Q.(a) Find the term independent of $x$ in the expansion of $\left(2x^2 + \dfrac{1}{x}\right)^{12}$.

(a) General term power $= 24-3r=0 \Rightarrow r=8$; term independent of $x$ is $\binom{12}{8}2^4 = 7920$. (b) Partial elasticities at $(3,7)$: $\tfrac{E_q}{E_{p_1}} = \tfrac{44}{19} \approx 2.316$, $\tfrac{E_q}{E_{p_2}} = \tfrac{56}{57} \approx 0.982$.

Part (a): Term independent of $x$ in $\left(2x^2 + \dfrac{1}{x}\right)^{12}$.

Step 1 — General term.

$T_{r+1} = \binom{12}{r}(2x^2)^{12-r}\left(\frac1x\right)^r = \binom{12}{r}2^{12-r}x^{2(12-r)}x^{-r} = \binom{12}{r}2^{12-r}x^{24-3r}.$

Step 2 — Independent of $x$: set exponent to 0.

$24 - 3r = 0 \;\Rightarrow\; r = 8.$

Step 3 — Evaluate $T_9$.

$T_9 = \binom{12}{8}2^{12-8} = \binom{12}{4}\cdot 2^4 = 495 \times 16 = 7920.$

Part (b): Partial elasticities of $q = 5 - 2p_1 + p_2 - p_1^2 p_2$ at $p_1=3,\ p_2=7$.

Step 1 — Value of $q$.

$q = 5 - 2(3) + 7 - (3)^2(7) = 5 - 6 + 7 - 63 = -57.$

Step 2 — Partial derivatives.

$\frac{\partial q}{\partial p_1} = -2 - 2p_1 p_2 = -2 - 2(3)(7) = -44,$

$\frac{\partial q}{\partial p_2} = 1 - p_1^2 = 1 - 9 = -8.$

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