Given that $t ^ { 3 } - 2 = 0$, which of the following is the equivalent of $\frac { 1 } { t ^ { 2 } + t + 1 }$ in terms of $t$? A) $t + 1$ B) $\mathrm { t } - 2$ C) $t - 1$ D) $t ^ { 2 } + 1$ E) $t ^ { 2 } + 3$
The geometric mean of numbers a and b is 3, and their arithmetic mean is 6. Accordingly, what is the arithmetic mean of $\mathrm { a } ^ { 2 }$ and $\mathrm { b } ^ { 2 }$? A) 67 B) 65 C) 63 D) 61 E) 57
Let x and y be real numbers such that $$\begin{aligned}
& x ^ { 3 } - 3 x ^ { 2 } y = 3 \\
& y ^ { 3 } - 3 x y ^ { 2 } = 11
\end{aligned}$$ Accordingly, what is the difference $x - y$? A) 3 B) 2 C) 1 D) - 2 E) - 3
For two-digit positive integers a and b $$\frac { a ! } { b ! } = 132$$ Given this, what is the sum $\mathbf { a } + \mathbf { b }$? A) 22 B) 23 C) 24 D) 25 E) 26
$$\frac { a ^ { 4 } - a ^ { 3 } } { a ^ { 4 } + a ^ { 2 } } \cdot \frac { a ^ { 2 } + 1 } { a ^ { 2 } - a }$$ Which of the following is the simplified form of this expression? A) $a - 1$ B) a C) 1 D) $a + 1$ E) $a ^ { 2 } + 1$
$$\begin{aligned}
& A = \left\{ n \in Z ^ { + } \mid n \leq 100 ; n \text{ is divisible by } 3 \right\} \\
& B = \left\{ n \in Z ^ { + } \mid n \leq 100 ; n \text{ is divisible by } 5 \right\}
\end{aligned}$$ The sets are given. Accordingly, how many elements does the difference set $\mathrm { A } \backslash \mathrm { B }$ have? A) 33 B) 32 C) 30 D) 28 E) 27
Let p and q be distinct prime numbers such that $$\begin{aligned}
& a = p ^ { 4 } \cdot q ^ { 2 } \\
& b = p ^ { 2 } \cdot q ^ { 3 }
\end{aligned}$$ Which of the following is the greatest common divisor of numbers a and b? A) $p ^ { 5 } \cdot q ^ { 4 }$ B) $p ^ { 4 } \cdot q ^ { 3 }$ C) $p ^ { 3 } \cdot q ^ { 4 }$ D) $p ^ { 2 } \cdot q ^ { 2 }$ E) $p ^ { 2 } \cdot q ^ { 3 }$
$$\begin{aligned}
& 2 ^ { x } \equiv 1 ( \bmod 7 ) \\
& 3 ^ { y } \equiv 4 ( \bmod 7 )
\end{aligned}$$ For the smallest positive integers x and y satisfying these congruences, what is the difference $y - x$? A) 5 B) 4 C) 3 D) 2 E) 1
$$\left. \begin{array} { l }
x ( 3 - x ) > 0 \\
( 2 x + 1 ) ( x - 2 ) < 0
\end{array} \right\}$$ If the solution set of the inequality system given above is the open interval $(\mathbf { a } , \mathbf { b })$, what is the difference $\mathbf { a - b }$? A) - 2 B) 0 C) 1 D) $\frac { 1 } { 2 }$ E) $\frac { 3 } { 2 }$
Let x be a real number with $| x | \leq 4$, and $$2 x + 3 y = 1$$ What is the sum of the integer values of y that satisfy this equation? A) - 1 B) 0 C) 1 D) 2 E) 3
Real coefficient polynomials $P ( x ) , Q ( x )$ and $R ( x )$ are given. For the polynomial $\mathrm { P } ( \mathrm { x } )$ whose constant term is nonzero, $$P ( x ) = Q ( x ) \cdot R ( x + 1 )$$ the equality is satisfied. If the constant term of P is twice the constant term of Q, what is the sum of the coefficients of R? A) $\frac { 2 } { 3 }$ B) $\frac { 1 } { 4 }$ C) $\frac { 3 } { 4 }$ D) 1 E) 2
The leading coefficient is 1, and the fourth-degree polynomial $\mathbf { P } ( \mathbf { x } )$ with real coefficients has roots $-i$ and $2i$. What is $\mathbf { P } ( \mathbf { 0 } )$? A) 2 B) 4 C) 6 D) 7 E) 8
$$P ( x ) = ( x + 2 ) ^ { 4 } + 3 ( x + 1 ) ^ { 3 }$$ In this polynomial, what is the coefficient of the $\mathbf { x }$ term? A) 41 B) 39 C) 37 D) 35 E) 33
From a group of 6 girls and 7 boys, 2 representatives are selected. What is the probability that one of the two selected representatives is a girl and the other is a boy? A) $\frac { 3 } { 4 }$ B) $\frac { 3 } { 8 }$ C) $\frac { 2 } { 13 }$ D) $\frac { 7 } { 13 }$ E) $\frac { 9 } { 13 }$
