RS Aggarwal Solutions Class 10 Maths Quadratic Equations (Exercise 4F)

 Page 1


 
1. Which of the following is a quadratic equation?
(a)
3
3 2 0 x x (b) 
2
1
x x
x
(c)  
2
2
1
5 x
x
(d)
2
2
2 5 1 x x x
Answer: (d) 
2
2
2 5 1 x x x
Sol:
A quadratic equation is the equation with degree 2.
2
2
2 2
2 2
2 5 1
2 5 2 1
2 5 2 1 0
x x x
x x x x
x x x x
2
3 1 0, x x which is a quadratic equation
2. Which of the following is a quadratic equation?
(a)
2
2
1 2 3 x x (b)
3
3 2
1 x x x
(c) 
2
2 3 5 2 3 x x x (d) None of these
Answer: (b) 
3
3 2
1 x x x
Sol:
3
3 2
3 2 3 2
1
3 3 1
x x x
x x x x x
2
2 3 1 0, x x which is a quadratic equation
3. Which of the following is not a quadratic equation?
(a) 
2 2
3 5 x x x (b) 
2
2
2 2 5 x x
(c)
2
2
2 3 2 6 x x (d)
2
2
1 3 2 x x x
Answer: (c) 
2
2
2 3 2 6 x x
Sol:
2
2
2
2 3 2 6
2 9 6 2 2 6
x x
x x x
6 2 3 0, x which is not a quadratic equation
Page 2


 
1. Which of the following is a quadratic equation?
(a)
3
3 2 0 x x (b) 
2
1
x x
x
(c)  
2
2
1
5 x
x
(d)
2
2
2 5 1 x x x
Answer: (d) 
2
2
2 5 1 x x x
Sol:
A quadratic equation is the equation with degree 2.
2
2
2 2
2 2
2 5 1
2 5 2 1
2 5 2 1 0
x x x
x x x x
x x x x
2
3 1 0, x x which is a quadratic equation
2. Which of the following is a quadratic equation?
(a)
2
2
1 2 3 x x (b)
3
3 2
1 x x x
(c) 
2
2 3 5 2 3 x x x (d) None of these
Answer: (b) 
3
3 2
1 x x x
Sol:
3
3 2
3 2 3 2
1
3 3 1
x x x
x x x x x
2
2 3 1 0, x x which is a quadratic equation
3. Which of the following is not a quadratic equation?
(a) 
2 2
3 5 x x x (b) 
2
2
2 2 5 x x
(c)
2
2
2 3 2 6 x x (d)
2
2
1 3 2 x x x
Answer: (c) 
2
2
2 3 2 6 x x
Sol:
2
2
2
2 3 2 6
2 9 6 2 2 6
x x
x x x
6 2 3 0, x which is not a quadratic equation
 
4. If 3 x is a solution of the equation
2
3 1 9 0, x k x then ? k
(a) 11 (b) 11 (c) 13 (d) 13
Answer: (b) 11
Sol:
It is given that 3 x is a solution of 
2
3 1 9 0; x k x therefore, we have:
2
3 3 1 3 9 0
27 3 1 9 0
3 1 36
1 12
11
k
k
k
k
k
5. If one root of the equation 
2
2 6 0 x a x is 2 then a=?
(a) 7 (b) -7 (c) 
7
2
(d) 
7
2
Answer: (b) -7
Sol:
It is given that one root of the equation 
2
2 6 0 x a x is 2.
2
2 2 2 6 0
2 14 0
7
a
a
a
6. The sum of the roots of the equation
2
6 2 0 x x is 
(a) 2 (b) -2 (c) 6 (d) -6
Answer: (b) -2
Sol:
Sum of the roots of the equation 
2
6 2 0 x x is
6
6,
1
b
a
where and are the roots of the equation.
7. If the product of the roots of the equation
2
3 10 x x k is -2 then the value of k is 
(a) -2 (b) -8 (c) 8 (d) 12
Answer: (c) 8 
Sol:
It is given that the product of the roots of the equation 
2
3 10 x x k is . 2
The equation can be rewritten as:
2
3 10 0 x x k
Page 3


