10th Maths 4.1
NCERT Class 10th solution of Exercise 4.2
NCERT Class 10th solution of Exercise 4.4
Exercise 4.1
Q1. Check whether the following are quadratic equations :
i) `(x+1)^2=2(x-3)`
ii) `x^2-2x=(-2)(3-x)`
iii) `(x-2)(x+1)=(x-1)(x+3)`
iv) `(x-3)(2x+1)=x(x+5)`
v) `(2x-1)(x-3)=(x+5)(x-1)`
vi) `x^2+3x+1=(x-2)^2`
vii) `(x+2)^3=2x(x^2-1)`
viii) `x^3-4x^2-x+1=(x-2)^3`
Sol. :
i)
`(x+1)^2=2(x-3)`
`x^2+2x+1=2x-6`
`x^2+2x+1-2x+6=0`
`x^2+7=0`
Answer :
Yes.
ii)
`x^2-2x=(-2)(3-x)`
`x^2-2x=-6+2x`
`x^2-2x+6-2x=0`
`x^2-4x+6=0`
Answer :
Yes.
iii)
`(x-2)(x+1)=(x-1)(x+3)`
`x(x+1)-2(x+1)=x(x+3)-1(x+3)`
`x^2+x-2x-2=x^2+3x-x-3`
`x^2-x-2=x^2+2x-3`
`x^2-x-2-x^2-2x+3=0`
`-3x+1=0`
`3x-1=0`
Answer :
No.
iv)
`(x-3)(2x+1)=x(x+5)`
`x(2x+1)-3(2x+1)=x^2+5x`
`2x^2+x-6x-3-x^2-5x=0`
`x^2-10x-3=0`
Answer :
Yes.
v)
`(2x-1)(x-3)=(x+5)(x-1)`
`2x(x-3)-1(x-3)=x(x-1)+5(x-1)`
`2x^2-6x-x+3=x^2-x+5x-5`
`2x^2-7x+3=x^2+4x-5`
`2x^2-7x+3-x^2-4x+5=0`
`x^2-11x+8=0`
Answer :
Yes.
vi)
`x^2+3x+1=(x-2)^2`
`x^2+3x+1=x^2-4x+4`
`x^2+3x+1-x^2+4x-4=0`
`7x+3=0`
Answer :
No.
vii)
`(x+2)^3=2x(x^2-1)`
`x^3+8+6x^2+12x=2x^3-2x`
`x^3+8+6x^2+12x-2x^3+2x=0`
`-x^3+6x^2+14x+8=0`
`x^3-6x^2+14x+8=0`
Answer :
No.
viii)
`x^3-4x^2-x+1=(x-2)^3`
`x^3-4x^2-x+1=x^3-8-6x^2+12x`
`x^3-4x^2-x+1-x^3+8+6x^2-12x=0`
`2x^2-13x+9=0`
Answer :
Yes.
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Q2. Represent the following situations in the form of quadratic equations :
i) The area of a rectangular plot is `528 m^2.` The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.
ii) The product of two consecutive positive integers is `306` we need to find the integers.
iii) Rohan's mother is `26` years older than him. The product of their ages (in years) three years from now will be `360`. We would like to find Rohan's present age.
iv) A train travels a distance of `480`km at a uniform speed. If the speed had been `8`km/hr less, then it would have taken `3` hours more to cover the same distance, we need to find the speed of the train.
Sol. :
i)
Let the breadth is `x`units,
and the length is `(x+1)`units.
According to question
Area of a rectangle = Length`times`breadth
`528=(2x+1)x`
`528=2x^2+x`
`2x^2+x-528=0`
Answer :
The quadratic equation `2x^2+x-528=0`.
ii)
Let first positive integer is `x`,
and second integer is `(x+1)`
According to question
`x(x+1)=306`
`x^2+x=306`
`x^2+x-306=0`
Answer :
The quadratic equation `x^2+x-306=0`.
iii)
Let the present age of Rohan is `x` years,
and the age of his mother `(x+26)`
According to question
`(x+3)(x+26+3)=360`
`x(x+29)+3(x+29)=360`
`x^2+29x+3x+87=360`
`x^2+32x+87-360=0`
`x^2+32x-273=0`
Answer :
The quadratic equation `x^2+32x-273=0`.
iv)
Let the speed of the train is `x`km/h.
According to question
Time taken to cover `480`km distance`=480/x`hr
`480/(x-8)=480/x=3`
`480/(x-8)-480/x=3`
`160/(x-8)-160/x=3`
`(160x-160(x-8))/(x(x-8))=1`
`1280=x^2-8x`
`x^2-8x-1280=0`
Answer :
The quadratic equation `x^2-8x-1280=0`.
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