Mathers

  Q) Sum of the areas of two squares is 468 m². If the difference of their perimeters are 24 m, find the sides of the two squares. Solution) Let the sides of the two squares be x m and y m. Thus, their perimeters will be 4x and 4y and areas will be x² and y². Hence, according to question – =4x-4y=24 ⇒x-y=6 ⇒x=y+6 And x²+y² = 468 Substituting value of x- (y+6)²+y²=468 ⇒36+y²+12y+y²=468 ⇒2y²+12y-432=0 ⇒y²+6y-216=0 ⇒y²+18y-12y-216=0 ⇒y(y+18)-12(y+18)=0 ⇒(y+18)(y-12)=0 ⇒y=-18 or 12 Since, side cannot be negative. Therefore, the sides of the square are 12 m and (12+6)m i.e 18 m.

Q) 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 speeds of the express train is 11 km/h more than that of the passenger train, find the average speed of the two trains. Solution)  Let the average speed of passenger train be x km/hr. So, Average speed of express train be (x+11) km/hr Since, speed cannot be negative. Therefore, the speed of the passenger train will be 33 km/h and thus, the speed of the express train will be 44 km/h . HSHH

  Q) Two water taps together can fill a tank in 75/8 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. Solution) Let the time taken by the smaller pipe to fill the tank be x hr .  So , time taken by larger pipe be ( x - 10 ) hr .  Part of the tank filled by smaller pipe in 1 hour is 1/x  Part of the tank filled by larger pipe in 1 hour is  1/x-10   So , according to the question   Case 1- If time taken by smaller pipe be 30/8 i.e 3.75 hours .  So , Time taken by larger pipe will be negative which is not possible .  Hence , this case is rejected .  Case 2- If the time taken by smaller pipe be 25. Then , time taken by larger pipe will be 15 hours .  Therefore , time taken by smaller pipe be 25 hours and time taken by larger pipe will be 15 hours .

 Q) A train travels 360 kmkm at a uniform speed. If the speed had been 5km/h more, it would have taken 1hour less for the same journey. Find the speed of the train. Solution) Let the speed of the train be x km / h .  Time taken to cover 360 km/h be 360/x According to question ( x + 5 ) (360/X --1 ) = 360 =  360 - x + 1600/x -5 = 360   = x² + 5x - 1800 = 0  = x² + 45x - 40x - 1800 = 0  = x ( x + 45 ) -40 ( x + 45 ) = 0  = ( x + 45 ) ( x - 40 ) = 0  ⇒x = 40 , -45 X  Since , the speed cannot be negative .  Therefore , the speed of the train is 40 km / h .

Q)  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. Solution) Let the larger number be x and smaller number be y  According to question x² - y² = 180 and y² = 8x  ⇒ x² - 8x = 180 x² - 8x - 180-0  ⇒ x² - 18x + 10x - 180 = 0  ⇒ x ( x - 18 ) + 10 ( x - 18 ) = 0  ⇒ ( x - 18 ) ( x + 10 ) = 0  ⇒x = 18 , -10  Since , larger cannot be negative as 8 times of the larger number will be negative and hence , the square of the smaller number will be negative which is not possible .  Therefore , the larger number will be 18 . .. y² = 8 ( 18 ) ⇒ y² = 144 y = ± 12  Hence , smaller number be ± 12 .  Therefore , the numbers are 18 and 12 or 18 and - 12 .

 Q)  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. Solution) Let the shorter side of the rectangle be x m . Thus , Larger side of the rectangle will be ( x + 30 ) m . Diagonal of the rectangle be√x² + ( x + 30 ) ² Hence , according to question x² + ( x + 30 ) ² = x + 60 ⇒ x² + ( x + 30 ) ² = ( x + 60 ) ² = x² + x² + 900 + 60x = x² + 3600 + 120x = x² - 60x - 2700-0 =x² - 90x + 30x - 2700 = 0 =x ( x - 90 ) + 30 ( x - 90 ) = 0 ⇒ ( x - 90 ) ( x + 30 ) = 0 ⇒x = 90 , -30 Therefore , the length of the shorter side of rectangle is 90 m . Hence , length of the larger side of the rectangle be 120 m . Since , side cannot be negative .