Тест по математике экзамена GRE

Each of the questions or incomplete statements below is followed by five suggested answers or completions. In each case, select the one that is best of the choices offered and then mark the corresponding space on the answer sheet.

Note: In this examination: (1) All logarithms with an unspecified base are natural logarithms, that is, with base e. (2) The set of all x such that $a\leq x\leq b$ is denoted by $[a; b]$ (3) The symbols Z, Q, R and C denote the sets of integers, rational numbers, real numbers, and complex numbers, respectively.

1. If $x^2+x=3$, then $x^4+x=$

(A) $x^2+3$     (B) $4(3-x)$     (C) $3(x-1)$     (D) $6(2-x)$     (E) $3x$

2. $\displaystyle\frac{d}{dx}\left(\frac{1}{x}-\frac{1}{\log x}\right)=$

(A) $\displaystyle\frac{x-(\log x)^2}{(x\log x)^2}$  (B) $\displaystyle\frac{x^2-2\log x}{2x^2\log x}$   (C) $\displaystyle\frac{x-1}{(x\log x)^2}$   (D) $\displaystyle\frac{2x-1}{2x^2\log x}$  (E) $\displaystyle\frac{2x-1}{(x\log x)^2}$

3. $\lim_{x\to 0}\displaystyle\frac{\sqrt{x+1}-1}{x}=$

(A) $-\frac{1}{2}$   (B) $\frac{1}{2}$   (C)  $1$   (D)  $2$   (E)  $\infty$

4. Let $A=\{\emptyset,a\}$, where $\emptyset$ is the empty set. If $B$ is the set of all subsets of $A$, what's the cardinality of the set $A\cup B$?

(A)  4     (B)  5     (C)   6     (D)   7     (E)   8

5. For how many values of $a$ is the integral $\int_{0}^{a}(12x^2+8x+1)dx$ equal to zero?

(A)  0     (B)   1     (C)   2     (D)   3     (E)   4

6. At how many points the graph of the curve $y=x^5+x^3+x-2000$ cross the x-axis?

(A)   1     (B)   2     (C)   3     (D)   4     (E)   5

7. Let $f(x,y)=xy(x-y)$, where $x(t,u)=t(t-u)$ and $y(t,u)=u(u-t)$. What is the vake of $\displaystyle\frac{\partial f}{\partial t}$ at the point $(t,u)=(1,-1)$?

(A)   -8     (B)   -4     (C)   2     (D)   4     (E)   8

8. If $\int_{0}^{b}\tan x dx = 2$, then $b$ could equal

(A)  $\arccos 2$     (B)  $\arccos(2e)$   (C) $arcsec 2$   (D)   $arcsec^2 e$    (E)  $arcsec(e^2)$

9. Let $a$ be the smallest positive value of $x$ at which the function $f(x)=(\cos x^2)(\sin x^2)$ has a critical point. What is the value of $f(a)$?

(A)  $\displaystyle\frac{\sqrt{2}}{4}$   (B)  $\displaystyle\frac{1}{2}$    (C)   $\displaystyle\frac{\sqrt{2}}{2}$   (D) $1$   (E)  $\sqrt{2}$

10. Find the equation of the integral curve that passes through the point $(2,3)$ of the differential equation $y'=\displaystyle\frac{2x}{y^2-1}$

(A) $x^2-2=\frac{1}{3}y^3+y$   (B)  $x^2-1=\frac{1}{3}y^3-2y$    (C)  $x^2+1=\frac{1}{3}y^3+y$   (D)   $x^2+2=\frac{1}{3}y^3-y$   (E) $2(x^2-1)=\frac{1}{3}y^3-y$

11. Let $\vec{i}=(1,0,0)$, $\vec{j}=(0,1,0)$, and $\vec{k}=(0,0,1)$. If $b$ and $c$ are vectors such that $\| b\|=3$,$b\cdot c=5$, $\|c\|=7$, and $b\times c=8\vec{j}+4\vec{k}$, which of the following vectors satisfies the equation $a\cdot [(a+c)\times b]=12$?

(A) $a=2\vec{i}+\vec{j}-2\vec{k}$   (B)  $a=\vec{i}-\vec{j}+2\vec{k}$    (C)  $a=\vec{i}-2\vec{j}+\vec{k}$   (D)   $a=2\vec{i}-\vec{j}+\vec{k}$   (E) $a=\vec{i}+2\vec{j}-2\vec{k}$

12. If one arch of the curve $y=\sin x$ is revolved around the x-axis, what's the volume of the generated solid?

(A) $\frac{\pi ^2}{4}$    (B)  $\frac{\pi^2}{2}$      (C) $2\pi$     (D) $\pi^2$      (E) $2\pi^2$

13.  If $a$ is a positive constant, what is the maximum value of the following function? $f(x)=\displaystyle\frac{\log x}{x^a}$

(A) $\displaystyle\frac{1}{ae}$   (B)  $a^{a-1}$     (C) $a^e$     (D)   $e^a$      (E)  $\displaystyle\frac{e^{a^2}}{a}$

14. Find the family of all nonzero solutions of the equation $\displaystyle\frac{d^2 y}{dx^2}=\left(\frac{dy}{dx}\right)^2$

(A)  $y=\frac{1}{x+c_1}+c_2$     (B)  $y=c_2+\log (c_1-x)$     (C) $y=c_2-\log(c_1-x^2)$     (D)   $y=c_2+\log(c_1+x)$     (E)    $y=c_2-\log(c_1+x)$

15.  For each integer $n>1$, let $a_n=\displaystyle\frac{1}{\log n}$. Which of the following statements is/are true?

I.  The sequence $(a_n)$ converges

II.  The series $\sum_{n=2}^{\infty} a_n$ converges

III.  The series $\sum_{n=2}^{\infty}(-1)^n a_{n}^2$ converges

(A)   I only     (B)   I  and  II only     (C) I and III only     (D)   II and III only    (E)  I, II and III