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 (p(x)=x^2-2ax+(a^2-7)) and (a) is rational, which statement about the zeroes is correct?ExpertLevel 25For which value of \(k\) will \(x^2-6x+k\) have real and irrational zeroes?ExpertLevel 25If the zeroes of a monic quadratic polynomial are (4+\sqrt{11}) and (4-\sqrt{11}), what will be the constant term?ExpertLevel 25If 2 + √13 is one zero of a quadratic polynomial with rational coefficients, what can the coefficient of x be?HardLevel 25If (\alpha=5+\sqrt{6}) and (\beta=5-\sqrt{6}), what is the value of (\alpha^2+\beta^2)?ExpertLevel 25If (p(x)=x^2-10x+17), what is the difference between its zeroes?ExpertLevel 25If (p(x)=3x^2-18x+21), what type of zeroes does it have?ExpertLevel 25If (\sqrt{3}) and (\sqrt{12}) are zeroes of a monic quadratic polynomial, what will be the coefficient of (x)?ExpertLevel 25Which of the following polynomials has zeros \(3+\sqrt{2}\) and \(3-\sqrt{2}\)?ExpertLevel 25If (p(x)=x^2-(\sqrt{5}+\sqrt{7})x+\sqrt{35}), which is the correct pair of zeroes?ExpertLevel 25If (p(x)=x^2-2\sqrt{10}x+10), which statement about its zeroes is correct?ExpertLevel 25If (p(x)=x^2-4x-1), what is (\frac{1}{\alpha}+\frac{1}{\beta}), where (\alpha) and (\beta) are its zeroes?ExpertLevel 25Which polynomial has rational coefficients and both zeroes irrational real?ExpertLevel 25If (\alpha=1+\sqrt{2}) and (\beta=1-\sqrt{2}), what is (\alpha^3+\beta^3)?ExpertLevel 25If the zeroes of \(x^2+px+q\) are \(7+2\sqrt{3}\) and \(7-2\sqrt{3}\), what is the value of \(p+q\)?ExpertLevel 25If p(x)=x²+2x−8 and q(x)=x²+2x−7, which comparison is correct?HardLevel 25If (p(\sqrt{5})=0) and (p(x)=x^2+ax-5), what is the value of (a)?ExpertLevel 25If (p(x)=2x^2-8x+1), which is the correct form of its zeroes?ExpertLevel 25If (p(x)=x^2-2x-4), what is the sum of squares of its zeroes?ExpertLevel 25If the zeroes of x²−2x+m are 1+√6 and 1−√6, what is m?MediumLevel 25