Question

The statement having truth value 'T' from the following is

• A. \(sin(x)\) is an even function.
• B. Every square matrix is non-singular.
• C. The product of complex number and its conjugate is purely imaginary.
• D. A square of any real number is non-negative.

Hint:

A statement is true only if it holds for every case; test each one with a quick example.

Answer 1

The Correct Option is D

Step 1: Approach
For each sentence look for one counter-example. A sentence that survives every test is true.

Step 2: Tests
- $\sin x$: take $x=\pi/6$. Then $\sin(-\pi/6)=-1/2\ne1/2$. Not even.
- Matrix $\begin{bmatrix}2&4\\1&2\end{bmatrix}$ has determinant $4-4=0$, so some square matrices are singular.
- $z\bar z=|z|^2\ge0$ is real, never purely imaginary for $z\ne0$.
- A real square is $x^2=|x|^2\ge0$ for every real $x$. No counter-example exists.

Step 3: Conclusion
Only (D) is true.

Final Answer:
Only the statement that the square of any real number is non-negative is always true, option (D). \[ \boxed{\text{(D)}} \]