NIMCET 2026 — Mathematics PYQ

To solve this problem, we use the standard formulas for the Arithmetic Mean (AM) and Harmonic Mean (HM) of two numbers $a$ and $b$.

1. Using the Arithmetic Mean:

The formula for the arithmetic mean is:

$$\frac{a+b}{2}=5$$

$$a+b=10\text{— (Equation 1)}$$

2. Using the Harmonic Mean:

The formula for the harmonic mean is:

$$\frac{2ab}{a+b}=3.2$$

Substitute $(a+b)=10$ from Equation 1 into this formula:

$$\frac{2ab}{10}=3.2$$

$$2ab=32$$

$$ab=16\text{— (Equation 2)}$$

3. Finding the numbers:

We need two numbers that sum to $10$ and multiply to $16$. We can form a quadratic equation $x^2-(a+b)x+ab=0$:

$$x^2-10x+16=0$$

Factor the quadratic:

$$(x-2)(x-8)=0$$

So, the values are $x=2$ and $x=8$.

Thus, the numbers are $a=2$ and $b=8$.

Correct Option: (d)