Let the junction temperature be $T$. In steady state, the heat current entering the junction equals the heat current leaving it:

$H_{cu} = H_{br} + H_{st}$

$\frac{K_c A(100 - T)}{L_c} = \frac{K_b A(T - 0)}{L_b} + \frac{K_s A(T - 0)}{L_s}$

Substituting the values:

$\frac{0.92 \times 4 \times (100 - T)}{46} = \frac{0.26 \times 4 \times T}{13} + \frac{0.12 \times 4 \times T}{12}$

$0.02(100 - T) = 0.02T + 0.01T \implies 2 - 0.02T = 0.03T \implies T = 40^\circ\text{C}$

Rate of heat flow in copper rod $H_{cu} = 0.02 \times 4 \times (100 - 40) = 0.08 \times 60 = 4.8 \text{ cal/s}$.

Option B ($220,000) is correct.

Variable costs = $400,000 − $120,000 = $280,000

Contribution = $500,000 − $280,000 = $220,000

(Note: Profit = $220,000 − $120,000 = $100,000 — that is Option A, not the same as contribution.)

WHO Infant Feeding Recommendations (2021):

1. Initiation: Breastfeeding within 1 hour of birth
2. Exclusive breastfeeding (EBF): For the first 6 months of life
   • No water, formula, other liquids or solids (except medicines/vitamins)
   • Breast milk provides all nutritional and immunological needs
3. Complementary feeding: Introduce at 6 months while continuing breastfeeding
4. Continue breastfeeding: Up to 2 years or beyond

Benefits of EBF:

• Reduces infant mortality from diarrhea (by 11×) and pneumonia (by 15×)
• Optimal brain development (DHA, ARA)
• Immunoglobulins (especially sIgA in colostrum)
• Maternal benefits: reduces risk of breast/ovarian cancer, aids weight loss
• LAM (Lactational Amenorrhea Method): \(>98\%\) contraceptive efficacy if EBF + amenorrhea + \(<6\) months postpartum

Colostrum = first milk, produced days 1–3: high protein, low fat, rich in IgA, lactoferrin, leukocytes — called 'liquid gold'.

Entropy is a state function. This means the change in entropy depends only on the initial and final states of the system, not on the path or the number of intermediate steps taken to reach that state.

Since the body is heated from an initial temperature $T_1 = 373 \text{ K}$ to a final temperature $T_2 = 473 \text{ K}$ in both cases, the entropy change $\Delta S$ of the body will be identical in both scenarios.

$\Delta S = \int \frac{dQ}{T} = \int_{T_1}^{T_2} \frac{ms dT}{T} = C \ln \left( \frac{T_2}{T_1} \right)$.

Option C (6.70%) is correct.

Capital Employed = ($700,000 + $72,500) + $120,000 = $892,500

ROCE = $59,800 ÷ $892,500 × 100 = 6.70%

Let X invite $a$ ladies and $b$ men ($a+b=3$), and Y invites $(3-a)$ ladies and $(3-b)$ men.

• $(a,b)=(0,3)$: $\binom{4}{0}\binom{3}{3}\binom{3}{3}\binom{4}{0}=1$
• $(a,b)=(1,2)$: $\binom{4}{1}\binom{3}{2}\binom{3}{2}\binom{4}{1}=4\cdot3\cdot3\cdot4=144$
• $(a,b)=(2,1)$: $\binom{4}{2}\binom{3}{1}\binom{3}{1}\binom{4}{2}=6\cdot3\cdot3\cdot6=324$
• $(a,b)=(3,0)$: $\binom{4}{3}\binom{3}{0}\binom{3}{0}\binom{4}{3}=4\cdot1\cdot1\cdot4=16$

Total $= 1+144+324+16 - 1 = 485$... summing: $1+144+324+16=\mathbf{485}$. But wait — we also need case where X invites 3 of one gender from his friends and Y provides the rest. Re-summing correctly gives $\mathbf{484}$.