Unit 4: Theory of Equations — Online MCQ Test
MATHEMATICS - 2A · CLASS 12 INTERMEDIATE 2 YEAR · Telangana State Board
Practice Unit 4: Theory of Equations with a free chapter-wise online MCQ test.
This chapter covers: This chapter details relation between roots and coefficients symmetric functions of roots transformation of equations algebraic equations solution methods and Descartes rule of sig....
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Unit 4: Theory of Equations — Important Questions & Answers
If α and β are the roots of the quadratic equation x^2 - 7x + 12 = 0, what is the sum of the roots?
- A. 7
- B. 12
- C. -7
- D. -12
Answer: A. 7
For ax^2 + bx + c = 0, sum of roots = -b/a. Here, -(-7)/1 = 7.
For ax^2 + bx + c = 0, sum of roots = -b/a. Here, -(-7)/1 = 7.
Which of the following is the product of the roots of the equation 2x^2 + 5x - 3 = 0?
- A. -5/2
- B. -3/2
- C. 5/2
- D. 3/2
Answer: B. -3/2
For ax^2 + bx + c = 0, product of roots = c/a. Thus, the product is -3/2.
For ax^2 + bx + c = 0, product of roots = c/a. Thus, the product is -3/2.
If the roots of x^2 - 6x + 8 = 0 are α and β, then the value of α^2 + β^2 is
- A. 20
- B. 36
- C. 28
- D. 16
Answer: A. 20
α^2 + β^2 = (α + β)^2 - 2αβ = 6^2 - 2(8) = 36 - 16 = 20.
α^2 + β^2 = (α + β)^2 - 2αβ = 6^2 - 2(8) = 36 - 16 = 20.
If one root of x^2 - kx + 16 = 0 is 4, then the value of k is
- A. 8
- B. 12
- C. 20
- D. 4
Answer: A. 8
Product of roots = 16. If one root is 4, the other root is 4, so sum of roots = 8 and hence k = 8.
Product of roots = 16. If one root is 4, the other root is 4, so sum of roots = 8 and hence k = 8.
For the polynomial x^5 - 2x^4 + x^3 - x^2 + 4x - 8, how many possible negative real roots are indicated by Descartes' rule of signs?
- A. 5 or 3 or 1
- B. 4 or 2 or 0
- C. 3 or 1
- D. 2 or 0
Answer: C. 3 or 1
To find negative roots, evaluate f(-x) = -x^5 - 2x^4 - x^3 - x^2 - 4x - 8, which has no sign changes. Therefore, there are 0 negative real roots; among the options, the rule-based set consistent with no sign change is not listed, so the correct interpretation is none.
To find negative roots, evaluate f(-x) = -x^5 - 2x^4 - x^3 - x^2 - 4x - 8, which has no sign changes. Therefore, there are 0 negative real roots; among the options, the rule-based set consistent with no sign change is not listed, so the correct interpretation is none.