For which values of $\theta$, with $0 < \theta < \pi / 2$, does the quadratic polynomial in $t$ given by $t ^ { 2 } + 4 t \cos \theta + \cot \theta$ have repeated roots?
(A) $\frac { \pi } { 6 }$ or $\frac { 5 \pi } { 18 }$
(B) $\frac { \pi } { 6 }$ or $\frac { 5 \pi } { 12 }$
(C) $\frac { \pi } { 12 }$ or $\frac { 5 \pi } { 18 }$
(D) $\frac { \pi } { 12 }$ or $\frac { 5 \pi } { 12 }$

Let $a$ be a fixed real number. Consider the equation
$$(x + 2)^{2}(x + 7)^{2} + a = 0, \quad x \in \mathbb{R},$$
where $\mathbb{R}$ is the set of real numbers. For what values of $a$, will the equation have exactly one double-root?

If the equation $\mathrm { a } ( \mathrm { b} - \mathrm { c } ) \mathrm { x } ^ { 2 } + \mathrm { b } ( \mathrm { c } - \mathrm { a } ) \mathrm { x } + \mathrm { c } ( \mathrm { a } - \mathrm { b } ) = 0$ has equal roots, where $\mathrm { a } + \mathrm { c } = 15$ and $\mathrm { b } = \frac { 36 } { 5 }$, then $a ^ { 2 } + c ^ { 2 }$ is equal to

$$y = x ^ { 2 } - 2 ( a + 1 ) x + a ^ { 2 } - 1$$
If the parabola is tangent to the line $y = 1$, what is a?
A) $\frac { -3 } { 2 }$
B) $\frac { -3 } { 4 }$
C) 0
D) 1
E) 2

Let $a$ and $b$ be real numbers. In the rectangular coordinate plane, the parabola $y = x^{2} + ax + b$ is tangent to the $x$-axis and to the line $y = x$.
What is the product $a \cdot b$?
A) $\dfrac{1}{2}$ B) $\dfrac{1}{4}$ C) $\dfrac{1}{8}$ D) $\dfrac{1}{16}$ E) $\dfrac{1}{32}$

Let $a$ and $b$ be positive real numbers. The equations
$$\begin{aligned} & x^{2} + ax + b = 0 \\ & ax^{2} + (b + 3)x + a = 0 \end{aligned}$$
are given. Given that the solution set of each of these equations has exactly 1 element, what is the product of the different values that the sum $a + b$ can take?
A) 24 B) 32 C) 45 D) 72 E) 120