10th Maths 2.2
NCERT Class 10th solution of Exercise 2.1
Exercise 2.2
Q1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
i) `x^2-2x-8` ii) `4s^2-4s+1` iii) `6x^2-3-7x`
iv) `4u^2+8u` v) `t^2-15` vi) `3x^2-x-4`
Sol. :
i) Given quadratic equation `x^2-2x-8` compare with `ax^2+bx+c=0`
where `a=1, b=-2` and `c=-8`
Let
`x^2-2x-8=0`
`x^2-4x+2x-8=0`
`x(x-4)+2(x-4)=0`
`(x-4)(x+2)=0`
`x-4=0`
`x=4`
`x+2=0`
`x=-2`
Now, the sum of the zeroes`= -b/a`
`alpha+beta=-((-2))/1`
`4-2=-((-2))/1`
`2=2`
and the product of the zeroes `=c/a`
`alpha times beta = -8/1`
`4 times (-2) = -8`
`-8 = -8`
Answer :
The required zeroes `4` or `-2` and the relation of zeroes & coefficient are verified.
ii) Given quadratic equation `4s^2-4s+1` compare with `ax^2+bx+c=0`
where `a=4, b=-4` and `c=-1`
Let
`4s^2-4s+1=0`
`4s^2-2s-2s+1=0`
`2s(2s-1)-1(2s-1)=0`
`(2s-1)(2s-1)=0`
`2s-1=0`
`s=1/2`
`2s-1=0`
`s=1/2`
Now, the sum of the zeroes `= -b/a`
`alpha+beta=-((-4)/4)`
`1/2+1/2=1`
`1=1`
and the product of the zeroes`c/a`
`alpha times beta= 1/4`
`1/2 times1/2 = 1/4`
`1/4 = 1/4`
Answer :
The required zeroes `1/4, 1/4` and the relation of zeroes & coefficient are verified.
iii) Given quadratic equation `6x^2-3-7x=6x^2-7x-3` compare with `ax^2+bx+c=0`
where `a=6, b=-7` and `c=-3`
Let
`6x^2-7x-3=0`
`6x^2-9x+2x-3=0`
`3x(2x-3)+1(2x-3)=0`
`(2x-3)(3x+1)=0`
`2x-3=0`
`x=3/2`
`3x+1=0`
`x=-1/3`
Now, the sum of the zeroes`= -b/a`
`alpha+beta= - ((-7)/6)`
`3/2 + (-1/3) = 7/6`
`7/6 = 7/6`
and the product of the zeroes`= c/a`
`alpha times beta=-3/6`
`3/2 times(-1/3) = -1/2`
`-1/2= -1/2`
Answer :
The required zeroes `2/3, -1/3` and the relation of zeroes & coefficient are verified.
iv) Given quadratic equation `4u^2+8u` compare with `ax^2+bx+c=0`
where `a=4, b=8` and `c=0`
Let
`4u^2+8u=0`
`4u(u+2)=0`
`4u=0`
`u=0`
`u+2=0`
`u=-2`
Now, sum of the zeroes`= -b/a`
`alpha+beta= - 8/4`
`0+(-2) = -2`
`-2 = -2`
and product of the zeroes`=c/a`
`alpha times beta=0/4`
`0times-2 = 0`
`0=0`
Answer :
The required zeroes are `0, -2`, and the relation of zeroes & coefficient are verified.
v) Given quadratic equation `t^2-15` compare with `ax^2+bx+c=0`
where `a=1, b=0` and `c=-15`
Let
`t^2-15=0`
`t^2-(sqrt15)^2=0`
`(t-sqrt15)(t+sqrt15)=0`
`t-sqrt15=0`
`t=sqrt15`
`t+sqrt15=0`
`t=-sqrt15`
Now, the sum of the zeroes `-b/a`
`alpha+beta=-(0/(-sqrt15))`
`sqrt15+(-sqrt15)=0`
`0=0`
and the product of the zeroes` = c/a`
`alpha times beta=-15`
`sqrt15 times (-sqrt15) = -15`
`-15 = -15`
Answer :
The required zeroes are `sqrt15, -sqrt15` and the relation of zeroes & coefficient are verified.
