The number of distinct real roots of the equation $x^4 - 4x^3 + 12x^2 + x - 1 = 0$ is: (1) 2 (2) 3 (3) 0 (4) 4

$\lim _ { x \rightarrow 0 } \frac { ( 27 + x ) ^ { \frac { 1 } { 3 } } - 3 } { 9 - ( 27 + x ) ^ { \frac { 2 } { 3 } } }$ equals
(1) $- \frac { 1 } { 6 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 1 } { 3 }$
(4) $- \frac { 1 } { 3 }$

For each $t \in R$, let $[ t ]$ be the greatest integer less than or equal to $t$. Then $\lim _ { x \rightarrow 0 ^ { + } } x \left( \left[ \frac { 1 } { x } \right] + \left[ \frac { 2 } { x } \right] + \ldots + \left[ \frac { 15 } { x } \right] \right)$
(1) does not exist (in $R$ )
(2) is equal to 0
(3) is equal to 15
(4) is equal to 120

$\lim _ { x \rightarrow 2 } \frac { 3 ^ { x } + 3 ^ { 3 - x } - 12 } { 3 ^ { - \frac { x } { 2 } } - 3 ^ { 1 - x } }$ is equal to

If $[ x ]$ be the greatest integer less than or equal to $x$, then $\sum _ { n = 8 } ^ { 100 } \left[ \frac { ( - 1 ) ^ { n } n } { 2 } \right]$ is equal to:
(1) 0
(2) 4
(3) - 2
(4) 2

The number of real roots of the equation $e ^ { 4 x } + 2 e ^ { 3 x } - e ^ { x } - 6 = 0$ is :
(1) 0
(2) 1
(3) 4
(4) 2

The number of distinct real roots of the equation $x ^ { 7 } - 7 x - 2 = 0$ is
(1) 5
(2) 7
(3) 1
(4) 3

The set of values of $a$ for which $\lim _ { x \rightarrow a } ( [ x - 5 ] - [ 2 x + 2 ] ) = 0$, where $[ \zeta ]$ denotes the greatest integer less than or equal to $\zeta$ is equal to
(1) $( - 7.5 , - 6.5 )$
(2) $( - 7.5 , - 6.5 ]$
(3) $[ - 7.5 , - 6.5 ]$
(4) $[ - 7.5 , - 6.5 )$

$\lim_{x \rightarrow \infty} \frac{(\sqrt{3x+1} + \sqrt{3x-1})^6 + (\sqrt{3x+1} - \sqrt{3x-1})^6}{\left(x + \sqrt{x^2-1}\right)^6 + \left(x - \sqrt{x^2-1}\right)^6} x^3$
(1) is equal to $\frac{27}{2}$
(2) is equal to 9
(3) does not exist
(4) is equal to 27

The equation $x ^ { 2 } - 4 x + [ x ] + 3 = x [ x ]$, where $[ x ]$ denotes the greatest integer function, has:
(1) exactly two solutions in $( - \infty , \infty )$
(2) no solution
(3) a unique solution in $( - \infty$, 1)
(4) a unique solution in $( - \infty , \infty )$

If $\alpha > \beta > 0$ are the roots of the equation $a x ^ { 2 } + b x + 1 = 0$, and $\lim _ { x \rightarrow \frac { 1 } { \alpha } } \left( \frac { 1 - \cos \left( x ^ { 2 } + b x + a \right) } { 2 ( 1 - \alpha x ) ^ { 2 } } \right) ^ { \frac { 1 } { 2 } } = \frac { 1 } { k } \left( \frac { 1 } { \beta } - \frac { 1 } { \alpha } \right)$, then $k$ is equal to
(1) $2 \beta$
(2) $\alpha$
(3) $2 \alpha$
(4) $\beta$

$\lim _ { x \rightarrow 0 } \left( \left( \frac { 1 - \cos ^ { 2 } ( 3 x ) } { \cos ^ { 3 } ( 4 x ) } \right) \left( \frac { \sin ^ { 3 } ( 4 x ) } { \left( \log _ { e } ( 2 x + 1 ) \right) ^ { 5 } } \right) \right)$ is equal to
(1) 15
(2) 9
(3) 18
(4) 24

The number of real solutions of the equation $x\left(x^2 + 3|x| + 5|x-1| + 6|x-2|\right) = 0$ is $\underline{\hspace{1cm}}$.