 
1. Which of the following is a quadratic equation?
(a)
3
3 2 0 x x (b) 
2
1
x x
x
(c)  
2
2
1
5 x
x
(d)
2
2
2 5 1 x x x
Answer: (d) 
2
2
2 5 1 x x x
Sol:
A quadratic equation is the equation with degree 2.
2
2
2 2
2 2
2 5 1
2 5 2 1
2 5 2 1 0
x x x
x x x x
x x x x
2
3 1 0, x x which is a quadratic equation
2. Which of the following is a quadratic equation?
(a)
2
2
1 2 3 x x (b)
3
3 2
1 x x x
(c) 
2
2 3 5 2 3 x x x (d) None of these
Answer: (b) 
3
3 2
1 x x x
Sol:
3
3 2
3 2 3 2
1
3 3 1
x x x
x x x x x
2
2 3 1 0, x x which is a quadratic equation
3. Which of the following is not a quadratic equation?
(a) 
2 2
3 5 x x x (b) 
2
2
2 2 5 x x
(c)
2
2
2 3 2 6 x x (d)
2
2
1 3 2 x x x
Answer: (c) 
2
2
2 3 2 6 x x
Sol:
2
2
2
2 3 2 6
2 9 6 2 2 6
x x
x x x
6 2 3 0, x which is not a quadratic equation
 
4. If 3 x is a solution of the equation
2
3 1 9 0, x k x then ? k
(a) 11 (b) 11 (c) 13 (d) 13
Answer: (b) 11
Sol:
It is given that 3 x is a solution of 
2
3 1 9 0; x k x therefore, we have:
2
3 3 1 3 9 0
27 3 1 9 0
3 1 36
1 12
11
k
k
k
k
k
5. If one root of the equation 
2
2 6 0 x a x is 2 then a=?
(a) 7 (b) -7 (c) 
7
2
(d) 
7
2
Answer: (b) -7
Sol:
It is given that one root of the equation 
2
2 6 0 x a x is 2.
2
2 2 2 6 0
2 14 0
7
a
a
a
6. The sum of the roots of the equation
2
6 2 0 x x is 
(a) 2 (b) -2 (c) 6 (d) -6
Answer: (b) -2
Sol:
Sum of the roots of the equation 
2
6 2 0 x x is
6
6,
1
b
a
where and are the roots of the equation.
7. If the product of the roots of the equation
2
3 10 x x k is -2 then the value of k is 
(a) -2 (b) -8 (c) 8 (d) 12
Answer: (c) 8 
Sol:
It is given that the product of the roots of the equation 
2
3 10 x x k is . 2
The equation can be rewritten as:
2
3 10 0 x x k
 
 
Product of the roots of a quadratic equation
c
a
2
c
a
10
2
1
8
k
k
8. The ratio of the sum and product of the roots of the equation 
2
7 12 18 0 x x is
(a) 7 :12 (b) 7 :18 (c) 3 : 2 (d) 2 : 3
Answer : (d) 2 : 3
Sol:
Given:
2
7 12 18 0 x x
12 18
,
7 7
and where and are the roots of the equation
Ratio of the sum and product of the roots
12 18
:
7 7
12:18
2:3
9. If one root of the equation 
2
3 10 3 0 x x is 
1
3
then the other root is 
(a) 
1
3
(b) 
1
3
(c) -3 (d)3
Answer: (d)3
Sol:
Given:
2
3 10 3 0 x x
One root of the equation is 
1
.
3
Let the other root be .
Product of the roots
c
a
1 3
3 3
3
10. If one root of 
2
5 13 0 x x k be the reciprocal of the other root then the value of k is
Page 4


 
1. Which of the following is a quadratic equation?
(a)
3
3 2 0 x x (b) 
2
1
x x
x
(c)  
2
2
1
5 x
x
(d)
2
2
2 5 1 x x x
Answer: (d) 
2
2
2 5 1 x x x
Sol:
A quadratic equation is the equation with degree 2.
2
2
2 2
2 2
2 5 1
2 5 2 1
2 5 2 1 0
x x x
x x x x
x x x x
2
3 1 0, x x which is a quadratic equation
2. Which of the following is a quadratic equation?
(a)
2
2
1 2 3 x x (b)
3
3 2
1 x x x
(c) 
2
2 3 5 2 3 x x x (d) None of these
Answer: (b) 
3
3 2
1 x x x
Sol:
3
3 2
3 2 3 2
1
3 3 1
x x x
x x x x x
2
2 3 1 0, x x which is a quadratic equation
3. Which of the following is not a quadratic equation?
(a) 
2 2
3 5 x x x (b) 
2
2
2 2 5 x x
(c)
2
2
2 3 2 6 x x (d)
2
2
1 3 2 x x x
Answer: (c) 
2
2
2 3 2 6 x x
Sol:
2
2
2
2 3 2 6
2 9 6 2 2 6
x x
x x x
6 2 3 0, x which is not a quadratic equation
 
