МF - Practice test 2

PRACTICE TEST 2

The value of the expression $\left(20^{{-2}}+0,6^{2}+1/5-\sqrt{(-7/16)^{2}}\right)^{{-2/3}}$ is:

(A) $-\frac{1}{4}$ ; (B) $4$ ; (C) $1$ ; (D) $1,6^{{-1/3}}$ ; (E) $0,42$ .
Point $M(-1,4)$ lies on the parabola $y=x^{2}-ax+7$ . The ordinate of the vertex of the parabola is

(A) $3$ ; (B) $5$ ; (C) $7$ ; (D) $17$ ; (E) $19$ .
Which quadratic equation has the solutions $x_{1}=\sqrt{4-\sqrt{7}}$ and $x_{2}=\sqrt{4+\sqrt{7}}$ ?

(A) $x^{2}-4x+\sqrt{14}=0$ ; (B) $x^{2}-\sqrt{14}x-3=0$ ; (C) $x^{2}-\sqrt{7}x+3=0$ ;
(D) $x^{2}-\sqrt{14}x+3=0$ ; (E) $x^{2}-\sqrt{2}x-3=0$ .
If $f(x)=\dfrac{x+3}{3x-2}$ , then $f(f(x))$ equals:

(A) $\dfrac{x+6}{9x-8}$ ; (B) $\dfrac{2x+5}{11-3x}$ ; (C) $\dfrac{7x+8}{8x-7}$ ; (D) $x$ ; (E) $\dfrac{10x-3}{13-3x}$ .
The minimum of the function $(2x-1)^{2}+\dfrac{1}{x^{2}-x}$ for $x>1$ is equal to:

(A) $0$ ; (B) $4$ ; (C) $5$ ; (D) $6$ ; (E) $8$ .
If the polynomial $x^{3}+ax^{2}+b$ is divisible by the polynomial $x^{2}+3x+1$ then $a+2b$ is equal to:

(A) $0$ ; (B) $1$ ; (C) $\frac{5}{3}$ ; (D) $2$ ; (E) $\frac{8}{3}$ .
If $x=1$ and $x=2$ are roots of the polynomial $x^{4}-6x^{2}+ax+b$ , then the smallest real root of this polynomial is:

(A) $\dfrac{3-\sqrt{5}}{2}$ ; (B) $-\dfrac{3+\sqrt{5}}{2}$ ; (C) $1$ ; (D) $1-\sqrt{3}$ ; (E) $-1-\sqrt{3}$ .
The solution of the equation $9^{{x-3}}=(1/3)^{{3x-1}}$ lies in the interval:

(A) $(-\infty,-\frac{1}{2})$ ; B) $[-\frac{1}{2},0)$ ; (C) $[0,\frac{1}{2})$ ; (D) $[\frac{1}{2},1)$ ; (E) $[1,+\infty)$ .
If $x$ is the solution of the equation $\log _{{2x}}{x}+\log _{{4x}}{2x}=1$ with $x>1$ , then $\log _{2}x$ is equal to:

(A) $\dfrac{\sqrt{5}-1}{2}$ ; (B) $\dfrac{3-\sqrt{5}}{2}$ ; (C) $1$ ; (D) $\log _{4}3$ ; (E) $\dfrac{\sqrt{5}}{2}$ .
If $\lambda$ is a positive real number and $\dfrac{\lambda+i}{1+\lambda i}+\dfrac{1}{2}i$ is real number, then $\lambda$ equals

(A) $\frac{3}{2}$ ; (B) $\sqrt{2}$ ; (C) $\sqrt{3}$ ; (D) $1$ ; (E) $2$ .
The imaginary part of the solution of the equation $\displaystyle\frac{z+i+1}{z}=2\cdot\frac{\bar{z}+i+1}{\bar{z}}$ is:

