GRE Algebra

Question

If x, y, z are real and distinct then x ^2 + 4y ^2 + 9z ^2 -6yz - 3zx - 2xy is always

Possible Answers

Selected	Possible Answer
	A. Positive
	B. Negative
	C. Non-negative
	D. Non-positive
	E. Zero

Question

The condition that one root of the equation ax ^2 + bx + c = 0 exceeds the other by p is

Possible Answers

Selected	Possible Answer
	A.
	B.
	C.
	D.
	E.

Question

If $\alpha, \beta$ are the roots of 3x ^2 + 5x - 7 = 0

then the value of $(\frac{1}{3\alpha + 5}) ^ 2 + (\frac{1}{3\beta + 5}) ^ 2$ is

Possible Answers

Selected	Possible Answer
	A. 67/63
	B. 109/63
	C. 109/441
	D. 67/441
	E. 63/67

Question

If the two equations $x ^2 - cx + d = 0 \quad \And \quad x ^2 - ax + b = 0$ have a common root and the second equation has equal roots, then

Possible Answers

Selected	Possible Answer
	A.
	B.
	C.
	D.
	E.

Question

If $\alpha \not = \beta \quad$ but $\quad \alpha ^2 = 2\alpha - 3; \beta ^2 = 2\beta - 3$ then the equations whose roots are $\frac{\alpha}{\beta} \quad \And \quad \frac{\beta}{\alpha} \quad is$

Possible Answers

Selected	Possible Answer
	A.
	B.
	C.
	D.
	E.

Question

If the roots of the equation 12x ^2 + mx + 5 = 0 are in the ratio 5 : 4, then m=?

Possible Answers

Selected	Possible Answer
	A. $6\sqrt{3}$
	B. $\pm 6\sqrt{3}$
	C. $9\sqrt{3}$
	D. $\pm 9\sqrt{3}$
	E. $6\sqrt{2}$

Question

If $\alpha, \beta$ are the roots of ax ^2 + bx + c = 0

then the value of $(\frac{\beta}{a\alpha + b}) ^ 4 - (\frac{\alpha}{a\beta + b}) ^ 4$ is

Possible Answers

Selected	Possible Answer
	A. $\frac{1-a}{a ^4}$
	B. $\frac{1+a}{a ^3}$
	C. $\frac{1+a}{a ^2}$
	D.
	E. None

Question

If the roots of $px ^2 + 2qx + r = 0 \quad and \quad qx ^2 - 2\sqrt{pr}x + q = 0$ are simultaneously real, then

Possible Answers

Selected	Possible Answer
	A.
	B. $2q = \sqrt{pr}$
	C. $p = q; r \not = 0$
	D.
	E.

Question

If the coefficient of x in the quadratic equation x ^2 + px + q = 0 was taken as 17 in place of 13, its roots were found to be -2 and -15 the roots of the original equation are

Possible Answers