SSC Linear Equations questions

If a - b = 5 and $$a^2 + b^2 = 45$$, then the value of ab is:

A and B are positive integers. If $$A + B + AB = 65$$, then what is the difference between $$A$$ and $$B (A, B \leq 15)?$$

If $$a^3 - b^3 = 1603$$ and $$(a - b) = 7$$, then $$(a + b)^2 - ab$$ is equal to:

If $$x + y + z = 22$$ and $$xy + yz + zx = 35$$, then what is the value of $$(x - y)^2 + (y - z)^2 + (z - x)^2$$?

If $$a^3 — b^3 = 208$$ and $$a - b = 4$$, then $$(a + b)^2 — ab$$ is equal to:

If $$\alpha$$ and $$\beta$$ are the roots of the equation $$x^2 + x - 1 = 0$$, then what is the equation whose roots are $$\alpha^5$$ and $$\beta^5$$?

If $$\frac{(x + y)}{z} = 2$$, then what is the value of $$\left[\frac{y}{(y - z)}\right] + \left[\frac{x}{(x - z)}\right]$$?

If $$x + \frac{1}{x} = 7$$, then $$x^3 + \frac{1}{x^3}$$ is equal to:

If $$\alpha$$ and $$\beta$$ are the roots of equation $$x^2 - 2x + 4 = 0$$, then what is the equation whose roots are $$\frac{\alpha^3}{\beta^2}$$ and $$\frac{\beta^3}{\alpha^2}$$?

If $$ab + bc + ca = 8$$ and $$a+b +c = 12$$ then $$(a^2 + b^2 + c^2)$$ is equal to: 

If one root of the equation $$Ax^2 +Bx + C = 0$$ is two and a half times the others, then which of the following is TRUE?

If $$x + (\frac{1}{x}) = \frac{(\surd3 + 1)}{2}$$, then what is the value of $$x^4 + (\frac{1}{x^4})$$?

If $$a + a^2 + a^3 - 1 = 0$$, then what is the value of $$a^3 + (\frac{1}{a})$$?

If $$x^2 - 12x + 33 = 0$$, then what is the value of $$(x - 4)^2 + \left[\frac{1}{(x - 4)^2}\right]$$?

If $$a - \left(\frac{1}{a}\right) = b, b - \left(\frac{1}{b}\right) = c$$ and $$c - \left(\frac{1}{c}\right) = a$$, then what is the value of $$\left(\frac{1}{ab}\right) + \left(\frac{1}{bc}\right) + \left(\frac{1}{ca}\right)$$?

If $$a^4 + 1 = \left[\frac{a^2}{b^2}\right] (4b^2 - b^4 - 1)$$, then what is the value of $$a^4 + b^4$$?

If $$3\sqrt{\frac{1-a}{a}} + 9 =19 - 3\sqrt{\frac{a}{1-a}}$$, then what is the value of $$a$$?

If the roots of the equation $$a(b - c)x^2 +b(c - a)x + c(a - b) = 0$$ are equal, then which of the following is true?

If $$a + b =10$$ and $$\sqrt{\frac{a}{b}} - 13 = -\sqrt{\frac{b}{a}} -11$$, then what is the value of $$3ab + 4a^2 + 5b^2$$

If $$[\surd(a^2 + b^2 + ab)] + [\surd(a^2 + b^2 - ab)] = 1$$, then what is the value of $$(1 - a^2)(1 - b^2)$$?