TRB BT Assistant - Transformation of Equations and Reciprocal Equations Online Test (MCQ Quiz)

1. If the roots of the equation x³ + px² + qx + r = 0 are α, β, γ, what are the roots of the equation x³ - px² + qx - r = 0?

A) -α, -β, -γ B) α, β, γ C) 1/α, 1/β, 1/γ D) -1/α, -1/β, -1/γ

2. Given the equation x² + 5x + 6 = 0 with roots α and β, find the equation whose roots are α+1 and β+1.

A) x² + 3x + 2 = 0 B) x² + 7x + 12 = 0 C) x² + 5x + 6 = 0 D) x² - 5x + 6 = 0

3. If α, β, γ are the roots of the cubic equation ax³ + bx² + cx + d = 0, what is the sum of the roots of the equation formed by replacing x with (1/x)?

A) -b/d B) b/d C) -c/d D) c/d

4. What is the condition for the equation ax³ + bx² + cx + d = 0 to be a reciprocal equation of the first type?

A) a = d and b = c B) a = -d and b = -c C) a = b and c = d D) a = c and b = d

5. Find the transformed equation whose roots are the squares of the roots of the equation x³ - 6x² + 11x - 6 = 0.

A) x³ + 30x² + 121x + 36 = 0 B) x³ - 30x² + 121x - 36 = 0 C) x³ + 30x² - 121x + 36 = 0 D) x³ - 30x² - 121x - 36 = 0

6. If α, β are the roots of x² + bx + c = 0, then the equation with roots α² and β² is:

A) x² - (b² - 2c)x + c² = 0 B) x² + (b² - 2c)x + c² = 0 C) x² - (b² + 2c)x + c² = 0 D) x² + (b² + 2c)x + c² = 0

7. Consider the equation x⁴ + 2x³ + 3x² + 2x + 1 = 0. This is an example of:

A) A reciprocal equation of the first type B) A reciprocal equation of the second type C) A standard quartic equation D) An incomplete equation

8. To solve a reciprocal equation of the form ax⁴ + bx³ + cx² + bx + a = 0, we typically divide by x² and substitute:

A) y = x + 1/x B) y = x - 1/x C) y = x² + 1/x² D) y = x² - 1/x²

9. If the roots of x³ + 2x² + 3x + 4 = 0 are α, β, γ, what is the sum of the roots of the equation whose roots are 2α, 2β, 2γ?

A) -4 B) 4 C) -8 D) 8

10. What is the condition for the equation ax⁴ + bx³ + cx² + dx + e = 0 to be a reciprocal equation of the second type?

A) a = -e and b = -d B) a = e and b = d C) a = -e and b = d D) a = e and b = -d

11. Given the equation x² - 5x + 6 = 0 with roots α and β, find the equation whose roots are 1/α and 1/β.

A) 6x² - 5x + 1 = 0 B) 6x² + 5x + 1 = 0 C) x² - 5x + 6 = 0 D) x² + 5x + 6 = 0

12. If α, β, γ are the roots of x³ - 3x² + 2x - 1 = 0, find the equation whose roots are α+2, β+2, γ+2.

A) x³ + 3x² + 4x + 1 = 0 B) x³ - 3x² + 4x - 1 = 0 C) x³ + 3x² - 4x + 1 = 0 D) x³ - 3x² - 4x - 1 = 0

13. The equation x⁵ + 2x⁴ + 3x³ + 3x² + 2x + 1 = 0 is:

A) A reciprocal equation of the first type (odd degree) B) A reciprocal equation of the second type (odd degree) C) A standard quintic equation D) A reciprocal equation of the first type (even degree)

14. If y = x + 1/x, then x² + 1/x² can be expressed in terms of y as:

A) y² - 2 B) y² + 2 C) y² - 1 D) y² + 1

15. Find the equation whose roots are the reciprocals of the roots of 2x³ - 3x² + 4x - 5 = 0.

A) 5x³ - 4x² + 3x - 2 = 0 B) 5x³ + 4x² + 3x + 2 = 0 C) 2x³ - 3x² + 4x - 5 = 0 D) 5x³ - 4x² - 3x + 2 = 0

16. If α, β, γ are the roots of x³ + px² + qx + r = 0, the equation whose roots are αβ, βγ, γα is:

A) x³ - qx² + (pr)x - r² = 0 B) x³ + qx² + (pr)x + r² = 0 C) x³ - qx² + (pr)x + r² = 0 D) x³ + qx² - (pr)x - r² = 0

17. What is the relationship between the coefficients of a reciprocal equation of the first type of odd degree?

A) Coefficients equidistant from the beginning and end are equal B) Coefficients equidistant from the beginning and end are equal in magnitude but opposite in sign C) All coefficients are zero D) The middle coefficient is zero

