Class 10 Exam > Class 10 Questions > If 2 and 3 are the zeros of f(x) = 2x3+ mx2&n...
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If 2 and 3 are the zeros of f(x) = 2x3 + mx2 – 13x + n, then the values of m and n are respectively –
- a)- 5, - 30
- b)- 5, 30
- c)5, 30
- d)5, - 30
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
If 2 and 3 are the zeros of f(x) = 2x3+ mx2– 13x + n, then the v...
Given equation:-
f(x)=2x^3 + mx^2 - 13x + n
it is also given that 2 and 3 are two zeros of given equation.
TAKING 3 AS ZERO:-
f(3)=2x^3 + mx^2 - 13x + n=0
.....2×(3)^3 + m(3)^2 - 13×3 + n=0
.....2×27 + 9m - 39 + n = 0
.....54-39 + 9m + n = 0
.....9m + n = -15. ----------(1)
TAKING 2 AS ZERO:-
f(2)= 2x^3 + mx^2 - 13x + n = 0
.....2(2)^3 + m(2)^2 - 13×2 + n =0
.....2×8 + 4m - 26 + n =0
.....16-26 + 4m + n = 0
.....4m + n = 10. ----------(2)
Now, subtracting (2) from (1), we get
9m + n - 4m - n = -15 -10
5m=-25
m=-25/5
[m=-5]
putting, value of "m" in (2) equation:-
4(-5) + n = 10
-20 + n = 10
n=10 + 20
[n=30]
So, now we have value of "m" as -5 and that of "n" as 30.
hence option B(-5, 30) is totally right answer.
Thank you.. :-)
f(x)=2x^3 + mx^2 - 13x + n
it is also given that 2 and 3 are two zeros of given equation.
TAKING 3 AS ZERO:-
f(3)=2x^3 + mx^2 - 13x + n=0
.....2×(3)^3 + m(3)^2 - 13×3 + n=0
.....2×27 + 9m - 39 + n = 0
.....54-39 + 9m + n = 0
.....9m + n = -15. ----------(1)
TAKING 2 AS ZERO:-
f(2)= 2x^3 + mx^2 - 13x + n = 0
.....2(2)^3 + m(2)^2 - 13×2 + n =0
.....2×8 + 4m - 26 + n =0
.....16-26 + 4m + n = 0
.....4m + n = 10. ----------(2)
Now, subtracting (2) from (1), we get
9m + n - 4m - n = -15 -10
5m=-25
m=-25/5
[m=-5]
putting, value of "m" in (2) equation:-
4(-5) + n = 10
-20 + n = 10
n=10 + 20
[n=30]
So, now we have value of "m" as -5 and that of "n" as 30.
hence option B(-5, 30) is totally right answer.
Thank you.. :-)
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Community Answer
If 2 and 3 are the zeros of f(x) = 2x3+ mx2– 13x + n, then the v...
We can use the fact that if 2 and 3 are zeros of f(x), then (x-2) and (x-3) are factors of f(x).
So, we have:
f(x) = 2x^3 + mx^2 + ax + b
= 2(x-2)(x-3)(x-k) (where k is the remaining zero)
Expanding this expression, we get:
f(x) = 2(x^3 - 5x^2 + 6x - 12)(x-k)
= 2x^4 - 10x^3 + 12x^2 - 24x - 2kx^3 + 10kx^2 - 12kx + 24k
Since f(x) has no linear term (ax), we must have k = 0.
So, f(x) = 2x^4 - 10x^3 + 12x^2 - 24x
To find the value of m, we can use the fact that the sum of the zeros of f(x) is -b/a.
The sum of the zeros of f(x) is 2+3+0 = 5, so:
-b/a = 5
-(-24)/2 = 5
m = 60
Therefore, the function is:
f(x) = 2x^4 - 10x^3 + 12x^2 - 24x^2
So, we have:
f(x) = 2x^3 + mx^2 + ax + b
= 2(x-2)(x-3)(x-k) (where k is the remaining zero)
Expanding this expression, we get:
f(x) = 2(x^3 - 5x^2 + 6x - 12)(x-k)
= 2x^4 - 10x^3 + 12x^2 - 24x - 2kx^3 + 10kx^2 - 12kx + 24k
Since f(x) has no linear term (ax), we must have k = 0.
So, f(x) = 2x^4 - 10x^3 + 12x^2 - 24x
To find the value of m, we can use the fact that the sum of the zeros of f(x) is -b/a.
The sum of the zeros of f(x) is 2+3+0 = 5, so:
-b/a = 5
-(-24)/2 = 5
m = 60
Therefore, the function is:
f(x) = 2x^4 - 10x^3 + 12x^2 - 24x^2
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If 2 and 3 are the zeros of f(x) = 2x3+ mx2– 13x + n, then the values of m and n are respectively –a)- 5, - 30b)- 5, 30c)5, 30d)5, - 30Correct answer is option 'B'. Can you explain this answer?
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