RS Aggarwal Solutions: Exercise 4A - Quadratic Equations Class 10 Notes | EduRev

Mathematics (Maths) Class 10

Class 10 : RS Aggarwal Solutions: Exercise 4A - Quadratic Equations Class 10 Notes | EduRev

 Page 1


 
 
1. Which of the following are quadratic equation in x?
(i) 
2
3 0 x x
(ii) 
2
5
2 3 0
2
x x
(iii) 
2
2 7 5 2 x x
(iv)  
2
1 1
2 0
3 5
x x
(v)  
2
3 4 0 x x x
(vi) 
6
3 x
x
(vii) 
2 2
2
x x
x
(viii) 
2
2
1
5 x
x
(ix) 
3
3
2 8 x x
(x) 2 3 3 2 6 1 2 x x x x
(xi) 
2
1 1
2 3 x x
x x
Sol:
(i)
2
3 x x is a quadratic polynomial
2
3 0 x x is a quadratic equation. 
(ii) Clearly, 
2
5
2 3
2
x x is a quadratic polynomial.
2
5
2 3 0
2
x x is a quadratic equation.
(iii) Clearly, 
2
2 7 5 2 x x is a quadratic polynomial.
2
2 7 5 2 0 x x is a quadratic equation. 
(iv) Clearly, 
2
1 1
2
3 5
x x is a quadratic polynomial.
2
1 1
2 0
3 5
x x is a quadratic equation. 
(v)
2
3 4 x x x contains a term with , x i.e.,
1
2
, x where 
1
2
is not a ineger.
Page 2


 
 
1. Which of the following are quadratic equation in x?
(i) 
2
3 0 x x
(ii) 
2
5
2 3 0
2
x x
(iii) 
2
2 7 5 2 x x
(iv)  
2
1 1
2 0
3 5
x x
(v)  
2
3 4 0 x x x
(vi) 
6
3 x
x
(vii) 
2 2
2
x x
x
(viii) 
2
2
1
5 x
x
(ix) 
3
3
2 8 x x
(x) 2 3 3 2 6 1 2 x x x x
(xi) 
2
1 1
2 3 x x
x x
Sol:
(i)
2
3 x x is a quadratic polynomial
2
3 0 x x is a quadratic equation. 
(ii) Clearly, 
2
5
2 3
2
x x is a quadratic polynomial.
2
5
2 3 0
2
x x is a quadratic equation.
(iii) Clearly, 
2
2 7 5 2 x x is a quadratic polynomial.
2
2 7 5 2 0 x x is a quadratic equation. 
(iv) Clearly, 
2
1 1
2
3 5
x x is a quadratic polynomial.
2
1 1
2 0
3 5
x x is a quadratic equation. 
(v)
2
3 4 x x x contains a term with , x i.e.,
1
2
, x where 
1
2
is not a ineger.
 
 
Therefore, it is not a quadratic polynomial.
2
3 4 0 x x x is not a quadratic equation.
(vi)
6
3 x
x
2
2
6 3
3 6 0
x x
x x
3
3 6 x x is not quadratic polynomial; therefore, the given equation is quadratic.  
(vii)
2 2
2
x x
x
2 3
3 2
2
2 0
x x
x x
3 2
2 x x is not a quadratic polynomial.
3 2
2 0 x x is not a quadratic equation.
(viii)
2
2
1
5 x
x
4 2
1 5 x x
4 2
5 1 0 x x
4 2
5 1 x x
is a polynomial with degree 4.
4 2
5 1 0 x x
is not a quadratic equation.
(ix)
3
3
2 8 x x
3 2 3
2
6 12 8 8
6 12 16 0
x x x x
x x
This is of the form 
2
0 a x b x c
Hence, the given equation is a quadratic equation.
(x)
2 3 3 2 6 1 2 x x x x
2 2
2 2
6 4 9 6 6 3 2
6 13 6 6 18 12
31 6 0
x x x x x
x x x x
x
This is of the form 
2
0 a x b x c
Hence, the given equation is not a quadratic equation.
(xi)
2
1 1
2 3 x x
x x
Page 3


 
 
