If `((1-3x)^(1//2)+(1-x)^(5//3))/(sqrt(4-x))` is approximately equal to `a+bx` for small values of `x`, then `(a,b) ` is (a) `(1,35/24)` (b) `(1/-35/24)` (c) `(2,35/12)` (d) `(2,-35/12)`

Options

A. `(1,(35)/(24))`
B. `(1,-(35)/(24))`
C. `(2,(35)/(12))`
D. `(2,-(35)/(12))`

Explanation

To solve the problem, we need to simplify the expression \(\frac{(1-3x)^{1/2} + (1-x)^{5/3}}{\sqrt{4-x}}\) for small values of \(x\) and find the coefficients \(a\) and \(b\) such that the expression is approximately equal to \(a + bx\).

### Step 1: Simplify the Denominator
The denominator is \(\sqrt{4-x}\). For small values of \(x\), we can use the binomial expansion:
\[ \sqrt{4-x} = \sqrt{4(1 - \frac{x}{4})} = 2\sqrt{1 - \frac{x}{4}} \approx 2\left(1 - \frac{x}{8}\right) = 2 - \frac{x}{4} \]

**Hint:** Use the binomial expansion for \((1-u)^n\) where \(u = \frac{x}{4}\) and \(n = \frac{1}{2}\).

### Step 2: Simplify the Numerator
Now, we simplify the numerator \((1-3x)^{1/2} + (1-x)^{5/3}\).

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

2. For \((1-x)^{5/3}\):
\[ (1-x)^{5/3} \approx 1 - \frac{5}{3}x \]

Combining these, we have:
\[ (1-3x)^{1/2} + (1-x)^{5/3} \approx \left(1 - \frac{3x}{2}\right) + \left(1 - \frac{5}{3}x\right) = 2 - \left(\frac{3}{2} + \frac{5}{3}\right)x \]

**Hint:** Use the binomial expansion for both terms and combine like terms.

### Step 3: Combine the Numerator and Denominator
Now we combine the simplified numerator and denominator:
\[ \frac{2 - \left(\frac{3}{2} + \frac{5}{3}\right)x}{2 - \frac{x}{4}} \]

Calculating \(\frac{3}{2} + \frac{5}{3}\):
\[ \frac{3}{2} = \frac{9}{6}, \quad \frac{5}{3} = \frac{10}{6} \quad \Rightarrow \quad \frac{3}{2} + \frac{5}{3} = \frac{19}{6} \]

Thus, the numerator becomes:
\[ 2 - \frac{19}{6}x \]

Now, substituting back:
\[ \frac{2 - \frac{19}{6}x}{2 - \frac{x}{4}} \approx \frac{2(1 - \frac{19}{12}x)}{2(1 - \frac{x}{8})} = \frac{1 - \frac{19}{12}x}{1 - \frac{x}{8}} \]

Using the expansion for \((1-u)^{-1}\):
\[ \approx \left(1 - \frac{19}{12}x\right)\left(1 + \frac{x}{8}\right) \approx 1 - \frac{19}{12}x + \frac{x}{8} \]

**Hint:** Use the formula \((1-u)(1+v) \approx 1 + (v-u)\) for small \(u\) and \(v\).

### Step 4: Combine Terms
Now, we need to combine the coefficients of \(x\):
\[ -\frac{19}{12} + \frac{1}{8} = -\frac{19}{12} + \frac{3}{24} = -\frac{38}{24} + \frac{3}{24} = -\frac{35}{24} \]

### Final Result
Thus, the expression simplifies to:
\[ 1 - \frac{35}{24}x \]
From this, we can identify \(a = 1\) and \(b = -\frac{35}{24}\).

### Conclusion
The values of \(a\) and \(b\) are:
\[ (a, b) = \left(1, -\frac{35}{24}\right) \]

### Answer
The correct option is (b) \((1, -\frac{35}{24})\).