18. Solve the reciprocal equation x⁴ - 5x³ + 6x² - 5x + 1 = 0.

A) Roots are 1, 1, (5 ± √21)/2 B) Roots are -1, -1, (5 ± √21)/2 C) Roots are 1, -1, (5 ± √21)/2 D) Roots are 1, 1, (5 ± √5)/2

19. If the roots of x² + ax + b = 0 are α and β, find the equation whose roots are α+k and β+k.

A) x² + (a-2k)x + (b-ak+k²) = 0 B) x² + (a+2k)x + (b+ak+k²) = 0 C) x² + (a-2k)x + (b+ak+k²) = 0 D) x² + (a+2k)x + (b-ak+k²) = 0

20. The equation x³ + 2x² + 2x + 1 = 0 is a reciprocal equation of:

A) First type, odd degree B) Second type, odd degree C) First type, even degree D) Second type, even degree

21. If α, β, γ are roots of x³ - 6x² + 11x - 6 = 0, find the equation whose roots are α-1, β-1, γ-1.

A) x³ - 3x² + 2x = 0 B) x³ + 3x² + 2x = 0 C) x³ - 3x² - 2x = 0 D) x³ + 3x² - 2x = 0

22. If the roots of ax² + bx + c = 0 are α and β, the equation with roots α+1/α and β+1/β is:

A) a²x² - (ab-bc)x + (b²-ac) = 0 B) a²x² + (ab-bc)x + (b²-ac) = 0 C) a²x² - (ab+bc)x + (b²-ac) = 0 D) a²x² - (ab-bc)x - (b²-ac) = 0

23. A reciprocal equation of the second type of odd degree always has a root:

A) 1 B) -1 C) 0 D) i

24. Transform the equation x³ + 6x² + 11x + 6 = 0 into an equation whose roots are the reciprocals of its roots.

A) 6x³ + 11x² + 6x + 1 = 0 B) 6x³ - 11x² + 6x - 1 = 0 C) x³ + 6x² + 11x + 6 = 0 D) 6x³ + 11x² - 6x + 1 = 0

25. If α, β are the roots of x² - 4x + 3 = 0, find the equation whose roots are α³ and β³.

A) x² - 37x + 27 = 0 B) x² + 37x + 27 = 0 C) x² - 37x - 27 = 0 D) x² + 37x - 27 = 0

26. What is the transformed equation if we replace x by (x+1) in the equation x³ - 2x² + 3x - 4 = 0?

A) x³ + x² + 2x - 2 = 0 B) x³ - x² + 2x - 2 = 0 C) x³ + x² - 2x - 2 = 0 D) x³ - x² - 2x - 2 = 0

27. The equation 2x⁵ - 3x⁴ + 4x³ - 4x² + 3x - 2 = 0 is a reciprocal equation of:

A) First type, odd degree B) Second type, odd degree C) First type, even degree D) Second type, even degree

28. If α, β, γ are the roots of x³ + 2x² - x + 3 = 0, find the equation whose roots are 1/α, 1/β, 1/γ.

A) 3x³ - x² + 2x + 1 = 0 B) 3x³ + x² - 2x + 1 = 0 C) 3x³ + x² + 2x + 1 = 0 D) 3x³ - x² - 2x + 1 = 0

29. Consider the reciprocal equation x⁴ + 3x³ + 4x² + 3x + 1 = 0. After dividing by x² and substituting y = x + 1/x, the equation in y is:

A) y² + 3y + 2 = 0 B) y² - 3y + 2 = 0 C) y² + 3y - 2 = 0 D) y² - 3y - 2 = 0

30. If α, β are the roots of x² - 6x + 8 = 0, find the equation whose roots are α+3 and β+3.

A) x² - 12x + 24 = 0 B) x² + 12x + 24 = 0 C) x² - 12x - 24 = 0 D) x² + 12x - 24 = 0

31. The equation x⁵ - 3x⁴ + 2x³ - 2x² + 3x - 1 = 0 is a reciprocal equation of:

A) First type, odd degree B) Second type, odd degree C) First type, even degree D) Second type, even degree

32. If α, β, γ are the roots of x³ + 2x + 1 = 0, find the equation whose roots are α², β², γ².

A) x³ + 4x² - 4x - 1 = 0 B) x³ - 4x² + 4x - 1 = 0 C) x³ + 4x² + 4x - 1 = 0 D) x³ - 4x² - 4x - 1 = 0