(A) $-\frac{9}{5}$ ; (B) $2$ ; (C) $\frac{2}{3}$ ; (D) $\frac{3}{5}$ ; (E) $3$ .
If $\sin x\neq 1$ , then the expression $\displaystyle\frac{1+\sin x}{1-\sin x}$ equals:

(A) $\dfrac{\:\mathrm{tg}\: x+1}{\:\mathrm{tg}\: x-1}$ ; (B) $\dfrac{\:\mathrm{tg}\: x-1}{\:\mathrm{tg}\: x+1}$ ; (C) $\dfrac{\:\mathrm{tg}\:\frac{x}{2}+1}{\:\mathrm{tg}\:\frac{x}{2}-1}$ ; (D) $\left(\dfrac{\:\mathrm{tg}\:\frac{x}{2}-1}{\:\mathrm{tg}\:\frac{x}{2}+1}\right)^{2}$ ;
(E) $\left(\dfrac{\:\mathrm{tg}\:\frac{x}{2}+1}{\:\mathrm{tg}\:\frac{x}{2}-1}\right)^{2}$ .
If $\sin x=\frac{5}{13}$ , then the possible values of the expression $\sin(x+\arccos(-\frac{3}{5}))$ are:

(A) $-\dfrac{33}{65}$ and $\dfrac{63}{65}$ ; (B) $\dfrac{33}{65}$ and $-\dfrac{63}{65}$ ; (C) $\pm\dfrac{33}{65}$ ; (D) $\pm\dfrac{63}{65}$ ; (E) $\pm\dfrac{33}{65}$ and $\pm\dfrac{63}{65}$ .
The set of all solutions of the inequality $2\cos^{2}x>1+\cos 108^{\circ}$ in the interval $[0,\pi)$ is:

(A) $(54^{\circ},126^{\circ})$ ; (B) $(0^{\circ},54^{\circ})$ ; (C) $(0^{\circ},54^{\circ})\cup(126^{\circ},180^{\circ})$ ; (D) $(0^{\circ},126^{\circ})$ ;
(E) $(54^{\circ},90^{\circ})\cup(126^{\circ},180^{\circ})$ .
An isosceles trapezoid with the acute angle $60^{\circ}$ and the shorter base $1$ is circumscribed about a circle. The length of a diagonal of this trapezoid is:

(A) $\sqrt{7}$ ; (B) $3$ ; (C) $\sqrt{3}$ ; (D) $\sqrt{5}$ ; (E) $2$ .
The edge of a cube is 2. Each diagonal of the cube is extended by 1 on both sides, producing a larger cube. The volume of the larger cube is:

(A) $8\left(1+\dfrac{\sqrt{3}}{3}\right)^{3}$ ; (B) $32$ ; (C) $64$ ; (D) $8\left(\dfrac{2+\sqrt{3}}{3}\right)^{3}$ ; (E) $24\sqrt{3}$
A cone of height $H=12$ is inscriped in a sphere of radius $r=8$ . The lateral area of this cone is:

(A) $64\sqrt{3}\pi$ ; (B) $32\sqrt{3}\pi$ ; (C) $96\pi$ ; (D) $108\pi$ ; (E) $32\sqrt{6}\pi$ .
The tangent to the circle $x^{2}+y^{2}-7x-y=0$ at point $A(3,4)$ meets $x$ axis at the point $(x_{\ast},0)$ . Then $x_{\ast}$ equals:

(A) $-13$ ; (B) $-22$ ; (C) $28$ ; (D) $31$ ; (E) $-25$ .
The sum of the first 10 terms of an arithmetic progression is equal to the sum of the first 15 terms. If the 10-th term is equal to 15, then the first term is:
(A) $24$ ; (B) $60$ ; (C) $39$ ; (D) $42$ ; (E) $52$ .
How many four-digit numbers with the digits arranged in the strictly increasing order are there?

(A) $432$ ; (B) $3024$ ; (C) $5040$ ; (D) $126$ ; (E) $210$ .