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 x = √8 − √2, what is x equal to?EasyLevel 27In which option is the correct conjugate chosen to rationalize an irrational denominator?ExpertLevel 27If (\alpha=7+\sqrt{6}) and (\beta=7-\sqrt{6}), what is (\alpha^2+\beta^2)?ExpertLevel 27If (x=2+\sqrt{7}), what is the value of (x+\frac{1}{x})?ExpertLevel 27If (x=2+\sqrt{7}), what is the correct value of (x+\frac{1}{x})?ExpertLevel 27If the zeroes of p(x) = x² − 2kx + 20 are k + √5 and k − √5, what is the positive value of k?HardLevel 27Which option is a correct counterexample showing that (\sqrt{a+b}=\sqrt{a}+\sqrt{b}) is false?ExpertLevel 27Which option is an irrational number between (3) and (4)?ExpertLevel 27If (m) is a positive integer and (\sqrt{m}) is rational, which conclusion about (m) is correct?ExpertLevel 27If \(x=\sqrt{11}-\sqrt{2}\), what is \(x^2\)?ExpertLevel 27If α = 5 + 2√6 and β = 5 − 2√6, what is αβ?MediumLevel 27Which option is the correct multiplicative inverse of (2+\sqrt{3})?ExpertLevel 27If (p(x)=x^2-14x+40), what is the type of its zeroes?ExpertLevel 27Which option is equal to (\sqrt{72}-\sqrt{50}+\sqrt{8})?ExpertLevel 27If (r) is a non-zero rational number and (s) is an irrational number, what type of number is (\frac{s}{r})?ExpertLevel 27What is the rationalized form of (\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}})?ExpertLevel 27What is the simplest rationalized form of (\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}})?ExpertLevel 27If \(x=3+\sqrt{8}\), which quadratic polynomial can have \(x\) as a zero?ExpertLevel 27Which option explains why (\sqrt{a^2}=a) is not always true?ExpertLevel 27If (x=\sqrt{6}+\sqrt{5}) and (y=\sqrt{6}-\sqrt{5}), what is (x^2-y^2)?ExpertLevel 27