1. Which of the following are quadratic equation in x?
(i) 
2
3 0 x x
(ii) 
2
5
2 3 0
2
x x
(iii) 
2
2 7 5 2 x x
(iv)  
2
1 1
2 0
3 5
x x
(v)  
2
3 4 0 x x x
(vi) 
6
3 x
x
(vii) 
2 2
2
x x
x
(viii) 
2
2
1
5 x
x
(ix) 
3
3
2 8 x x
(x) 2 3 3 2 6 1 2 x x x x
(xi) 
2
1 1
2 3 x x
x x
Sol:
(i)
2
3 x x is a quadratic polynomial
2
3 0 x x is a quadratic equation. 
(ii) Clearly, 
2
5
2 3
2
x x is a quadratic polynomial.
2
5
2 3 0
2
x x is a quadratic equation.
(iii) Clearly, 
2
2 7 5 2 x x is a quadratic polynomial.
2
2 7 5 2 0 x x is a quadratic equation. 
(iv) Clearly, 
2
1 1
2
3 5
x x is a quadratic polynomial.
2
1 1
2 0
3 5
x x is a quadratic equation. 
(v)
2
3 4 x x x contains a term with , x i.e.,
1
2
, x where 
1
2
is not a ineger.
 
 
Therefore, it is not a quadratic polynomial.
2
3 4 0 x x x is not a quadratic equation.
(vi)
6
3 x
x
2
2
6 3
3 6 0
x x
x x
3
3 6 x x is not quadratic polynomial; therefore, the given equation is quadratic.  
(vii)
2 2
2
x x
x
2 3
3 2
2
2 0
x x
x x
3 2
2 x x is not a quadratic polynomial.
3 2
2 0 x x is not a quadratic equation.
(viii)
2
2
1
5 x
x
4 2
1 5 x x
4 2
5 1 0 x x
4 2
5 1 x x
is a polynomial with degree 4.
4 2
5 1 0 x x
is not a quadratic equation.
(ix)
3
3
2 8 x x
3 2 3
2
6 12 8 8
6 12 16 0
x x x x
x x
This is of the form 
2
0 a x b x c
Hence, the given equation is a quadratic equation.
(x)
2 3 3 2 6 1 2 x x x x
2 2
2 2
6 4 9 6 6 3 2
6 13 6 6 18 12
31 6 0
x x x x x
x x x x
x
This is of the form 
2
0 a x b x c
Hence, the given equation is not a quadratic equation.
(xi)
2
1 1
2 3 x x
x x
 
 
2
2 2
1 1
2 3
x x
x x
2
2 2 2
4 2 3 2
4 3 2
1 2 1 3
2 1 2 2 3
2 2 1 0
x x x x
x x x x x
x x x x
This is not of the form
2
0 a x b x c
Hence, the given equation is not a quadratic equation.
2. Which of the following are the roots of 
2
3 2 1 0? x x
(i) 1 (ii)
1
3
(iii)
1
2
Sol:
The given equation is 
2
3 2 1 0 . x x
(i) 1 x
L.H.S.
2
2 1 x x
2
3 1 2 1 1
3 2 1
0
. . . R H S
Thus, 1 is a root of 
2
3 2 1 0 . x x
(ii) On subtracting 
1
3
x in the given equation, we get:
L.H.S. 
2
3 2 1 x x
2
1 1
3 2 1
3 3
1 2
3 1
9 3
1 2 3
3
0
3
0
. . . R H S
Thus,
1
3
is a root of
2
3 2 1 0 x x
Page 4


 
 
1. Which of the following are quadratic equation in x?
(i) 
2
3 0 x x
(ii) 
2
5
2 3 0
2
x x
(iii) 
2
2 7 5 2 x x
(iv)  
2
1 1
2 0
3 5
x x
(v)  
2
3 4 0 x x x
(vi) 
6
3 x
x
(vii) 
2 2
2
x x
x
(viii) 
2
2
1
5 x
x
(ix) 
3
3
2 8 x x
(x) 2 3 3 2 6 1 2 x x x x
(xi) 
2
1 1
2 3 x x
x x
Sol:
(i)
2
3 x x is a quadratic polynomial
2
3 0 x x is a quadratic equation. 
(ii) Clearly, 
2
5
2 3
2
x x is a quadratic polynomial.
2
5
2 3 0
2
x x is a quadratic equation.
(iii) Clearly, 
2
2 7 5 2 x x is a quadratic polynomial.
2
2 7 5 2 0 x x is a quadratic equation. 
(iv) Clearly, 
2
1 1
2
3 5
x x is a quadratic polynomial.
2
1 1
2 0
3 5
x x is a quadratic equation. 
(v)
2
3 4 x x x contains a term with , x i.e.,
1
2
, x where 
1
2
is not a ineger.
 
