台灣人工智慧學校新竹分校第一期技術領袖培訓班資格考試考古題

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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

Calculus

1. Let the function
   $f(x)=a{x}^3+b{x}^2+cx+d$. 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
   $f^{\prime }(x)=0$, and the point of inflection of
   $f$ satify
   $f^{″}(x)=0$.
   (
   $A$)
   $a=1$,
   $b=5$,
   $c=8$,
   $d=6$
   (
   $B$)
   $a=13$,
   $b=-3$,
   $c=-5$,
   $d=4$
   *(
   $C$)
   $a=3$,
   $b=-9$,
   $c=0$,
   $d=4$
   (
   $D$)
   $a=6$,
   $b=11$,
   $c=1$,
   $d=-2$

$f(x)=a{x}^3+b{x}^2+cx+d$

$f^{\prime }(x)=3a{x}^2+2bx+c$

$f^{″}(x)=6ax+2b$
Then solve:

$f(0)=d=4$…(1)

$f(1)=a+b+c+d=-2$…(2)

$f^{\prime }(0)=c=0$…(3)

$f^{″}(1)=6a+2b=0$…(4)

2. 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
   $d$?
   (
   $A$)
   $(x,y)=(3,\pm 4)$,
   $d=2$
   (
   $B$)
   $(x,y)=(8,17)$,
   $d=23$
   (
   $C$)
   $(x,y)=(3,0)$,
   $d=\sqrt{3}$
   *(
   $D$)
   $(x,y)=(3,\pm 1)$,
   $d=\sqrt{5}$

The points given in (A)(B)(C) doesn't meet the equation.

3. 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

4. Let
   $f(x,y)=sin(x^2-y)$, find
   ${\displaystyle \frac{{\partial }^{2}f(2,4)}{\partial x\partial y}}$
   (
   $A$) 5 (
   $B$) 9 (
   $C$) 4 *(
   $D$) 0

${\displaystyle \frac{{\partial }^{2}f(2,4)}{\partial x\partial y}}=\sin (x^2-y)\ast 2x=0$

5. Let
   $z=f(x-y,y-x)$, what is
   ${\displaystyle \frac{\partial z}{\partial x}}+{\displaystyle \frac{\partial z}{\partial y}}$?
   (
   $A$) 1 *(
   $B$) 0 (
   $C$) 6 (
   $D$) 2

Always results in pairs of postive and negative terms that have same values.

Linear Algebra

6. Let
   $A=\left[\begin{array}{ccc}1& 2& 1\\ 0& 1& 2\\ 1& 3& 2\end{array}\right]$, find a matrix
   $B$ such that
   $AB=A^2+2A$
   (
   $A$)
   $\left[\begin{array}{ccc}3& 2& 1\\ 0& 7& 2\\ 1& 1& 6\end{array}\right]$* (
   $B$)
   $\left[\begin{array}{ccc}3& 2& 1\\ 0& 3& 2\\ 1& 3& 4\end{array}\right]$ (
   $C$)
   $\left[\begin{array}{ccc}3& 7& 8\\ 3& 3& 7\\ 13& 5& 4\end{array}\right]$ (
   $D$)
   $\left[\begin{array}{ccc}9& 2& 1\\ 7& 2& 4\\ 6& 3& 6\end{array}\right]$

$B=A^{-1}(A^2+2A)=A+2I$
where

$I$ denotes the identity matrix.

7. What is the rank of
   $\left[\begin{array}{ccccc}1& 2& 3& 4& 5\\ 1& 0& 0& 1& 0\\ 1& 1& 1& 1& 1\\ 2& 2& 3& 5& 5\\ 5& 5& 7& 11& 11\end{array}\right]$?
   (
   $A$) 5 (
   $B$) 6 (
   $C$) 2 *(
   $D$) 3

Please checkout the definition of rank

8. Find the eigenvalues of the following matrix

   $\left[\begin{array}{cccc}1& 1& 2& 2\\ 1& 1& 2& 2\\ 2& 2& 1& 1\\ 2& 2& 1& 1\end{array}\right]$?
   Hint: Try Gaussian elimination first.
   (
   $A$) 0,1,-6 (
   $B$) 3,5,-10 *(
   $C$) 0,6,-2 (
   $D$) 0,6,12

