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
Hide Solution
Verified By Collegedunia
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.
Was this answer helpful?
0
0
Top Questions on Sets
- If $A = \{x \in \mathbb{R} \mid \sin^{-1}(\sqrt{x^2+x+1}) \in [-\frac{\pi}{2}, \frac{\pi}{2}]\}$ and $B = \{y \in \mathbb{R} \mid y = \sin^{-1}(\sqrt{x^2+x+1}), x \in A\}$, then
- Given the sets $ A = \{ x \mid |x - 2|<3 \} $ and $ B = \{ x \mid |x + 1| \leq 4 \} $, find $ A \cap B $.
- If \[ S(x) = (1 + x) + 2(1 + x)^2 + 3(1 + x)^3 + \ldots + 60(1 + x)^{60}, \, x \neq 0, \] and \[ (60)^2 S(60) = a(b)^b + b, \] where $a, b \in \mathbb{N}$, then $(a + b)$ is equal to ________.
- The distance of the point \( (2, 3) \) from the line \( 2x - 3y + 28 = 0 \), measured parallel to the line \( \sqrt{3}x - y + 1 = 0 \), is equal to
- Two finite sets have $ m $ and $ n $ number of elements respectively. The total number of subsets of the first set is 112 more than the total number of subsets of the second set. Then the values of $ m $ and $ n $ are respectively.
View More Questions
Questions Asked in JEE Main exam
- The ratio of the power of a light source \( S_1 \) to that of the light source \( S_2 \) is 2. \( S_1 \) is emitting \( 2 \times 10^{15} \) photons per second at 600 nm. If the wavelength of the source \( S_2 \) is 300 nm, then the number of photons per second emitted by \( S_2 \) is \_\_\_\_\_ \( \times 10^{14} \).
- JEE Main - 2025
- Units and measurement
- In a group of 3 girls and 4 boys, there are two boys \( B_1 \) and \( B_2 \). The number of ways in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but \( B_1 \) and \( B_2 \) are not adjacent to each other, is:
- JEE Main - 2025
- Permutations
Let \( T_r \) be the \( r^{\text{th}} \) term of an A.P. If for some \( m \), \( T_m = \dfrac{1}{25} \), \( T_{25} = \dfrac{1}{20} \), and \( \displaystyle\sum_{r=1}^{25} T_r = 13 \), then \( 5m \displaystyle\sum_{r=m}^{2m} T_r \) is equal to:
- JEE Main - 2025
- Arithmetic and Geometric Progressions
- Let $ A(4, -2), B(1, 1) $ and $ C(9, -3) $ be the vertices of a triangle ABC. Then the maximum area of the parallelogram AFDE, formed with vertices D, E and F on the sides BC, CA and AB of the triangle ABC respectively, is __________.
- JEE Main - 2025
- Geometry
- The value of \( (\sin 70^\circ)(\cot 10^\circ \cot 70^\circ - 1) \) is:
- JEE Main - 2025
- Trigonometric Identities
View More Questions
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 "|".