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 \(p(x)=x^2+ax-6\) is \(-\sqrt{6}\) and the other is \(\sqrt{6}\), what is the value of \(a\)?HardLevel 28Which polynomial has zeros \(1+\sqrt{2}\) and \(1-\sqrt{2}\)?HardLevel 28Which of the following polynomials has zeros \,(\sqrt{2}+\sqrt{3}) and \,(\sqrt{2}-\sqrt{3})?HardLevel 28If a quadratic polynomial has rational coefficients and one zero is \\(4+\sqrt{11}\\), what is the sum of its zeroes?HardLevel 28If \(p(x)=x^2+2x-1\), what are its zeroes?HardLevel 28If \(p(x)=x^2-2x-3\), are both of its zeroes real?HardLevel 28In which case will the zeroes of (x^2-5x+c) be real and irrational?HardLevel 28If \(x=\sqrt{2}+\sqrt{5}\), which polynomial equation is satisfied by \(x\)?HardLevel 28If \(p(x)=x^2-2\sqrt{3}x+2\), what is the product of its zeros?HardLevel 28If zeroes of (p(x)=x^2-mx+9) are (3\sqrt{2}) and (\frac{3}{\sqrt{2}}), what is (m)?HardLevel 28If (\alpha=\sqrt{12}) and (\beta=-\sqrt{3}), what is (\alpha+\beta)?HardLevel 26For which polynomial does substituting (x=\sqrt{3}) give value 0?HardLevel 26If the zeros of \(f(x)=x^2+px+q\) are \(7+\sqrt{13}\) and \(7-\sqrt{13}\), what is the value of \(p+q\)?HardLevel 26If the zeros of \(p(x)=x^2-6x+n\) are \(3+\sqrt{m}\) and \(3-\sqrt{m}\), and \(n=5\), what is the value of \(m\)?HardLevel 26If \(p(x)=x^2-4\sqrt{2}x+8\), in which form can it be written?HardLevel 26If zeroes of (x^2-2ax+(a^2-5)=0) are (a+\sqrt{5}) and (a-\sqrt{5}), why is it correct?HardLevel 26If \(p(x)=x^2-2x-2\), what is the product of its zeroes?HardLevel 26If \(p(x)=x^2-2\sqrt{7}x+7\), what is the number and nature of its zeroes?HardLevel 26If \(p(x)=x^2+2\sqrt{3}x+3\), what are its zeros?HardLevel 26If the zeros of a polynomial are \(2\sqrt{2}\) and \(3\sqrt{2}\) and the leading coefficient is 1, what is its constant term?HardLevel 26