Let $\alpha _ { \theta }$ and $\beta _ { \theta }$ be the distinct roots of $2 x ^ { 2 } + ( \cos \theta ) x - 1 = 0 , \theta \in ( 0,2 \pi )$. If m and M are the minimum and the maximum values of $\alpha _ { \theta } ^ { 4 } + \beta _ { \theta } ^ { 4 }$, then $16 ( M + m )$ equals :
(1) 24
(2) 25
(3) 17
(4) 27

If $\alpha , \beta$ are roots of quadratic equation $\lambda \mathrm { x } ^ { 2 } - ( \lambda + 3 ) \mathrm { x } + 3 = 0$ and $\alpha < \beta$ such that $\frac { 1 } { \alpha } - \frac { 1 } { \beta } = \frac { 1 } { 3 }$ ，then find sum of all possible values of $\lambda$ ． （A） 3 （B） 2 （C） 1 （D） 4

Let $a, b, c$ be nonzero real numbers, and the two roots of the quadratic equation $ax^2 + bx + c = 0$ both lie between 1 and 3. Select the equation whose two roots must lie between 4 and 5.
(1) $a(x-2)^2 + b(x-2) + c = 0$
(2) $a(x+2)^2 + b(x+2) + c = 0$
(3) $a(2x-7)^2 + b(2x-7) + c = 0$
(4) $a\left(\frac{x+7}{2}\right)^2 + b\left(\frac{x+7}{2}\right) + c = 0$
(5) $a(3x-11)^2 + b(3x-11) + c = 0$

The quadratic expression $x^2 - 14x + 9$ factorises as $(x - \alpha)(x - \beta)$, where $\alpha$ and $\beta$ are positive real numbers.
Which quadratic expression can be factorised as $(x - \sqrt{\alpha})(x - \sqrt{\beta})$?
A $x^2 - \sqrt{10}x + 3$
B $x^2 - \sqrt{14}x + 3$
C $x^2 - \sqrt{20}x + 3$
D $x^2 - 178x + 81$
E $x^2 - 176x + 81$
F $x^2 + 196x + 81$

$$(3x-1)(x+1)+(3x-1)(x-2)=0$$
What is the sum of the real numbers $x$ that satisfy the equation?
A) $\frac{2}{3}$
B) $\frac{3}{4}$
C) $\frac{3}{5}$
D) $\frac{5}{6}$
E) $\frac{7}{6}$

Let $a$ and $b$ be positive real numbers. For each of the equations
$$\begin{aligned} & ax^{2} - 2x + b = 0 \\ & bx^{2} - 3bx + a = 0 \end{aligned}$$
the sum of roots is 1 more than the product of roots.
Which of the following could be the quadratic equation whose roots are $a$ and $b$?
A) $9x^{2} + 8x + 18 = 0$ B) $9x^{2} - 14x + 8 = 0$ C) $9x^{2} - 18x + 14 = 0$ D) $9x^{2} - 8x + 14 = 0$ E) $9x^{2} - 18x + 8 = 0$