MF - Practice test 4

PRACTICE TEST 4

Given that $x+y=a$ and $x^{3}+y^{3}=b$ , what is $x^{2}+y^{2}$ ?

(А) $\dfrac{a+b}{2}$ ; (B) $\sqrt{ab}$ ; (C) $\dfrac{a+2b}{3}$ ; (D) $\dfrac{a^{2}}{3}+\dfrac{2b}{3a}$ ; (Е) $\dfrac{a^{2}+2b}{3}$ .
The graph of the function $y=(a+1)x^{2}+(a+2)x-(a+3)$ contains point $M(-1,1)$ if

(А) $a=1$ ; (B) $a=5$ ; (C) $a=-5$ ; (D) $a=3$ ; (Е) $a=-1$ .
The sum of squares of the equation $x^{2}-3x+2m=1$ equals $7$ . What is $m$ ?

(А) $1$ ; (B) $4$ ; (C) $\frac{5}{2}$ ; (D) $-2$ ; (Е) $\frac{5}{3}$ .
The set of solutions of the inequality $x<x\sqrt{x^{2}+7x+13}$ is:

(А) $(-\infty,-4)\cup(-3,\infty)$ ; (B) $(-4,-3)$ ; (C) $(-\infty,-4)\cup(-4,-3)$ ;
(D) $(-\infty,-4)$ ; (Е) $(-4,-3)\cup(0,\infty)$ .
If $f(x-1)=x^{2}-3x-3$ , then $f(x+1)$ equals:

(А) $x^{2}-3x-1$ ; (B) $x^{2}+x-5$ ; (C) $x^{2}+x-3$ ; (D) $x^{2}+x-6$ ; (Е) $x^{2}+3x-3$ .
The domain of the function $f(x)=\arcsin\frac{x+1}{2x-1}$ is:

(А) $(-\infty,0]\cup[2,\infty)$ ; (B) $[2,\infty)$ ; (C) $(\frac{1}{2},\infty)$ ; (D) $(-\infty,0]\cup(\frac{1}{2},\infty)$ ; (Е) $(-\infty,\frac{1}{2})\cup[2,\infty)$ .
The product of all values of the real number $a$ for which the polynomials $x^{2}+ax+6$ and $x^{2}-ax-14$ have a real root in common is:

(А) $40$ ; (B) $10$ ; (C) $5$ ; (D) $-10$ ; (Е) $-25$ .
The number of the solutions of the equation $x^{{5-x}}=x^{{((5-x)^{{x-1}})}}$ in the interval $(0,5)$ is:

(А) $0$ ; (B) $1$ ; (C) $2$ ; (D) $3$ ; (Е) greater than $3$ .
If $x$ are $y$ the solutions of the system of equations $\log _{x}(x+y)=3$ , $\log _{y}(3x)=1$ , then $xy$ is equal to:

(А) $16$ ; (B) $12$ ; (C) $8$ ; (D) $6$ ; (Е) $4$ .
What is the value of $\left(1+i\sqrt{5}\right)^{5}+\left(1-i\sqrt{5}\right)^{5}$ ?

(А) $2$ ; (B) $72$ ; (C) $152$ ; (D) $252$ ; (Е) $352$ .
If $0<x<\pi/2$ and $\:\mathrm{tg}\: 2x=2$ , then $\:\mathrm{tg}\: x$ equals:

(А) $1$ ; (B) $\dfrac{1}{2}$ ; (C) $\dfrac{\sqrt{3}}{3}$ ; (D) $\dfrac{\sqrt{5}+1}{2}$ ; (Е) $\dfrac{\sqrt{5}-1}{2}$ .
Which of the following expressions is identically equal to $\dfrac{4(\sin^{5}x-\cos^{5}x)}{\sin x-\cos x}$ ?

(А) $1+\sin{2x}-\sin^{2}{2x}$ ; (B) $4+2\sin{2x}-\sin^{2}{2x}$ ;
(C) $4+2\sin{2x}+\sin^{2}{2x}$ ; (D) $1+\sin{2x}+\sin^{2}{2x}$ ;
(Е) $4-2\sin{2x}+\sin^{2}{2x}$ .
The sum of the three smallest positive solutions of the equation $\sin 8x=\sin 2x$ is:

(А) $\frac{11}{15}\pi$ ; (B) $\frac{2}{5}\pi$ ; (C) $2\pi$ ; (D) $\pi$ ; (Е) $\frac{9}{10}\pi$ .
The equilateral triangle $ABC$ has the side length $a$ . Line $p$ is parallel to side $AB$ and distinct from it, and is tangent to the inscribed circle of triangle $ABC$ . The length of the part of the line $p$ inside the triangle is:

(А) $\dfrac{a}{2\sqrt{2}}$ ; (B) $\dfrac{a}{2}$ ; (C) $\dfrac{a}{3}$ ; (D) $\dfrac{a}{4}$ ; (Е) $\dfrac{a}{2\sqrt{3}}$ .
A regular octagon of area 4 is inscribed in a circle. The radius of the circle is:

(А) $1$ ; (B) $\sqrt{1+\sqrt{2}}$ ; (C) $\dfrac{\sqrt{2}+1}{2}$ ; (D) $\sqrt[4]{2}$ ; (Е) $\sqrt{2}$ .
If the height of a regular tetrahedron is $\sqrt{3}$ , then its surface area is:

(А) $3\sqrt{3}$ ; (B) $\dfrac{9{\sqrt{3}}}{4}$ ; (C) $\dfrac{9\sqrt{3}}{2}$ ; (D) $\dfrac{9\sqrt{2}}{\sqrt{3}}$ ; (Е) $\dfrac{9\sqrt{6}}{4}$ .
A right-angled trapezoid with the bases $a=20cm$ and $b=8cm$ , and the shorter leg $c=5cm$ can rotate about either base, thus producing two different solids. The ratio of the surface areas of these two solids is:

(А) $1:1$ ; (B) $1:2$ ; (C) $2:3$ ; (D) $3:4$ ; (Е) $1:3$ .
The distance between the two tangents to the ellipse $x^{2}+2y^{2}=3$ parallel to the line $x=2y$ is:

(А) $3$ ; (B) $6$ ; (C) $2\sqrt{5}$ ; (D) $3/\sqrt{5}$ ; (Е) $6/\sqrt{5}$ .
The sum of the second, third and fourth term of a geometric progression is 3, and the sum of the squares of these three terms is 5. Then the sum of the first and the fifth term of the progression is:

(А) $\frac{41}{4}$ ; (B) $\frac{41}{6}$ ; (C) $6$ ; (D) $\frac{27}{5}$ ; (Е) $\frac{27}{4}$ .
In how many ways can one choose three cards of pairwise distinct values and distinct colors from the standard 52-card deck?

(А) $6864$ ; (B) $1144$ ; (C) $41184$ ; (D) $1716$ ; (Е) $10296$ .