1. Where has the सूत्रम् 5-4-113 बहुव्रीहौ सक्थ्यक्ष्णोः स्वाङ्गात्‌ षच् (used in step 6) been used in the first five verses of Chapter Eleven of the गीता?
Answer: The सूत्रम् 5-4-113 बहुव्रीहौ सक्थ्यक्ष्णोः स्वाङ्गात्‌ षच् has been used in the form कमलपत्राक्ष (प्रातिपदिकम् ’कमलपत्राक्ष’, पुंलिङ्गे सम्बुद्धिः) in the following verse of the गीता –

भवाप्ययौ हि भूतानां श्रुतौ विस्तरशो मया ।
त्वत्तः कमलपत्राक्ष माहात्म्यमपि चाव्ययम्‌ ॥ 11-2 ॥

The लौकिक-विग्रहः is –
कमलपत्रे इवाक्षिणी यस्य सः = कमलपत्राक्षः – He (Lord Śrī Kṛṣṇa) who has eyes like lotus petals
Note: ’कमलपत्र’ itself is a षष्ठी-तत्पुरुषः compound. कमलस्य पत्रम्‌ = कमलपत्रम्‌ – petal of a lotus.

The derivation of the compound प्रातिपदिकम् ’कमलपत्राक्ष’ is similar to the derivation of the compound प्रातिपदिकम् ‘विशालाक्ष’ as shown in the post.

The gender of a बहुव्रीहि: compound matches that of which it qualifies. In the present example ’कमलपत्राक्ष’ is qualifying श्रीकृष्णः। Hence we assign the masculine gender to the compound प्रातिपदिकम् ’कमलपत्राक्ष’। It declines like राम-शब्द:।

2. Commenting on the सूत्रम् 5-4-113 बहुव्रीहौ सक्थ्यक्ष्णोः स्वाङ्गात्‌ षच् the सिद्धान्तकौमुदी says – स्‍वाङ्गात्‍किम्? दीर्घसक्थि शकटम्। स्‍थूलाक्षा वेणुयष्‍टिः। ‘अक्ष्णोऽदर्शनात्‌’ इत्यच्। Please explain.
Answer: स्‍वाङ्गात्‍किम्? To understand the importance of the condition स्वाङ्गात्‌, consider the following example दीर्घं सक्थि यस्य तत् = दीर्घसक्थि (शकटम्) – that (a carriage) which has a long shaft. As per the सूत्रम् 5-4-113 बहुव्रीहौ सक्थ्यक्ष्णोः स्वाङ्गात्‌ षच् – Following a बहुव्रीहि: compound ending in either i) ‘सक्थि’ (when denoting a limb of one’s body) or ii) ‘अक्षि’ (when denoting a limb of one’s body) the तद्धित: affix षच् is prescribed and this affix becomes the ending member of the compound. In the present example, ‘सक्थि’ does not denote a limb of one’s body and hence the समासान्त-प्रत्यय: षच् (prescribed by the सूत्रम् 5-4-113) does not apply. If the सूत्रम् 5-4-113 did not mention the condition स्वाङ्गात्‌, here the compound ‘दीर्घसक्थि’ would have taken the affix ‘षच्’ which is undesirable.

Similarly in the example – स्थूलान्यक्षीणि (पर्वग्रन्थय:) यस्या: सा = स्‍थूलाक्षा (वेणुयष्‍टिः) – that (a bamboo-stick) which has large joints. Here the term ’अक्षि’ does not denote a limb of one’s body and hence the समासान्त-प्रत्यय: षच् (prescribed by the सूत्रम् 5-4-113) does not apply. (Instead the compound ‘स्‍थूलाक्षि’ takes the तद्धित: affix अच् prescribed by the सूत्रम् 5-4-76 अक्ष्णोऽदर्शनात्‌।) If the सूत्रम् 5-4-113 did not mention the condition स्वाङ्गात्‌, here the compound would have taken the affix ‘षच्’ which is undesirable.