 
Therefore, it is not a quadratic polynomial.
2
3 4 0 x x x is not a quadratic equation.
(vi)
6
3 x
x
2
2
6 3
3 6 0
x x
x x
3
3 6 x x is not quadratic polynomial; therefore, the given equation is quadratic.  
(vii)
2 2
2
x x
x
2 3
3 2
2
2 0
x x
x x
3 2
2 x x is not a quadratic polynomial.
3 2
2 0 x x is not a quadratic equation.
(viii)
2
2
1
5 x
x
4 2
1 5 x x
4 2
5 1 0 x x
4 2
5 1 x x
is a polynomial with degree 4.
4 2
5 1 0 x x
is not a quadratic equation.
(ix)
3
3
2 8 x x
3 2 3
2
6 12 8 8
6 12 16 0
x x x x
x x
This is of the form 
2
0 a x b x c
Hence, the given equation is a quadratic equation.
(x)
2 3 3 2 6 1 2 x x x x
2 2
2 2
6 4 9 6 6 3 2
6 13 6 6 18 12
31 6 0
x x x x x
x x x x
x
This is of the form 
2
0 a x b x c
Hence, the given equation is not a quadratic equation.
(xi)
2
1 1
2 3 x x
x x
 
 
2
2 2
1 1
2 3
x x
x x
2
2 2 2
4 2 3 2
4 3 2
1 2 1 3
2 1 2 2 3
2 2 1 0
x x x x
x x x x x
x x x x
This is not of the form
2
0 a x b x c
Hence, the given equation is not a quadratic equation.
2. Which of the following are the roots of 
2
3 2 1 0? x x
(i) 1 (ii)
1
3
(iii)
1
2
Sol:
The given equation is 
2
3 2 1 0 . x x
(i) 1 x
L.H.S.
2
2 1 x x
2
3 1 2 1 1
3 2 1
0
. . . R H S
Thus, 1 is a root of 
2
3 2 1 0 . x x
(ii) On subtracting 
1
3
x in the given equation, we get:
L.H.S. 
2
3 2 1 x x
2
1 1
3 2 1
3 3
1 2
3 1
9 3
1 2 3
3
0
3
0
. . . R H S
Thus,
1
3
is a root of
2
3 2 1 0 x x
 
 
(iii) On subtracting 
1
2
x in the given equation, we get
L.H.S.
2
3 2 1 x x
2
1 1
3 2 1
2 2
1
3 1 1
4
3
2
4
3 8
4
5
0
4
Thus, . . . . . L H S R H S
Hence, 
1
2
is a solution of 
2
3 2 1 0 . x x
3. Find the value of k for which 1 x is a root of the equation
2
3 0. x k x
Sol:
It is given that 1 x is a root of 
2
3 0 . x k x
Therefore, 1 x must satisfy the equation.
2
1 1 3 0
4 0
4
k
k
k
Hence, the required value of k is 4.
4. Find the value of a and b for which
3
4
x and 2 x are the roots of the equation
2
6 0 a x b x
Sol:
It is given that 
3
4
is a root of 
2
6 0; a x b x therefore, we have:
2
3 3
6 0
4 4
a b
Page 5


 
 
1. Which of the following are quadratic equation in x?
(i) 
2
3 0 x x
(ii) 
2
5
2 3 0
2
x x
(iii) 
2
2 7 5 2 x x
(iv)  
2
1 1
2 0
3 5
x x
(v)  
2
3 4 0 x x x
(vi) 
6
3 x
x
(vii) 
2 2
2
x x
x
(viii) 
2
2
1
5 x
x
(ix) 
3
3
2 8 x x
(x) 2 3 3 2 6 1 2 x x x x
(xi) 
2
1 1
2 3 x x
x x
Sol:
(i)
2
3 x x is a quadratic polynomial
2
3 0 x x is a quadratic equation. 
(ii) Clearly, 
2
5
2 3
2
x x is a quadratic polynomial.
2
5
2 3 0
2
x x is a quadratic equation.
(iii) Clearly, 
2
2 7 5 2 x x is a quadratic polynomial.
2
2 7 5 2 0 x x is a quadratic equation. 
(iv) Clearly, 
2
1 1
2
3 5
x x is a quadratic polynomial.
2
1 1
2 0
3 5
x x is a quadratic equation. 
(v)
2
3 4 x x x contains a term with , x i.e.,
1
2
, x where 
1
2
is not a ineger.
 