33. What is the sum of the roots of the equation obtained by replacing x with (x-1) in x³ - 5x² + 6x - 1 = 0?

A) 2 B) 3 C) 4 D) 5

34. The equation 3x⁴ - 5x³ + 7x² - 5x + 3 = 0 is a reciprocal equation of:

A) First type, even degree B) Second type, even degree C) First type, odd degree D) Second type, odd degree

35. If α, β, γ are roots of x³ + 2x² + 3x + 4 = 0, find the equation whose roots are α+1, β+1, γ+1.

A) x³ + 5x² + 8x + 5 = 0 B) x³ - 5x² + 8x + 5 = 0 C) x³ + 5x² - 8x + 5 = 0 D) x³ + 5x² + 8x - 5 = 0

36. For the equation ax⁴ + bx³ + cx² + dx + e = 0 to be a reciprocal equation of the first type, which condition MUST hold?

A) a = e and b = d B) a = -e and b = -d C) a = 0 and e = 0 D) b = 0 and d = 0

37. Find the equation whose roots are the negatives of the roots of x³ - 2x² + 3x - 4 = 0.

A) x³ + 2x² + 3x + 4 = 0 B) x³ - 2x² + 3x + 4 = 0 C) x³ + 2x² - 3x + 4 = 0 D) x³ + 2x² + 3x - 4 = 0

38. If α, β, γ are roots of x³ + px² + qx + r = 0, the equation whose roots are α+β, β+γ, γ+α is:

A) x³ + px² + (q²-2pr)x - (r(p²-2q)) = 0 B) x³ - px² + (q²-2pr)x - (r(p²-2q)) = 0 C) x³ + px² - (q²-2pr)x - (r(p²-2q)) = 0 D) x³ + px² + (q²-2pr)x + (r(p²-2q)) = 0

39. The equation x³ - 2x² - 2x + 1 = 0 is a reciprocal equation of:

A) Second type, odd degree B) First type, odd degree C) Second type, even degree D) First type, even degree

40. If α, β are roots of x² + 5x + 6 = 0, find the equation whose roots are α-2 and β-2.

A) x² + x + 2 = 0 B) x² - x + 2 = 0 C) x² + x - 2 = 0 D) x² - x - 2 = 0

41. Transform the equation x³ - 3x² + 2x - 1 = 0 into an equation whose roots are 3 times the roots of the original equation.

A) x³ - 9x² + 18x - 27 = 0 B) x³ + 9x² + 18x - 27 = 0 C) x³ - 9x² - 18x - 27 = 0 D) x³ - 9x² + 18x + 27 = 0

42. The reciprocal equation x⁵ + 1 = 0 has roots:

A) The roots of x+1=0 and the roots of x⁴-x³+x²-x+1=0 B) The roots of x-1=0 and the roots of x⁴+x³+x²+x+1=0 C) The roots of x+1=0 and the roots of x⁴+x³+x²+x+1=0 D) The roots of x-1=0 and the roots of x⁴-x³+x²-x+1=0

43. If α, β, γ are roots of x³ + 6x² + 11x + 6 = 0, find the equation whose roots are α+1, β+1, γ+1.

A) x³ + 3x² + 2x = 0 B) x³ - 3x² + 2x = 0 C) x³ + 3x² - 2x = 0 D) x³ - 3x² - 2x = 0

44. For a reciprocal equation of the second type of even degree, like ax⁴ + bx³ + cx² - bx + a = 0, what is a guaranteed root?

A) 1 B) -1 C) 0 D) i

45. If α, β are the roots of x² - 3x + 2 = 0, find the equation whose roots are α² and β².

A) x² - 5x + 4 = 0 B) x² + 5x + 4 = 0 C) x² - 5x - 4 = 0 D) x² + 5x - 4 = 0

46. What is the sum of the roots of the equation formed by replacing x with (x+2) in x³ - 4x² + 5x - 1 = 0?

A) 10 B) 9 C) 8 D) 7

47. Which type of equation is x⁴ - 5x³ + 6x² - 5x + 1 = 0?

A) Reciprocal equation of the first type, even degree B) Reciprocal equation of the second type, even degree C) Reciprocal equation of the first type, odd degree D) Reciprocal equation of the second type, odd degree

48. If α, β, γ are the roots of x³ - 2x² + 3x - 4 = 0, find the equation whose roots are α-1, β-1, γ-1.

A) x³ - 5x² + 7x - 10 = 0 B) x³ + 5x² + 7x - 10 = 0 C) x³ - 5x² - 7x - 10 = 0 D) x³ - 5x² + 7x + 10 = 0

49. The sum of the roots of a reciprocal equation of the first type of degree n is always:

A) 0 B) 1 C) -1 D) Equal to the coefficient of x^(n-1)

Transformation of Equations and Reciprocal Equations - Online Test

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