Quadratic Equations in one variable

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Exercise 1.1

1. I write quadratic polynomials from the following polynomials by understanding it.
(i) x2-7x+2                       
(ii) 7x5-x(x+5)+1                    
(iii) 2x (x+5)+1                     
(iv) 2x-1

2.Which of the following equations can be written in the form of ax2+bx+c=0, where c, b, c are real numbers and a≠0, let us write it.

3. Let us determine the power of the variable for which the equation x6-x3-2=0 will become a quadratic equation?

4.(i) Let us determine the value of ‘a’ for which the equation (a-2)x2+3x+5=0 will not be a quadratic equation?

4.(ii) If x/4-x = 1/3x, (x≠0, x≠4) be expressed in the form of ax2+bx+c= (a≠0), then let us determine the co-efficient of x.

4.(iii) Let us express the equation 3x2+7x+23 = (x+4) (x+3) +2 in the form of the quadratic equation ax2+bx+c=0 (a≠0)

4.(iv) Let us express the equation (x+2)3 = x(x2-1) in the form of ax2+bx+c=0 (a≠0) and write the co-efficient of x2, x and x0.

5. Let us construct the quadratic equation in one variable from the following statements.
(i) Divide 42 into two part such that one part is equal to the square of the other part.

5.(ii) The product of two consecutive positive odd numbers is 143.

5.(iii) The sum of the squares of two consecutive number is 313.

6. let us construct the quadratic equations in one variable from the following statements.
(i) The length of the diagonal of a rectangular area is 15 m and the length exceeds its breadth by 3 m.

6.(ii) One person bought some kg sugar in ₹80. If he would get 4 kg more sugar with that money, then the price of 1 kg sugar would be less by ₹ 1.

6.(iii) The distance between two stations is 300 km. A train went to second station from first station with uniform velocity. If the velocity of the train is 5km/hour more, then the time taken by the train to reach the second station would be lesser by 2 hours.

6.(iv) A clock seller sold a clock by producing it at ₹ 336. The amount of his profit percentage is as much as the amount with which he bought the clock.

6.(v) If the velocity of the stream is 2 kms/hr, then the time taken by Ratanmajhi to cover 21 kms in down stream and upstream is 10 hours.

6.(vi) The time taken to clean out garden by Majid is 3 hours more than Mahim. Both of them together can complete the work in 2 hours.

6.(vii) The unit digit of a two digit number exceeds its tens’ digit by 6 and the product of two digits is less by 12 from the number.

6.(viii) There is a road of equal width around the outside of a rectangular playground having the length 45 m and breadth 40 m and the area of the road is 450 sqm.

Exercise 1.2

1.In each of the following cases, let us justify & write whether the given values are the root of the given quadratic equation:

2.(i) Let us calculate and write the value of k for which 2/3 will be a root the quadratic equation 7x2+x+1 = 0.

2.(ii) Let us
calculate and write the value of k for which -a will be a root the quadratic equation
x2+3ax+k = 0.

3. If 2/3 and -3 are the two roots of the quadratic equation ax2+7x+b = 0, then let me calculate the values of a and b.

4. Let us solve:

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Exercise 1.3

1. The difference of two positive whole numbers is 3 and sum of their squares is 117, by calculating, let us write the two numbers.

2. The base of the triangle is 18m more than two times of its heights, if the area of the triangle is 360 sq.m, then let us determine the height of it.

3. If 5 times of a positive whole numbers is less by 3 than twice of its square, then let us determine the number.

4. The distance between two places is 200 km, the time taken by motor car from one place to another is less by 2 hrs than the time if the speed of the motor car is 5 kms/hr more than the speed of the zeep car, then by calculating let us write the speed of the motor car.

5. The area of the Amita’s rectangular land is 2000sq.m and perimeter of it is 180 m by calculating, let us write the length and breadth of the Amita’s land.

6. The tens digit of a two-digit number is less by 3 than the unit digit. If the product of the two digits are subtracted from the number, the result is 15. Let us write the unit digit of the number by calculation.

7. There are two pipes in a water reservoir of our school, two pipes together take 11 1/9 minutes to fill the reservoir. If the two popes are opened separately, then one pope would take 5 minutes more time than the other pope. Let us write by calculating, the time taken to fill the reservoir separately by each of the popes.

8. Porna and Pijush together complete a work in 4 days. If they work separately, then the time taken by porna would by 6 days more than the time taken by Pijush. Let us write by calculating, the time taken by Porna alone to complete the work.

9. If the price of 1 dozen pen is reduced by ₹6, then 3 more pens will be got in ₹ 30. Before the reduction of price, let us calculate the prize of 1 dozen pen.

