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If p and q are the roots of the quadratic equation ax2 + bx + c = 0, where a*b*c ≠ 0, is the product of p and q greater than 0?
(1) |p + q| = |p| + |q|
(2) ac > 0
- a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
- b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
- c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
- d)EACH statement ALONE is sufficient to answer the question asked.
- e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
If p and q are the roots of the quadratic equation ax2 + bx + c = 0, w...
Steps 1 & 2: Understand Question and Draw Inferences
- ax2+bx+c=0
- where a*b*c ≠ 0
- This means, a ≠ 0, b ≠ 0 and c ≠ 0
- p and q are roots of the equation
To Find: pq > 0?
- Is pq > 0?
- pq > 0, if p and q are of the same signs
- pq < 0, if p and q are of the opposite signs
- Also, Product of roots = pq = c/a
- So, the question becomes: Is c/a>0?
- c/a>0? , if c and a are of the same signs
- c/a<0 , if c and a are of the opposite signs
So, we need to find the signs of either (p and q) of (c and a).
Step 3: Analyze Statement (1) independently
(1) |p + q| = |p| + |q|
We know, pq = c/a , where a ≠ 0 and c ≠ 0, So p ≠ 0 and q ≠ 0
Now, we know that |x| = x if x ≥ 0 and |x| = -x if x < 0.Since, we have |p| and |q|, following cases are possible:
- p > 0 and q >0
- As p > 0 and q > 0, we have |p| = p and |q| = q
- So, p + q > 0
- Hence, we can write |p+q| = p +q and |p| +|q| = p + q.
- So, |p+q| = |p| + |q|
- Thus, the equation in statement-1 holds true for p > 0 and q > 0
- In this case pq > 0
- p > 0 and q < 0
- As p > 0, we have |p| = p and as q < 0, we have |q| = -q
- Now, |p+q| = p + q, if p + q > 0 or –(p+q), if p + q < 0
- Also, |p| +|q| = p - q
- So, for both the values of |p+q|, we have |p + q| ≠ |p| + |q|
- Thus, the equation in statement-1 does not hold true for p >0 and q < 0
- p < 0 and q >0
- As p < 0, we have |p| = -p and as q >0, we have |q| = q
- Now, |p+q| = p + q, if p +q ≥ 0 or –(p+q) if p + q < 0
- Also, |p| + |q| = -p +q
- So, for both the values of |p+q|, we have |p + q| ≠ |p| + |q|
- Thus, the equation in statement-1 does not hold true for p < 0 and q >0
- p < 0 and q < 0
- As p < 0, we have |p| = -p and as q < 0, we have |q| = -q
- Now, |p+q| = - (p+q) because p + q < 0
- Also, |p| + |q| = -p – q = - (p+q)
- So, |p+q| = |p| + |q|
- Thus, the equation in statement-1 holds true for p < 0 and q < 0
- In this case pq > 0
So, |p+q| = |p| +|q| is possible only when p, q > 0 or p,q < 0
So in both the cases pq will be greater than 0.
Hence Sufficient to answer
Step 4: Analyze Statement (2) independently
(2) ac > 0
Tells us that a and c have the same signs. Thus c/a>0
and hence pq > 0.
Sufficient to answer
Step 5: Analyze Both Statements Together (if needed)
As we have a unique answer from step 3 and step 4, this step is not required.
Answer: D
Learning
- |a+b| = |a| + |b| if a and b have same sign
- |a+b| < |a| + |b| if a and b have different sign
Alternatively,
- |a-b| = |a| - |b| if a and b have same sign
- |a-b| > |a| -|b| if a and b have different sign
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If p and q are the roots of the quadratic equation ax2 + bx + c = 0, where a*b*c ≠ 0, is the product of p and q greater than 0?(1) |p + q| = |p| + |q|(2) ac > 0a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the question asked.e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.Correct answer is option 'D'. Can you explain this answer?
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If p and q are the roots of the quadratic equation ax2 + bx + c = 0, where a*b*c ≠ 0, is the product of p and q greater than 0?(1) |p + q| = |p| + |q|(2) ac > 0a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the question asked.e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.Correct answer is option 'D'. Can you explain this answer? for GMAT 2024 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about If p and q are the roots of the quadratic equation ax2 + bx + c = 0, where a*b*c ≠ 0, is the product of p and q greater than 0?(1) |p + q| = |p| + |q|(2) ac > 0a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the question asked.e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for GMAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If p and q are the roots of the quadratic equation ax2 + bx + c = 0, where a*b*c ≠ 0, is the product of p and q greater than 0?(1) |p + q| = |p| + |q|(2) ac > 0a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the question asked.e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.Correct answer is option 'D'. Can you explain this answer?.
If p and q are the roots of the quadratic equation ax2 + bx + c = 0, where a*b*c ≠ 0, is the product of p and q greater than 0?(1) |p + q| = |p| + |q|(2) ac > 0a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the question asked.e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.Correct answer is option 'D'. Can you explain this answer? for GMAT 2024 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about If p and q are the roots of the quadratic equation ax2 + bx + c = 0, where a*b*c ≠ 0, is the product of p and q greater than 0?(1) |p + q| = |p| + |q|(2) ac > 0a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the question asked.e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for GMAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If p and q are the roots of the quadratic equation ax2 + bx + c = 0, where a*b*c ≠ 0, is the product of p and q greater than 0?(1) |p + q| = |p| + |q|(2) ac > 0a)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.b)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.c)BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.d)EACH statement ALONE is sufficient to answer the question asked.e)Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.Correct answer is option 'D'. Can you explain this answer?.
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