MF - Practice test 5

PRACTICE TEST 5

The value of the expression $\dfrac{\sqrt{12}-\sqrt{27}+\sqrt{75}}{\sqrt{12}+\sqrt{27}+\sqrt{75}}$ is:

(А) $0,4$ ; (B) $0,5$ ; (C) $0,6$ ; (D) $\sqrt{\frac{10}{19}}$ ; (Е) $\frac{10}{19}$ .
The codomain of the function $f(x)=x^{4}-4x^{2}+2$ is:

(А) $[-6,\infty)$ ; (B) $(-\infty,-2]$ ; (C) $[2,\infty)$ ; (D) $[-2,\infty)$ ; (Е) $(-\infty,2]$ .
The sum of the absolute values of the solutions of the equation $x^{2}=|5x-6|$ is:

(А) $0$ ; (B) $5$ ; (C) $7$ ; (D) $10$ ; (Е) $12$ .
The set of real solutions of the inequality $\sqrt{2x-10}<\sqrt{x+7}$ is:

(А) $(-\infty,17)$ ; (B) $(5,17)$ ; (C) $[5,17)$ ; (D) $[5,\infty)$ ; (Е) $(17,\infty)$ .
The domain of the function $f(x)=\ln\dfrac{2x+3}{x-2}$ is:

(А) $\mathbb{R}\setminus\{ 2\}$ ; (B) $(-\infty,-\frac{3}{2})\cup(2,\infty)$ ; (C) $(-\infty,-\frac{3}{2}]\cup(2,\infty)$ ;
(D) $(2,\infty)$ ; (Е) $\mathbb{R}\setminus\{-\frac{3}{2},2\}$ .
What is the minimum of the function $f(x)=3|x-2|-|x+3|$ is:

(А) $0$ ; (B) $-3$ ; (C) $-5$ ; (D) $-6$ ; (Е) there is no minimum.
If $a$ , $b$ and $c$ are the zeros of the polynomial $x^{3}+x^{2}-4x+1$ , what is $a^{2}+b^{2}+c^{2}$ ?

(А) $1$ ; (B) $5$ ; (C) $8$ ; (D) $-7$ ; (Е) $9$ .
How many integers in the interval $[-10,10]$ are there that satisfy $|\frac{x}{3}-2|^{{5x-x^{2}}}\geqslant 1$ ?

(А) $2$ ; (B) $4$ ; (C) $6$ ; (D) $8$ ; (Е) $10$ .
The imaginary part of the complex number $\dfrac{1+2i}{3-i}$ is:

(А) $\frac{7}{10}$ ; (B) $\frac{3}{5}$ ; (C) $\frac{1}{10}$ ; (D) $\frac{1}{2}$ ; (Е) $-\frac{3}{10}$ .
The real part of the solution of the equation $z+|z+2i|=2i+1$ is:

(А) $5$ ; (B) $-\frac{15}{2}$ ; (C) $2$ ; (D) $\frac{17}{2}$ ; (Е) $-\frac{17}{2}$ .
What is $\cos 105^{\circ}+\sqrt{3}\sin 105^{\circ}$ ?

(А) $2\sin 75^{\circ}$ ; (B) $\sqrt{3}$ ; (C) $\frac{1}{2}\sqrt{2}$ ; (D) $\sqrt{2}$ ; (Е) $2$ .
What is the fundamental period of the function $4\cos^{3}{2x}-3\cos{2x}$ ?

(А) $\pi$ ; (B) $\pi/2$ ; (C) $\pi/3$ ; (D) $2\pi/3$ ; (Е) $\pi/6$ .
The set of solutions of the inequality $\sin 3x<2\sin x$ in the interval $[0,2\pi)$ is:

(А) $(\frac{\pi}{6},\frac{5\pi}{6})\cup(\pi,\frac{7\pi}{6})\cup(\frac{11\pi}{6},2\pi)$ ; (B) $(\frac{\pi}{6},\frac{\pi}{2})\cup(\pi,\frac{7\pi}{6})\cup(\frac{11\pi}{6},2\pi)$ ;
(C) $(\frac{\pi}{6},\frac{\pi}{2})\cup(\frac{5\pi}{6},\frac{7\pi}{6})\cup(\frac{11\pi}{6},2\pi)$ ; (D) $(\frac{\pi}{6},\frac{5\pi}{6})\cup(\frac{7\pi}{6},\frac{11\pi}{6})$ ;
(Е) $(0,\frac{\pi}{6})\cup(\frac{5\pi}{6},\frac{7\pi}{6})\cup(\frac{11\pi}{6},2\pi)$ .
The angles of a triangle are $\alpha=15^{\circ}$ and $\beta=45^{\circ}$ , whereas the radius of its circumscribed circle is 1. What is the area of this triangle?

(А) $\dfrac{3-\sqrt{3}}{2}$ ; (B) $\dfrac{3-\sqrt{3}}{4}$ ; (C) $\dfrac{\sqrt{3}}{3}$ ; (D) $\dfrac{\sqrt{3}}{6}$ ; (Е) $\dfrac{\sqrt{3}-1}{4}$ .
The chords $AB$ and $CD$ of a circle with center $O$ are perpendicular and intersect at point $E$ . If $AB=6$ , $CD=7$ и $OE=3$ , then the radius of the circle is:

(А) $\frac{7}{4}\sqrt{5}$ ; (B) $\sqrt{13}$ ; (C) $3\sqrt{2}$ ; (D) $4$ ; (Е) $\frac{11}{4}\sqrt{2}$ .
The base of a pyramid is a right-angled triangle with the legs $a=35cm$ и $b=12cm$ . Each lateral face of the pyramid forms the angle of $60^{\circ}$ with the base. The area of this pyramid is:

(А) $450\sqrt{2}cm^{2}$ ; (B) $1260cm^{2}$ ; (C) $630cm^{2}$ ; (D) $450\sqrt{3}cm^{2}$ ; (Е) $945cm^{2}$ .
The distance from the intersection point of the lines $3x-2y-5=0$ and $x+2y-7=0$ to the line $3x-4y+15=0$ equals:

(А) $3$ ; (B) $\frac{16}{5}$ ; (C) $\frac{18}{5}$ ; (D) $4$ ; (Е) $\frac{24}{5}$ .
The sum of the slopes of the tangents to the circle $x^{2}+y^{2}=2$ that pass through the intersection point of the lines $x-y-1=0$ and $x+y-3=0$ is:

(А) $4$ ; (B) $-\sqrt{6}$ ; (C) $\sqrt{6}$ ; (D) $2$ ; (Е) $2\sqrt{6}$ .
The second term of an infinite decreasing geometric progression is 2, whereas the sum of all its terms is 9. Find the first term, knowing that it is less than 6.

(А) $3$ ; (B) $\sqrt{12}$ ; (C) $\frac{8}{3}$ ; (D) $4$ ; (Е) $5$ .
A fair coin is tossed three times. What is the probability that the same face is up all three times?

(А) $1/8$ ; (B) $1/6$ ; (C) $1/3$ ; (D) $1/2$ ; (Е) $1/4$ .