NIMCET 2026 — Mathematics PYQ

To solve this, we express both sides with common bases.

Step 1: Simplify the equation using base 2 and base 3

The given equation is:

$$2^{x+1/2}\times (2^2{)}^{y-5/6}=3^{x-1/2}\times (3^2{)}^{y-1/3}$$

Apply the power of a power rule $(a^m{)}^n=a^{mn}$:

$$2^{x+1/2}\times 2^{2y-5/3}=3^{x-1/2}\times 3^{2y-2/3}$$

Step 2: Combine exponents

Use the product rule $a^m\times a^n=a^{m+n}$:

$$2^{(x+1/2)+(2y-5/3)}=3^{(x-1/2)+(2y-2/3)}$$

$$2^{x+2y+(3/6-10/6)}=3^{x+2y+(-3/6-4/6)}$$

$$2^{x+2y-7/6}=3^{x+2y-7/6}$$

Step 3: Solve for the variables

For the equality $2^k=3^k$ to hold for real numbers, the exponent $k$ must be zero, because the only solution to $a^k=b^k$ (for $a\ne b$) is $k=0$.

Therefore:

$$x+2y-\frac{7}{6}=0$$

Multiplying the entire equation by $6$ to clear the denominator:

$$6x+12y-7=0$$

Thus, the correct relation is $6x+12y-7=0$.