Answer

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**Hint:**We solve this question by using the basic multiplication concept. This involves the process of two-digit multiplication. We need to find out the value of the digit K. In order to do so, let us multiply the two numbers given in terms of K and obtain the answer. Then comparing it with the answer K2K, we get the value of the digit K.

**Complete step by step answer:**

In order to solve this question, let us multiply the two given digits using the basic multiplication of two digits method. This is done as follows,

$\begin{align}

& \Rightarrow \text{ }1K \\

& \text{ }\underline{\times K1} \\

& \\

\end{align}$

We now multiply the units place digit of the second number with both the digits of the first number. Then we put a cross in the next step for the units place indicating that no digit is supposed to be written here. Then we multiply the second digit of the second number with both the digits of the first number. Then, we sum up both the results to obtain the final answer.

$\begin{align}

& \Rightarrow \text{ }1K \\

& \text{ }\underline{\times K1} \\

& \text{ }1K \\

& \text{ }\underline{K{{K}^{2}}\times \text{ }} \\

& K\left( {{K}^{2}}+1 \right)K \\

\end{align}$

This can be simplified by comparing this result with the result K2K. Equating the two middle terms,

$\Rightarrow {{K}^{2}}+1=2$

Subtracting both sides by 1,

$\Rightarrow {{K}^{2}}=2-1$

Subtracting the terms on the right-hand side,

$\Rightarrow {{K}^{2}}=1$

Taking square root on both sides,

$\Rightarrow K=\sqrt{1}=\pm 1$

Since K is a digit here and not the value of a number, we neglect the sign altogether. The value of K is therefore obtained to be 1.

We can verify this as $11\times 11=121.$ Since the product is right, the value of K is correct.

**So, the correct answer is “Option A”.**

**Note:**We need to know the concept of the very basic multiplication of two-digit numbers. Another way to solve this sum would be to substitute the value of K from the options given and check by multiplying the terms to see if it would equal the right-side of the equation. If so, the assumed option is correct.

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