स् । अनुबन्ध-लोपः by 1-3-2 उपदेशेऽजनुनासिक इत्, 1-3-9 तस्य लोपः।
= युवान् + स् । By 6-4-8 सर्वनामस्थाने चासम्बुद्धौ – The penultimate letter of the अङ्गम् (base) ending in the letter ‘न्’ gets elongated if it is followed by a non-vocative affix having the designation सर्वनामस्थानम्।
= युवान् । By 6-1-68 हल्ङ्याब्भ्यो दीर्घात् सुतिस्यपृक्तं हल्। Now ‘युवान्’ gets पद-सञ्ज्ञा by 1-4-14 सुप्तिङन्तं पदम् with the help of 1-1-62 प्रत्ययलोपे प्रत्ययलक्षणम्। This allows 8-2-7 to apply in the next step.
= युवा । By 8-2-7 नलोपः प्रातिपदिकान्तस्य।

2. Which सूत्रम् is applied to perform the following operation (in the commentary) – कुर्वन् + जीवन् = कुर्वञ्जीवन्?
Answer: This सन्धिः operation (replacing the letter ‘न्’ by the letter ‘ञ्’) is performed by the सूत्रम् 8-4-40 स्तोः श्चुना श्चु: – When the letter ‘स्’ or a letter of the त-वर्ग: (‘त्’, ‘थ्’, ‘द्’, ‘ध्’, ‘न्’) comes in contact with either the letter ‘श्’ or a letter of the च-वर्ग: (‘च्’, ‘छ्’, ‘ज्’, ‘झ्’, ‘ञ्’), then it is replaced respectively by ‘श्’, च-वर्ग: (‘च्’, ‘छ्’, ‘ज्’, ‘झ्’, ‘ञ्’)।