Exercise 1.4

1.(i) Let us write by understanding whether Sridhara Acharyya’s formula is applicable or not applicable to solve the equation 4x2+(2x-1) (2x+1) = 4x(2x-1)

1.(ii) Let us write by understanding what type of equation can be solved with the help of Sridhara Acharyya’s formula.

1.(iii) By applying Sridhara Acharyya’s formula in the equation 5x2+2x-7 = 0, it is found that x = (k+12)/10, let us write by calculating, what will be the value of k.

2. If the following quadratic equations have real roots, then let us determine them with the help of Sridhara Acharyya’s formula.

(i) 3x2+11x-4 = 0

2.(ii) (x-2) (x+4) +0 = 0

2. (iii) (4x-3)2-2(x+3) = 0

2. (iv) 3x2+2x-1 = 0

2.(v) 3x2+2x+1 = 0 

2.(vi) 10x2-x-3 = 0

2.(vii) 10x2-x+3 = 0

2.(viii) 25x2-30x+7 = 0

2.(ix) (4x-2)2+6x = 25

3. Let us express the following mathematical problems in the form of quadratic equation with one variable and solve them by applying Shridhara Acharyya’s formula or with the help of factorization.

(i) Sathi has drawn a right-angled triangle whose length of the hypotenuse is 6 cm more than the twice of the shortest side. If the length of the third side is 2 cm. less than the length of the hypotenuse, then by calculating, let us write the length of three sides of the right-angled triangle drawn by Sathi.

(ii) If a two-digit positive number is multiplied by its unit digit, then the product is 189 and if the tens digit is twice of the unit digit then let us calculate the unit digit.

(iii) The speed of Salma is 1m/second more than the speed of Anik in a 180 m run, Salma reaches 2 seconds before than Anik. Let us write by calculating the speed of Anik in m/sec.

(iv) There is a squared park in our locality. The area of a rectangular Park is 78 sqm less than the twice of the area of that square shaped Park whose length is 5 m. more than the length of the side of that park and the breadth is 3 m. less than the length of the side of that park. Let us right by calculating, the length of the side of the square shaped Park.

(v) In our village, proloy babu bought 350 Chili plants for planting in his rectangular land. when he put the plant in rows, he noticed that, if he would put 24 rows more than the number of plants in each row, 10 plants would remain excess. Let let us right by calculating the number of plants he put in each row.

(vi) Joseph and Kuntal work in a factory. Joseph takes 5 minutes less time than Kuntal to make a product. Joseph make 6 products more than Kuntal while working for 6 hours. Let us right by calculating the number of products Kuntal makes during that time.

(vii) The speed of a boat in still water is 8 kms/ hr. If the boat can go 15 kms. downstream and 22 kms. upstream in 5 hours, then let us write by calculating, the speed of the stream.

(viii) A superfast train runs having the speed 15 km/hr more than that of an express train. Leaving same station, the superfast train reached at a station of 180 kms distance 1 hour before than the express train. Let us determine the speed of the super-fast train in km/hr.

(ix) Rehana went to the market and saw that the price of dal of 1kg is ₹ 20 and the price of rice of 1 kg is ₹ 40 less than that of fish of 1 kg the quantity of fish and that of dal in ₹ 240 is equal to the quantity of rice in ₹ 280 let us calculate the cost price of 1 kg fish.

Exercise 1.5

1. Let us write the nature of two roots of the following quadratic equations:

2. By calculating, let us write the value/values of k for which each of the following quadratic equations has real and equal roots-

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3. let us form the quadratic equations form two roots given below-

(i) 4, 2  (ii) -4, -3 
(iii) -4, 3  (iv) 5, -3

4. What value of m for which two roots of the quadratic equation 4x2+4(3m-1)x+(m+7)=0 are reciprocal to each other.

5. If two roots of the quadratic equation (b-c) x2+(c-a)x+(a-b)=0 are equal, then let us prove that, 2b=a+c

6. If two roots of
the quadratic equation (a2+b2) x2-2(ac+bd) x+(c2+d2)
=0 are equal, then let us prove that, a/b=c/d

7. Let us prove that, the quadratic equation 2(a2+b2) x2+2(a+b)x+1=0 has no real root, if a≠b.

8. If two roots of the quadratic equation 5x2+2x-3=0 is α and β, the let us determine the value of,

8. 
         

9. If one root of the equation ax2+bx+c=0 if twice of the other, then let us show that, 2b2=9ac.

10. Let us form the equation whose roots
are reciprocals to the roots of the equation x2+px+1=0.

11. Let us determine the equation whose roots are square of the roots of the equation x2+x+1=0.

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