Question : Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each given of the following : (i) p(x) = \(x^3 – 3x^2 + 5x – 3\), g(x) = \(x^2 – 2\) (ii) p(x) = \(x^4 – 3x^2 + 4x + 5\), g(x) = \(x^2 + 1 – x\) …
Question : Find a quadratic polynomial whose sum of zeroes and product of zeroes are respectively. (i) \(1\over 4\), -1 (ii) \(\sqrt{2}\), \(1\over 3\) (iii) 0, \(\sqrt{5}\) (iv) 1, 1 (v) \(-1\over 4\), \(1\over 4\) (vi) 4, 1 Solution : Let the polynomial be \(ax^2 + bx + c\) and its zeroes be \(\alpha\) and …
Find a quadratic polynomial whose sum of zeroes and product of zeroes are respectively. Read More »
Question : Find the zeroes of the quadratic polynomials and verify a relationship between zeroes and its coefficients. (i) \(x^2 – 2x – 8\) (ii) \(4s^2 – 4s + 1\) (iii) \(6x^2 – 3 – 7x\) (iv) \(4u^2 + 8u\) (v) \(t^2 – 15\) (vi) \(3x^2 – x – 4\) Solution : (i) \(x^2 – …
Question : The graph of y = p(x) are given below, for some polynomial p(x). Find the number of zeroes of p(x), in each case. Solution : (i) There are no zeroes as the graph does not intersect the x-axis. (ii) The number of zeroes is one as the graph intersect the x-axis at one point …
Question : The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational and of the form \(p\over q\), what can you say about the prime factors of q ? (i) 43.123456789 (ii) 0.120120012000120000….. (iii) 43.\(\overline{123456789}\) Solution : (i) 43.123456789 is terminating. …
The following real numbers have decimal expansions as given below. Read More »
Question : Write down the decimal expansion of these given rational numbers which have terminating decimal expansions. (i) \(13\over 3125\) (ii) \(17\over 8\) (iii) \(64\over 455\) (iv) \(15\over 1600\) (v) \(29\over 343\) (vi) \(23\over {2^3 5^2}\) (vii) \(129\over {2^2 5^7 7^5}\) (viii) \(6\over 15\) (ix) \(35\over 50\) (x) \(77\over 210\) Solution : (i) \(13\over 3125\) …
Question : Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion : (i) \(13\over 3125\) (ii) \(17\over 8\) (iii) \(64\over 455\) (iv) \(15\over 1600\) (v) \(29\over 343\) (vi) \(23\over {2^3 5^2}\) (vii) \(129\over {2^2 5^7 7^5}\) (viii) \(6\over 15\) (ix) …
Question : Prove that the following are irrationals : (i) \(1\over \sqrt{2}\) (ii) \(7\sqrt{5}\) (iii) \(6 + \sqrt{2}\) Solution : (i) Let us assume, to the contrary, that \(1\over \sqrt{2}\) is rational. That is, we can find co-prime integers p and q (q \(\ne\) 0) such that \(1\over \sqrt{2}\) = \(p\over q\) or \(1\times \sqrt{2}\over …
Solution : Let us assume, to the contrary, that \(3 + 2\sqrt{5}\) is an irrational number. Now, let \(3 + 2\sqrt{5}\) = \(a\over b\), where a and b are coprime and b \(ne\) 0. So, \(2\sqrt{5}\) = \(a\over b\) – 3 or \(\sqrt{5}\) = \(a\over 2b\) – \(3\over 2\) Since a and b are integers, …
Solution : Suppose that \(\sqrt{5}\) is an irrational number. Then \(\sqrt{5}\) can be expressed in the form \(p\over q\) where p, q are integers and have no common factor, q \(ne\) 0. \(\sqrt{5}\) = \(p\over q\) Squaring both sides, we get 5 = \(p^2\over q^2\) \(\implies\) \(p^2\) = \(5q^2\) …
Prove that \(\sqrt{5}\) is an irrational number by contradiction method. Read More »