4. If 3 x is a solution of the equation
2
3 1 9 0, x k x then ? k
(a) 11 (b) 11 (c) 13 (d) 13
Answer: (b) 11
Sol:
It is given that 3 x is a solution of 
2
3 1 9 0; x k x therefore, we have:
2
3 3 1 3 9 0
27 3 1 9 0
3 1 36
1 12
11
k
k
k
k
k
5. If one root of the equation 
2
2 6 0 x a x is 2 then a=?
(a) 7 (b) -7 (c) 
7
2
(d) 
7
2
Answer: (b) -7
Sol:
It is given that one root of the equation 
2
2 6 0 x a x is 2.
2
2 2 2 6 0
2 14 0
7
a
a
a
6. The sum of the roots of the equation
2
6 2 0 x x is 
(a) 2 (b) -2 (c) 6 (d) -6
Answer: (b) -2
Sol:
Sum of the roots of the equation 
2
6 2 0 x x is
6
6,
1
b
a
where and are the roots of the equation.
7. If the product of the roots of the equation
2
3 10 x x k is -2 then the value of k is 
(a) -2 (b) -8 (c) 8 (d) 12
Answer: (c) 8 
Sol:
It is given that the product of the roots of the equation 
2
3 10 x x k is . 2
The equation can be rewritten as:
2
3 10 0 x x k
 
 
Product of the roots of a quadratic equation
c
a
2
c
a
10
2
1
8
k
k
8. The ratio of the sum and product of the roots of the equation 
2
7 12 18 0 x x is
(a) 7 :12 (b) 7 :18 (c) 3 : 2 (d) 2 : 3
Answer : (d) 2 : 3
Sol:
Given:
2
7 12 18 0 x x
12 18
,
7 7
and where and are the roots of the equation
Ratio of the sum and product of the roots
12 18
:
7 7
12:18
2:3
9. If one root of the equation 
2
3 10 3 0 x x is 
1
3
then the other root is 
(a) 
1
3
(b) 
1
3
(c) -3 (d)3
Answer: (d)3
Sol:
Given:
2
3 10 3 0 x x
One root of the equation is 
1
.
3
Let the other root be .
Product of the roots
c
a
1 3
3 3
3
10. If one root of 
2
5 13 0 x x k be the reciprocal of the other root then the value of k is
 
(a) 0 (b) 1 (c)2(d)5
Answer: (d)5
Sol:
Let the roots of the equation 
2
3
be and 
1
.
Product of the roots 
c
a
1
5
5
5
k
k
a
k
11. If the sum of the roots of the equation 
2
2 3 0 k x x k is equal to their product then the value
of k
(a) 
1
3
(b) 
1
3
(c)
2
3
(d)
2
3
Answer: (d) 
2
3
Sol:
Given:
2
2 3 0 k x x k
Sum of the roots = Product of the roots
2 3
3 2
2
3
k
k k
k
k
12. The roots of a quadratic equation are 5 and -2. Then, the equation is
(a) 
2
3 10 0 x x (b) 
2
3 10 0 x x (c) 
2
3 10 0 x x (d)
2
3 10 0 x x
Answer: (b) 
2
3 10 0 x x
Sol:
It is given that the roots of the quadratic equation are 5 and . 2
Then, the equation is:
2
2
5 2 5 2 0
3 10 0
x x
x x
13. If the sum of the roots of a quadratic equation is 6 and their product is 6, the equation is
Page 5


 
1. Which of the following is a quadratic equation?
(a)
3
3 2 0 x x (b) 
2
1
x x
x
(c)  
2
2
1
5 x
x
(d)
2
2
2 5 1 x x x
Answer: (d) 
2
2
2 5 1 x x x
Sol:
A quadratic equation is the equation with degree 2.
2
2
2 2
2 2
2 5 1
2 5 2 1
2 5 2 1 0
x x x
x x x x
x x x x
2
3 1 0, x x which is a quadratic equation
2. Which of the following is a quadratic equation?
(a)
2
2
1 2 3 x x (b)
3
3 2
1 x x x
(c) 
2
2 3 5 2 3 x x x (d) None of these
Answer: (b) 
3
3 2
1 x x x
Sol:
3
3 2
3 2 3 2
1
3 3 1
x x x
x x x x x
2
2 3 1 0, x x which is a quadratic equation
3. Which of the following is not a quadratic equation?
(a) 
2 2
3 5 x x x (b) 
2
2
2 2 5 x x
(c)
2
2
2 3 2 6 x x (d)
2
2
1 3 2 x x x
Answer: (c) 
2
2
2 3 2 6 x x
Sol:
2
2
2
2 3 2 6
2 9 6 2 2 6
x x
x x x
6 2 3 0, x which is not a quadratic equation
 
