The mean proportional of 8a² and 18⁄a⁴, where a is a positive number, is ______.

Correct — C. Rule: the mean proportional of x and y is √(xy) — the m for which x : m = m : y.

m = √(8a² × 18/a⁴)
8 × 18 = 144, and a² ÷ a⁴ = 1/a²
m = √(144/a²) = 12/a → option (c)

The stem states a is positive, so the root is taken as +12/a and no ± is needed.

Why the others are wrong

• (a)
  12/a² keeps the power the product had. Taking a square root halves every exponent, so a⁻² under the root becomes a⁻¹ outside it.
• (b)
  12a² has the exponent's sign wrong as well as its size. The a⁴ sits in the denominator of 18/a⁴, so a² ÷ a⁴ leaves a negative power, never a positive one.
• (d)
  The number 12 is right and the letter is on the wrong side of the line. √(1/a²) = 1/a, so a belongs in the denominator.

The mean proportional — the geometric mean — of x and y is √(xy), defined by the proportion x : m = m : y. It is not the arithmetic mean (x + y)/2, and SSC relies on that confusion.

Under the root, multiply the coefficients and halve the exponents. Here 8 × 18 = 144 is a perfect square and a² × a⁻⁴ = a⁻² has an even exponent, which is how the answer comes out clean.

Key facts

• The mean proportional of x and y is √(xy), the m satisfying x : m = m : y.
• √144 = 12, and for a > 0, √(a⁻²) = a⁻¹.
• The third proportional to x and y is y²/x, a different construction SSC also sets.

Common traps

• Taking the arithmetic mean (x + y)/2 in place of the geometric mean √(xy).
• Halving the coefficient instead of the exponent — √144 is 12, not 72.
• Hunting for a ± answer after the stem has already restricted a to positive values.

No directly related past PYQ was found.