NIMCET 2023 — Mathematics PYQ

1. Symmetry Rules for Functions

• If $f(-x)=f(x)$, the function is Even (symmetric about the Y-axis).

• If $f(-x)=-f(x)$, the function is Odd (symmetric about the Origin).

2. Test the Function for $f(-x)$

Given:

$$f(x)={\log }_e\left(x^3+\sqrt{x^6+1}\right)$$

Substitute $x$ with $-x$:

$$f(-x)={\log }_e\left((-x{)}^3+\sqrt{(-x{)}^6+1}\right)$$

$$f(-x)={\log }_e\left(-x^3+\sqrt{x^6+1}\right)$$

3. Simplify using Rationalization

To relate $f(-x)$ back to $f(x)$, multiply and divide the terms inside the logarithm by its conjugate $\left(\sqrt{x^6+1}+x^3\right)$:

$$f(-x)={\log }_e\left(\frac{\left(\sqrt{x^6+1}-x^3\right)\left(\sqrt{x^6+1}+x^3\right)}{\sqrt{x^6+1}+x^3}\right)$$

Apply the algebraic identity $(a-b)(a+b)=a^2-b^2$ in the numerator:

$$f(-x)={\log }_e\left(\frac{(x^6+1)-x^6}{\sqrt{x^6+1}+x^3}\right)$$

$$f(-x)={\log }_e\left(\frac{1}{x^3+\sqrt{x^6+1}}\right)$$

4. Apply Logarithmic Property

Using the property $\log \left(\frac{1}{M}\right)=-\log (M)$:

$$f(-x)={\log }_e{\left(x^3+\sqrt{x^6+1}\right)}^{-1}$$

$$f(-x)=-{\log }_e\left(x^3+\sqrt{x^6+1}\right)$$

$$f(-x)=-f(x)$$

Since $f(-x)=-f(x)$, the function is an odd function. Therefore, its graph is symmetric about the Origin.