MF - Practice test 3

PRACTICE TEST 3

The number $\sqrt[3]{2-\sqrt{5}}+\sqrt[3]{2+\sqrt{5}}$ is equal to

(A) undefined; (B) $\sqrt[3]{4}$ ; (C) $\frac{1}{2}\sqrt{5}$ ; (D) $\sqrt{5}$ ; (Е) $1$ .
The minimum of the function $f(x)=\dfrac{1}{x^{2}-6x+2}$ on the set $[3,5]\cup[7,9]$ is:

(A) $-\frac{1}{7}$ ; (B) $-\frac{1}{3}$ ; (C) $-\frac{1}{6}$ ; (D) $\frac{1}{9}$ ; (Е) $\frac{1}{29}$ .
The roots of the equation $x^{2}+(p+2)x+2(p+1)$ differ by $1$ if and only if:

(A) $p\in\{ 1,5\}$ ; (B) $p=5$ ; (C) $p\in\{-1,5\}$ ; (D) $p=-1$ ; (Е) $p\in\{-1,1,5\}$ .
How many real solutions does the equation $\displaystyle\sqrt[4]{x}-2\sqrt[8]{x}=\frac{1}{2}$ have?

(A) $1$ ; (B) $0$ ; (C) $2$ ; (D) $3$ ; (Е) $4$ .
If $f(x)=\sqrt{x}$ and $g(x)=\log _{{1/2}}(-\log _{{1/2}}x)$ , then $g(f(\sqrt{5}-1))-g(\sqrt{5}-1)$ is equal to:

(A) $-\frac{1}{2}$ ; (B) $-1$ ; (C) $0$ ; (D) $1$ ; (Е) $\frac{1}{2}\log _{2}5$ .
The difference between the maximum and the minimum of the function $f(x)=x^{3}-2x^{2}+x$ on the segment [-1,1] is:

(A) $\frac{112}{27}$ ; (B) $4$ ; (C) $\frac{40}{9}$ ; (D) $5$ ; (Е) $\frac{16}{3}$ .
A polynomial gives the remainder 1 when divided by by $x$ , and $3$ when divided by $x-1$ . What is the remainder of this polynomial when divided by $x^{2}-x$ ?

(A) $3$ ; (B) $x+1$ ; (C) $3x$ ; (D) $x+4$ ; (Е) $2x+1$ .
What is the sum of solutions of the equation $2^{{2x-1}}=5\cdot 2^{x}-2$ ?

(A) $1$ ; (B) $2$ ; (C) $\log _{2}5$ ; (D) $3$ ; (Е) $4$ .
How many real solutions does the equation $\log _{3}(3^{x}-1)\cdot\log _{3}(3^{{x+1}}-3)=6$ have?

(A) $0$ ; (B) $1$ ; (C) $2$ ; (D) $3$ ; (Е) $4$ .
The complex number $(\sqrt{3}+i)^{{30}}$ is equal to:

(A) $1$ ; (B) $4^{{15}}$ ; (C) $3^{{15}}$ ; (D) $-4^{{15}}$ ; (Е) $-3^{{15}}$ .
If $\:\mathrm{tg}\: x=-\sqrt{8}$ and ${\pi}/2<x<{3\pi}/2$ , what is $\sin x$ ?

(A) $-\dfrac{\sqrt{8}}{3}$ ; (B) $\dfrac{\sqrt{8}}{3}$ ; (C) $-\dfrac{1}{3}$ ; (D) $\dfrac{1}{3}$ ; (Е) $\dfrac{\sqrt{8}}{8}$ .
Which of the following expressions is identically equal to $\:\mathrm{tg}\: x\:\mathrm{ctg}\: 2x$ ?

(A) $1-\frac{1}{2}\sec^{2}x$ ; (B) $\frac{1}{2}-\frac{1}{2}\sec^{2}x$ ; (C) $\frac{1}{2}-\frac{1}{4}\sec^{2}x$ ; (D) $1-\sec^{2}x$ ;
(Е) $\frac{1}{2}+\frac{1}{2}\sec^{2}x$ .
How many solutions does the equation $\cos 2x+\sin 3x=1$ have in $(0,2\pi)$ ?

(A) $1$ ; (B) $2$ ; (C) $3$ ; (D) $4$ ; (Е) $5$ .
In a right-angled triangle, one leg is 5 times the radius of the inscribed circle, and the other leg has length 1. What is the length of the hypotenuse?

(A) $\frac{5}{4}$ ; (B) $\frac{13}{12}$ ; (C) $\frac{5}{3}$ ; (D) $\frac{13}{5}$ ; (Е) $\frac{17}{8}$ .
A trapezoid $ABCD$ with the bases $AB=3$ and $CD=1$ is circumscribed about a circle of diameter $\frac{12}{7}$ . What is the length of the smaller leg of the trapezoid?

(A) $2-\frac{\sqrt{2}}{4}$ ; (B) $\sqrt{5}$ ; (C) $2$ ; (D) $\frac{13}{7}$ ; (Е) $\frac{15}{7}$ .
The area of the smaller diagonal cross-section of a regular hexagonal prism is $S$ . The lateral area of the prism is equal to:

(A) $S\sqrt{3}$ ; (B) $3S$ ; (C) $4S\sqrt{3}$ ; (D) $6S$ ; (Е) $2S\sqrt{3}$ .
The total area of a cone is twice the area of the sphere inscribed in that cone. Then the volumes ot the cone and the sphere are in ratio:

(A) $4:\sqrt{3}$ ; (B) $2\pi:3$ ; (C) $2:1$ ; (D) $6:\pi$ ; (Е) $3\pi:4$ .
If the circles $x^{2}+y^{2}-1=0$ and $x^{2}+y^{2}-3x-4y+a=0$ are externally tangent, then the value of $a$ is:

(A) $3$ ; (B) $4$ ; (C) $5$ ; (D) $6$ ; (Е) $7$ .
The first, third and sixth term of an arithmetic progression are also three consecutive terms of some geometric progression. If the first term of the arithmetic progression is 1, what is the sum of its first six terms?

(A) $6$ ; (B) $8$ ; (C) $\frac{32}{3}$ ; (D) $\frac{39}{4}$ ; (Е) $\frac{17}{2}$ .
In how many ways can we place four people and two (distinct) elephants into three trucks so that the two elephants are not in the same truck?

(A) $486$ ; (B) $729$ ; (C) $720$ ; (D) $360$ ; (Е) $120$ .