Chapter Exam

Unit 4: Theory of Equations — Online MCQ Test

Q: If α and β are the roots of the quadratic equation x^2 - 7x + 12 = 0, what is the sum of the roots?

Answer: A. 7 — For ax^2 + bx + c = 0, sum of roots = -b/a. Here, -(-7)/1 = 7.

Q: Which of the following is the product of the roots of the equation 2x^2 + 5x - 3 = 0?

Answer: B. -3/2 — For ax^2 + bx + c = 0, product of roots = c/a. Thus, the product is -3/2.

Q: If the roots of x^2 - 6x + 8 = 0 are α and β, then the value of α^2 + β^2 is

Answer: A. 20 — α^2 + β^2 = (α + β)^2 - 2αβ = 6^2 - 2(8) = 36 - 16 = 20.

Q: If one root of x^2 - kx + 16 = 0 is 4, then the value of k is

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.

Q: 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?

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.

MATHEMATICS - 2A · CLASS 12 INTERMEDIATE 2 YEAR · Telangana State Board

Practice Unit 4: Theory of Equations with a free chapter-wise online MCQ test for Telangana State Board CLASS 12 INTERMEDIATE 2 YEAR MATHEMATICS - 2A. 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.... AI-generated questions from basic to board-exam level, with instant results and explanations.

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Unit 4: Theory of Equations — Important Questions & Answers (FAQ)

Frequently asked questions from Telangana State Board CLASS 12 INTERMEDIATE 2 YEAR MATHEMATICS - 2A — Unit 4: Theory of Equations, with answers and explanations. These are sample questions; the exam has its own separate question set.

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.

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.

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.

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.

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.

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