Câu 1 (Choose 1 answer)
Find the domain of the function $f(x)=ln(3x+6)$
A. (i) $[-2,\infty)$
B. (ii) $[-6,0)$
C. (iii) $(-2,\infty)$
D. (iv) $(6,\infty)$
E. (v) $(-\infty,\infty)$
F. None of the other choices is correct
Câu 2 (Choose 1 answer)
Describe how the graph of $y=f(3x)+1$ is obtained from graph of $y=f(x)$
A. Compress horizontally by a factor of 3, and then shift 1 unit upward
B. Compress vertically by a factor of 3, and then shift 1 unit upward
C. Stretch horizontally by a factor of 3, and then shift 1 unit upward
D. Stretch vertically by a factor of 3, and then shift 1 unit upward
E. None of the other choices is correct
Câu 3 (Choose 1 answer)
Find $lim_{x\rightarrow3}\frac{\sqrt{x+13}-10}{x+3}$
A. 1
B. 2
C. 1/2
D. 1/4
E. None of the others.
Câu 4 (Choose 1 answer)
Write ONE equation that expresses the fact that a function $f$ is continuous at the number 4.
A. (i) $lim_{x\rightarrow4^{-}}f(x)=-f(4)$
B. (ii) $lim_{x\rightarrow4}f(x)=f(4)$
C. (iii) $lim_{x\rightarrow4^{-}}f(x)=f(4)$
D. (iv) $lim_{x\rightarrow4^{+}}f(x)=lim_{x\rightarrow4^{+}}f(x)$
E. (v) $lim_{x\rightarrow4^{+}}f(x)=f(4)$
Câu 5 (Choose 1 answer)
Simplify the expression $\frac{f(x+h)-f(x)}{h}$ for $f(x)=x^{2}+5x$
A. 2h+x+5
B. h+2x-5
C. h+2x+5
D. 2h+x-5
E. None of the other choices is correct
Câu 6 (Choose 1 answer)
Find $y'(1)$ for $y=\frac{x^{2}+2x-8}{3x+9}$
A. 61/144
B. 33/72
C. 63/144
D. 31/72
E. None of the other choices is correct
Câu 7 (Choose 1 answer)
A rocket is fired vertically upward from the ground. The distances in feet that the rocket travels from the ground after t seconds is given by $s(t)=-16t^{2}+560t$. Find the acceleration ($ft/s^{2}$) of the rocket 2 seconds after being fired.
A. -32
B. 1077
C. 496
D. -496
E. None of the other choices is correct
Câu 8 (Choose 1 answer)
Find $\frac{d^{3}y}{dx^{3}}$ for $y=\frac{1}{x+1}$
A. (iii) $(x+1)^{-4}$
B. (i) $6(x+1)^{-4}$
C. (ii) $-6(x+1)^{-4}$
D. (iv) $-(x+1)^{-4}$
E. None of the other choices is correct
Câu 9 (Choose 1 answer)
Find $dy/dx$ by implicit differentiation: $2y-x+xy=6$
A. $(y-1)/(2+x)$
B. $(1+y)/(2+x)$
C. $(y+1)/(x+2)$
D. None of the other choices is correct
E. $-(y+1)/(x+2)$
Câu 10 (Choose 1 answer)
Volume of a spherical balloon is decreasing at rate $9 cm^{3}/s$. How fast is its diameter decreasing when the volume is $6 cm^{3}$?
A. 1.127 cm/s
B. 2.127 cm/s
C. 3.127 cm/s
D. 0.127 cm/s
E. None of the other choices is correct
Câu 11 (Choose 1 answer)
Let $y=ax+b$ be the linear approximation for $f(x)=x^{2/3}$ at $x=1$. Find b.
A. 1/3
B. -1/3
C. 2/3
D. -2/3
E. None of the other choices is correct
Câu 12 (Choose 1 answer)
Find the critical numbers of the function $y=\frac{x^{5}}{5}-\frac{2x^{3}}{3}-3x+2$
A. (i) -1; 1
B. (ii) -1; $-\sqrt{3}$
C. (iii) $\sqrt{3}$; 1
D. (iv) $\sqrt{3}$; $-\sqrt{3}$
E. None of the other choices is correct
Câu 13 (Choose 1 answer)
Determine where the function $f(x)=x^{3}-12x-1$ is increasing and where it is decreasing.
