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 (\alpha=3+\sqrt{11}) and (\beta=3-\sqrt{11}), what is (\alpha^2+\beta^2)?HardLevel 27If (p(x)=x^2-2mx+(m^2-3)), in what form will its zeroes be?HardLevel 27Which statement is always true if a quadratic polynomial has rational coefficients and one zero is (\sqrt{13})?HardLevel 27If p(x)=5x^2-5, what are its zeroes and their type?MediumLevel 27If (p(x)=x^2-12x+31), what is the difference between its zeroes?HardLevel 27If \(\alpha\) and \(\beta\) are zeroes of \(x^2-7x+5\), what type of numbers are \(\alpha\) and \(\beta\)?HardLevel 27Which polynomial has a rational sum of zeroes but both zeroes are irrational?HardLevel 27If (p(x)=x^2+2\sqrt{5}x+5), what is its real zero?HardLevel 27If one zero of (p(x)=x^2+ax+7) is (\sqrt{7}) and (a) is rational, what is (a)?HardLevel 27If (p(x)=x^2-3x-\sqrt{2}), which statement about the coefficients of (p(x)) is correct?HardLevel 27If p(x) = x² − (√2 + √3)x + √6, what are the zeroes?HardLevel 27If (p(x)=x^2-14x+45) and (q(x)=x^2-14x+40), which statement is correct?HardLevel 27If (p(x)=x^2-14x+45) and (q(x)=x^2-14x+43), which statement is correct?HardLevel 27If (\alpha=2+\sqrt{5}) and (\beta=2-\sqrt{5}), what is (\frac{1}{\alpha}+\frac{1}{\beta})?HardLevel 27If the zeroes of (x^2+bx+12) are (2+\sqrt{7}) and (2-\sqrt{7}), what is the error?HardLevel 27If \(p(x)=x^2-2x+5\), why are its zeroes not real?HardLevel 27Which option gives the correct simplified form of (\sqrt{12}), useful in simplifying polynomial zeroes?HardLevel 27If p(x)=x²−2√6x+5, what are the product and sum of its zeroes?MediumLevel 27Which statement is correct for (x^2-2ax+(a^2-b)), where (b>0) and (b) is not a perfect square?HardLevel 27If (p(x)=x^2-9x+14) and (q(x)=x^2-9x+15), which statement about the types of zeroes is correct?HardLevel 27