9. Find the solution set for the following linear matrix equation
   

   $Ax=\left[\begin{array}{cccc}1& 0& 1& 0\\ 2& 2& 0& 3\\ 0& 4& -4& 5\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\\ x_4\end{array}\right]=\left[\begin{array}{c}2\\ 1\\ 7\end{array}\right]$
   *(
   $A$)
   $\left\{\begin{array}{c}\left[\begin{array}{c}-t+2\\ t-3\\ t\\ 1\end{array}\right]|t\in F\end{array}\right\}$ (
   $B$)
   $\left\{\begin{array}{c}\left[\begin{array}{c}t+2\\ 10t\\ t\\ 3t+1\end{array}\right]|t\in F\end{array}\right\}$ (
   $C$)
   $\left\{\begin{array}{c}\left[\begin{array}{c}t\\ 3t\\ -t-10\\ 1\end{array}\right]|t\in F\end{array}\right\}$
   (
   $D$)
   $\left\{\begin{array}{c}\left[\begin{array}{c}3t\\ t\\ t+10\\ 5\end{array}\right]|t\in F\end{array}\right\}$

10. For which

    $x$ is
    $A=LU$ decomposiition impossible?
    
    $A=\left[\begin{array}{ccc}1& 2& 0\\ 3& x& 1\\ 0& 1& 1\end{array}\right]$
    *(
    $A$)
    $x=6$ (
    $B$)
    $x=4$ (
    $C$)
    $x=12$ (
    $D$)
    $x=0$

If

$A$ is invertible, then it admits an LU factorization if and only if all its leading principal minors are nonzero.

Statistics & Probability

11. 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

12. Assume that

    $P(A)=0.4$ and
    $P(B)=0.3$, and
    $P(A$ or
    $B)=0.7$,
    $P(A)\ast P(B)=0.12$. 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

13. 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

14. 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

15. Assuming

    $P(A1)=0.3$,
    $P(A2)=0.7$,
    $P(B|A1)=0.2$, and
    $P(B|A2)=0.4$,
    $\{A1,A2\}$ is a partition of
    $U$, then
    $P(A1|B)$?
    (
    $A$) 0.111 *(
    $B$) 0.177 (
    $C$) 0.272 (
    $D$) 0.323 (
    $E$) 0.37

16. 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

17. 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.

18. 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.
    *(
    $C$)
    $y={\beta }_{1}x+{\beta }_2$, Set I.
    $r^2=0$ and Set II.
    $r^2>0$
    (
    $D$)
    $y={\beta }_{1}x+{\beta }_2$, Set I.
    $r^2<0$ and Set II.
    $r^2=0$
    (
    $E$) none above.

19. 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

$s^2=\frac{n(1-p)\ast (2s{)}^2}{n}$

$1=(1-p)\ast 4$

$p=75\mathrm{\%}$

20. Which following table is a valid probability distribution of a discrete random variable?

(

$A$)

x	-2	0	2	4	6
P(x)	0.2	0.5	0.2	0.1	0.1
( $B$)
x	0	1	2	3	4
–-	–-	–-	–-	–-	–-
P(x)	0.2	0.2	0.2	0.1	0.1
( $C$)
x	0.5	0.25	0.3	0.4	0.7
–-	–-	–-	–-	–-	–-
P(x)	0.2	-0.3	0.2	-0.1	0.1
*( $D$)
x	-1	0	1	3	5
–-	–-	–-	–-	–-	–-
P(x)	0.2	0.3	0.2	0.2	0.1
( $E$) none above

Programming

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]

1. Define a function which satisfies the following requirement.
   Given an integer
   $N\ge 1$, 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;
}

2. (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

$\{x_n\}$, a list of unsorted numbers, return the sum of the first quartile

$Q1$ and the third quartile

$Q3$.
Hints:

$$Q_x=\{\begin{array}{ll}{a}_{z+1}& \text{if }z+1>{\displaystyle \frac{n\ast x}{4}}>z\\ {\displaystyle \frac{a_z+a_{z+1}}{2}}& \text{if }{\displaystyle \frac{n\ast x}{4}}=z\end{array}$$

1. $z$ is an integer
2. $n$ is the length of given list of numbers
3. $\{a_n\}$ 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;
}