Quadratic Equations

1

If the roots of the equation (a2 - bc)x2 + 2(b2 - ac)x + (c2 - ab) = 0 are equal, where b ≠ 0, then which one of the following is correct?

  1. a3 + b3 + c3 = 0
  2. a3 + b3 + c3 = 3abc
  3. a2 + b2 + c2 = 0
  4. a + b + c = abc
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2

If α and β are the roots of the equation x2 - x - 1 = 0, then what is (α2 + β2)/((α2 - β2)(α - β))

  1. 2/5
  2. 4/5
  3. 3/5
  4. 1/5
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3

If x2 = 6 + √(6 + √(6 + √(6 +....∞)), then what is one of the values of x equal to?

  1. 3
  2. 4
  3. 5
  4. 6
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4

The expression 2x3 + x2 - 2x - 1 = 0 is divisible by

  1. 2x - 1
  2. 2x + 1
  3. x + 2
  4. x - 2
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5

In solving a problem, one student makes a mistake in the coefficient of the first degree term and obtains -9 and -1 for the roots. Another student makes a mistake in the constant term of the equation and obtains 8 and 2 for the roots. The correct equation was:

  1. x2 + 10x + 9 = 0
  2. x2 - 10x + 16 = 0
  3. x2 - 10x + 9 = 0
  4. x2 + 10x + 16 = 0
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6

The sign of the quadratic polynomial ax2 + bx + c is always positive if

  1. a is positive and b2 - 4ac ≤ 0.
  2. a can be any real number and b2 - 4ac ≤ 0.
  3. a can be any real number and b2 - 4ac ≥ 0.
  4. a is positive and b2 - 4ac ≥ 0.
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7

If a - b = 4 and a2 + b2 = 40 where a and b are positive integers, where a and b are positive integers, then a3 + b6 is equal to

  1. 264
  2. 280
  3. 300
  4. 324
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8

If the roots of the equation Ax2 + Bx + C = 0 are -1 and 1, then which one of the following is correct?

  1. A and B are both positive
  2. A and C are both negative
  3. A and C are of opposite sign
  4. A and C are both zero
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9

The value of x2 - 4x + 11 can never be less than

  1. 7
  2. 8
  3. 11
  4. 22
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10

Aman and Alok attempted to solve a quadratic equation. Aman made a mistake in writing down the constant term and ended up in roots (4, 3). Alok made a mistake in writing down the coefficient of x to get roots (3, 2). The correct roots of the equation are

  1. –4, –3
  2. 6, 1
  3. 4, 3
  4. –6, –1
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11

If α and β are the roots of the equation x2 + px + q = 0, then what is α2 + β2 equal to?

  1. p2 – 2q
  2. q2 – 2p
  3. p2 + 2q
  4. q2 – q
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