gist僅為備份用檔案,完整rendered題目請見hackmd
參考解答會在選項前以星號(*)標記,不過目前並不保證一定正確,各位高手可以自行編輯(需登入)更正答案或提供各題詳解。 感謝一同討論解題的各位: Paul, 陳彥吉, 游聲峰Robert, Sean, Moony Hsieh, johnson, 怡中
- Let the function
$f(x)=ax^3+bx^2+cx+d$ . Suppose that$f(0)=4$ is a critical point of$f$ and$f(1)=-2$ is a point of inflection, find$a$ ,$b$ ,$c$ and$d$ . Hint: The critical point of$f$ is the point such that$f'(x)=0$ , and the point of inflection of$f$ satify$f''(x)=0$ . ($A$ )$a=1$ ,$b=5$ ,$c=8$ ,$d=6$ ($B$ )$a=13$ ,$b=-3$ ,$c=-5$ ,$d=4$ *($C$ )$a=3$ ,$b=-9$ ,$c=0$ ,$d=4$ ($D$ )$a=6$ ,$b=11$ ,$c=1$ ,$d=-2$
$f(x)=ax^3+bx^2+cx+d$ $f'(x)=3ax^2+2bx+c$ $f''(x)=6ax+2b$ Then solve:$f(0)=d=4$ ...(1)$f(1)=a+b+c+d=-2$ ...(2)$f'(0)=c=0$ ...(3)$f''(1)=6a+2b=0$ ...(4)
- Find the
$(x,y)\in{(x,y)|2x^2-y^2=1}$ which minimizes the distance from$(3,0)$ , and what is the minimal distance$d$ ? ($A$ )$(x,y)=(3,\pm4)$ ,$d=2$ ($B$ )$(x,y)=(8,17)$ ,$d=23$ ($C$ )$(x,y)=(3,0)$ ,$d=\sqrt{3}$ *($D$ )$(x,y)=(3,\pm1)$ ,$d=\sqrt{5}$
The points given in (A)(B)(C) doesn't meet the equation.
- Find the equation of the tangent line to
$y=2^x$ at$(1,2)$ *($A$ )$y=x\ln{4}-\ln{4}+2$ ($B$ )$y=8x\exp^x-4x+3$ ($C$ )$y=-\sqrt[3]{x}+1$ ($D$ )$y=\ln{6^x}+9x$
Only the equation listed in (A) is a linear equation
- Let
$f(x,y)=sin(x^2-y)$ , find$\dfrac{\partial^2 f(2,4)}{\partial x \partial y}$ ($A$ ) 5 ($B$ ) 9 ($C$ ) 4 *($D$ ) 0
$\dfrac{\partial^2 f(2,4)}{\partial x \partial y}=\sin (x^2-y)*2x=0$
- Let
$z=f(x-y,y-x)$ , what is$\dfrac{\partial z}{\partial x}+\dfrac{\partial z}{\partial y}$ ? ($A$ ) 1 *($B$ ) 0 ($C$ ) 6 ($D$ ) 2
Always results in pairs of postive and negative terms that have same values.
- Let $A=\begin{bmatrix}
1 & 2 & 1 \
0 & 1 & 2\
1 & 3 & 2
\end{bmatrix}$, find a matrix
$B$ such that$AB=A^2+2A$ ($A$ ) $\begin{bmatrix} 3 & 2 & 1 \ 0 & 7 & 2\ 1 & 1 & 6 \end{bmatrix}$* ($B$ ) $\begin{bmatrix} 3 & 2 & 1 \ 0 & 3 & 2\ 1 & 3 & 4 \end{bmatrix}$ ($C$ ) $\begin{bmatrix} 3 & 7 & 8 \ 3 & 3 & 7\ 13 & 5 & 4 \end{bmatrix}$ ($D$ ) $\begin{bmatrix} 9 & 2 & 1 \ 7 & 2 & 4\ 6 & 3 & 6 \end{bmatrix}$
$B=A^{-1}(A^2+2A)=A+2I$ where$I$ denotes the identity matrix.
- What is the rank of $\begin{bmatrix}
1 & 2 & 3 & 4 & 5 \
1 & 0 & 0 & 1 & 0 \
1 & 1 & 1 & 1 & 1 \
2 & 2 & 3 & 5 & 5 \
5 & 5 & 7 & 11 & 11 \
\end{bmatrix}$?
