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SubjectsMathematics
वर्ग और वर्गमूल
This Class 9 Mathematics topic helps students understand squares and square roots through clear methods, patterns, and examples. They learn to find perfect squares, identify square numbers, estimate square roots, and use prime factorisation or long-division methods where appropriate. The topic also strengthens calculation skills and supports problem-solving in numerical questions. As part of the Common Questions section, it provides focused explanations for reviewing key ideas and addressing frequent doubts about Squares and Square Roots.
TOPIC PRACTICE
Up to 7 questions from this page. Select your focus, then start.
A set-builder description must include every listed element and exclude numbers not in the set. The numbers 2, 3, 5, and 7 are all prime natural numbers less than 10, so option A is exact. Option B would contain only 2, 4, 6, and 8. Option C is too broad because it also includes 1 and 9, while option D gives 1, 4, and 9.
Positive perfect squares below 20 are obtained from 1², 2², 3², and 4²: 1, 4, 9, and 16. The next square is 5² = 25, which is not less than 20. Thus the complete roster is {1, 4, 9, 16}. Option B lists even numbers, C lists initial natural numbers, and D includes 25, so A is correct.
The members of K are successive squares: 1=1², 4=2², 9=3², 16=4², and 25=5². Every one is less than 30, while the next square, 6²=36, is not less than 30. Hence K is precisely the set of perfect squares below 30. The numbers are not all prime or even, and most are not multiples of 5, so option A is correct.
The governing concept is filtering a set using two conditions. The natural perfect squares below 36 are 1²=1, 2²=4, 3²=9, 4²=16, and 5²=25. The number 36 is excluded because the condition says less than 36, and 4 is removed by x≠4. Thus V={1,9,16,25}, which is option A.
The governing concept is solving a square equation over the integers and then counting distinct set elements. From x²=16, we obtain x=4 or x=-4 because both values have square 16. Thus C={-4,4}, which contains two distinct elements. The value 16 is the square on the right side, not the cardinality, and 4 is only one solution. Therefore option B is correct.
Since \((5a)^2=5^2\times a^2=25a^2\), the correct expression is \(5a\). The square of \(25a\) is \(625a^2\), while the square of \(5a^2\) is \(25a^4\). Exam tip: to find the square root of a perfect-square monomial, take the square root of the coefficient and halve the exponent of each variable.
The governing concept is evaluation of principal square roots before carrying out addition. Since 4 = 2², √4 = 2, and since 9 = 3², √9 = 3. Therefore, √4 + √9 = 2 + 3 = 5, so option B is correct. A common error is to combine the radicands and write √(4 + 9) = √13; however, the identity √a + √b = √(a+b) is not generally valid. Thus option A applies an incorrect rule. Option C adds 4 and 9 but does not take their roots, so it does not represent the given expression. Option D has no valid calculation supporting it. The answer is especially clear because both radicands are perfect squares and their roots are whole numbers.
QUIZ COMPLETE
