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Consider the following two curves in the x-y plane
y = x3 + x2 + 5
y = x2 + x + 5
Which of the following statements is true for −2 ≤ x ≤ 2?
y = x3 + x2 + 5
y = x2 + x + 5
Which of the following statements is true for −2 ≤ x ≤ 2?
- a)The two curves intersect once
- b)The two curves intersect twice.
- c)The two curves do not intersect.
- d)The two curves intersect thrice.
- e)None of these
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
Consider the following two curves in thex-yplaney = x3+ x2+ 5y = x2+ x...
y=x³ + x² + 5
y = x² + x + 5
We equate each other.
x³ + x² + 5 = x² + x + 5
Subtract x² both the side
x³ + 5 = x + 5
Subtract 5 both the side
x³ = x
x³ - x = 0
x(x² - 1) = 0
x(x + 1)(x - 1) = 0
We get
x = 0 & x - 1 = 0 & x + 1 = 0
x = 0, x = 1 & x = -1
We get the three values of x in interval [-2,2]
So, both the curves cuts each other exactly three times.
y = x² + x + 5
We equate each other.
x³ + x² + 5 = x² + x + 5
Subtract x² both the side
x³ + 5 = x + 5
Subtract 5 both the side
x³ = x
x³ - x = 0
x(x² - 1) = 0
x(x + 1)(x - 1) = 0
We get
x = 0 & x - 1 = 0 & x + 1 = 0
x = 0, x = 1 & x = -1
We get the three values of x in interval [-2,2]
So, both the curves cuts each other exactly three times.
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Community Answer
Consider the following two curves in thex-yplaney = x3+ x2+ 5y = x2+ x...
To determine which statement is true for the given curves, we need to find their points of intersection.
Setting the equations equal to each other, we have:
x^3 + x^2 - 5y = x^2 + x - 5
Simplifying, we get:
x^3 - 6y = 0
This equation represents a curve with a single branch. To find its points of intersection with the y-axis, we set x = 0:
0 - 6y = 0
y = 0
Thus, the curve intersects the y-axis at the point (0,0).
Next, let's find the points of intersection with the x-axis. Setting y = 0, we have:
x^3 - 6(0) = 0
x^3 = 0
x = 0
Therefore, the curve also intersects the x-axis at the point (0,0).
Since both curves intersect at the point (0,0), the statement "The curves intersect at a single point." is true.
Setting the equations equal to each other, we have:
x^3 + x^2 - 5y = x^2 + x - 5
Simplifying, we get:
x^3 - 6y = 0
This equation represents a curve with a single branch. To find its points of intersection with the y-axis, we set x = 0:
0 - 6y = 0
y = 0
Thus, the curve intersects the y-axis at the point (0,0).
Next, let's find the points of intersection with the x-axis. Setting y = 0, we have:
x^3 - 6(0) = 0
x^3 = 0
x = 0
Therefore, the curve also intersects the x-axis at the point (0,0).
Since both curves intersect at the point (0,0), the statement "The curves intersect at a single point." is true.
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Consider the following two curves in thex-yplaney = x3+ x2+ 5y = x2+ x + 5Which of the following statements is true for−2 ≤ x ≤ 2?a)The two curves intersect onceb)The two curves intersect twice.c)The two curves do not intersect.d)The two curves intersect thrice.e)None of theseCorrect answer is option 'D'. Can you explain this answer?
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Consider the following two curves in thex-yplaney = x3+ x2+ 5y = x2+ x + 5Which of the following statements is true for−2 ≤ x ≤ 2?a)The two curves intersect onceb)The two curves intersect twice.c)The two curves do not intersect.d)The two curves intersect thrice.e)None of theseCorrect answer is option 'D'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Consider the following two curves in thex-yplaney = x3+ x2+ 5y = x2+ x + 5Which of the following statements is true for−2 ≤ x ≤ 2?a)The two curves intersect onceb)The two curves intersect twice.c)The two curves do not intersect.d)The two curves intersect thrice.e)None of theseCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following two curves in thex-yplaney = x3+ x2+ 5y = x2+ x + 5Which of the following statements is true for−2 ≤ x ≤ 2?a)The two curves intersect onceb)The two curves intersect twice.c)The two curves do not intersect.d)The two curves intersect thrice.e)None of theseCorrect answer is option 'D'. Can you explain this answer?.
Consider the following two curves in thex-yplaney = x3+ x2+ 5y = x2+ x + 5Which of the following statements is true for−2 ≤ x ≤ 2?a)The two curves intersect onceb)The two curves intersect twice.c)The two curves do not intersect.d)The two curves intersect thrice.e)None of theseCorrect answer is option 'D'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Consider the following two curves in thex-yplaney = x3+ x2+ 5y = x2+ x + 5Which of the following statements is true for−2 ≤ x ≤ 2?a)The two curves intersect onceb)The two curves intersect twice.c)The two curves do not intersect.d)The two curves intersect thrice.e)None of theseCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following two curves in thex-yplaney = x3+ x2+ 5y = x2+ x + 5Which of the following statements is true for−2 ≤ x ≤ 2?a)The two curves intersect onceb)The two curves intersect twice.c)The two curves do not intersect.d)The two curves intersect thrice.e)None of theseCorrect answer is option 'D'. Can you explain this answer?.
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