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)(x+1)2=2(x−3)
ii) x2-2x=(-2)(3-x)x2−2x=(−2)(3−x)
iii) (x-2)(x+1)=(x-1)(x+3)(x−2)(x+1)=(x−1)(x+3)
iv) (x-3)(2x+1)=x(x+5)(x−3)(2x+1)=x(x+5)
v) (2x-1)(x-3)=(x+5)(x-1)(2x−1)(x−3)=(x+5)(x−1)
vi) x2+3x+1=(x-2)2x2+3x+1=(x−2)2
vii) (x+2)3=2x(x2-1)(x+2)3=2x(x2−1)
viii) x3-4x2-x+1=(x-2)3x3−4x2−x+1=(x−2)3
Sol. :
i)
(x+1)2=2(x-3)(x+1)2=2(x−3)
x2+2x+1=2x-6x2+2x+1=2x−6
x2+2x+1-2x+6=0x2+2x+1−2x+6=0
x2+7=0x2+7=0
Answer :
Yes.
ii)
x2-2x=(-2)(3-x)x2−2x=(−2)(3−x)
x2-2x=-6+2xx2−2x=−6+2x
x2-2x+6-2x=0x2−2x+6−2x=0
x2-4x+6=0x2−4x+6=0
Answer :
Yes.
iii)
(x-2)(x+1)=(x-1)(x+3)(x−2)(x+1)=(x−1)(x+3)
x(x+1)-2(x+1)=x(x+3)-1(x+3)x(x+1)−2(x+1)=x(x+3)−1(x+3)
x2+x-2x-2=x2+3x-x-3x2+x−2x−2=x2+3x−x−3
x2-x-2=x2+2x-3x2−x−2=x2+2x−3
x2-x-2-x2-2x+3=0x2−x−2−x2−2x+3=0
-3x+1=0−3x+1=0
3x-1=03x−1=0
Answer :
No.
iv)
(x-3)(2x+1)=x(x+5)(x−3)(2x+1)=x(x+5)
x(2x+1)-3(2x+1)=x2+5xx(2x+1)−3(2x+1)=x2+5x
2x2+x-6x-3-x2-5x=02x2+x−6x−3−x2−5x=0
x2-10x-3=0x2−10x−3=0
Answer :
Yes.
v)
(2x-1)(x-3)=(x+5)(x-1)(2x−1)(x−3)=(x+5)(x−1)
2x(x-3)-1(x-3)=x(x-1)+5(x-1)2x(x−3)−1(x−3)=x(x−1)+5(x−1)
2x2-6x-x+3=x2-x+5x-52x2−6x−x+3=x2−x+5x−5
2x2-7x+3=x2+4x-52x2−7x+3=x2+4x−5
2x2-7x+3-x2-4x+5=02x2−7x+3−x2−4x+5=0
x2-11x+8=0x2−11x+8=0
Answer :
Yes.
vi)
x2+3x+1=(x-2)2x2+3x+1=(x−2)2
x2+3x+1=x2-4x+4x2+3x+1=x2−4x+4
x2+3x+1-x2+4x-4=0x2+3x+1−x2+4x−4=0
7x+3=07x+3=0
Answer :
No.
vii)
(x+2)3=2x(x2-1)(x+2)3=2x(x2−1)
x3+8+6x2+12x=2x3-2xx3+8+6x2+12x=2x3−2x
x3+8+6x2+12x-2x3+2x=0x3+8+6x2+12x−2x3+2x=0
-x3+6x2+14x+8=0−x3+6x2+14x+8=0
x3-6x2+14x+8=0x3−6x2+14x+8=0
Answer :
No.
viii)
x3-4x2-x+1=(x-2)3x3−4x2−x+1=(x−2)3
x3-4x2-x+1=x3-8-6x2+12xx3−4x2−x+1=x3−8−6x2+12x
x3-4x2-x+1-x3+8+6x2-12x=0x3−4x2−x+1−x3+8+6x2−12x=0
2x2-13x+9=02x2−13x+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 528m2.528m2. 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 306306 we need to find the integers.
iii) Rohan's mother is 2626 years older than him. The product of their ages (in years) three years from now will be 360360. We would like to find Rohan's present age.
iv) A train travels a distance of 480480km at a uniform speed. If the speed had been 88km/hr less, then it would have taken 33 hours more to cover the same distance, we need to find the speed of the train.
Sol. :
i)
Let the breadth is xxunits,
and the length is (x+1)units.
According to question
Area of a rectangle = Length×breadth
528=(2x+1)x
528=2x2+x
2x2+x-528=0
Answer :
The quadratic equation 2x2+x-528=0.
ii)
Let first positive integer is x,
and second integer is (x+1)
According to question
x(x+1)=306
x2+x=306
x2+x-306=0
Answer :
The quadratic equation x2+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
x2+29x+3x+87=360
x2+32x+87-360=0
x2+32x-273=0
Answer :
The quadratic equation x2+32x-273=0.
iv)
Let the speed of the train is xkm/h.
According to question
Time taken to cover 480km distance=480xhr
480x-8=480x=3
480x-8-480x=3
160x-8-160x=3
160x-160(x-8)x(x-8)=1
1280=x2-8x
x2-8x-1280=0
Answer :
The quadratic equation x2-8x-1280=0.
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