Each question is followed by two statements, A and B. Choose the most appropriate options given below.Roots of x3 + ax2 + bx + c = 0 are integers and are in arithmetic progression. If a, b, and c are also integers, then what is the value of a + b + c?A. a2 = 3(b + 4) and 4a b 6aB. 5a = b + 4 and c = 7aa)Question can be answered by using statement A alone but not by using statement B alone.b)Question can be answered by using statement B alone but not by using statement A alone.c)Question can be answered by using either statement alone.d)Question can be answered by using both the statements together but not by either of the statements alone.e)Question cannot be answered on the basis of the two statements.Correct answer is option 'D'. Can you explain this answer?

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Each question is followed by two statements, A and B. Choose the most appropriate options given below.

Roots of x³ + ax² + bx + c = 0 are integers and are in arithmetic progression. If a, b, and c are also integers, then what is the value of a + b + c?

A. a² = 3(b + 4) and 4a < b < 6a

B. 5a = b + 4 and c = 7a

a)
Question can be answered by using statement A alone but not by using statement B alone.
b)
Question can be answered by using statement B alone but not by using statement A alone.
c)
Question can be answered by using either statement alone.
d)
Question can be answered by using both the statements together but not by either of the statements alone.
e)
Question cannot be answered on the basis of the two statements.

Correct answer is option 'D'. Can you explain this answer?

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Each question is followed by two statements, A and B. Choose the most ...

As roots of the equation are in A.R, hence let the three roots be p - r, p and p + r.
Here p and r are integers.
Now, p - r + p + p + r = - a. a = - 3p, b = p(p - r) + p(p + r) + (p - r)(p + r) = 3p² - r². c = - p( p²- r² )

Considering statement A alone: a² = 3 (b + 4). 9p² = 3(3p² - r² + 4).
r= 2
Also, as a²is multiple of 3 and a is an integer, hence a is also a multiple of 3.

Also, 4a < b < 6a Multiplying the expression by 3 and adding 12,

Now, a is a multiple of 3, hence we can conclude that a can either be 15 or 18.
Correspondingly, p = -5 or p = -6 We get two corresponding values of (a + b + c).
Statement A alone is insufficient to solve the question.

Consider statement B alone: 5a = b + 4 and c = 7a

But p is integer, hence p = -5 is the only valid solution.
Using, p² - r² = 21, r = 2. The value of (a + b + c) corresponding to p = -5 and r= 2 can be found.
Case (ii): If p = 0, a = 0, b = -4, r = 2 and hence a + b + c can be found.
However, we find two values of (a + b + c).
Hence, statement B alone is also not sufficient to answer the question.
Considering both statements A and B together: p = - 5 and r - 2. A unique value of (a + b + c) corresponding to this can be found.
Hence, option 4.

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Each question is followed by two statements, A and B. Choose the most appropriate options given below.Roots of x3 + ax2 + bx + c = 0 are integers and are in arithmetic progression. If a, b, and c are also integers, then what is the value of a + b + c?A. a2 = 3(b + 4) and 4a b 6aB. 5a = b + 4 and c = 7aa)Question can be answered by using statement A alone but not by using statement B alone.b)Question can be answered by using statement B alone but not by using statement A alone.c)Question can be answered by using either statement alone.d)Question can be answered by using both the statements together but not by either of the statements alone.e)Question cannot be answered on the basis of the two statements.Correct answer is option 'D'. Can you explain this answer?

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