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CAT Previous Year Questions - Square Root and Cube Root | Quantitative Aptitude (Quant) PDF Download
2024
Q1: If (a + b√n) is the positive square root of (29 - 12√5), where a and b are integers, and n is a natural number, then the maximum possible value of (a + b + n) is
(a) 4
(b) 18
(c) 6
(d) 22
Ans: b
Sol: So 29 - 12√5 is the positive square root of (29 - 12√5).
So 29 - 12√5 = (a + b√n)²
29 - 12√5 = a² + b²n + 2ab√n
a² + b²n = 29
ab√n = -6√5
a²b²n² = 180
b²n = 180 / a²
Substituting this in the above equation:
a² + b²n = 29
a² + (180 / a²) = 29
a⁴ - 29a² + 180 = 0
a² = (29 ± √(29² - 4*180)) / 2
a² = 9 or 20
That means one of a² or b²n is 9 or 20.
We also have ab√n = -6√5; that means one of a or b should be negative.
And also the fact that this is a positive root, and we need to maximize the value of a, b, and n.
We can have a = -3, b = 1, and n = 20.
This satisfies all the above equations, and the value of a + b + n = 18
Q2: For any natural number n, let an be the largest integer not exceeding √n. Then the value of a1 + a2 + a3 + ... + a50 is
Ans: 217
Sol:
2023
Q1: If 
then 
is equal to
(a) 4√5
(b) 2√7
(c) 3√31
(d) 3√7
Ans: d
Sol:
Q2: For some positive and distinct real numbers x,y and z, if 
is the arithmetic mean of 
and 
then the relationship which will always hold true, is
(a) √x, √y and √z are in arithmetic progression
(b) √x, √z and √y are in arithmetic progression
(c) y, x and z are in arithmetic progression
(d) x, y and z are in arithmetic progression
Ans: c
Sol:
x is the Arithmetic Mean of z & y, therefore, z, x, y form an A.P.
It goes without saying that y, x, z also forms an A.P.
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FAQs on CAT Previous Year Questions - Square Root and Cube Root - Quantitative Aptitude (Quant)
| 1. What are square roots and cube roots, and how are they calculated? |  |
Ans. Square roots are values that, when multiplied by themselves, give the original number. For example, the square root of 9 is 3 because 3 x 3 = 9. Cube roots are similar but involve multiplying a number by itself twice. For instance, the cube root of 27 is 3 because 3 x 3 x 3 = 27. To calculate square roots, you can use methods like prime factorization or estimation, while cube roots can be calculated by finding the number that, when cubed, equals the target number.
| 2. Why are square roots and cube roots important in competitive exams like the CAT? | |
Ans. Square roots and cube roots are fundamental concepts in mathematics that frequently appear in competitive exams like the CAT. They are essential for solving problems related to algebra, geometry, and number theory. Understanding these concepts helps in tackling quantitative aptitude questions efficiently, as they often require quick calculations and simplifications.
| 3. How can I practice square root and cube root questions for the CAT exam? | |
Ans. To practice square root and cube root questions for the CAT exam, you can use various resources such as online mock tests, practice books, and previous year question papers. Websites and apps that focus on quantitative reasoning often have sections dedicated to these topics. Additionally, joining study groups or forums can help you exchange tips and solve problems collaboratively.
| 4. What are some common tricks to solve square root and cube root problems quickly? | |
Ans. Common tricks include memorizing squares and cubes of integers up to a certain number (e.g., 1-20), which can help you recognize patterns quickly. For square roots, knowing that √a * √b = √(ab) can simplify calculations. For cube roots, understanding that (a*b)³ = a³ * b³ can also aid in finding answers faster. Estimation techniques, like rounding numbers to the nearest perfect square or cube, can further speed up problem-solving.
| 5. Are there any formulas related to square roots and cube roots that I should know for the CAT? | |
Ans. Yes, there are several important formulas. For square roots, the formula √(a*b) = √a * √b is useful, as is the property that √(a²) = a. For cube roots, the formula (a*b)³ = a³ * b³ is important to remember. Understanding these properties can help simplify complex problems and enhance calculation speed during the exam.
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