Variation

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  1. Corresponding values of two variables A and B are:

    If there is any relation of variation between A and B, let us determine it and write the value of variation constant.

 

2. Corresponding value of two variables X and Y. If there is any relation of variation between x and y, let us write it by understanding.

3.(i) A taxi of Bipin uncle travels 14 km path in 25 minutes. Let us calculate by applying theory of variation how much path he will go in 5 hours by driving taxi with same speed.

3.(ii) A box of sweets is divided among 24 children of class one of our school, they will get 5 sweets each. Let us calculate by applying theory of variation how many sweets would each get, if the number of the children is reduced by 4.

3.(iii) 50 villagers had taken 18 days to dig a pond. Let us calculate by using theory of variation how many extra persons will be required to dig the pond in 15 days.

4.(i) y varies directly with square root of ax and y=9 when x=9. Let us find the value of x when y=6.

4.(ii) X varies directly with y and inversely with z. When y=5, x=9, then x=3. Again. If y=16, z=30, let us write by calculating the value of x.

4.(iii) x varies directly with y and inversely with z. When y=5, z=9, then x=1/6. Let us find the relation among three variables x, y and z and if y=6 and z=1/5, let us write by calculating the value of x.

5.(i) if X ∝ Y, let us show that x+y ∝ x-y

5.(ii) A ∝ 1/C, C ∝ 1/B let u show that A ∝ B.

5.(iii) If a ∝ b, b ∝ 1/c and c ∝ d, let us write the relation of variation between a and d.

5.(iv) If x ∝ y, y ∝ z and z ∝ x. Let us find the relation among three constants of variation.

6.If x+y ∝ x-y, let us show that 
(i) x2+y∝ xy

6.(ii) x3+Y3 ∝ x3-y3

6.(iii) ax+by ∝ px+qy [Where a, b, p, q are non zero constant]

7.(i) If a2+b2 ∝ ab, let us prove that a+b ∝ a-b

7.(ii) If x3+y3 ∝ x3-y3, let us prove that x+y ∝ x-y

8. If 15 farmers can cultivate 18 bighas of land in 5 days, let us determine by using theory of variation the number of days required by 10 farmers to cultivate 12 bighas of land.

9. Volume of a sphere varies directly with the cube of its radius. Three solid spheres having length of 1 1/2, 2 and 2 ½ meter diameter of the new sphere. [let us consider that the volume of sphere remains same before and after melting]

10.Y is a sum of two variables, one of which varies directly with x and another varies inversely with x. When x=1, then y=1 and when x=3, then y=5. Let us find the relation between x and y.

11. If a ∝ b, b ∝ c let us show that a3b3+b3c3+c3a∝ abc(a3+b3+c3)

12. To dig a well of x dcm deep. one part of the total expenses the varies directly with x and other part caries directly with x2. If the expenses of digging wells of 100 dcm and 200 dcm depths are ₹ 5000 and ₹ 12000 respectively, let us write by calculating the expenses of digging a well of 250 dcm depth.

13.Volume of a cylinder is in joint variations with square of the length of radius of base and its height. Ratio of radii of bases of two cylinders is 2 : 3 and ratio of their heights is 5 : 4, let us find the ratio of their volumes.

14. An agricultural Co-operative Society of village of Pachla has purchased a tractor. Previously 2400 bighas of land were cultivated by 25 ploughs in 36 days. Now half of the land can be cultivated only by that-tractor in 30 days. Let us calculate by using the theory of variations, the number of plough work equally with one tractor.

15. Volume of sphere varies directly with cube of length of its radius and surface area of sphere varies directly with the square of the length of radius. Let us prove that the square of volume of sphere varies directly with cube of its surface area.

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