A. (i) Increasing on $(-\infty, -2)$ and $(2,\infty)$, decreasing on $(-2,2)$
B. (ii) Increasing on $(-\infty, -4)$ and $(4,\infty)$, decreasing on $(-4,4)$
C. (iii) Decreasing on $(-\infty, -2)$ and $(2,\infty)$, increasing on $(-2,2)$
D. (iv) Decreasing on $(-\infty, -4)$ and $(4,\infty)$, increasing on $(-2,2)$
E. None of the other choices is correct
Câu 14 (Choose 1 answer)
Find the limit $lim_{x\rightarrow\infty}\frac{3x\sqrt{x+2}+5}{(4x-1)\sqrt{x}}$
A. 3/4
B. 4/3
C. 0
D. infinity
E. 5/4
F. None of the other choices is correct
Câu 15 (Choose 1 answer)
Find the minimum of the sum of two positive numbers whose product is 324.
A. 36
B. 24
C. 42
D. 20
E. None of the other choices is correct
Câu 16 (Choose 1 answer)
Use Newton's method with the specified initial approximation $x_1=2$ to find $x_3$ of the following equation: $ln(x^{2}+4)-2x=0$
A. 0.76054
B. 0.71963
C. 0.76070
D. 0.71696
E. None of the other choices is correct
Câu 17 (Choose 1 answer)
Estimate the area under the graph of $f(x)=25-x^{2}$ on $[0, 5]$ using 5 rectangles and right endpoints.
A. 55
B. 50
C. None of the other choices is correct
D. 60
E. 70
Câu 18 (Choose 1 answer)
Express the following limit as a definite integral: $lim_{n\rightarrow\infty}\sum_{i=1}^{n}\frac{2+4i/n}{1+(2+4i/n)^{5}}.\frac{4}{n}$
A. (i) $\int_{2}^{6}\frac{2+4x}{1+(2+4x)^{5}}dx$
B. (ii) $\int_{2}^{6}\frac{2+x}{1+(2+x)^{5}}dx$
C. (iii) $\int_{2}^{6}\frac{x}{1+x^{5}}dx$
D. (iv) $\int_{1}^{4}\frac{2+x}{1+(2+x)^{5}}dx$
E. None of the other choices is correct
Câu 19 (Choose 1 answer)
Find $\frac{dy}{dx}$ for $y=\int_{1}^{2x}tdt$
A. 3x
B. 2x
C. x
D. 4x
E. 1
F. None of the other choices is correct
Câu 20 (Choose 1 answer)
Evaluate $\int_{1}^{3}\frac{x^{5}-x^{-1}}{x^{2}}dx$
A. 731/36
B. 704/9
C. None of the other choices is correct
D. 176/9
E. 1403/72
Câu 21 (Choose 1 answer)
Evaluate $\int te^{-7t^{2}}dt$
A. None of the other choices is correct
B. (i) $-7e^{-7t^{2}}+C$
C. (iii) $-(1/14)te^{-7t^{2}}+C$
D. (iv) $-e^{-7t^{2}}+C$
E. (ii) $-(1/14)e^{-7t^{2}}+C$
Câu 22 (Choose 1 answer)
Find $\int 4x ln(2x)dx$
A. (i) $x^{2}(2 ln(2x)-1)+C$
B. (ii) $x^{2}(ln(2x)+1)+C$
C. (iii) $2x^{2}(ln(2x)-1)+C$
D. (iv) $2x^{2}(ln(2x)+1)+C$
E. None of the other choices is correct
Câu 23 (Choose 1 answer)
Use Midpoint Rule with $n=4$ to approximate $\int_{0}^{8}f(x)dx$ based on the table (x values: 0, 1, 2, 3, 4, 5, 6, 7, 8; f(x) values: 3, 1, 2, 5, 3, 8, 7, 6, 2).
A. 40
B. 20
C. 30
D. 34
E. 28
F. None of the other choices is correct
Câu 24 (Choose 1 answer)
Calculate the following integral (if it converges): $\int_{2}^{\infty}\frac{1}{x^{2}}dx$
A. -0.5
B. 0.5
C. 0.25
D. It is divergent
E. None of the other choices is correct
Câu 25 (Choose 1 answer)
For a system of 4 equations in 4 unknowns, which of the following statements are true?
(i) The system can be inconsistent.
(ii) The system can have a unique solution.
(iii) The system can have infinitely many solutions.
A. (i), (ii)
B. (iii)
C. All of (i), (ii) and (iii)
D. (i), (ii) [repeated option in source]
E. (i), (iii)
Câu 26 (Choose 1 answer)
Find conditions on $(a,b,c)$ such that the following system is inconsistent:
$\begin{cases}x+2y-z=a\\ 2x+y+3z=b\\ x-4y+9z=c\end{cases}$
A. 2c-(2b-3a) is nonzero
B. c-(2b-3a) is nonzero
C. c-2b-3a
D. 2c-b-3a
E. None of the other choices is correct
Câu 27 (Choose 1 answer)
Consider a homogeneous system of 5 linear equations in 6 unknowns. Which of the following is true?
