73
(a) Find two consecutive odd integers, sum of whose squares is $290$.
OR
(b) A charity trust decides to build a rectangular hall having an area of $300\text{ m}^2$. The length of the hall is one metre more than twice its width. Find the length and breadth of the hall.
OR
(b) A charity trust decides to build a rectangular hall having an area of $300\text{ m}^2$. The length of the hall is one metre more than twice its width. Find the length and breadth of the hall.
Show SolutionHide Solution↓
(a) Let the two consecutive odd integers be $x$ and $x + 2$ [$\frac{1}{2}$ mark]
$x^2 + (x + 2)^2 = 290$ [$1\frac{1}{2}$ marks]
$2x^2 + 4x - 286 = 0$ or $x^2 + 2x - 143 = 0$ [$1\frac{1}{2}$ marks]
$(x - 11)(x + 13) = 0$
$x = 11$ [$1$ mark]
Required odd integers are $11$ and $13$ [$\frac{1}{2}$ mark]
OR
(b) Let width be $x\text{ m}$ and length be $(2x + 1)\text{ m}$ [$\frac{1}{2}$ mark]
A.T.Q. $(2x + 1)x = 300$ [$1\frac{1}{2}$ marks]
$2x^2 + x - 300 = 0$ [$1\frac{1}{2}$ marks]
$(x - 12)(2x + 25) = 0$
$x = 12$ [$1$ mark]
(Rejecting $x = -\frac{25}{2}$)
$\text{length} = 25\text{ m and width} = 12\text{ m}$ [$\frac{1}{2}$ mark]
$x^2 + (x + 2)^2 = 290$ [$1\frac{1}{2}$ marks]
$2x^2 + 4x - 286 = 0$ or $x^2 + 2x - 143 = 0$ [$1\frac{1}{2}$ marks]
$(x - 11)(x + 13) = 0$
$x = 11$ [$1$ mark]
Required odd integers are $11$ and $13$ [$\frac{1}{2}$ mark]
OR
(b) Let width be $x\text{ m}$ and length be $(2x + 1)\text{ m}$ [$\frac{1}{2}$ mark]
A.T.Q. $(2x + 1)x = 300$ [$1\frac{1}{2}$ marks]
$2x^2 + x - 300 = 0$ [$1\frac{1}{2}$ marks]
$(x - 12)(2x + 25) = 0$
$x = 12$ [$1$ mark]
(Rejecting $x = -\frac{25}{2}$)
$\text{length} = 25\text{ m and width} = 12\text{ m}$ [$\frac{1}{2}$ mark]