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 a quadratic polynomial has zeroes with sum \(5\sqrt{2}\) and product \(12\), which polynomial represents it?HardLevel 26If \(p(x)=x^2-3x+\sqrt{2}\), what is the product of its zeroes?HardLevel 26If (p(x)=\sqrt{2}x^2-4x+\sqrt{2}), what is the sum of its zeroes?HardLevel 26If p(x)=\sqrt{3}x^2-6x+2\sqrt{3}, what is the product of its zeroes?HardLevel 26If (p(x)=x^2-(\sqrt{2}+\sqrt{3})x+\sqrt{6}), what can its zeroes be?HardLevel 26If (p(x)=x^2+(\sqrt{5}-2)x-2\sqrt{5}), which pair can be its zeroes?HardLevel 26If (\sqrt{3}) is a repeated zero of (x^2+kx+3), what is (k)?HardLevel 26If the zeros of the quadratic \(x^2-4x+r\) are rational, is the statement true when \(r=3\)?HardLevel 26Which of the following polynomials has rational coefficients and irrational conjugate zeros?HardLevel 26If (\alpha=2+\sqrt{7}) and (\beta=2-\sqrt{7}), what is (\alpha^2+\beta^2)?HardLevel 26If (\alpha=5+\sqrt{6}) and (\beta=5-\sqrt{6}), what is (\frac{1}{\alpha}+\frac{1}{\beta})?HardLevel 26If (p(x)=x^2-10x+19), what is (\frac{1}{\alpha}+\frac{1}{\beta}), where (\alpha,\beta) are its zeroes?HardLevel 26If one zero of a quadratic polynomial with rational coefficients is (6-2\sqrt{5}), what is one possible form of that polynomial?HardLevel 26If (p(x)=x^2-2), which statement about the zeroes of (p(x)) is correct?HardLevel 27If the zeroes of a quadratic polynomial are (3+\sqrt{5}) and (3-\sqrt{5}), what is the type of their sum?HardLevel 27If (p(x)=x^2-6x+4), what type of zeroes does it have?HardLevel 27If (2+\sqrt{3}) is a zero of a quadratic polynomial with rational coefficients, what will the other zero be?HardLevel 27Which quadratic polynomial has zeroes (\sqrt{7}) and (-\sqrt{7})?HardLevel 27If (p(x)=x^2-10x+1), what type of number is the product of its zeroes?HardLevel 27If the zeros of a quadratic polynomial are (5+\sqrt{2}) and (5-\sqrt{2}), what is the polynomial in standard form?HardLevel 27