3. Commenting on (the affix ’षच्’ prescribed by) the same सूत्रम् the तत्त्वबोधिनी says – षो ङीषार्थ:। Please explain.
Answer: What is the purpose of having the letter ‘ष्’ as a इत् in the तद्धितः affix ’षच्’? The तत्त्वबोधिनी explains षो ङीषार्थ: – This is to facilitate the generation of the correct feminine forms by the application of the सूत्रम् 4-1-41 षिद्गौरादिभ्यश्च – In the feminine gender the affix ‘ङीष्’ is prescribed following a प्रातिपदिकम् which either (i) contains the letter ‘ष्’ as a ‘इत्’ or (ii) belongs to the class of words beginning with ‘गौर’।
Note: The letter ‘ष्’ of the तद्धितः affix ’षच्’ has the इत्-सञ्ज्ञा by 1-3-6 षः प्रत्ययस्य।

4. In which word in the verses has the affix खल् been used?
Answer: The affix खल् has been used in the form दुर्जयः (प्रातिपदिकम् ‘दुर्जय’, पुंलिङ्गे प्रथमा-एकवचनम्)।

दु:खेन जीयते = दुर्जयः – difficult to conquer

The form ‘ दुर्जय’ is derived from the verbal root √जि (जि अभिभवे १. १०९६) in composition with the उपसर्गः ‘दुर्’/’दुस्’ as follows –
दुर्/दुस् जि + खल् । By 3-3-126 ईषद्‌दु:सुषु कृच्छ्राकृच्छ्रार्थेषु खल् – The affix खल् is used following a verbal root when in composition with either ‘ईषत्’ or ‘दुर्’/’दुस्’ or ‘सु’ used in the sense of ‘difficulty’ or ‘ease.’
Note: In the सूत्रम् 3-3-126 ईषद्‌दु:सुषु कृच्छ्राकृच्छ्रार्थेषु खल्, the term ईषद्‌दु:सुषु ends in the seventh (locative) case. Hence ‘दुर्/दुस्’ gets the उपपद-सञ्ज्ञा here by 3-1-92 तत्रोपपदं सप्तमीस्थम्‌।
= दुर्/दुस् जि + अ । अनुबन्ध-लोपः by 1-3-3 हलन्त्यम्, 1-3-8 लशक्वतद्धिते, 1-3-9 तस्य लोपः।
= दुर्/दुस् जे + अ । By 7-3-84 सार्वधातुकार्धधातुकयोः।
= दुर्/दुस् जय । By 6-1-78 एचोऽयवायावः।
Now we form the compound between ‘दुर्/दुस्’ (which is the उपपदम्) and ‘जय’ using the सूत्रम् 2-2-19 उपपदमतिङ्। Note: The उपपदम् ‘दुर्/दुस्’ gets the designation उपसर्जनम् as per the सूत्रम् 1-2-43 प्रथमानिर्दिष्टं समास उपसर्जनम् because in the सूत्रम् 2-2-19 उपपदमतिङ् (which prescribes the compounding) the term उपपदम् ends in the nominative case. And hence ‘दुर्/दुस्’ is placed in the prior position in the compound as per the सूत्रम् 2-2-30 उपसर्जनं पूर्वम्‌।
= दुर्/दुरुँ जय । By 8-2-66 ससजुषो रुः।
= दुर् + जय । अनुबन्ध-लोपः by 1-3-2 उपदेशेऽजनुनासिक इत्, 1-3-9 तस्य लोपः।
= दुर्जय ।

‘दुर्जय’ gets प्रातिपदिक-सञ्ज्ञा by 1-2-46 कृत्तद्धितसमासाश्च। This is an adjective. It is qualifying लक्ष्मणः here. Hence it is masculine in gender. It declines like ‘राम’-शब्दः। प्रथमा-एकवचनम् is दुर्जयः।

5. Which सूत्रम् prescribes the substitution ‘आ’ in the form महात्मना (प्रातिपदिकम् ‘महात्मन्’, पुंलिङ्गे तृतीया-एकवचनम्)?
Answer: The substitution ‘आ’ in the form महात्मना is prescribed by the सूत्रम् 6-3-46 आन्महतः समानाधिकरणजातीययोः – The ending letter (‘त्’) of ‘महत्’ is substituted by ‘आ’ when ‘महत्’ is followed by either –
i) a latter member (of a compound) which has समानाधिकरणम् – refers to the same item as does ‘महत्’ or
ii) affix ‘जातीय’ (ref. 5-3-69 प्रकारवचने जातीयर्)।
Note: As per the परिभाषा-सूत्रम् 1-1-52 अलोऽन्त्यस्य only the ending letter ‘त्’ of ‘महत्’ is substituted by ‘आ’।

