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61. $(\frac{1}{1.4} + \frac{1}{4.7} + \frac{1}{7.10} + \frac{1}{10.13} + \frac{1}{13.16})$ is equal to = ?

A. $\frac{1}{3}$

B. $\frac{5}{16}$

C. $\frac{3}{8}$

D. $\frac{41}{7280}$

Discuss

Solution:

(\frac{1}{1.4} + \frac{1}{4.7} + \frac{1}{7.10} + \frac{1}{10.13} + \frac{1}{13.16})

Formula :

\frac{1}{Difference denominator value} \times

[\frac{1}{First value} - \frac{1}{Last value}]

= \frac{1}{3} \times

[1 - \frac{1}{4} + \frac{1}{4} - \frac{1}{7} + \frac{1}{7} - \frac{1}{10} + \frac{1}{10} - \frac{1}{13} + \frac{1}{13} - \frac{1}{16}]

\begin{aligned} = \frac{1}{3} \times [1 - \frac{1}{16}] \\ = \frac{1}{3} \times \frac{15}{16} \\ = \frac{5}{16} \end{aligned}

62. Given that $\sqrt{5}$ = 2.236 and $\sqrt{3}$ = 1.732, then the value of $\frac{1}{\sqrt{5} + \sqrt{3}} is = ?$

A. 0.564

B. 0.504

C. 0.252

D. 0.202

Discuss

Solution:

\begin{aligned} \frac{1}{\sqrt{5} + \sqrt{3}} \times \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} - \sqrt{3}} \\ = \frac{\sqrt{5} - \sqrt{3}}{5 - 3} \\ = \frac{2.236 - 1.732}{2} \\ = \frac{0.504}{2} \\ = 0.252 \end{aligned}

63. If ${(\frac{3}{5})}^{3} {(\frac{3}{5})}^{- 6} = {(\frac{3}{5})}^{2 x - 1}$ then x is equal to ?

A. -2

B. -1

C. 1

D. 2

Discuss

Solution:

\begin{aligned} {(\frac{3}{5})}^{3} {(\frac{3}{5})}^{- 6} = {(\frac{3}{5})}^{2 x - 1} \\ \Rightarrow {(\frac{3}{5})}^{(3 - 6)} = {(\frac{3}{5})}^{2 x - 1} \\ \Rightarrow {(\frac{3}{5})}^{- 3} = {(\frac{3}{5})}^{2 x - 1} \\ \Rightarrow 2 x - 1 = - 3 \\ \Rightarrow 2 x = - 2 \\ \Rightarrow x = - 1 \end{aligned}

64. If $5 \sqrt{5} \times 5^{3} \div 5^{- \frac{3}{2}} = 5^{a + 2}$ then the value of a is = ?

A. 4

B. 5

C. 6

D. 8

Discuss

Solution:

\begin{aligned} 5 \sqrt{5} \times 5^{3} \div 5^{- \frac{3}{2}} = 5^{a + 2} \\ \Rightarrow \frac{5 \times 5^{\frac{1}{2}} \times 5^{3}}{5^{- \frac{3}{2}}} = 5^{a + 2} \\ \Rightarrow 5^{(1 + \frac{1}{2} + 3 + \frac{3}{2})} = 5^{a + 2} \\ \Rightarrow 5^{6} = 5^{a + 2} \\ \Rightarrow a + 2 = 6 \\ \Rightarrow a = 4 \end{aligned}

65. if $2^{x} = \sqrt[3]{32}$ then x is equal to = ?

A. 5

B. 3

C. $\frac{3}{5}$

D. $\frac{5}{3}$

Discuss

Solution:

\begin{aligned} 2^{x} = \sqrt[3]{32} \\ \Rightarrow 2^{x} = {(32)}^{\frac{1}{3}} = {(2^{5})}^{\frac{1}{3}} \\ \Rightarrow x = \frac{5}{3} \end{aligned}

66. $\sqrt{12 + \sqrt{12 + \sqrt{12 + . . . . .}}}$ is equal to = ?