For complex numbers $z = a + b i ( b \neq 0 )$ and $w = c + d i$, if the sum $\mathbf { Z } + \mathbf { W }$ and the product $\mathbf { Z } \cdot \mathbf { W }$ are both real numbers, then I. $z$ and $w$ are conjugates of each other. II. $\mathrm { z } - \mathrm { w }$ is real. III. $z ^ { 2 } + w ^ { 2 }$ is real. Which of the following statements are true? A) Only I B) Only II C) I and III D) II and III E) I, II and III
The function f on the set of complex numbers is $$f ( z ) = \sum _ { k = 0 } ^ { 101 } z ^ { k }$$ is defined in this form. Accordingly, what is the value of f(i)? A) $1 + i$ B) $1 - \mathrm { i }$ C) i D) - i E) 1
$$\log _ { 9 } \left( x ^ { 2 } + 2 x + 1 \right) = t \quad ( x > - 1 )$$ Given this equation, which of the following is the expression for x in terms of t? A) $3 ^ { t } - 1$ B) $3 ^ { \mathrm { t } - 1 }$ C) $3 - 2 ^ { t }$ D) $2 \cdot 3 ^ { \mathrm { t } - 1 }$ E) $3 ^ { t } - 2$
$$f ( x ) = \arcsin \left( \frac { x } { 3 } + 2 \right)$$ Which of the following is the inverse function $\mathbf { f } ^ { \mathbf { - 1 } } ( \mathbf { x } )$ of this function? A) $2 \sin ( x ) - 6$ B) $2 \sin ( x ) + 3$ C) $3 \sin ( x ) - 6$ D) $\sin ( 2 x - 6 )$ E) $\sin ( 2 x ) - 3$
The graph of the function $f ( x ) = x ^ { 2 } - 2 x + 3$ is translated $a$ units to the right and $b$ units downward to obtain the graph of the function $g ( x ) = x ^ { 2 } - 8 x + 14$. Accordingly, what is the value of the expression $| \mathbf { a } | + | \mathbf { b } |$? A) 4 B) 5 C) 6 D) 7 E) 8
The triangle ABC is drawn on unit squares as shown above. What is the tangent of angle $B$? A) $\frac { 25 } { 4 }$ B) $\frac { 34 } { 5 }$ C) $\frac { 40 } { 9 }$ D) 4 E) 5
The graph of the function $f$ is given below. Given that $\mathbf { g } ( \mathbf { x } ) = \mathbf { 3 } - \mathbf { f } ( \mathbf { x } - \mathbf { 2 } )$, what is the sum $\mathbf { g } ( - \mathbf { 2 } ) + \mathbf { g } ( \mathbf { 5 } )$? A) - 3 B) - 1 C) 1 D) 2 E) 3
For points $(x, y)$ on the boundary of the bounded region between the parabola $y = x ^ { 2 }$ and the line $y = 2 - x$, what is the maximum value of the expression $\mathbf { x } ^ { \mathbf { 2 } } + \mathbf { y } ^ { \mathbf { 2 } }$? A) 25 B) 20 C) 17 D) 13 E) 10
The piecewise function $f : R \rightarrow R$ is defined as $f ( x ) = \left\{ \begin{array} { c l } 3 x + 1 , & x \text { is rational } \\ x ^ { 2 } , & x \text { is irrational } \end{array} \right.$ Accordingly, which of the following is $( f \circ f ) \left( \frac { \sqrt { 2 } } { 2 } \right)$? A) $3 \sqrt { 2 } + 2$ B) $\sqrt { 2 } + 2$ C) $\frac { 1 } { 4 }$ D) $\frac { 5 } { 2 }$ E) $\frac { 7 } { 2 }$
The function f satisfies the equation $$f ( n ) = 2 \cdot f ( n - 1 ) + 1$$ for integers $n \geq 1$. Given that $f ( 0 ) = 1$, what is $f ( 2 )$? A) 8 B) 7 C) 6 D) 5 E) 4
The sequence $(a _ { k })$ is defined as $$\begin{aligned}
& a _ { 1 } = 40 \\
& a _ { k + 1 } = a _ { k } - k \quad ( k = 1,2,3 , \ldots )
\end{aligned}$$ Accordingly, what is the term $\mathrm { a } _ { 8 }$? A) 4 B) 7 C) 12 D) 15 E) 19