The लौकिक-विग्रहः is as follows –
महानात्मा यस्य सः = महात्मा – high-souled

The derivation of the compound प्रातिपदिकम् ‘महात्मन्’ is similar to the derivation of the compound प्रातिपदिकम् ‘महाबाहु’ as shown in answer to question 1 in the following comment – https://avg-sanskrit.org/2015/10/29/दत्तराज्यः-mns/#comment-35681. Except that we need to apply 6-1-101 अकः सवर्णे दीर्घः a second time to get
महा + आत्मन् = महात्मन् as the final compound प्रातिपदिकम्।

The gender of a बहुव्रीहि: compound matches that of which it qualifies. In the present example महात्मना is qualifying रामेण। Hence we assign the masculine gender to the compound प्रातिपदिकम् ‘महात्मन्’। It declines like आत्मन्-शब्द:। तृतीया-एकवचनम् is महात्मना।

6. How would you say this in Sanskrit?
“In Laṅkā Śrī Hanumān saw many demons with deformed eyes.” Construct a बहुव्रीहि: compound for ‘those who have deformed eyes’ = ‘विरूपाणि (विकृतरूपाणि) अक्षीणि येषां ते’।
Answer: लङ्कायाम् श्रीहनुमान् बहून् विरूपाक्षान् राक्षसान् ददर्श = लङ्कायां श्रीहनुमान् बहून् विरूपाक्षान् राक्षसान् ददर्श।

Easy questions:
1. Which सूत्रम् prescribes the substitution ‘अनङ्’ in the form भ्राता (पुंलिङ्ग-प्रातिपदिकम् ‘भ्रातृ’, प्रथमा-एकवचनम्)?
Answer: The सूत्रम् 7-1-94 ऋदुशनस्पुरुदंसोऽनेहसां च prescribes the substitution ‘अनङ्’ in the form भ्राता।

The derivation is as follows –
भ्रातृ + सुँ । By 4-1-2 स्वौजसमौट्छष्टाभ्याम्भिस्ङेभ्याम्भ्यस्ङसिभ्याम्भ्यस्ङसोसाम्ङ्योस्सुप्‌। Note: The affix ‘सुँ’ has the designation सर्वनामस्थानम् here as per the सूत्रम् 1-1-43 सुडनपुंसकस्य। This allows 6-4-8 to apply below.
भ्रातृ + स् । अनुबन्ध-लोप: by 1-3-2 उपदेशेऽजनुनासिक इत् and 1-3-9 तस्य लोपः।
= भ्रात् अनँङ् + स् । By 7-1-94 ऋदुशनस्पुरुदंसोऽनेहसां च – ‘ऋत्’ (short ‘ऋ’) ending terms, as well as the terms ‘उशनस्’, ‘पुरुदंसस्’ and ‘अनेहस्’ get the ‘अनँङ्’ replacement when the ‘सुँ’ suffix, that is not सम्बुद्धिः, follows. By 1-1-53 ङिच्च, a ङित् substitute (substitute which has the letter ‘ङ्’ as a marker), even if it is अनेकाल् (having more than one letter), takes the place of only the last letter of the term being substituted.
= भ्रातन् + स् । अनुबन्ध-लोप: by 1-3-2 उपदेशेऽजनुनासिक इत्, 1-3-3 हलन्त्यम् and 1-3-9 तस्य लोपः।
= भ्रातान् + स् । By 6-4-8 सर्वनामस्थाने चासम्बुद्धौ।
= भ्रातान् । By 6-1-68 हल्ङ्याब्भ्यो दीर्घात् सुतिस्यपृक्तं हल्। Now ‘भ्रातान्’ 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.
= भ्राता । By 8-2-7 नलोपः प्रातिपदिकान्तस्य।

2. Where has the सूत्रम् 6-1-111 ऋत उत्‌ been used in the verses?
Answer: The सूत्रम् 6-1-111 ऋत उत्‌ has been used in the form भर्तुः ( पुंलिङ्ग-प्रातिपदिकम् ‘भर्तृ’, षष्ठी-एकवचनम्)।

Please see the answer to question 5 in the following comment for the derivation – http://avg-sanskrit.org/2012/10/30/धनञ्जयः-mns/#comment-5749