**Answer:**

To find the standard deviation of a dataset, we need to calculate the square root of the variance. The variance of a dataset is the average of the squared differences from the mean.

Given that the mean of dataset X is 'a' and the standard deviation is 'b', we can use this information to find the variance of X.

**Variance of X:**

The formula for variance is:

Var(X) = (1/n) * Σ(xi - μ)^2

Where:
- n is the number of data points in the dataset
- xi is the i-th data point
- μ is the mean of the dataset

In this case, since the mean of X is 'a', the formula becomes:

Var(X) = (1/n) * Σ(xi - a)^2

**Variance of -X:**

To find the variance of -X, we need to calculate the squared differences from the mean of -X.

The mean of -X is simply the negative of the mean of X:

Mean(-X) = -a

Using the same formula for variance, the variance of -X becomes:

Var(-X) = (1/n) * Σ(xi - (-a))^2
= (1/n) * Σ(xi + a)^2

**Simplifying the expression:**

Expanding the squared term, we have:

Var(-X) = (1/n) * Σ(xi^2 + 2(xi)(a) + a^2)

Since we know that Var(X) = (1/n) * Σ(xi^2 - 2(xi)(a) + a^2), we can substitute this expression into the equation:

Var(-X) = (1/n) * Σ(xi^2 + 2(xi)(a) + a^2)
= (1/n) * Σ(xi^2 - 2(xi)(a) + a^2 + 4(xi)(a))
= (1/n) * Σ(xi^2 - 2(xi)(a) + a^2) + (4a/n) * Σ(xi)

The first part of the expression is simply Var(X), and the second part is a constant multiplied by the sum of the data points. Since the sum of the data points is n times the mean (Σ(xi) = n * a), this simplifies to:

Var(-X) = Var(X) + (4a/n) * n * a
= Var(X) + 4a^2

**Standard Deviation of -X:**

To find the standard deviation of -X, we take the square root of the variance:

SD(-X) = √(Var(-X))
= √(Var(X) + 4a^2)

Since we know that Var(X) = b^2, we can substitute this into the equation:

SD(-X) = √(b^2 + 4a^2)

Therefore, the standard deviation of -X is √(b^2 + 4a^2).

Answer: None of the given options (a), (b), (c), or (d) are correct.

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