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Quadratic Equations

Long Answer Type

31. In a class, the sum of Shefali’s marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product would have been 210. Find her marks in the two subjects.

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32. The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field.

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33. The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.

Let the smaller number = x
and the larger number = y
Then, $straight x squared space equals space 8 straight y$ ...(i)
and $straight y squared minus straight x squared space equals space 180$ ...(ii)
From (i) and (ii), we get
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$rightwards double arrow$ $straight y squared minus 8 straight y minus 180 space equals space 0$
which is a quadratic equation in y. We can solve it by factorisation method.
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$rightwards double arrow$ $straight y squared minus 8 straight y minus 180 space equals space 0$
which is a quadratic equation in y. We can solve it by factorisation method.
$rightwards double arrow space space space straight y squared minus 18 straight y plus 10 straight y minus 180 equals 0 rightwards double arrow space space straight y left parenthesis straight y minus 18 right parenthesis space plus space 10 left parenthesis straight y minus 18 right parenthesis space equals space 0 rightwards double arrow space space space space space space space space space space space space space space left parenthesis straight y minus 18 right parenthesis space left parenthesis straight y plus 10 right parenthesis space equals space 0 rightwards double arrow space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight y minus 18 space equals space 0 space space space space space or space space straight y space plus space 10 space equals 0 rightwards double arrow space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight y space equals space 18 space space space or space space straight y space equals space minus 10$
When y = -10, then $straight x squared equals negative 80$
$rightwards double arrow$ $straight x equals plus-or-minus square root of negative 80 end root space not an element of space straight R$
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When y = 18, then $straight x equals 8 cross times 18 space equals space 144$
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If smaller number = 12
Then, larger number = $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
If smaller number = -12
Then, larger number = $fraction numerator left parenthesis negative 12 right parenthesis squared over denominator 8 end fraction equals 144 over 8 equals 18 space Ans. space$
Hence, the required numbers are 18, 12 or 18, –12. Ans.

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34. A train travels 360 km at a uniform speed. If the speed had been 5 km/hr more, would have taken 1 hour less for the same journey. Find the speed of the train.

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35. Two water taps together can fill a tank in

space 9 3 over 8

hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

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36. An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is 11 km/hr more than that of the passenger train, find the average speed of the two trains.

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