NDA 2026 — Mathematics PYQ

Step 1: Simplify the left side of the equation

We know that $1={\log }_{10}10$. Substituting this:

$${\log }_{10}10-{\log }_{10}2={\log }_{10}(5^x+4^x+3^x+2^x+1)$$

Using the quotient property ${\log }_{b}M-{\log }_{b}N={\log }_b(\frac{M}{N})$:

$${\log }_{10}\left(\frac{10}{2}\right)={\log }_{10}(5^x+4^x+3^x+2^x+1)$$

$${\log }_{10}5={\log }_{10}(5^x+4^x+3^x+2^x+1)$$

Step 2: Equate the arguments

Since the logarithmic bases are the same, we can equate the arguments:

$$5=5^x+4^x+3^x+2^x+1$$

$$5^x+4^x+3^x+2^x=4$$

Step 3: Test the given options

Let's test the values provided:

• If $x=0$:

  $5^0+4^0+3^0+2^0=1+1+1+1=4$.

  This matches the equation $5^x+4^x+3^x+2^x=4$.

• If $x=1$:

  $5^1+4^1+3^1+2^1=5+4+3+2=14\ne 4$.

Since the equation holds true for $x=0$, this is the correct value.

Conclusion: The correct option is (d) 0.