vi) Given quadratic equation `3x^2-x-4` compare with `ax^2+bx+c=0`
where `a=3, b=-1` and `c=-4`
Let
`3x^2-x-4=0`
`3x^2-4x+3x-4=0`
`x(3x-4)+1(3x-4)=0`
`(3x-4)(x+1)=0`
`3x-4=0`
`x=4/3`
`x+1=0`
`x=-1`
Now, sum of the zeroes`= -b/a`
`alpha+beta= - (-1/3)`
`4/3+(-1)=1/3`
`1/3=1/3`
and product of the zeroes`=c/a`
`alpha times beta= -4/3`
`4/3times (-1) = -4/3`
`-4/3 = -4/3`
Answer :
The required zeroes are `4/3, -1`, and the relation of zeroes & coefficient are verified.
Q2. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
i) `1/4, -1` ii) `sqrt2, 1/3` iii) `0, sqrt5`
iv) `1, 1` v) `-1/4, 1/4` vi) `4, 1`
Sol. :
i) Let the required quadratic polynomial is `p(x)=ax^2+bx+c` and its zeroes are `alpha` and `beta`. Then we have (according to question)
`alpha+beta=-b/a=1/4=-(-1/4)`
`alpha times beta=c/a=(-4)/4=-1`
`a=4, b=-1,` and `c=-4`
`p(x)=4x^2-x-4`
Answer :
Thus, the required quadratic polynomial is `4x^2-x-4.`
ii) Let the required quadratic polynomial is `p(x)=ax^2+bx+c` and its zeroes are `alpha` and `beta`. Then we have (according to question)
`alpha+beta=-b/a=sqrt2=-((-3sqrt2)/3)`
`alpha times beta=c/a=1/3`
`a=3, b=-3sqrt2`, and `c=1`
`p(x)=3x^2-3sqrt2x+1`
Answer :
Thus, the required quadratic polynomial is `3x^2-3sqrt2x+1`.
iii) Let the required quadratic polynomial is `p(x)=ax^2+bx+c` and its zeroes are `alpha` and `beta`. Then we have (according to question)
`alpha+beta=-b/a=-0/1=0`
`alpha times beta=c/a=sqrt5=sqrt5/1`
`a=1, b=0`, and `c=sqrt5`
`p(x)=x^2+0.x+sqrt5`
`p(x)=x^2+sqrt5`
Answer :
Thus, the required quadratic polynomial is `x^2+sqrt5`.
iv) Let the required quadratic polynomial is `p(x)=ax^2+bx+c` and its zeroes are `alpha` and `beta`. Then we have (according to question)
`alpha+beta=-b/a=1=-(-1)/1.`
`alpha times beta= c/a =1=1/1`
`a=1, b=1`, and `c=1`
`p(x)=x^2-x+1`
Answer :
Thus, the required quadratic polynomial is `x^2-x+1`.
v) Let the required quadratic polynomial is `p(x)=ax^2+bx+c` and its zeroes are `alpha` and `beta`. Then we have (according to question)
`alpha+beta=-b/a=-1/4`
`alpha times beta=c/a=1/4`
`a=4, b=1`, and `c=1`
`p(x)=4x^2+x+1`
Answer :
Thus, the required quadratic polynomial is `4x^2+x+1`
vi) Let the required quadratic polynomial is `p(x)=ax^2+bx+c` and its zeroes are `alpha` and `beta`. Then we have (according to question)
`alpha+beta=-b/a=4=-((-4)/1)`
`alpha times beta=c/a=1=1/1`
`a=1, b=-4`, and `c=1`
`p(x)=x^2-4x+1`
Answer :
Thus, the required quadratic polynomial is `x^2-4x+1`.
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