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Quadratic Equations

Long Answer Type

191. The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm. Find the other two sides.

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192. The sum of the areas two squares is 640 m². If the difference in their perimeter be 64 m, Find the sides of the two squares.

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193. If twice the area of a smaller square is subtracted from the area of a larger square, the result is 14 sq cm. However, if twice the area of larger square is added to three times the area of the smaller square, the result is 203 sq. cm. Find the sides of the two squares.

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Short Answer Type

194. The hypotenuse of a right triangle is

3 square root of 5 space cm.

If the smaller side is tripled and the larger side is doubled, the new hypotenuse will be 15 cm. Find the length of each side.

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Long Answer Type

195. A farmer wishes to grow a 100 m²rectangular vegetable garden. Since he has with the only 30 m barbed wire, he fences three sides of the rectangular garden letting compound wall of his house act as the four the side-fence. Find the dimension of his garden.

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196. There is a square field whose side is 44 m. A square flower bed is prepared in its centre leaving a gravel path all round the flower bed. The total cost of laying the flower bed and gravelling the path at Rs. 2.75 and Rs. 1.50 per square metre, respectively, is Rs. 4904. Find the width of the gravel path.

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197. Two pipes running together can fill a cistern in $space space 3 1 over 13 space minutes.$ If one pipe takes 3 minutes more than the other to fill it, find the time in which each pipe would fill the cistern.

Suppose the time taken by the faster pipe is x minutes.
∴ Time taken by the slower pipe = (x + 3) minutes
Now,
Portion of cistern filled by the faster pipe $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$ and portion of cistern filled by the slower pipe = $space space fraction numerator 1 over denominator straight x plus 3 end fraction$ and portion of the cistern filled by the both in 1 minute

 $equals 1 over straight x plus fraction numerator 1 over denominator straight x plus 3 end fraction equals space fraction numerator straight x plus 3 plus straight x over denominator straight x left parenthesis straight x plus 3 right parenthesis end fraction equals fraction numerator 2 straight x plus 3 over denominator straight x left parenthesis 2 straight x plus 3 right parenthesis end fraction space... space left parenthesis straight i right parenthesis$
It is given that,
Portion of the cistern filled by both 1 minute $equals 13 over 40$ ...(ii)
Comparing (i) and (ii), we get
 $fraction numerator 2 straight x plus 3 over denominator straight x left parenthesis straight x plus 3 right parenthesis end fraction space equals space 13 over 40$

$rightwards double arrow$ $space fraction numerator 2 straight x plus 3 over denominator straight x squared plus 3 straight x end fraction equals 13 over 40$
$rightwards double arrow$ $40 left parenthesis 2 straight x plus 3 right parenthesis space equals space 13 left parenthesis straight x squared plus 3 straight x right parenthesis$
$rightwards double arrow$ $80 straight x plus 120 space equals space 13 straight x squared plus 39 straight x$
$rightwards double arrow$ $13 straight x squared minus 41 straight x minus 120 space equals space 0$
$rightwards double arrow space space 13 straight x squared minus 65 straight x plus 24 straight x minus 120 space equals space 0$

$space rightwards double arrow space 13 straight x left parenthesis straight x minus 5 right parenthesis space plus space 24 left parenthesis straight x minus 5 right parenthesis space equals space 0 rightwards double arrow space space space space space space space space space space left parenthesis 13 straight x plus 24 right parenthesis space left parenthesis straight x minus 5 right parenthesis space equals space 0 rightwards double arrow space space space space space space space space space space space space space space space space space space space 13 straight x plus 24 space equals space 0 space space space space space space space or space straight x minus 5 space equals space 0 rightwards double arrow space space space space space space space space space space space space space space space space space space space space space space space space 13 straight x space equals space minus 24 space space space rightwards double arrow space space space straight x space equals space 5 rightwards double arrow space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space straight x equals fraction numerator negative 24 over denominator 13 end fraction$

(which is rejected) Hence, time taken by the faster pipe = x = 5 minutes and time taken by the slower pipe = (x + 3) = 5 + 3 = 8 minutes.

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198. A person on tour has Rs. 360 for his expenses. If he extends his tour for 4 days, he has to cut down his daily expenses by Rs. 3. Find the original duration of the tour.

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199. Rs. 6500 were divided equally among a certain number of persons. Had there been 15 more persons, each would have got Rs. 30 less. Find the original number of persons.

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200. A shopkeeper buys a number of books for Rs. 80. If he had bought 4 more books for the same amount, each book would have cost Rs. 1 less. How many books did he buy?

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