Let for $f ( x ) = 7 \tan ^ { 8 } x + 7 \tan ^ { 6 } x - 3 \tan ^ { 4 } x - 3 \tan ^ { 2 } x , \quad \mathrm { I } _ { 1 } = \int _ { 0 } ^ { \pi / 4 } f ( x ) \mathrm { d } x$ and $\mathrm { I } _ { 2 } = \int _ { 0 } ^ { \pi / 4 } x f ( x ) \mathrm { d } x$. Then $7 \mathrm { I } _ { 1 } + 12 \mathrm { I } _ { 2 }$ is equal to:
(1) 2
(2) 1
(3) $2 \pi$
(4) $\pi$

Let $f$ be a real valued continuous function defined on the positive real axis such that $g ( x ) = \int _ { 0 } ^ { x } \mathrm { t } f ( \mathrm { t } ) \mathrm { dt }$. If $\mathrm { g } \left( x ^ { 3 } \right) = x ^ { 6 } + x ^ { 7 }$, then value of $\sum _ { r = 1 } ^ { 15 } f \left( \mathrm { r } ^ { 3 } \right)$ is :
(1) 270
(2) 340
(3) 320
(4) 310

If $I(m, n) = \int_0^1 x^{m-1}(1-x)^{n-1}\,dx$, $m, n > 0$, then $I(9, 14) + I(10, 13)$ is
(1) $I(19, 27)$
(2) $I(9, 1)$
(3) $I(1, 13)$
(4) $I(9, 13)$

If $\mathrm { I } = \int _ { 0 } ^ { \frac { \pi } { 2 } } \frac { \sin ^ { \frac { 3 } { 2 } } x } { \sin ^ { \frac { 3 } { 2 } } x + \cos ^ { \frac { 3 } { 2 } } x } \mathrm {~d} x$, then $\int _ { 0 } ^ { 2\mathrm{I} } \frac { x \sin x \cos x } { \sin ^ { 4 } x + \cos ^ { 4 } x } \mathrm {~d} x$ equals :
(1) $\frac { \pi ^ { 2 } } { 12 }$
(2) $\frac { \pi ^ { 2 } } { 4 }$
(3) $\frac { \pi ^ { 2 } } { 16 }$
(4) $\frac { \pi ^ { 2 } } { 8 }$

Let $f(x) = \frac{2^{x+2} + 16}{2^{2x+1} + 2^{x+4} + 32}$. Then the value of $8\left(f\left(\frac{1}{15}\right) + f\left(\frac{2}{15}\right) + \ldots + f\left(\frac{59}{15}\right)\right)$ is equal to
(1) 92
(2) 118
(3) 102
(4) 108

The value of $\int _ { e ^ { 2 } } ^ { e ^ { 4 } } \frac { 1 } { x } \left( \frac { e ^ { \left( \left( \log _ { e } x \right) ^ { 2 } + 1 \right) ^ { - 1 } } } { e ^ { \left( \left( \log _ { e } x \right) ^ { 2 } + 1 \right) ^ { - 1 } } + e ^ { \left( \left( 6 - \log _ { e } x \right) ^ { 2 } + 1 \right) ^ { - 1 } } } \right) d x$ is
(1) 2
(2) $\log _ { e } 2$
(3) 1
(4) $e ^ { 2 }$

Let for some function $\mathrm { y } = f ( x ) , \int _ { 0 } ^ { x } t f ( t ) d t = x ^ { 2 } f ( x ) , x > 0$ and $f ( 2 ) = 3$. Then $f ( 6 )$ is equal to
(1) 1
(2) 3
(3) 6
(4) 2

If $\int _ { - \frac { \pi } { 2 } } ^ { \frac { \pi } { 2 } } \frac { 96 x ^ { 2 } \cos ^ { 2 } x } { \left( 1 + e ^ { x } \right) } \mathrm { d } x = \pi \left( \alpha \pi ^ { 2 } + \beta \right) , \alpha , \beta \in \mathbb { Z }$, then $( \alpha + \beta ) ^ { 2 }$ equals
(1) 64
(2) 196
(3) 144
(4) 100

If $24 \int _ { 0 } ^ { \frac { \pi } { 4 } } \left( \sin \left| 4 x - \frac { \pi } { 12 } \right| + [ 2 \sin x ] \right) \mathrm { d } x = 2 \pi + \alpha$, where $[ \cdot ]$ denotes the greatest integer function, then $\alpha$ is equal to $\_\_\_\_$ .

Let $f:(0,\infty) \rightarrow \mathbf{R}$ be a twice differentiable function. If for some $\mathrm{a} \neq 0$, $\int_0^1 f(\lambda x)\,\mathrm{d}\lambda = \mathrm{a}f(x)$, $f(1) = 1$ and $f(16) = \frac{1}{8}$, then $16 - f'\left(\frac{1}{16}\right)$ is equal to \_\_\_\_ .

