Question

The coefficient of x\(^{48}\) in \(1(1+x)+2(1+x)^2+3(1+x)^3 +.....+100(1+x)^{100}\) is:

• A. \(^{100}\)C\(_{50}\) \(^{101}\)C\(_{50}\) – \(^{101}\)C\(_{49}\)
• B. 100\(^{101}\)C\(_{49}\) – \(^{101}\)C\(_{50}\)
• C. \(^{101}\)C\(_{46}\) – 100
• D. \(^{101}\)C\(_{47}\) – \(^{101}\)C\(_{46}\)

Hint:

Finding coefficients in complex series can often be simplified by first finding a closed-form expression for the sum of the series. For AGP, the method of subtracting a multiple of the series from itself is standard.

Answer 1

The Correct Option is B

Analysis: 1. **Spring Constant (k):** From \(F = kx\), \(k = F/x\). Dimensions: \([MLT^{-2}] / [L] = [MT^{-2}]\). Matches (iii). 2. **Thermal Conductivity (k or \(\lambda\)):** From power \(P = \frac{kA\Delta T}{\ell}\), \(k = \frac{P\ell}{A\Delta T}\). Dimensions: \([ML^2T^{-3}][L] / [L^2][K] = [MLT^{-3}K^{-1}]\). Matches (ii). 3. **Boltzmann Constant (\(k_B\)):** Relates energy to temperature (\(E \approx kT\)). Dimensions: \([ML^2T^{-2}] / [K] = [ML^2T^{-2}K^{-1}]\). Matches (i). 4. **Inductance (L):** From Energy \(U = \frac{1}{2}LI^2\). Dimensions: \([ML^2T^{-2}] / [A^2] = [ML^2T^{-2}A^{-2}]\). Matches (iv). Sequence: (1)→(iii), (2)→(ii), (3)→(i), (4)→(iv). \[ \boxed{\text{Correct option is (4)}} \]