(
$A$ ) 5 ($B$ ) 6 ($C$ ) 2 *($D$ ) 3
Please checkout the definition of rank
-
Find the eigenvalues of the following matrix $\begin{bmatrix} 1 & 1 & 2 & 2 \ 1 & 1 & 2 & 2 \ 2 & 2 & 1 & 1 \ 2 & 2 & 1 & 1 \ \end{bmatrix}$? Hint: Try Gaussian elimination first. (
$A$ ) 0,1,-6 ($B$ ) 3,5,-10 *($C$ ) 0,6,-2 ($D$ ) 0,6,12 -
Find the solution set for the following linear matrix equation $Ax=\begin{bmatrix} 1 & 0 & 1 & 0 \ 2 & 2 & 0 & 3 \ 0 & 4 & -4 & 5 \ \end{bmatrix}\begin{bmatrix} x_1 \ x_2 \ x_3 \ x_4 \ \end{bmatrix}=\begin{bmatrix} 2 \ 1 \ 7 \end{bmatrix}$ *(
$A$ ) $\begin{Bmatrix} \left. \begin{bmatrix} -t+2 \ t-3 \ t \ 1 \ \end{bmatrix} \right\rvert t\in F \end{Bmatrix}$ ($B$ ) $\begin{Bmatrix} \left. \begin{bmatrix} t+2 \ 10t \ t \ 3t+1 \ \end{bmatrix} \right\rvert t\in F \end{Bmatrix}$ ($C$ ) $\begin{Bmatrix} \left. \begin{bmatrix} t \ 3t \ -t-10 \ 1 \ \end{bmatrix} \right\rvert t\in F \end{Bmatrix}$ ($D$ ) $\begin{Bmatrix} \left. \begin{bmatrix} 3t \ t \ t+10 \ 5 \ \end{bmatrix} \right\rvert t\in F \end{Bmatrix}$ -
For which
$x$ is$A=LU$ decomposiition impossible? $A=\begin{bmatrix} 1 & 2 & 0 \ 3 & x & 1 \ 0 & 1 & 1 \ \end{bmatrix}$ *($A$ )$x=6$ ($B$ )$x=4$ ($C$ )$x=12$ ($D$ )$x=0$
If
$A$ is invertible, then it admits an LU factorization if and only if all its leading principal minors are nonzero.
-
Which of the following statements are true? I. Qualitative variables could be multiplied. II. Categorical variables could be continuous variables. III. Quantitative variables could be discrete variables. (
$A$ ) I only ($B$ ) II only ($C$ ) III only ($D$ ) I and II *($E$ ) I and III -
Assume that
$P(A)=0.4$ and$P(B)=0.3$ , and$P(A$ or$B)=0.7$ ,$P(A)*P(B)=0.12$ . Which of following statements are true? I.$A$ and$B$ are mutually exclusive II.$P(A$ and$B)=0.7$ III.$A$ and$B$ are independent event ($A$ ) I only ($B$ ) II only ($C$ ) III only ($D$ ) I and II *($E$ ) I and III -
A variable follow normal distribution. It has a mean value of
$80$ and a standard deviation of$15$ . If a z-score is$2$ , what's value on the normal distribution? ($A$ ) 68 ($B$ ) 95 ($C$ ) 99 *($D$ ) 110 ($E$ ) 125 -
A distribution that skewness value above
$2.5$ ($SK>2.5$ ), whicch of following statements are true? *($A$ ) mean > median > mode ($B$ ) mode > mean > median ($C$ ) mode > median > mean ($D$ ) mean = median = mode ($E$ ) none above -
Assuming
$P(A1)=0.3$ ,$P(A2)=0.7$ ,$P(B\vert A1)=0.2$ , and$P(B\vert A2)=0.4$ ,${A1,A2}$ is a partition of$U$ , then$P(A1\vert B)$ ? ($A$ ) 0.111 *($B$ ) 0.177 ($C$ ) 0.272 ($D$ ) 0.323 ($E$ ) 0.37 -
Assume that
$X$ is a random variable and its$E(X)=100$ and$\sigma^2(X)=10$ . The variable$Y$ is a linear function of$X$ ,$Y=2X+50$ . That$E(Y)$ and$\sigma^2(Y)$ , which of following statements are true? I.$E(Y)=100$ II.$E(Y)=200$ III.$\sigma^2(Y)=10$ IV.$\sigma^2(Y)=40$ ($A$ ) I only ($B$ ) II only ($C$ ) III only *($D$ ) IV only ($E$ ) none above -
In each case state whether you expect the two variables
$x$ and$y$ indicated to have positive, negative, or zero correlation. Which of following statements is negative? ($A$ ) The number$x$ of pafes in a book and the age$y$ of the author. ($B$ ) The number$x$ of pafes in a book and the age$y$ of the intended reader. *($C$ ) The weight$x$ of an automobile and the fuel economy$y$ in miles per gallon. ($D$ ) The weight$x$ of an automobile and the reading$y$ on its odometer. ($E$ ) The amount$x$ of a sedative a person took an hour ago and the time$y$ it takes him to respond to a stimulus. -
As the figure, which of following staements are true? (
$A$ ) Set I. SSE < Set II. SSE (SSE: sum of the squared errors) ($B$ ) A random pattern of residuals supports a linear model. *($C$ )$y=\beta_1x+\beta_2$ , Set I.$r^2=0$ and Set II.$r^2>0$ ($D$ )$y=\beta_1x+\beta_2$ , Set I.$r^2<0$ and Set II.$r^2=0$ ($E$ ) none above. -
A sample of size
$n=150$ has mean$x=30$ and standard deviation$s=3$ . Without knowing anything else about the sample, what can be said about the number of observations that lie in the interval$(24,36)$ *($A$ ) At least 75% ($B$ ) At least 85% ($C$ ) At least 90% ($D$ ) At least 95% ($E$ ) none above
$s^2=\frac{n(1-p)*(2s)^2}{n}$ $1=(1-p)*4$ $p=75%$
- Which following table is a valid probability distribution of a discrete random variable?
