Solving for the Two-Digit Number
Let the two-digit number be represented as $10x + y$, where $x$ is the digit in the tens place and $y$ is the digit in the units place.
Equation from Sum of Digits
The problem states that the sum of the digits is 12:
$x + y = 12 \quad (1)$
Equation from Reversed Digits
The number formed by reversing the digits is $10y + x$. This new number is 54 greater than the original number:
$ (10y + x) - (10x + y) = 54 $
Simplify the equation:
$ 10y + x - 10x - y = 54 $
$ 9y - 9x = 54 $
Divide by 9:
$ y - x = 6 \quad (2) $
Solving the System of Equations
We now have a system of two linear equations:
- $x + y = 12$
- $-x + y = 6$
Add equation (1) and equation (2):
$ (x + y) + (-x + y) = 12 + 6 $
$ 2y = 18 $
$ y = \frac{18}{2} $
$ y = 9 $
Substitute the value of $y$ back into equation (1):
$ x + 9 = 12 $
$ x = 12 - 9 $
$ x = 3 $
Determining the Original Number
The original number is $10x + y$. Substituting the values $x=3$ and $y=9$:
$ \text{Original Number} = 10(3) + 9 = 30 + 9 = 39 $
Verification
Check if the conditions are met:
- Sum of digits: $3 + 9 = 12$ (Correct).
- Reversed number: $93$.
- Difference: $93 - 39 = 54$ (Correct).
The original number is 39.
Important Questions from Numerical Computation
$A$ is an ($n \times n$) matrix. Consider the following two statements
Statement 1: Columns of matrix $A$ are linearly independent
Statement 2: Inverse of matrix $A$ exists
Which one of the following statements is TRUE?
- Consider the function $G(x, y, z) = 0$. This function allows us to implicitly define each of three variables as a function of the other two variables. Assume that all partial derivatives of the function $G(x, y, z)$ exist everywhere. Then, the value of $\left(\frac{\partial z}{\partial x} \times \frac{\partial x}{\partial y} \times \frac{\partial y}{\partial z}\right)$ is _________ (in integer)
- Levenshtein distance is used to measure the minimum edit distance between two strings by counting the minimum number of editing operations (such as, insertions, deletions, substitutions) required to transform one string to another. Consider an alternative version of the Levenshtein distance in which each insertion and each deletion has a cost of 1 and substitutions are not allowed.
Based on this alternative version, the Levenshtein distance between the strings, word and work is ______ (Answer in integer). - In a given collection of documents, let $N$ be the total number of documents and let $d$ be the number of documents in which the term $t$ occurs.
Which ONE of the following fractions is used to define the inverse document frequency (idf) of the term $t$? - The total number of unique character bigrams in the string thoughtsoftheghosts (which contains 19 characters) is ______.
