118. The graph of $f(x) = \dfrac{-2x^2 + 3x}{ax^2 + bx + c}$ has asymptotes $y = -1$, $y = -2$, $x = -2$, and $x = 1$. What is $f(-1)$?
(1) $1.25$ (2) $1.5$ (3) $1.75$ (4) $-1.5$

118. The function $f(x) = \dfrac{ax^3 - bx^2 + 2}{ax^3 - bx + 2}$ is discontinuous at exactly two points and has exactly two asymptotes parallel to the coordinate axes. What are the values of $a$ and $b$?
(1) $a = 0,\ b = 2$ (2) $a = 8,\ b = 10$
(3) $a = -2,\ b = 0$ (4) $a = -8,\ b = -6$

124-- The locus of the intersection of the asymptotes of the hyperbola $y=\dfrac{ax+3}{(a+1)x+(a-1)}$ is $y=\dfrac{3}{2}x^{2}+x+\dfrac{5}{6}$. The graph of this hyperbola intersects the $x$-axis at which length?
(1) $3$ (2) $-3$ (3) $\dfrac{3}{2}$ (4) $-\dfrac{3}{2}$

16. For a specific value of $k$, the function $$f(x) = \begin{cases} |x - [-x]| & x \in [x] \text{ even} \\ x - [x] + k & x \in [x] \text{ odd} \end{cases}$$ is continuous at $x = n$ and $x = -n$. Which case is correct regarding $n$ specifically? $(k, n \in \mathbb{N})$

• [(1)] $n$ even
• [(2)] $n$ odd
• [(3)] $f$ is continuous for all values of $n$.
• [(4)] $f$ is not continuous for any value of $n$.

%% Page 6

Draw the graph (on plain paper) of $f(x) = \min\{|x|-1, |x-1|-1, |x-2|-1\}$.

Let $h(x) = \frac{x^4}{(1-x)^4}$ and $g(x) = f(h(x)) = h(x) + 1/h(x)$. Sketch the graph of $g(x)$ and show that $g(x)$ has a root between $0$ and $1$.

Find all values of $c$ for which the equation $\log_2 x = cx$ has solutions.

The number of solutions of the equation $\sin ( \cos \theta ) = \theta$, $- 1 \leq \theta \leq 1$, is
(a) 0
(b) 1
(c) 2
(d) 3

Find the number of intersection points of $y = \log x$ and $y = x^2$.

Let $A = \{(x,y) : x^4 + y^2 \leq 1\}$ and $B = \{(x,y) : x^6 + y^4 \leq 1\}$. Which of the following is true?
(A) $A = B$
(B) $A \subset B$ (A is a proper subset of B)
(C) $B \subset A$ (B is a proper subset of A)
(D) Neither $A \subset B$ nor $B \subset A$

In the interval $( - 2 \pi , 0 )$, the function $f ( x ) = \sin \left( \frac { 1 } { x ^ { 3 } } \right)$
(A) never changes sign
(B) changes sign only once
(C) changes sign more than once, but finitely many times
(D) changes sign infinitely many times

Which of the following graphs represents the function $$f ( x ) = \int _ { 0 } ^ { \sqrt { x } } e ^ { - u ^ { 2 } / x } d u , \quad \text { for } \quad x > 0 \quad \text { and } \quad f ( 0 ) = 0 ?$$ (A), (B), (C), (D) as given by the respective graphs.

The equation $x^3 y + x y^3 + x y = 0$ represents
(A) a circle
(B) a circle and a pair of straight lines
(C) a rectangular hyperbola
(D) a pair of straight lines

In the interval $( 0,2 \pi )$, the function $\sin \left( \frac { 1 } { x ^ { 3 } } \right)$
(a) never changes sign
(b) changes sign only once
(c) changes sign more than once, but finitely many times
(d) changes sign infinitely many times.

