Mathematics
Irrational numbers and real numbers
अपरिमेय संख्याएँ और वास्तविक संख्याएँ
In this Class 10 Mathematics topic, students build a clear understanding of real numbers as the collection of rational and irrational numbers. They learn to identify irrational numbers, compare and represent real numbers on the number line, and interpret terminating, recurring, and non-terminating non-recurring decimals. The topic also develops confidence with properties and operations involving real numbers, providing useful foundations for reading polynomial expressions, coefficients, and real zeros in the surrounding Polynomials chapter.
Practice questions
If one zero of a polynomial is (\sqrt{11}) and the coefficients are rational, which zero should also occur?ExpertLevel 26If (p(x)=x^2-10x+23), what type of zeroes does it have?ExpertLevel 26Which option is equal to (\sqrt{50}-\sqrt{32}+\sqrt{2})?ExpertLevel 26Which number is an irrational number lying between (3) and (4)?ExpertLevel 26If (\alpha) and (\beta) are zeroes of a quadratic polynomial where (\alpha+\beta=8) and (\alpha\beta=11), which are the possible zeroes?ExpertLevel 26Which statement is correct about (\sqrt{a^2}), where (a) is a real number?ExpertLevel 26If (x=\sqrt{5}+\sqrt{2}) and (y=\sqrt{5}-\sqrt{2}), what is (x^2-y^2)?ExpertLevel 26Which option will have a non-terminating non-recurring decimal expansion?ExpertLevel 26If (x) is irrational, what is ((x+\sqrt{2})-(x-\sqrt{2})) equal to?ExpertLevel 26In which option is the product of two different irrational numbers rational?ExpertLevel 26If (2+\sqrt{3}) is a zero of a polynomial with rational coefficients, which linear factor is expected to accompany it?ExpertLevel 26Which option is equal to ((\sqrt{12}+\sqrt{27})^2)?ExpertLevel 26If (\sqrt{12}+\sqrt{27}) is simplified first, what will its square be?ExpertLevel 26Which option has all rational coefficients and zeroes (6+\sqrt{11}) and (6-\sqrt{11})?ExpertLevel 26If (\frac{1}{\sqrt{7}+\sqrt{6}}) is rationalized, what is its value?ExpertLevel 26Which option is a correct counterexample showing that the sum of two irrational numbers is not always irrational?ExpertLevel 26If (p) and (q) are rational numbers and (p+q\sqrt{5}) is rational, what is true about (q)?ExpertLevel 26What is the rationalized form of (\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}})?ExpertLevel 26If the zeroes of (p(x)=x^2-2kx+9) are (k+\sqrt{7}) and (k-\sqrt{7}), what is the value of (k)?ExpertLevel 26Which option incorrectly simplifies an irrational number as if it were rational?ExpertLevel 26