Robert Bosch Algebra Quiz 2

Explanation:

\frac{a}{3}= \frac{b}{2} =>\frac{a}{b}= \frac{3}{2}

2a + \frac{3b}{3a} - 2b = 2 \times \frac{a}{b} + \frac{3}{3}x \frac{a}{b} - 2

= 2 \times \frac{3}{2} + \frac{3}{3} \times \frac{3}{2} - 2

 =\frac{6}{\frac{9-4}{2}}

= \frac{12}{5}

 

Explanation:

 

Arithmetic mean (AM) = \frac{a+b}{2}

Geometric mean (GM) = ab

As AM > GM

\frac{a+b}{2}  > \sqrt{ab}

Explanation:

x^{\frac{1}{3}} = y^{\frac{1}{4}}

=>   x^{\frac{1}{3}} ^{12} =x^{\frac{1}{3}} ^{12}

x^{4} = y^{3}

x^{ 4}^{5} = y^{3}^{5} 

=>  x^{ 20} = y^ {15}

Explanation:

We know that   a^{°} = 1 

  a^{ 2x+2 }= 1 = a^{°} 

Þ 2x +2 = 0 

=> - x = - 2/ 2 = 1

Explanation:

For expression ax^{2} + bx + c, a > 0,   the minimum value is given by 

\frac{4ac - b^{2}}{4a} 

Here, for x^{ 2} – x + 1   

a = 1, b = –1, c = 1

Minimum value =  \frac{4*1*1 - 1}{4*1} = \frac{3}{4} 

Explanation:

(125)^{x} = 3125 

Þ (5^{3} )^{x} = 5^5  

Þ 3x = 5

Þ x =5/3

 

Explanation:

  2^{2x–y}  = 16 = 2^{4}  

Þ 2x – y = 4 ............... (i)

2^{x+y} = 32 = 2^{5} 

Þ x + y = 5 ............... (ii)

On adding equations (i) and (ii),

3x = 9 

Þ x = 3

From equation (ii),

y = 5– x = 5 – 3 = 2

\therefore xy = 3 * 2 = 6  

Explanation:

 

  a^{ 2} – b^{ 2} = 19  

Þ 10^{2} – 9^{2} = 19  

Þ a = 10

 

Explanation:

4^{4x + 1} = \frac{1}{64}= (\frac{1}{4}) ^{3}   

Þ  4^{4x + 1}= 4^{–3}  

 Þ 4x + 1 = – 3

 Þ 4x = – 4

 Þ x = – 1

Explanation:

x + \frac{1}{x}= \frac{1}{5}   

Þ  x^ {2} – 5x + 1 = 0   

Þ  3x ^{2} – 15x + 3 = 0  

  \frac{2x}{3x ^{2} } – 15x + 3 = \frac{2x}{15x} -5x  

  \frac{2x}{10x} = \frac{1}{5}