МF-Practice test 1

PRACTICE TEST 1

For $x>1$ , the expression $\displaystyle\frac{1}{x\sqrt{x-1}-(x-1)\sqrt{x}}-\frac{1}{(x+1)\sqrt{x}-x\sqrt{x+1}}$ is identically equal to:

(A) $(x-1)^{{-1/2}}+(x+1)^{{-1/2}}$ ; (B) $(x+1)^{{-1/2}}-(x-1)^{{-1/2}}$ ;
(C) $(x-1)^{{-1/2}}-(x+1)^{{-1/2}}$ ; (D) $(x-1)^{{1/2}}-(x+1)^{{1/2}}$ ;
(E) $(x+1)^{{1/2}}-(x-1)^{{1/2}}$ .
The sum of the minimum and the maximum of $f(x)=-x^{2}+8x+1$ on segment $[1,7]$ equals:

(A) $14$ ; (B) $17$ ; (C) $18$ ; (D) $23$ ; (E) $25$ .
The value of $mx^{2}+x+1$ is strictly positive for all real $x$ if and only if

(A) $m>0$ ; (B) $m>\frac{1}{4}$ ; (C) $m<\frac{1}{4}$ ; (D) $0<m<\frac{1}{4}$ ; (E) $m=\frac{1}{4}$ .
If $f\left(\sqrt{\frac{2x-1}{x+2}}\right)=x$ , then $f(-2)$ equals:

(A) $\frac{1}{2}$ ; (B) $3$ ; (C) $0$ ; (D) $-\frac{9}{2}$ ; (E) $-\frac{7}{6}$ .
The set of all real solutions of $2|x+1|-|x-3|\leqslant 10$ is:

(A) $[-15,5]$ ; (B) $(-\infty,5]$ ; (C) $[-\infty,4]$ ; (D) $[-10,4]$ ; (E) $[-5,5]$ .
When the polynomial $x^{4}+x^{2}$ is divided by $x^{2}-x-1$ the remainder is:

(A) $x^{2}+x+3$ ; (B) $x+3$ ; (C) $2x-2$ ; (D) $2x+1$ ; (E) $4x+3$ .
Polynomial $P(x)=x^{3}-ax^{2}+x+2$ has three distinct real zeros $p,q,r$ . If $p+q=1$ , then $r$ is equal to:

(A) $-1$ or $2$ ; (B) $2$ ; (C) $-1$ ; (D) $1$ or $2$ ; (E) $1$ or $-2$ .
If $\log _{5}{3}=a$ and $\log _{5}{4}=b$ , then $\log _{{15}}2$ is equal to:

(A) $\dfrac{2b}{a+1}$ ; (B) $\dfrac{a+b}{ab}$ ; (C) $\dfrac{2b-a}{2b}$ ; (D) $\dfrac{b}{2a+2}$ ; (E) $\dfrac{2a+2}{b}$ .
The set of all real numbers satisfying $\log _{x}{\frac{1}{2}}>-1$ is:

(A) $(1,2)$ ; (B) $(0,2)$ ; (C) $(0,1)\cup(2,+\infty)$ ; (D) $(0,1)\cup(1,2)$ ; (E) $(2,+\infty)$ .
The modulus of the complex number $\frac{1+i}{1-i}+\frac{1-i}{1+i}$ is:

(A) $0$ ; (B) $\frac{1}{2}$ ; (C) $\frac{1}{\sqrt{2}}$ ; (D) $1$ ; (E) $\sqrt{2}$ .
The sum of absolute values of the real parts of the solutions of $z^{4}+1=0$ is:

(A) $2$ ; (B) $2\sqrt{2}$ ; (C) $\sqrt{2}$ ; (D) $4$ ; (E) $\sqrt{2}+1$ .
If $x=\cos 40^{\circ}+\cos 80^{\circ}+\cos 160^{\circ}$ , then:

(A) $x<-\frac{1}{2}$ ; (B) $-\frac{1}{2}\leqslant x<0$ ; (C) $x=0$ ; (D) $0<x\leqslant\frac{1}{2}$ ; (E) $x>\frac{1}{2}$ .
The minimum of function $f(x)=5\cos x+\cos 2x$ is:

(A) $-4$ ; (B) $-\frac{33}{8}$ ; (C) $-5$ ; (D) $-6$ ; (E) $-\frac{25}{8}$ .
What is the smallest positive solution of the equation $\sin^{2}x+\sin x\cos x=\frac{2-\sqrt{2}}{4}$ ?

(A) ${\pi/6}$ ; (B) ${\pi/8}$ ; (C) ${\pi/12}$ ; (D) ${\pi/18}$ ; (E) ${\pi/24}$ .
A rhombus has side length $10$ and area $80$ . What is the length of its longer diagonal?

(A) $12.5$ ; (B) $15$ ; (C) $4\sqrt{5}$ ; (D) $8\sqrt{5}$ ; (E) $4\sqrt{10}$ .
The base of an isosceles triangle is $2$ , and the radius of its inscribed circle is $\frac{1}{3}$ . What is the perimeter of the triangle?

(A) $2+\sqrt{7}$ ; (B) $2+\sqrt{8}$ ; (C) $5$ ; (D) $4.5$ ; (E) $4.8$ .
The base of a pyramid is a parallelogram with sides $10\,\mathrm{cm}$ and $18\,\mathrm{cm}$ and area $90\,\mathrm{cm^{2}}$ . The height of the pyramid is $6\,\mathrm{cm}$ and its foot is the intersection point of the diagonals of the base. The lateral area of the pyramid is:

(A) $192\,\mathrm{cm^{2}}$ ; (B) $2\left(9\sqrt{61}+5\sqrt{117}\right)\,\mathrm{cm^{2}}$ ; (C) $18\sqrt{61}\,\mathrm{cm^{2}}$ ; (D) $196\,\mathrm{cm^{2}}$ ; (E) $9224\,\mathrm{cm^{2}}$ .
Given the points $A(-2,-1)$ and $B(2,2)$ , the slope of the perpendicular bisector of segment $AB$ is:

(A) $-1$ ; (B) $\frac{3}{4}$ ; (C) $-\frac{3}{4}$ ; (D) $\frac{4}{3}$ ; (E) $-\frac{4}{3}$ .
The sum of the first 3 terms of an arithmetic progression is 5, and the sum of the first 5 terms is 7. What is the sum of first 7 terms?

(A) $\frac{42}{5}$ ; (B) $\frac{131}{15}$ ; (C) $\frac{119}{15}$ ; (D) $9$ ; (E) $\frac{25}{3}$ .
Consider all three-digit numbers composed of digits from the set $\{ 1,2,3,4,5,6\}$ . How many of them are greater than $333$ ?

(A) $80$ ; (B) $72$ ; (C) $333$ ; (D) $120$ ; (E) $60$ .