52. Quadratic Equations Problems Part 4

151. If \(x\) be real, show that \(\frac{x}{x^2 - 5x + 9}\) lies between \(1\) and \(-\frac{1}{11}\).

152. If \(p > 1\) and \(x\) is real, show that \(\frac{x^2 - 2x + p^2}{x^2 + 2x + p^2}\) lies between \(\frac{p -1}{p + 1}\) and \(\frac{p + 1}{p - 1}\)

153. If \(x\) be real, find the greatest and the least values of \(\frac{x^2 + 2x + 1}{x^2 + 2x + 7}\)

154. If \(x\) be real, find the greatest and the least values of \(\frac{x^2 + 14x + 9}{x^2 + 2x + 3}\)

155. If \(x\) be real, find the greatest and the least values of \(\frac{x^2 -x + 1}{x^2 + x + 1}\)

156. If \(x\) be real, find the greatest and the least values of \(\frac{x^2 - 3x - 3}{2x^2 + 2x + 1}\)

157. If \(x\) be real, find the greatest and the least values of \(\frac{2x^2 + x - 1}{x^2 + 4x + 2}\)

158. If \(x\) be real, find the greatest and the least values of \(\frac{x^2 - 4x + 9}{x^2 + 4x + 9}\)

159. If \(x\) be real, find the greatest and the least values of \(\frac{x^2 + 7x + 16}{x^2 - 5x + 16}\)

160. If \(x\) be real, find the greatest and the least values of \(\frac{6x^2 - 2x + 3}{2x^2 - 2x + 1}\)

161. If \(x\) be real, prove that the expression \(\frac{(x - 1)(x + 3)}{(x - 2)(x + 4)}\) does not lie between \(\frac{4}{9}\) and \(1\).

162. If \(x\) is real, show that the expression \(\frac{2x^2 - 2x + 4}{x^2 - 4x + 3}\) cannot lie between \(1\) and \(-7\).

163. If \(x\) be real, prove that the expression \(\frac{x^2 + 2x - 11}{-x - 3}\) can have all numerical values except those between \(4\) and \(12\).

164. If \(x\) be real, prove that \(\frac{x}{x^2 + 1}\) can never be greater than \(\frac{1}{2}\)

165. If \(a^2 + c^2 > ab\) and \(b^2 > 4c^2\) for real \(x\), show that \(\frac{x + a}{x^2 + bx + c^2}\) cannot lie between two limits.

166. Show that if \(x\) real, the expression \(\frac{x^2 - bc}{2x - b - c}\) has no real value between \(b\) and \(c\).

167. If \(x\) is real, show that \(\frac{1}{x + 1} + \frac{1}{3x + 1} - \frac{1}{(x + 1)(3x + 1)}\) can have any numerical value except those lying between \(1\) and \(4\).

168. If \(x\) be real, show that \(\frac{2x^2 + x - 3}{3x + 1}\) can have any real value.

169. If \(x\) be real, show that \(\frac{2x^2 + 4x + 1}{x^2 + 4x + 2}\) can have any real value.

170. Prove that for real values of \(x\) the expression \(\frac{ax^2 + 3x - 4}{3x - 4x^2 + a}\) may have any value provided \(a\) lies between \(1\) and \(7\).

171. Show that the expression \(\frac{m^2}{1 + x} - \frac{n^2}{1 - x}\) can have any real value for real values of \(x\).

172. If \(x\) is real show that \(\left|\frac{4x}{x^2 + 16}\right| \le \frac{1}{2}\).

173. Show that no real values of \(x\) and \(y\) besides \(4\) can satisfy the equation \(x^2 - xy + y^2 - 4x - 4y + 16 = 0\)

174. Prove that if \(x^2 + 12xy + 4y^2 + 4x + 8y + 20 = 0\) is satisfied by real values of \(x\) and \(y\), \(x\) cannot lie between \(-2\) and \(1\) whereas \(y\) cannot lie between \(-1\) and \(\frac{1}{2}\).

175. Find the values of \(m\) so that the expression \(x^2 - 5mx + 4m^2 + 1\) is always positive for real values of \(x\).

