2010000813
Level:
C
Assuming \(x\neq -3\), \(x\neq 4\), simplify the expression:
\[\frac{x^{3} + 27} {x^2-x-12}\]
\(\frac{x^2-3x+9}
{x-4}\)
\(\frac{x^2+3x+9}
{x-4}\)
\(\frac{x^2+6x+9}
{x-4}\)
\(\frac{x^2-6x+9}
{x-4}\)
Polynomials and fractions
2010000812
Level:
A
Assuming \( y \neq 1\), \(x\neq \pm y\), simplify the expression:
\[\frac{y^2-2xy+x^2}{(1-y)(y-x)}\cdot\frac{3y^2-6y+3}{x^2-y^2}\]
\(\frac{3(y-1)}{x+y}\)
\(\frac{3(1-y)}{x+y}\)
\(\frac{3}{x+y}\)
\( 3\)
2010000811
Level:
A
Evaluate the following expression at \(x = 9\).
\[\frac{\frac{1}{x}+\frac{1}{x^2}}{\frac{1}{\sqrt{x}}}\]
\( \frac{10}{27}\)
\( -\frac{10}{9}\)
\(-30\)
\(30\)
2010000810
Level:
A
Evaluate the following expression at \(x = 4\).
\[\frac{\frac{1}{\sqrt{x}}}{\frac{1}{x^2}-\frac{1}{x}}\]
\(- \frac{8}{3}\)
\(\frac{31}{3}\)
\( \frac{8}{3}\)
\( 6\)
2010000809
Level:
A
Assuming \( x \notin \{-4;0;3;4\}\), simplify the following expression.
\[\frac{x^2-3x}{x^2-16}:\frac{x-3}{x^2+4x}\]
\( \frac{x^2}{x-4} \)
\( \frac{x-4}{x^2} \)
\( \frac{x-4}{x} \)
\( \frac{x}{x-4} \)
2010000808
Level:
A
Assuming \( x \notin \{0;1;3\}\), simplify the following expression.
\[\frac{x^2-9}{x^2-x}:\frac{x^2-3x}{x-1}\]
\( \frac{x+3}{x^2} \)
\( \frac{x-3}{x^2}\)
\( \frac{x+3}{2x}\)
\( \frac{x+3}{x} \)
2010000807
Level:
A
Assuming \(x\neq 0\),
\(y\neq 0\),
\(x\neq -y\),
simplify the following expression.
\[ {
\frac{1}
{y^{2}} - \frac{1}
{x^{2}}
\over -\frac{1}
{x} - \frac{1}
{y}}
\]
\(\frac{y-x}
{xy} \)
\(\frac{x-y}
{xy} \)
\(x-y\)
\(y-x\)
2010000806
Level:
C
Express \( d_1 \) from the formula \( v=\frac{v_1v_2(d_1+d_2 )}{d_1v_2+d_2v_1} \).
\( d_1=-\frac{d_2v_1(v-v_2)}{v_2(v-v_1)} \)
\( d_1=\frac{d_2v_1(v-v_2)}{v_2(v-v_1)} \)
\( d_1=-\frac{d_2v_1(v_2-v)}{v_2(v-v_1)} \)
\( d_1=\frac{d_2v_1(v_2-v)}{v_2(v_1-v)} \)
2010000805
Level:
C
In a parallel circuit the total resistance \( R \) of three components with the resistances \( R_1 \), \( R_2 \), \( R_3 \) is given by the formula \( \frac1R=\frac1{R_1}+\frac1{R_2}+\frac1{R_3} \). Express \( R_2 \) from this formula.
\( R_2=\frac{RR_1R_3}{R_1R_3-R(R_1+R_3)} \)
\( R_2=\frac{R-R_1-R_3}{RR_1R_3} \)
\( R_2=R-\frac{R_1R_3}{R_1+R_3} \)
\( R_2=\frac{R_1R_3-R(R_1+R_3)}{RR_1R_3} \)
2010000804
Level:
C
Factoring the polynomial \(-3x^5-24x^4-48x^3\) you get:
\( -3x^3(x+4)(x+4)\)
\( -3x^3(x-4)(x-4)\)
\( 3x^3(4-x)(4-x)\)
\( 3x^3(x+4)(x-4)\)