 
Therefore, it is not a quadratic polynomial.
2
3 4 0 x x x is not a quadratic equation.
(vi)
6
3 x
x
2
2
6 3
3 6 0
x x
x x
3
3 6 x x is not quadratic polynomial; therefore, the given equation is quadratic.  
(vii)
2 2
2
x x
x
2 3
3 2
2
2 0
x x
x x
3 2
2 x x is not a quadratic polynomial.
3 2
2 0 x x is not a quadratic equation.
(viii)
2
2
1
5 x
x
4 2
1 5 x x
4 2
5 1 0 x x
4 2
5 1 x x
is a polynomial with degree 4.
4 2
5 1 0 x x
is not a quadratic equation.
(ix)
3
3
2 8 x x
3 2 3
2
6 12 8 8
6 12 16 0
x x x x
x x
This is of the form 
2
0 a x b x c
Hence, the given equation is a quadratic equation.
(x)
2 3 3 2 6 1 2 x x x x
2 2
2 2
6 4 9 6 6 3 2
6 13 6 6 18 12
31 6 0
x x x x x
x x x x
x
This is of the form 
2
0 a x b x c
Hence, the given equation is not a quadratic equation.
(xi)
2
1 1
2 3 x x
x x
 
 
2
2 2
1 1
2 3
x x
x x
2
2 2 2
4 2 3 2
4 3 2
1 2 1 3
2 1 2 2 3
2 2 1 0
x x x x
x x x x x
x x x x
This is not of the form
2
0 a x b x c
Hence, the given equation is not a quadratic equation.
2. Which of the following are the roots of 
2
3 2 1 0? x x
(i) 1 (ii)
1
3
(iii)
1
2
Sol:
The given equation is 
2
3 2 1 0 . x x
(i) 1 x
L.H.S.
2
2 1 x x
2
3 1 2 1 1
3 2 1
0
. . . R H S
Thus, 1 is a root of 
2
3 2 1 0 . x x
(ii) On subtracting 
1
3
x in the given equation, we get:
L.H.S. 
2
3 2 1 x x
2
1 1
3 2 1
3 3
1 2
3 1
9 3
1 2 3
3
0
3
0
. . . R H S
Thus,
1
3
is a root of
2
3 2 1 0 x x
 
 
(iii) On subtracting 
1
2
x in the given equation, we get
L.H.S.
2
3 2 1 x x
2
1 1
3 2 1
2 2
1
3 1 1
4
3
2
4
3 8
4
5
0
4
Thus, . . . . . L H S R H S
Hence, 
1
2
is a solution of 
2
3 2 1 0 . x x
3. Find the value of k for which 1 x is a root of the equation
2
3 0. x k x
Sol:
It is given that 1 x is a root of 
2
3 0 . x k x
Therefore, 1 x must satisfy the equation.
2
1 1 3 0
4 0
4
k
k
k
Hence, the required value of k is 4.
4. Find the value of a and b for which
3
4
x and 2 x are the roots of the equation
2
6 0 a x b x
Sol:
It is given that 
3
4
is a root of 
2
6 0; a x b x therefore, we have:
2
3 3
6 0
4 4
a b
 
 
9 3
6
16 4
9 12
6
16
9 12 96 0
a b
a b
a b
3 4 32 .......... a b i
Again, 2 is a root of 
2
6 0; a x b x therefore, we have:
2
2 2 6 0
4 2 6
2 3 .........
a b
a b
a b i i
On multiplying (ii) by 4 and adding the result with (i), we get:
3 4 8 4 32 12
11 44
a b a b
a
4 a
Putting the value of a in (ii), we get:
2 4 3
8 3
5
b
b
b
Hence, the required values of a and b are 4 and 5, respectively.
5. 2 3 3 1 0 x x
Sol:
2 3 3 1 0 x x
2 3 0 3 1 0
2 3 3 1
3 1
2 3
x or x
x or x
x or x
Hence the roots of the given equation are 
3 1
.
2 3
a n d
6.
2
4 5 0 x x
Sol:
2
4 5 0 x x
4 5 0 x x
0 4 5 0 x o r x
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