A. The system always has infinitely many solutions.
B. The system has only the trivial solution, or infinitely many solutions.
C. The system has between 0 and 5 solutions.
D. The system has only the trivial solution.
E. The system can have no solution.
Câu 28 (Choose 1 answer)
Let $x=\begin{bmatrix}2\\ 1\end{bmatrix}$ and $y=\begin{bmatrix}4\\ 3\end{bmatrix}$. Assume that $2x+z=5y$. Find z.
A. (iii) $\begin{bmatrix}-16\\ -13\end{bmatrix}$
B. (i) $\begin{bmatrix}24\\ 17\end{bmatrix}$
C. (ii) $\begin{bmatrix}-24\\ -17\end{bmatrix}$
D. (iv) $\begin{bmatrix}16\\ 13\end{bmatrix}$
E. None of the other choices is correct
Câu 29 (Choose 1 answer)
Which statements are TRUE?
(i) $\begin{bmatrix}-2\\ 3\end{bmatrix}$ is a linear combination of $\begin{bmatrix}0\\ 1\end{bmatrix}$ and $\begin{bmatrix}1\\ 0\end{bmatrix}.$
(ii) If $x_1$ and $x_2$ are solutions to the system $Ax=b$ then $x_1-x_2$ is a solution to $Ax=0$.
A. (i)
B. (ii)
C. (i) and (ii)
D. None of the other choices is correct
Câu 30 (Choose 1 answer)
Find the (2,3)-entry of the product: $\begin{bmatrix}1\\ -2\\ 3\end{bmatrix} \begin{bmatrix}0 & 5 & -4\end{bmatrix}$
A. -14
B. 15
C. 8
D. -22
E. None of the other choices is correct
Câu 31 (Choose 1 answer)
Let A, B be $2\times2$ matrices with $A^{-1} = \dots$ and $B^{-1} = \dots$. Find the (1, 1)-entry of the matrix $(AB^{T})^{-1}$.
A. 5
B. -2
C. 2
D. 4
E. None of the other choices is correct
Câu 32 (Choose 1 answer)
Let $A=(a_{ij})$ be the matrix of rotation in the plane through $\pi/3$. Find $a_{12}$.
A. (i) $1/2$
B. (ii) $-1/2$
C. (iii) $\frac{\sqrt{3}}{2}$
D. (iv) $-\frac{\sqrt{3}}{2}$
E. None of the other choices is correct
Câu 33 (Choose 1 answer)
Let $A=\begin{bmatrix}2&*&*&*\\ 0&-3&*&*\\ 0&0&5&*\\ 0&0&0&6\end{bmatrix}$. Find $det(4A)$.
A. -46080
B. -46800
C. -720
D. 720
E. None of the others
Câu 34 (Choose 1 answer)
If A is $2\times3$ [Note: likely typo in source, possibly $3\times3$ or $2\times2$] and $det(2A^{-1})=-4$, then detA is...
A. -2
B. 2
C. -1/2
D. 1/2
E. None of the others.
Câu 35 (Choose 1 answer)
Find A when $(A^{-1}-2I)^{T}=-2\begin{bmatrix}2&3\\ 1&1\end{bmatrix}.$
A. (i) $\begin{bmatrix}0&-1/6\\ -1/2&1/6\end{bmatrix}$
B. (ii) $\begin{bmatrix}0&1/6\\ 1/2&1/6\end{bmatrix}$
C. (iii) $\begin{bmatrix}0&-1/6\\ -1/2&-1/6\end{bmatrix}$
D. (iv) $\begin{bmatrix}0&1/6\\ 1/2&-1/6\end{bmatrix}$
E. None of the other choices is correct
Câu 36 (Choose 1 answer)
Given that $\lambda=0$ is an eigenvalue of the matrix $\begin{bmatrix}1&1&0\\ 1&1&2\\ 1&1&2\end{bmatrix}$. Find a set of basic eigenvectors corresponding to this eigenvalue 0.