176. If \(x\) be real find the maximum value of \(-3x^2 + x + 2\) and the corresponding value of \(x\).

177. Show that the least value of the sum of a positive number and its reciprocal is \(2\).

178. A rectangular field, one of whose sides is a straight edge of a river is to be enclosed by \(600\) meters of fencing on the remaining three sides. What should be the length and breadth of the rectangle if the enclosed area is to be as large as possible.

179. Find the condition that the expression \(ax^2 _ 2hxy + by^2\) may have two factors of the form \(y - mx\) and \(my + x\).

180. For what values of \(k\) the expression \((4 - k)x^2 + 2(k + 2)x + 8k + 1\) will be a perfect square?

181. For what values of \(m\), \(3x^2 - xy - 2y^2 + mx + y + 1\) will resolve into two linear factors?

182. For what values of \(m\), \(6x^2 - 7xy - 3y^2 + mx + 17y - 20\) will resolve into two linear factors?

183. If \(\alpha\) is a common root between \(a_1x^2 + b_1x + c_1\) and \(a_2x^2 + b_2x + c_2\) then prove that \(\alpha(a_1 - a_2) = b_2 - b_1\)

184. Resolve \(6x^2 + 7xy + 2y^2 + 11x + 7y + 3\) into two linear factors.

185. Resolve \(x^2 - 5xy + 4y^2 + x + 2y -2\) into two linear factors.

186. Resolve \(2x^2 + 5xy - 3y^2 + x + 17y - 10\) into two linear factors.

187. Resolve \(3x^2 + 5xy - 2y^2 - 3x + 8y - 6\) into two linear factors.

188. if \(3x^2 + 2\alpha xy + 2y^2 + 2ax - 4y + 1\) can be resolved into two linear factors, prove that \(\alpha\) is a root of the equation \(x^2 + 4ax + 2a^2 + 6 = 0\).

189. If \(p, q , r, s\) are real and \(pr > 4(q + s)\) then show that at least one of the equations \(x^2 + px + q = 0\) and \(x^2 + rx + s = 0\) has real roots.

190. If \(P(x) = ax^2 + bx + c\) and \(Q(x) = -ax^2 + bx + c,\) where \(ac \ne 0,\) show that the equation \(P(x).Q(x) =0\) has at least two real roots.

191. Prove that the roots of the equation \(bx^2 + (b - c)x + b - c - a = 0\) are real if those of equation \(ax^2 + 2bx + b = 0\) are imaginary and vice-versa, where \(a, b, c \in R.\)

192. If \(a, b, c\) are odd numbers, show that the roots of equation \(ax^2 + bx + c = 0\) cannot be rational.

193. If roots of the equation \(ax^2 + 2bx + c = 0\) are real and distinct, then show that the roots of the equation \((a + c)(ax^2 + 2bx + c) = 2(ac - b^2)(x^2 + 1)\) are non-real complex numbers and vice-versa.

194. If \(n\) and \(r\) are positive integers such that \(0 < r < n,\) then show that the roots of the quadratic equation \(^nC_{r - 1}x^2 + ^nC_rx + ^nC_{r + 1} = 0\) are real and distinct.

195. find the integral values of \(m\) for which the roots of equation \(mx^2 + (2m - 1)x + m - 2 = 0\) are rational.

196. Show that the equation \(e^{\sin x} - e^{-\sin x} - 4 = 0\) has no real solutions.

197. If \(a, b, c\) are non-zero, real numbers and the equation \(az^2 + bz + c + i = 0\) have purely imaginary roots then prove that \(a = b^2c.\)

198. If \(a\) and \(b\) are integers and the roots of the equation \(x^2 + ax + b = 0\) are rational, show that they will be integers.

199. Show that the quadratic equation \(x^2 + 7x - 14(q^2 + 1) = 0,\) where \(q\) is an integer, has no integral roots.

200. Solve the equation \(a^3(b - c)(x - b)(x - c) + b^3(c - a)(x - a)(x - c) + c^3(a - b)(x - a)(x - b) = 0\). Also show that roots are equal if \(\frac{1}{\sqrt{a}}\pm \frac{1}{\sqrt{b}}\pm \frac{1}{c} = 0\)