Consider the function $$f(x) = \sin x + \frac{\sin 2x}{2} + \frac{\sin 3x}{3}$$ on the interval $0 \leqq x \leqq \pi$. We are to show that $f(x) > 0$ on $0 < x < \pi$, and to find the area $S$ of the region bounded by the graph of $y = f(x)$ and the $x$-axis.
(1) For $\mathbf{K}$, $\mathbf{N}$, $\mathbf{Q}$, $\mathbf{R}$ in the following sentences, choose the correct answer from the following two choices: (0) increasing, (1) decreasing, and for the other blanks, enter the correct number.
When we differentiate $f(x)$, we have $$f'(x) = (\mathbf{A}\cos^2 x - \mathbf{B})(\mathbf{C}\cos x + \mathbf{D}).$$ Hence, over the range $0 \leqq x \leqq \pi$, there are three $x$'s at which $f'(x) = 0$, and when they are arranged in ascending order, they are given accordingly.
Next, looking at whether $f(x)$ is increasing or decreasing, the behaviour is described accordingly.
Also, we have $$f(0) = 0, \quad f(\pi) = 0, \quad f\left(\frac{\mathbf{L}}{\mathbf{M}}\pi\right) = \frac{\sqrt{\mathbf{S}}}{\mathbf{T}} > 0.$$ Hence we see that $f(x) > 0$ on $0 < x < \pi$.
(2) The area $S$ of the region bounded by the graph of $y = f(x)$ and the $x$-axis is $$S = \frac{\mathbf{UV}}{\mathbf{W}}.$$

Let $F(x)$ and $f(x)$ both be polynomial functions with real coefficients. Given that $F'(x) = f(x)$, select the correct options.
(1) If $a \geq 0$, then $F(a) - F(0) = \int_{0}^{a} f(t)\, dt$
(2) If $F(x)$ divided by $x$ has quotient $Q(x)$, then $Q(0) = f(0)$
(3) If $f(x)$ is divisible by $x + 1$, then $F(x) - F(0)$ is divisible by $(x+1)^{2}$
(4) If $F(x) \geq \frac{x^{2}}{2}$ holds for all real numbers $x$, then $f(x) \geq x$ also holds for all real numbers $x$
(5) If $f(x) \geq x$ holds for all $x > 0$, then $F(x) \geq \frac{x^{2}}{2}$ also holds for all $x > 0$

Find the value of the limit $\lim _ { n \rightarrow \infty } \frac { 10 ^ { 10 } } { n ^ { 10 } } \left[ 1 ^ { 9 } + 2 ^ { 9 } + 3 ^ { 9 } + \cdots + ( 2 n ) ^ { 9 } \right]$ .
(1) $10 ^ { 9 }$
(2) $10 ^ { 9 } \times \left( 2 ^ { 10 } - 1 \right)$
(3) $2 ^ { 9 } \times \left( 10 ^ { 10 } - 1 \right)$
(4) $10 ^ { 9 } \times 2 ^ { 10 }$
(5) $2 ^ { 9 } \times 10 ^ { 10 }$

There is a wooden block where $ACFD$ and $ABED$ are two congruent isosceles trapezoids, and $BCFE$ is a rectangle. Let the projection of point $A$ on line $BC$ be $M$ and its projection on plane $BCFE$ be $P$. Given that $\overline{AD} = 30$, $\overline{CF} = 40$, $\overline{AP} = 15$, and $\overline{BC} = 10$. Place plane $BCFE$ on a horizontal table, and call any plane parallel to $BCFE$ a horizontal plane. The intersection of a horizontal plane at distance $x$ from $A$ (where $0 < x < 15$) with the wooden block is a rectangle of area $20x + \frac{4}{9}x^2$. Divide the line segment $\overline{AP}$ into $n$ equal parts, and denote the division points along the direction of vector $\overrightarrow{AP}$ as $A = P_0, P_1, \ldots, P_{n-1}, P_n = P$. For each segment $\overline{P_{k-1}P_k}$, consider the rectangular prism formed by taking the rectangle formed by the intersection of the horizontal plane passing through $P_k$ with this wooden block as the base and $\overline{P_{k-1}P_k}$ as the height. Please use this slicing method to write down the Riemann sum estimating the volume of this wooden block (no need to simplify), express the volume of this wooden block as a definite integral, and find its value. (Non-multiple choice question, 6 points)