(
x | -2 | 0 | 2 | 4 | 6 |
---|---|---|---|---|---|
P(x) | 0.2 | 0.5 | 0.2 | 0.1 | 0.1 |
( |
|||||
x | 0 | 1 | 2 | 3 | 4 |
--- | --- | --- | --- | --- | --- |
P(x) | 0.2 | 0.2 | 0.2 | 0.1 | 0.1 |
( |
|||||
x | 0.5 | 0.25 | 0.3 | 0.4 | 0.7 |
--- | --- | --- | --- | --- | --- |
P(x) | 0.2 | -0.3 | 0.2 | -0.1 | 0.1 |
*( |
|||||
x | -1 | 0 | 1 | 3 | 5 |
--- | --- | --- | --- | --- | --- |
P(x) | 0.2 | 0.3 | 0.2 | 0.2 | 0.1 |
( |
本部份共有兩大題(合計三小題),每題依序為6分,8分,6,分,合計20分。程式題請使用偽代碼(pseudocode) 作答。 Pseudocode is a simple way of writing programming code in English. Pseudocode is not actual programming language. It uses short phrases to write code for programs before you actually create it in a specific language. The purpose of using pseudocode is that it is easier for people to understand the logic behind the algorithms.
・ Rules for pseudo code, Write only one statement per line ・ Available keywords: IF, ELSE, ENDIF, WHILE, ENDWHILE, REPEAT, UNTIL, FUNCTION, FOR, PRINT, LENGTH
example 1:
function example1(x){
y<-"hello, ";
print(y,x);
}
X <- "AI";
example1(X);
The output of example 1: hello, AI
example 2:
function example2(n){
var y[n];
for (i from 0 to n-1){
y[i] <- i+1;
}
return y;
}
a <- 0;
while(a<4){
a <- a+1;
if(a != 2)
print(example2(a));
print(",");
}
The output of example 2: [1],[1,2,3],[1,2,3,4]
- Define a function which satisfies the following requirement.
Given an integer
$N \geq 1$ , please return all integers$X$ , betwenn$1$ and$N$ , which are indivisible by 5. For example, input fun(6) and get the output$1,2,3,4,6$ .
參考解答:
function func(n) {
var y[n-n/5];
var c = 1;
var i = 0;
while ( i < n-n/5-1){
if (c%5 != 0) {
y[i] = c;
}
}
return y;
}
- (1) Write down the output of the following codes.
function func1(x,i,j) {
var a;
a <- x[i];
x[i] <- x[j];
x[j] <- a;
}
function func2(data) {
var i, j;
for (i from 0 to length(data)-1){
for (j from 0 to length(data)-1-i){
if (data[j] > data[j+1])
func1(data, j, j+1);
}
}
}
參考解答: [1, 1, 5, 6, 8, 9, 13, 22]
Please check Bubble sort.
(2) Define a function which satisfies the following staement.
Given
$$ Q_x= \left{ \begin{array}{l} a_{z+1} & \text{if } z+1>\dfrac{nx}{4}>z\ \dfrac{a_{z}+a_{z+1}}{2} & \text{if } \dfrac{nx}{4}=z \end{array} \right . $$
-
$z$ is an integer -
$n$ is the length of given list of numbers -
${a_n}$ is the sorted list of${x_n}$
參考解答:
function swap(x,i,j) {
var a;
a <- x[i];
x[i] <- x[j];
x[j] <- a;
}
function sort(data) {
var i, j;
for (i from 0 to length(data)-1){
for (j from 0 to length(data)-1-i){
if (data[j] > data[j+1])
func1(data, j, j+1);
}
}
}
function func(data) {
var len = length(data);
var x[len] = sort(data);
var q1, q3;
if (len%4 == 0) {
q1 = (x[len/4-1] + x[len/4])/2;
q3 = (x[(len*3)/4-1] + x[(len*3)%4])/2;
}
else{
q1 = x[len/4] ;
q3 = x[(len*3)/4] ;
}
return q1+q3;
}