A. (i) [1 1 0]
B. (ii) $[1 -1 0]^T$
C. (iii) $[1 0 -1]^T$
D. (iv) $[1 -1 0]^T$ and $[1 0 -1]^T$
E. None of the other choices is correct
Câu 37 (Choose 1 answer)
Let $A=\begin{bmatrix}2&-4\\ -1&-1\end{bmatrix}$. Find a diagonal form D of the matrix A (if exists)
A. None of the other choices is correct
B. (i) $\begin{bmatrix}2&0\\ 0&3\end{bmatrix}$
C. (ii) $\begin{bmatrix}2&0\\ 0&-3\end{bmatrix}$
D. (iii) $\begin{bmatrix}-2&0\\ 0&3\end{bmatrix}$
Câu 38 (Choose 1 answer)
Let $A(3,-1,-1), B(1,-2,0), C(1,-1,2)$ be three vertices of the parallelogram with two adjacent sides AB and AC. Find the fourth vertex.
A. (-1, 2, 3)
B. None of the other choices is correct
C. (-1,-2,-3)
D. (1,-2,3)
E. (-1,-2, 3)
Câu 39 (Choose 1 answer)
Let A be the point on the line $x=t, y=2t, z=1-t$ that is closest to the point (1, 2, 3). Determine the first coordinate of A.
A. 0
B. 1
C. None of the other choices is correct
D. -1
E. 3
Câu 40 (Choose 1 answer)
Find the second coordinate of the projection of the vector $\vec{u}=\begin{bmatrix}1 & 2 & -1\end{bmatrix}^T$ on the vector $\vec{v}=\begin{bmatrix}0 & 2 & 1\end{bmatrix}^T$
A. 2/5
B. All of the other choices are incorrect
C. 6/5
D. 3/5
Câu 41 (Choose 1 answer)
Find the volume of the tetrahedron with vertices (0, 0, 0), (1, 2, 3), (2, 1, 3), and (3, 1, 2).
A. 6
B. 1
C. 7/3
D. 14
E. None of the other choices is correct
Câu 42 (Choose 1 answer)
Which of the followings are subspaces of $R^3$?
(i) $U=\{(x,y,z+1)|x,y \text{ and } z \text{ in } R\}$
(ii) $U=\{(x,y,z)|x+2y-3z=0\}$
(iii) $U=\{(x,y,z)|x^2+y^2+z^2=1\}$
A. (i)
B. (ii)
C. (ii) and (iii)
D. (i) and (ii)
Câu 43 (Choose 1 answer)
Find the dimension of the subspace spanned by the set $\{(1,-1,2), (0,2,-3), (1,-3,5), (1,1,-1)\}$
A. 1
B. 3
C. 2
D. 0
E. 4
Câu 44 (Choose 1 answer)
Let $U=\{(a,b,c,d)| a+2d=3b+c\}$ be a subspace in $R^4$. Find the dimension of U.
A. 0
B. 1
C. 2
D. 3
E. 4
Câu 45 (Choose 1 answer)
Find a-2b if the set $\{[1, 0, 1], [-2, 1, 2], [1, a, b]\}$ is orthogonal.
A. 1
B. 6
C. 3
D. 4
Câu 46 (Choose 1 answer)
Let A be a $30\times70$ matrix having rank A=20. Find $dim(Col(A))$, $dim(Row(A))$ and $dim(Null(A))$.
A. 70, 30, 20
B. 20, 20, 40
C. 20, 30, 50
D. None of the other choices is correct.
Câu 47 (Choose 1 answer)
Find c such that the graph of $y=a+bx+cx^2$ passes through (0, 5), (-1, -7), (2,11).
A. -3
B. 3
C. -9
D. 9
E. None of the other choices is correct
Câu 48 (Choose 1 answer)
Let $T:R^{3}\rightarrow R^{2}$ be a linear transformation such that $T(1,1,1)=(0,1), T(1,1,0)=(1,0)$ and $T(1,0,0)=(1,1)$. Find $T(3,2,1)$.
A. (2,2)
B. (1,1)
C. (1,2)
D. (2,1)
E. None of the other choices is correct
Câu 49 (Choose 1 answer)
Find a parametric equation of the line which goes through $P(2,3,5)$ and perpendicular to the plane $2x+4y+6z-7=0$.
A. $x=2+t, y=3+2t, z=5+3t$.
B. $x=2+2t, y=4+3t, z=6+5t$.
C. $x=2t, y=4t, z=6t-7$.
D. None of the other choices is correct
Câu 50 (Choose 1 answer)
Let $U=[1,1,1], V=[0,1,2], W=[2,0,3]$ and $K=[0,0,1]$. The vector K can be written uniquely as $K=aU+bV+cW$ where a, b, c are numbers. Find a.
A. 2/5
B. -1/5
C. -2/5
D. 1/5
E. None of the other choices is correct
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