An equilateral triangle ABC with side length 1 unit has points D and E marked on sides AB and AC respectively, where these sides are divided into three equal parts. Let K be the midpoint of the line segment DE. A new equilateral triangle is drawn with one vertex at K and the opposite side on BC, and the same process is applied to the newly drawn equilateral triangles. What is the sum of the areas of all nested triangular regions drawn in this manner, in square units? A) $\frac { \sqrt { 3 } } { 3 }$ B) $\frac { 3 \sqrt { 3 } } { 4 }$ C) $\frac { 8 \sqrt { 3 } } { 9 }$ D) $\frac { 5 \sqrt { 3 } } { 16 }$ E) $\frac { 9 \sqrt { 3 } } { 32 }$
$$\prod _ { n = 1 } ^ { 7 } ( 3 n + 2 )$$ If this number is divisible by $10 ^ { \mathbf { m } }$, what is the maximum integer value that m can take? A) 2 B) 3 C) 4 D) 5 E) 6
$$\lim _ { x \rightarrow 0 } \frac { x + \arcsin x } { \sin 2 x }$$ What is the value of this limit? A) 0 B) 1 C) $\frac { 2 } { 3 }$ D) $\frac { 4 } { 3 }$ E) $\frac { 1 } { 6 }$
$$f ( x ) = \sin ^ { 2 } \left( 3 x ^ { 2 } + 2 x + 1 \right)$$ Given this, what is the value of $f ^ { \prime } ( 0 )$? A) $2 \cos 2$ B) $2 \cos 3$ C) $6 \sin 1$ D) $4 \sin 2$ E) $2 \sin 2$
$$\begin{aligned}
& f ^ { \prime } ( x ) = 3 x ^ { 2 } + 4 x + 3 \\
& f ( 0 ) = 2
\end{aligned}$$ Given this, what is the value of $\mathbf { f } ( - \mathbf { 1 } )$? A) - 2 B) - 1 C) 0 D) 1 E) 2
$$\begin{aligned}
& f ( x ) = 2 x - 1 \\
& g ( x ) = \frac { x } { 2 } - \frac { 1 } { x }
\end{aligned}$$ Given this, what is the value of $\lim _ { x \rightarrow 2 } \frac { f ( g ( x ) ) } { x - 2 }$? A) 0 B) 1 C) 3 D) $\frac { 1 } { 2 }$ E) $\frac { 3 } { 2 }$
At what point does the tangent line to the curve $\mathbf { y } = \sin ( \pi \mathrm { x } ) + \mathrm { e } ^ { \mathrm { x } }$ at $\mathrm { x } = 1$ intersect the y-axis? A) $- \pi$ B) - 1 C) 0 D) $e - 1$ E) $\pi$
Below is the graph of the derivative of a function f defined on the interval $[ - 5,5 ]$. According to this graph, I. The function f is decreasing for $x > 0$. II. $f ( - 2 ) > f ( 0 ) > f ( 2 )$. III. The function f has local extrema at $x = - 2$ and $x = 2$. Which of the following statements are true? A) Only I B) Only II C) I and II D) I and III E) I, II and III
A line $d$ with negative slope passing through the point $(1,2)$ forms a triangular region with the coordinate axes. What is the minimum area of this triangular region in square units? A) 2 B) 3 C) 4 D) $\frac { 9 } { 2 }$ E) $\frac { 7 } { 2 }$
The slope of the tangent line to the graph of a function f at $\mathrm { x } = \mathrm { a }$ is $1$, and the slope of the tangent line at $x = b$ is $\sqrt { 3 }$. Given that the second derivative function $\mathbf { f } ^ { \prime \prime } ( \mathbf { x } )$ is continuous on the interval $[ \mathbf { a } , \mathbf { b } ]$, what is the value of $$\int _ { b } ^ { a } f ^ { \prime } ( x ) \cdot f ^ { \prime \prime } ( x ) d x$$ ? A) - 1 B) 1 C) 2 D) $\frac { 1 } { 3 }$ E) $\frac { 2 } { 3 }$
In the graph below, the line $y = k$ is drawn such that the areas of regions A and B are equal. Accordingly, what is the value of k? A) 2 B) 3 C) 4 D) $\frac { 9 } { 4 }$ E) $\frac { 11 } { 2 }$
$$\int _ { 1 } ^ { e } \ln ^ { 3 } x \, d x = 6 - 2 e$$ Given this, what is the value of the integral $\int _ { 1 } ^ { e } \ln ^ { 4 } x \, d x$? A) $7 e - 16$ B) $8 e - 18$ C) $9 e - 24$ D) $10 e - 26$ E) $11 e - 28$
In the integral $\int \frac { \ln \sqrt { x } } { \sqrt { x } } d x$, if the substitution $u = \sqrt { x }$ is made, which of the following integrals is obtained? A) $\int \ln u \, d u$ B) $\int 2 \ln u \, d u$ C) $\int \frac { \ln u } { u } d u$ D) $\int \frac { \ln u } { 2 u } d u$ E) $\int u \ln u \, d u$