A rational number whose reciprocal does not exist is
A. 1
B.-1
C. 0
D.10

Hint:This is a simple problem of application of rational numbers. Rational numbers can be defined in a number of ways. Such as, a rational Number can be made by dividing two integers. In maths, rational numbers are represented in \[\dfrac{p}{q}\] form where q is not equal to zero. It is also a type of real number. Any fraction with non-zero denominators is a rational number.

Complete step-by-step solution:
Now, for solving this problem we should know about two things
1) The reciprocal of a number is 1 divided by the number. The reciprocal of a number is also called its multiplicative inverse. The product of a number and its reciprocal is 1.
2) Meaning of undefined numbers i.e. an expression in mathematics which does not have meaning and so which is not assigned an interpretation.
We have to check if a given number’s reciprocal exists or not.
In first option, Reciprocal of 1 = \[\dfrac{1}{1}\] = 1
So, Reciprocal of 1 exists that is equal to 1.

In second option, Reciprocal of -1 = \[\dfrac{1}{{ - 1}}\] = -1
So, Reciprocal of -1 exists that is equal to -1.

In third option, Reciprocal of 0 = \[\dfrac{1}{0}\]
\[\dfrac{1}{0}\] is said to be undefined because division is defined in terms of multiplication
Thus \[\dfrac{1}{0}\] does not exist, or is not defined, or is undefined.
All numbers except 0 have a reciprocal.

In fourth option, Reciprocal of 10 = \[\dfrac{1}{{10}}\]
So, Reciprocal of 1 exists that is equal to \[\dfrac{1}{{10}}\].

Hence, Answer is Option C .

Note:All numbers except 0 have a reciprocal. As already discussed for finding the reciprocal of any number just divide it by 1. So, it can be said that the reciprocal of 0 is a very large number that is beyond the scope of study and hence great mathematicians referred to it to be as indefinite or just undefined.Multiplicative inverse of any number can also be found out by taking reciprocal of that number.
e.g. Multiplicative inverse of 2 is $\dfrac{1}{2}$