4. If 3 x is a solution of the equation
2
3 1 9 0, x k x then ? k
(a) 11 (b) 11 (c) 13 (d) 13
Answer: (b) 11
Sol:
It is given that 3 x is a solution of 
2
3 1 9 0; x k x therefore, we have:
2
3 3 1 3 9 0
27 3 1 9 0
3 1 36
1 12
11
k
k
k
k
k
5. If one root of the equation 
2
2 6 0 x a x is 2 then a=?
(a) 7 (b) -7 (c) 
7
2
(d) 
7
2
Answer: (b) -7
Sol:
It is given that one root of the equation 
2
2 6 0 x a x is 2.
2
2 2 2 6 0
2 14 0
7
a
a
a
6. The sum of the roots of the equation
2
6 2 0 x x is 
(a) 2 (b) -2 (c) 6 (d) -6
Answer: (b) -2
Sol:
Sum of the roots of the equation 
2
6 2 0 x x is
6
6,
1
b
a
where and are the roots of the equation.
7. If the product of the roots of the equation
2
3 10 x x k is -2 then the value of k is 
(a) -2 (b) -8 (c) 8 (d) 12
Answer: (c) 8 
Sol:
It is given that the product of the roots of the equation 
2
3 10 x x k is . 2
The equation can be rewritten as:
2
3 10 0 x x k
 
 
Product of the roots of a quadratic equation
c
a
2
c
a
10
2
1
8
k
k
8. The ratio of the sum and product of the roots of the equation 
2
7 12 18 0 x x is
(a) 7 :12 (b) 7 :18 (c) 3 : 2 (d) 2 : 3
Answer : (d) 2 : 3
Sol:
Given:
2
7 12 18 0 x x
12 18
,
7 7
and where and are the roots of the equation
Ratio of the sum and product of the roots
12 18
:
7 7
12:18
2:3
9. If one root of the equation 
2
3 10 3 0 x x is 
1
3
then the other root is 
(a) 
1
3
(b) 
1
3
(c) -3 (d)3
Answer: (d)3
Sol:
Given:
2
3 10 3 0 x x
One root of the equation is 
1
.
3
Let the other root be .
Product of the roots
c
a
1 3
3 3
3
10. If one root of 
2
5 13 0 x x k be the reciprocal of the other root then the value of k is
 
(a) 0 (b) 1 (c)2(d)5
Answer: (d)5
Sol:
Let the roots of the equation 
2
3
be and 
1
.
Product of the roots 
c
a
1
5
5
5
k
k
a
k
11. If the sum of the roots of the equation 
2
2 3 0 k x x k is equal to their product then the value
of k
(a) 
1
3
(b) 
1
3
(c)
2
3
(d)
2
3
Answer: (d) 
2
3
Sol:
Given:
2
2 3 0 k x x k
Sum of the roots = Product of the roots
2 3
3 2
2
3
k
k k
k
k
12. The roots of a quadratic equation are 5 and -2. Then, the equation is
(a) 
2
3 10 0 x x (b) 
2
3 10 0 x x (c) 
2
3 10 0 x x (d)
2
3 10 0 x x
Answer: (b) 
2
3 10 0 x x
Sol:
It is given that the roots of the quadratic equation are 5 and . 2
Then, the equation is:
2
2
5 2 5 2 0
3 10 0
x x
x x
13. If the sum of the roots of a quadratic equation is 6 and their product is 6, the equation is
 
(a)
2
6 6 0 x x (b)
2
6 6 0 x x (c)
2
6 6 0 x x (d) 
2
6 6 0 x x
Answer: (a) 
2
6 6 0 x x
Sol:
Given:
Sum of roots = 6
Product of roots = 6
Thus, the equation is:
2
6 6 0 x x
14. If a n d are the roots of the equation 
2
3 8 2 0 x x then 
1 1
?
(a) 
3
8
(b)
2
3
(c) -4 (d) 4 
Answer: (c) -4
Sol:
It is given that and are the roots of the equation 
2
3 8 2 0 x x
8 2
3 3
and
8
1 1
3
4
2
3
15. The roots of the equation 
2
0 a x b x c will be reciprocal each other if
(a) a = b (b) b = c (c) c = a (d) none of these
Answer: (c) c = a
Sol:
Let the roots of the equation 
2
0 a x b x c be and 
1
.
Product of the roots 
1
1
1
c
a
c a
16. If the roots of the equation 
2
0 a x b x c are equal then c=?