The number of real solutions of the equation $x | x + 5 | + 2 | x + 7 | - 2 = 0$ is $\_\_\_\_$

Let the positive integers be written in the form : If the $k ^ { \text {th} }$ row contains exactly $k$ numbers for every natural number $k$, then the row in which the number 5310 will be, is $\_\_\_\_$

Let $\mathrm { f } ( \mathrm { x } ) = 2 ^ { \mathrm { x } } - \mathrm { x } ^ { 2 } , \mathrm { x } \in \mathrm { R }$. If m and n are respectively the number of points at which the curves $\mathrm { y } = \mathrm { f } ( \mathrm { x } )$ and $\mathrm { y } = \mathrm { f } ^ { \prime } ( \mathrm { x } )$ intersects the x-axis, then the value of $\mathrm { m } + \mathrm { n }$ is

Let $\mathrm{A} = \left\{ x \in (0, \pi) - \left\{ \frac{\pi}{2} \right\} : \log_{(2/\pi)} |\sin x| + \log_{(2/\pi)} |\cos x| = 2 \right\}$ and $\mathrm{B} = \{ x \geqslant 0 : \sqrt{x}(\sqrt{x} - 4) - 3|\sqrt{x} - 2| + 6 = 0 \}$. Then $\mathrm{n}(\mathrm{A} \cup \mathrm{B})$ is equal to:
(1) 4
(2) 8
(3) 6
(4) 2

The number of solutions of the equation $\left(\frac{9}{x} - \frac{9}{\sqrt{x}} + 2\right)\left(\frac{2}{x} - \frac{7}{\sqrt{x}} + 3\right) = 0$ is:
(1) 2
(2) 3
(3) 1
(4) 4

If $\lim _ { x \rightarrow \infty } \left( \left( \frac { \mathrm { e } } { 1 - \mathrm { e } } \right) \left( \frac { 1 } { \mathrm { e } } - \frac { x } { 1 + x } \right) \right) ^ { x } = \alpha$, then the value of $\frac { \log _ { \mathrm { e } } \alpha } { 1 + \log _ { \mathrm { e } } \alpha }$ equals:
(1) $e ^ { - 1 }$
(2) $\mathrm { e } ^ { 2 }$
(3) $e ^ { - 2 }$
(4) e

Let $[t]$ be the greatest integer less than or equal to $t$. Then the least value of $p \in \mathbf{N}$ for which $$\lim_{x \rightarrow 0^+}\left(x\left(\left[\frac{1}{x}\right] + \left[\frac{2}{x}\right] + \ldots + \left[\frac{\mathrm{p}}{x}\right]\right) - x^2\left(\left[\frac{1}{x^2}\right] + \left[\frac{2^2}{x^2}\right] + \ldots + \left[\frac{9^2}{x^2}\right]\right)\right) \geq 1$$ is equal to \_\_\_\_ .

Q84. If $\lim _ { x \rightarrow 1 } \frac { ( 5 x + 1 ) ^ { 1 / 3 } - ( x + 5 ) ^ { 1 / 3 } } { ( 2 x + 3 ) ^ { 1 / 2 } - ( x + 4 ) ^ { 1 / 2 } } = \frac { \mathrm { m } \sqrt { 5 } } { \mathrm { n } ( 2 \mathrm { n } ) ^ { 2 / 3 } }$, where $\operatorname { gcd } ( \mathrm { m } , \mathrm { n } ) = 1$, then $8 \mathrm {~m} + 12 \mathrm { n }$ is equal to

Q88. If $S = \{ a \in \mathbf { R } : | 2 a - 1 | = 3 [ a ] + 2 \{ a \} \}$, where $[ t ]$ denotes the greatest integer less than or equal to $t$ and $\{ t \}$ represents the fractional part of $t$, then $72 \sum _ { a \in S } a$ is equal to $\_\_\_\_$

11. Let $f(x)$ be a real cubic polynomial with leading coefficient 1. Given that $f(1) = 1, f(2) = 2, f(5) = 5$, in which of the following intervals must $f(x) = 0$ have a real root?
(1) $(-\infty, 0)$
(2) $(0, 1)$
(3) $(1, 2)$
(4) $(2, 5)$
(5) $(5, \infty)$

Part Two: Fill-in Questions (45 points)

Instructions: 1. For questions A through I, mark your answers on the "Answer Sheet" at the row numbers indicated (12–41).
2. Each completely correct answer receives 5 points. Wrong answers do not result in deduction. Incomplete answers receive no points.
A. Let real number $x$ satisfy $0 < x < 1$ and $\log_x 4 - \log_2 x = 1$. Then $x = $ (12). (Express as a fraction in lowest terms)
B. In $\triangle ABC$ on the coordinate plane, $P$ is the midpoint of side $\overline{BC}$, and $Q$ is on side $\overline{AC}$ such that $\overline{AQ} = 2\overline{QC}$. Given that $\overrightarrow{PA} = (4, 3)$ and $\overrightarrow{PQ} = (1, 5)$, then $\overrightarrow{BC} = ($ (14) (15), (16) (17) $)$.
C. In a certain talent competition, to avoid excessive subjective influence from individual judges on contestants' scores, the

$$\lim_{x \rightarrow 1} \frac{(1 - \sqrt{x}) \cdot (\sqrt[3]{x} - 2)}{-x^{2} + 9x - 8}$$
What is the value of this limit?
A) 1 B) $\dfrac{1}{2}$ C) $\dfrac{1}{7}$ D) $\dfrac{1}{14}$ E) $\dfrac{1}{18}$