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Q5.    Is it possible to design a rectangular park of perimeter 80 m and area 400 m2? If so, find its length and breadth.

Let us assume the length and breadth of the park be  respectively. Then, the perimeter will be  The area of the park is: Given :  Comparing to get the values of a, b and c. The value of the discriminant  As . Therefore, this equation will have two equal roots. And hence the roots will be: Therefore, the length of the park,  and breadth of the park .  

Q4.    Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.

Let the age of one friend be  and the age of another friend will be:  4 years ago, their ages were,  and . According to the question, the product of their ages in years was 48.   or   Now, comparing to get the values of . Discriminant value  As . Therefore, there are no real roots possible for this given equation and hence, This situation is NOT possible.    

Q3.    Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m2? If so, find its length and breadth.

Let the breadth of mango grove be . Then the length of mango grove will be . And the area will be: Which will be equal to  according to question. Comparing to get the values of . Finding the discriminant value: Here,  Therefore, the equation will have real roots. And hence finding the dimensions: As negative value is not possible, hence the value of breadth of mango grove will be...

Q2.    Find the values of k for each of the following quadratic equations, so that they have two equal roots.

                (ii)    kx(x-2) + 6 = 0

For two equal roots for the quadratic equation:  The value of the discriminant . Given equation:  Can be written as:  Comparing and getting the values of a,b, and, c. The value of  But  is NOT possible because it will not satisfy the given equation. Hence the only value of  is 6 to get two equal roots.  

Q2.    Find the values of k for each of the following quadratic equations, so that they have two equal roots.

                (i)    2x^2 + kx + 3 = 0

For two equal roots for the quadratic equation:  The value of discriminant . Given equation:  Comparing and getting the values of a,b, and, c. The value of  Or,   

Q1.    Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:

                (iii)    2x^2 - 6x + 3 = 0

The value of the discriminant  The discriminant > 0. Therefore the given quadratic equation has two distinct real root roots are So the roots are  

Q1.    Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:

                (ii)    3x^2 - 4\sqrt3x + 4 = 0

Here the value of discriminant =0, which implies that roots exist and the roots are equal. The roots are given by the formula So the roots are

Q1.    Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:

                (i)    2x^2 - 3x +5 = 0

For a quadratic equation,   the value of discriminant determines the nature of roots and is equal to: If D>0 then roots are distinct and real. If D<0 then no real roots. If D= 0 then there exists two equal real roots. Given the quadratic equation,  . Comparing with general to get the values of a,b,c. Finding the discriminant: Here D is negative hence there are no real roots possible for the...

Q11.    Sum of the areas of two squares is 468 m2 . If the difference of their perimeters is 24 m, find the sides of the two squares.

Let the sides of the squares be .              (NOTE: length are in meters) And the perimeters will be:  respectively. Areas  respectively. It is given that,                     .................................(1)                        .................................(2) Solving both equations:   or    putting in equation (1), we obtain Solving by the factorizing method: Here the roots...

Q10.    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 11km/h more than that of the passenger train, find the average speed of the two trains.

Let the average speed of the passenger train be . Given the average speed of the express train  also given that the time taken by the express train to cover 132 km is 1 hour less than the passenger train to cover the same distance. Therefore,  Can be written as quadratic form: Roots are:  As the speed cannot be negative. Therefore, the speed of the passenger train will be  and  The speed...

Q9.    Two water taps together can fill a tank in 9\frac{3}{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.

Let the time taken by the smaller pipe to fill the tank be  Then, the time taken by the larger pipe will be: . The fraction of the tank filled by a smaller pipe in 1 hour:   The fraction of the tank filled by the larger pipe in 1 hour. Given that two water taps together can fill a tank in  hours. Therefore, Making it a quadratic equation: Hence the roots are  As the time taken cannot...

Q8.    A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.

Let the speed of the train be  Then, time taken to cover  will be:  According to the question, Making it a quadratic equation. Now, solving by the factorizing method: However, the speed cannot be negative hence, The speed of the train is .

Q7.    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.

Given the difference of squares of two numbers is 180. Let the larger number be 'x' and the smaller number be 'y'. Then, according to the question:   and   On solving these two equations: Solving by the factorizing method: As the negative value of x is not satisfied in the equation:  Hence, the larger number will be 18 and a smaller number can be found by,  putting x = 18, we...

Q6.    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.

Let the shorter side of the rectangle be x m. Then, the larger side of the rectangle wil be . Diagonal of the rectangle: It is given that the diagonal of the rectangle is 60m more than the shorter side. Therefore,  Solving by the factorizing method: Hence, the roots are:  But the side cannot be negative. Hence the length of the shorter side will be: 90 m  and the length of the larger...

Q5.    In a class test, 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 of their marks would have been 210. Find her marks in the two subjects.

Let the marks obtained in Mathematics be 'm' then, the marks obtain in English will be '30-m'. Then according to the question: Simplifying to get the quadratic equation: Solving by the factorizing method: We have two situations when, The marks obtained in Mathematics is 12, then marks in English will be 30-12 = 18.   Or, The marks obtained in Mathematics is 13, then marks in English will...

Q4.    The sum of the reciprocals of Rehman’s ages, (in years) 3 years ago and 5 years from now is \frac{1}{3} . Find his present age.

Let the present age of Rehman be  years. Then, 3 years ago, his age was  years. and 5 years later, his age will be  years. Then according to the question we have, SImplifying it to get the quadratic equation: Hence the roots are:  However, age cannot be negative Therefore, Rehman is 7 years old in present.  

Q3.    Find the roots of the following equations:

                (ii)    \frac{1}{x+4} - \frac{1}{x- 7} = \frac{11}{30},\ x\neq -4,7

Given equation:  So, simplifying it,     or     Can be written as: Hence the roots of the given equation are:

Q3.    Find the roots of the following equations:

                (i)    x - \frac{1}{x} = 3, x\neq 0

Given equation:  So, simplifying it, Comparing with the general form of the quadratic equation: , we get Now, applying the quadratic formula to find the roots: Therefore, the roots are        

Q2.    Find the roots of the quadratic equations given in Q.1 above by applying the quadratic formula.

(i)  The general form of a quadratic equation is : , where a, b, and c are arbitrary constants. Hence on comparing the given equation with the general form, we get And the quadratic formula for finding the roots is: Substituting the values in the quadratic formula, we obtain Therefore, the real roots are:    (ii)  The general form of a quadratic equation is : , where a, b, and c are...

Q2.    Find the roots of the following quadratic equations, if they exist, by the method of completing the square:

                (iv)    2x^2 + x + 4 = 0

Given equation:  On dividing both sides of the equation by 2, we obtain Adding and subtracting   in the equation, we get Here the real roots do not exist (in the higher studies we will study how to find the root of such equations).  
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