Question

If the vectors \(a\bar{i} + \bar{j} + \bar{k}\), \(\bar{i} + b\bar{j} + \bar{k}\), \(\bar{i} + \bar{j} + c\bar{k}\) (\(a \ne b \ne c \ne 1\)) are coplanar, then \(\frac{1}{1 - a} + \frac{1}{1 - b} + \frac{1}{1 - c} =\)

• A. \(0\)
• B. \(1\)
• C. \(2\)
• D. \(-1\)

Hint:

The scalar triple product of vectors \(\bar{a}\), \(\bar{b}\), and \(\bar{c}\) is \(\bar{a} \cdot (\bar{b} \times \bar{c})\).

Answer 1

The Correct Option is B

We want to find the coefficient of \(x\) in the expansion of 

\[ \frac{(1 - 4x)^2 (1 - 2x^2)^{1/2}}{(4 - x)^{3/2}}. \]

We can write

\[ \frac{(1 - 4x)^2 (1 - 2x^2)^{1/2}}{(4 - x)^{3/2}} = (1 - 4x)^2 (1 - 2x^2)^{1/2} \cdot 4^{-3/2} \left( 1 - \frac{x}{4} \right)^{-3/2} \] \[ = \frac{1}{8} (1 - 8x + 16x^2) (1 - x^2 + \ldots) \left( 1 + \frac{3}{8} x + \ldots \right). \]

We are only interested in the coefficient of \(x\), so we can ignore terms of degree 2 or higher. Then

\[ \frac{(1 - 4x)^2 (1 - 2x^2)^{1/2}}{(4 - x)^{3/2}} = \frac{1}{8} (1 - 8x) \left( 1 + \frac{3}{8} x \right) + O(x^2) \] \[ = \frac{1}{8} \left( 1 - 8x + \frac{3}{8} x - 3x^2 \right) + O(x^2) \] \[ = \frac{1}{8} \left( 1 - \frac{61}{8} x \right) + O(x^2) \] \[ = \frac{1}{8} - \frac{61}{64} x + O(x^2). \]

Therefore, the coefficient of \(x\) is \(-\frac{61}{64}\).

To solve this problem, we first perform the partial fraction decomposition of the given expression:

\[ \frac{4x^2 + 5}{(x - 2)^4} = \frac{A}{(x - 2)} + \frac{B}{(x - 2)^2} + \frac{C}{(x - 2)^3} + \frac{D}{(x - 2)^4}. \]

Step 1: Clear the denominator

Multiply both sides by \( (x - 2)^4 \) to eliminate the denominators:

\[ 4x^2 + 5 = A(x - 2)^3 + B(x - 2)^2 + C(x - 2) + D. \]

Step 2: Expand the right-hand side

\[ A(x - 2)^3 = A(x^3 - 6x^2 + 12x - 8), \] \[ B(x - 2)^2 = B(x^2 - 4x + 4), \] \[ C(x - 2) = Cx - 2C, \] \[ D = D. \]

Combine these terms:

\[ A(x^3 - 6x^2 + 12x - 8) + B(x^2 - 4x + 4) + Cx - 2C + D. \]

Step 3: Equate coefficients

Compare the expanded form with the left-hand side \( 4x^2 + 5 \).

Step 4: Solve for \( A, B, C, D \)

\[ A = 0, \quad B = 4, \quad C = 16, \quad D = 21. \]

Step 5: Compute the expression

\[ \sqrt{\frac{A}{C} + \frac{B}{C} + \frac{D}{C}} = \frac{5}{4}. \]

Final Answer: \(\frac{5}{4}\)