### If the roots of the equation (a2 - bc)x2 + 2(b2 - ac)x + (c2 - ab) = 0 are equal

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

### If α and β are the roots of the equation x^2 - x - 1 = 0

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

### If x^2 = 6 + √(6 + √(6 + √(6 + ....∞)), then what is one of the values

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

### The expression 2x^3 + x^2 - 2x - 1 = 0 is divisible by

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

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

### In solving a problem, one student makes a mistake in the coefficient

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

### The sign of the quadratic polynomial ax^2 + bx + c is always

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.

### If a - b = 4 and a^2 + b^2 = 40 where a and b are positive integers

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

### If the roots of the equation Ax^2 + Bx + C = 0 are -1 and 1

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

### The value of x^2 - 4x + 11 can never be less than

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

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

### Aman and Alok attempted to solve a quadratic equation. Aman made a mistake

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

### If α and β are the roots of the equation x^2 + px + q = 0

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