What is the limit
$$\lim _ { n \rightarrow \infty } \frac { 3 } { n ^ { 2 } } \left( \sqrt { 4 n ^ { 2 } + 9 \times 1 ^ { 2 } } + \sqrt { 4 n ^ { 2 } + 9 \times 2 ^ { 2 } } + \cdots + \sqrt { 4 n ^ { 2 } + 9 \times ( n - 1 ) ^ { 2 } } \right)$$
which of the following definite integrals can represent?
(1) $\int _ { 0 } ^ { 3 } \sqrt { 1 + x ^ { 2 } } d x$
(2) $\int _ { 0 } ^ { 3 } \sqrt { 1 + 9 x ^ { 2 } } d x$
(3) $\int _ { 0 } ^ { 3 } \sqrt { 4 + x ^ { 2 } } d x$
(4) $\int _ { 0 } ^ { 3 } \sqrt { 4 + 9 x ^ { 2 } } d x$
(5) $\int _ { 0 } ^ { 3 } \sqrt { 4 x ^ { 2 } + 9 } d x$

Let $f ( x )$ be a real-coefficient cubic polynomial. The function $y = f ( x )$ has a local minimum at $x = - 3$ and a local maximum at $x = 1$. Given that the slope of the tangent line at the inflection point of the graph of $y = f ( x )$ is 4, so that $f ^ { \prime } ( x )$ is as found in question 14.
Find the value of $\int _ { - 3 } ^ { 1 } f ^ { \prime } ( x ) \, d x$.

$$f''(x) = 6x - 2, \quad f'(0) = 4, \quad f(0) = 1$$
For the function $f$ that satisfies these conditions, what is the value of $f(1)$?
A) 4
B) 5
C) 6
D) 7
E) 8

$$f(x) = \begin{cases} 3 - x, & x < 2 \\ 2x - 3, & x \geq 2 \end{cases}$$
What is the value of the integral $\displaystyle\int_{1}^{3} f(x+1)\, dx$?
A) 2
B) 4
C) 6
D) 8
E) 10

$$\begin{aligned} & f ^ { \prime } ( x ) = 3 x ^ { 2 } + 4 x + 3 \\ & f ( 0 ) = 2 \end{aligned}$$
Given this, what is the value of $\mathbf { f } ( - \mathbf { 1 } )$?
A) - 2
B) - 1
C) 0
D) 1
E) 2

Below, the graph of the derivative of a function f is given. Given that $f ( 0 ) = 1$, what is the value of $f ( 2 )$?
A) $\frac { 3 } { 2 }$
B) $\frac { 5 } { 2 }$
C) $\frac { 4 } { 3 }$
D) $\frac { - 1 } { 2 }$
E) $\frac { - 1 } { 3 }$

For a continuous function f defined on the set of real numbers,
$$\int _ { 1 } ^ { 3 } f ( x ) d x = 5$$
is known. Accordingly,
$$\int _ { 0 } ^ { 1 } ( 4 + f ( 2 x + 1 ) ) d x$$
What is the value of this integral?
A) 1
B) 2
C) 3
D) $\frac { 5 } { 2 }$
E) $\frac { 13 } { 2 }$

Let $n$ be a natural number,
$$\begin{aligned} & f _ { n } : [ n , n + 1 ) \rightarrow \left[ 0 , \frac { 1 } { 2 ^ { n } } \right) \\ & f _ { n } ( x ) = \frac { ( x - n ) ^ { 2 } } { 2 ^ { n } } \end{aligned}$$
The regions between the functions defined in this form and the x-axis are given shaded in the figure below.
Accordingly, what is the sum of the areas of all shaded regions in square units?
A) $\frac { 2 } { 3 }$
B) $\frac { 3 } { 4 }$
C) $\frac { 5 } { 6 }$
D) $\frac { 8 } { 9 }$
E) $\frac { 11 } { 12 }$

The derivative of a function f that is defined and differentiable on the set of real numbers is given as
$$f ^ { \prime } ( x ) = \begin{cases} 1 , & \text{if } x \leq 1 \\ x , & \text{if } x > 1 \end{cases}$$
Given that $f ( 1 ) = 1$, what is the value of $f ( 0 ) + f ( 3 )$?
A) 2
B) 3
C) 4
D) 5
E) 6

$f ( x ) = \begin{cases} 2 x - 4 , & \text{if } 0 \leq x < 1 \\ - 2 , & \text{if } 1 \leq x < 4 \\ x - 6 , & \text{if } 4 \leq x \leq 6 \end{cases}$
Given this, what is the value of the integral $\int _ { 0 } ^ { 6 } f ( x ) d x$?
A) - 11
B) - 10
C) - 9
D) - 8
E) - 7

Let a be a positive real number. For every second-degree polynomial $P ( x )$ with real coefficients and leading coefficient 1,
$$\int _ { - 1 } ^ { 1 } \mathrm { P } ( \mathrm { x } ) \mathrm { dx } = \mathrm { P } ( \mathrm { a } ) + \mathrm { P } ( - \mathrm { a } )$$
the equality is satisfied. Accordingly, what is the value of a?
A) $\sqrt { 2 }$
B) $\sqrt { 3 }$
C) $\sqrt { 6 }$
D) $\frac { \sqrt { 2 } } { 2 }$
E) $\frac { \sqrt { 3 } } { 3 }$