In the interval $( 0,2 \pi )$, the function $\sin \left( \frac { 1 } { x ^ { 3 } } \right)$
(a) never changes sign
(b) changes sign only once
(c) changes sign more than once, but finitely many times
(d) changes sign infinitely many times.

The equation $x ^ { 3 } y + x y ^ { 3 } + x y = 0$ represents
(a) a circle
(b) a circle and a pair of straight lines
(c) a rectangular hyperbola
(d) a pair of straight lines.

In the interval $( - 2 \pi , 0 )$, the function $f ( x ) = \sin \left( \frac { 1 } { x ^ { 3 } } \right)$
(A) never changes sign
(B) changes sign only once
(C) changes sign more than once, but finitely many times
(D) changes sign infinitely many times

Let the function $f : [0,1] \rightarrow \mathbb{R}$ be defined as $$f(x) = \max\left\{\frac{|x - y|}{x + y + 1} : 0 \leq y \leq 1\right\} \text{ for } 0 \leq x \leq 1$$ Then which of the following statements is correct?
(A) $f$ is strictly increasing on $\left[0, \frac{1}{2}\right]$ and strictly decreasing on $\left[\frac{1}{2}, 1\right]$.
(B) $f$ is strictly decreasing on $\left[0, \frac{1}{2}\right]$ and strictly increasing on $\left[\frac{1}{2}, 1\right]$.
(C) $f$ is strictly increasing on $\left[0, \frac{\sqrt{3}-1}{2}\right]$ and strictly decreasing on $\left[\frac{\sqrt{3}-1}{2}, 1\right]$.
(D) $f$ is strictly decreasing on $\left[0, \frac{\sqrt{3}-1}{2}\right]$ and strictly increasing on $\left[\frac{\sqrt{3}-1}{2}, 1\right]$.

Let $f ( x )$ be a degree 4 polynomial with real coefficients. Let $z$ be the number of real zeroes of $f$, and $e$ be the number of local extrema (i.e., local maxima or minima) of $f$. Which of the following is a possible $( z , e )$ pair?
(A) $( 4,4 )$
(B) $( 3,3 )$
(C) $( 2,2 )$
(D) $( 0,0 )$

Let $f : (0, \infty) \rightarrow \mathbb{R}$ be defined by $$f(x) = \lim_{n \rightarrow \infty} \cos^{n}\left(\frac{1}{n^{x}}\right)$$
(a) Show that $f$ has exactly one point of discontinuity.
(b) Evaluate $f$ at its point of discontinuity.

The locus of points ( $x , y$ ) in the plane satisfying $\sin ^ { 2 } ( x ) + \sin ^ { 2 } ( y ) = 1$ consists of
(A) A circle that is centered at the origin.
(B) infinitely many circles that are all centered at the origin.
(C) infinitely many lines with slope $\pm 1$.
(D) finitely many lines with slope $\pm 1$.

The number of real solutions of the equation $x ^ { 2 } = e ^ { x }$ is:
(A) 0
(B) 1
(C) 2
(D) 3 .

The number of distinct real roots of the equation $x \sin ( x ) + \cos ( x ) = x ^ { 2 }$ is
(A) 0
(B) 2
(C) 24
(D) none of the above.

The number of real solutions of $e ^ { x } = \sin ( x )$ is
(A) 0
(B) 1
(C) 2
(D) infinite.

The range of values that the function $$f ( x ) = \frac { x ^ { 2 } + 2 x + 4 } { 2 x ^ { 2 } + 4 x + 9 }$$ takes as $x$ varies over all real numbers in the domain of $f$ is:
(A) $\frac { 3 } { 7 } < f ( x ) \leq \frac { 1 } { 2 }$
(B) $\frac { 3 } { 7 } \leq f ( x ) < \frac { 1 } { 2 }$
(C) $\frac { 3 } { 7 } < f ( x ) \leq \frac { 4 } { 9 }$
(D) $\frac { 3 } { 7 } \leq f ( x ) \leq \frac { 1 } { 2 }$