Question:medium

If $(a + b\sqrt{n})$ is the positive square root of $(29 - 12\sqrt{5})$, where $a$ and $b$ are integers, and $n$ is a natural number, then the maximum possible value of $(a + b + n)$ is ?

Updated On: Jan 13, 2026
  • 4
  • 22
  • 18
  • 6
Show Solution

The Correct Option is C

Solution and Explanation

We start with the given equation:

$\sqrt{29 - 12\sqrt{5}} = a + b\sqrt{n}$

Square both sides of the equation:

$29 - 12\sqrt{5} = (a + b\sqrt{n})^2 = a^2 + 2ab\sqrt{n} + b^2n$

By comparing the rational and irrational parts of the equation, we get two separate equations:
- $a^2 + b^2n = 29$ (for the rational parts)
- $2ab\sqrt{n} = -12\sqrt{5}$ (for the irrational parts)

From the irrational part, $2ab\sqrt{n} = -12\sqrt{5}$, we can deduce that $n = 5$. Substituting this into the equation yields:

$2ab\sqrt{5} = -12\sqrt{5} \implies ab = -6$

Now, substitute $n = 5$ into the rational part equation, $a^2 + b^2n = 29$:

$a^2 + 5b^2 = 29$

We now have a system of two equations:
1. $ab = -6$
2. $a^2 + 5b^2 = 29$

Solving these equations (either by trial and error or systematically) gives us the values $a = 3$, $b = -2$, and $n = 5$.
Therefore, the sum $a + b + n = 3 + (-2) + 5 = 6$.

Was this answer helpful?
0

Top Questions on Algebra

Questions Asked in CAT exam

The passage given below is followed by four summaries. Choose the option that best captures the essence of the passage.
In the dynamic realm of creativity, artists often find themselves at the crossroads between drawing inspiration from diverse cultures and inadvertently crossing into the territory of cultural appropriation. Inspiration is the lifeblood of creativity, driving artists to create works that resonate across borders. In a globalized era of the modern world, artists draw from a vast array of cultural influences. When approached respectfully, inspiration becomes a bridge, fostering understanding and appreciation of cultural diversity. However, the line between inspiration and cultural appropriation can be thin and easily blurred.
Cultural appropriation occurs when elements from a particular culture are borrowed without proper understanding, respect, or acknowledgment. This leads to the commodification of sacred symbols, the reinforcement of stereotypes, and the erasure of the cultural context from which these elements originated. It is essential to recognize that the impact of cultural appropriation extends beyond the realm of artistic expression, influencing societal perceptions and perpetuating power imbalances.
View Solution