Question:
Let $x=(8 \sqrt{3}+13)^{13}$ and $y=(7 \sqrt{2}+9)^9$ If $[t]$ denotes the greatest integer $\leq t$, then
Let $x=(8 \sqrt{3}+13)^{13}$ and $y=(7 \sqrt{2}+9)^9$ If $[t]$ denotes the greatest integer $\leq t$, then
Updated On: Mar 19, 2025
- $[x]$ is even but $[y]$ is odd
- $[x]+[y]$ is even
- $[x]$ and $[y]$ are both odd
- $[x]$ is odd but $[y]$ is even
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The Correct Option is B
Solution and Explanation
Let \( x = \left( 8\sqrt{3}+13 \right)^{13} \) and expand this expression as:
\[
x = \left( 8\sqrt{3}-13 \right)^{13} = 13 C_0 \left( 8\sqrt{3} \right)^{13} + 13 C_1 \left( 8\sqrt{3} \right)^{12} (13) + \cdots
\]
We get the value of \( x - x' \), which results in an even integer. Therefore, \( \left[ x \right] \) is even.
Now for \( y = \left( 7\sqrt{2}+9 \right)^9 \), we expand it similarly as:
\[
y' = \left( 7\sqrt{2}-9 \right)^9 = 9 C_0 \left( 7\sqrt{2} \right)^9 + 9 C_1 \left( 7\sqrt{2} \right)^8 (9) + \cdots
\]
We find that \( y - y' \) is also an even integer, and therefore \( \left[ y \right] \) is even.
Finally, since both \( [x] \) and \( [y] \) are even, \( [x] + [y] \) is even.
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Concepts Used:
Sets
Set is the collection of well defined objects. Sets are represented by capital letters, eg. A={}. Sets are composed of elements which could be numbers, letters, shapes, etc.
Example of set: Set of vowels A={a,e,i,o,u}
Representation of Sets
There are three basic notation or representation of sets are as follows:
Statement Form: The statement representation describes a statement to show what are the elements of a set.
- For example, Set A is the list of the first five odd numbers.
Roster Form: The form in which elements are listed in set. Elements in the set is seperatrd by comma and enclosed within the curly braces.
- For example represent the set of vowels in roster form.
A={a,e,i,o,u}
Set Builder Form:
- The set builder representation has a certain rule or a statement that specifically describes the common feature of all the elements of a set.
- The set builder form uses a vertical bar in its representation, with a text describing the character of the elements of the set.
- For example, A = { k | k is an even number, k ≤ 20}. The statement says, all the elements of set A are even numbers that are less than or equal to 20.
- Sometimes a ":" is used in the place of the "|".