A. 3

B. 4

C. 6

D. 2

Discuss

Solution:

\sqrt{12 + \sqrt{12 + \sqrt{12 + . . . . .}}}

(3, 4) are the factor of 4
If there is '+' in

\sqrt{}

Answer is Highest value
If there is '-' in

\sqrt{}

Answer is lowest value.
So the answer is 4

67. If $5^{(x + 3)} = 2 5^{(3 x - 4)}$ then the value of x is = ?

A. $\frac{5}{11}$

B. $\frac{11}{5}$

C. $\frac{11}{3}$

D. $\frac{13}{5}$

Discuss

Solution:

\begin{aligned} 5^{(x + 3)} = 2 5^{(3 x - 4)} \\ \Rightarrow 5^{(x + 3)} = {(5^{2})}^{(3 x - 4)} \\ \Rightarrow 5^{(x + 3)} = {5^{2}}^{(3 x - 4)} \\ \Rightarrow 5^{(x + 3)} = 5^{(6 x - 8)} \\ \Rightarrow x + 3 = 6 x - 8 \\ \Rightarrow 5 x = 11 \\ \Rightarrow x = \frac{11}{5} \end{aligned}

68. $\frac{2^{n + 4} - 2 (2^{n})}{2 (2^{n + 3})}$ when simplified is = ?

A. $2^{n + 1} - \frac{1}{8}$

B. -2⁽ⁿ⁺¹⁾

C. 1 - 2ⁿ

D. $\frac{7}{8}$

Discuss

Solution:

\begin{aligned} \frac{2^{n + 4} - 2 (2^{n})}{2 (2^{n + 3})} \\ = \frac{2^{n + 4} - 2^{n + 1}}{2^{n + 4}} \\ = \frac{2^{n + 4}}{2^{n + 4}} - \frac{2^{n + 1}}{2^{n + 4}} \\ = 1 - 2^{(n + 1) - (n + 4)} \\ = 1 - 2^{- 3} \\ = 1 - \frac{1}{8} \\ = \frac{7}{8} \end{aligned}

69. If $a = \frac{\sqrt{5} + 1}{\sqrt{5} - 1}$ and $b = \frac{\sqrt{5} - 1}{\sqrt{5} + 1}$ then the value of $(\frac{a^{2} + a b + b^{2}}{a^{2} - a b + b^{2}})$ is = ?

A. $\frac{3}{4}$

B. $\frac{4}{3}$

C. $\frac{3}{5}$

D. $\frac{5}{3}$

Discuss

Solution:

\begin{aligned} a + b = \frac{\sqrt{5} + 1}{\sqrt{5} - 1} + \frac{\sqrt{5} - 1}{\sqrt{5} + 1} \\ = \frac{{(\sqrt{5} + 1)}^{2} + {(\sqrt{5} - 1)}^{2}}{(\sqrt{5} - 1) (\sqrt{5} + 1)} \\ = \frac{2 [{(\sqrt{5})}^{2} + 1]}{5 - 1} \\ = \frac{2 (5 + 1)}{4} \\ = 3 \\ a . b = \frac{\sqrt{5} + 1}{\sqrt{5} - 1} \times \frac{\sqrt{5} - 1}{\sqrt{5} + 1} = 1 \\ Put value in expression \\ \frac{a^{2} + a b + b^{2}}{a^{2} - a b + b^{2}} \\ = \frac{{(a + b)}^{2} - a b}{{(a + b)}^{2} - 3 a b} \\ = \frac{3^{2} - 1}{3^{2} - 3} \\ = \frac{9 - 1}{9 - 3} \\ = \frac{4}{3} \end{aligned}

70. $6^{1.2} \times 36^{?} \times 30^{2.4} \times 25^{1.3} = 30^{5}$

A. 0.1

B. 0.7

C. 1.4

D. 2.6

Discuss

Solution:

\begin{aligned} Let 6^{1.2} \times 36^{x} \times 30^{2.4} \times 25^{1.3} = 30^{5} \\ Then, 6^{1.2} \times (6^{2})^{x} \times (6 \times 5)^{2.4} \times (5^{2})^{1.3} = 30^{5} \\ \Leftrightarrow 6^{1.2} \times 6^{2 x} \times 6^{2.4} \times 5^{2.4} \times 5^{2.6} = (6 \times 5)^{5} \\ \Leftrightarrow 6^{(1.2 + 2 x + 2.4)} \times 5^{(2.4 + 2.6)} = 6^{5} \times 5^{5} \\ \Leftrightarrow 6^{(3.6 + 2 x)} \times 5^{5} = 6^{5} \times 5^{5} \\ \Leftrightarrow 3.6 + 2 x = 5 \\ \Leftrightarrow 2 x = 1.4 \\ \Leftrightarrow x = 0.7 \end{aligned}

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