Simplification MCQ Quiz - Objective Question with Answer for Simplification - Download Free PDF

Given:

(√13 − 1)/(√13 + 1) = a + b√13

Formula used:

Rationalize denominator: (a − b)/(a + b) = [(a − b)(a − b)] / [(a + b)(a − b)]

Use: (a + b)(a − b) = a² − b²

Calculation:

(√13 − 1)/(√13 + 1)

⇒ Multiply numerator and denominator by (√13 − 1)

⇒ [(√13 − 1) × (√13 − 1)] / [(√13 + 1) × (√13 − 1)]

⇒ (13 − 2√13 + 1) / (13 − 1)

⇒ (14 − 2√13) / 12

⇒ 14/12 − (2√13)/12

⇒ 7/6 − (√13)/6

∴ a = 7/6 and b = −1/6

Given:

\(\frac{2}{5}, \frac{5}{4}, \frac{6}{8}, \frac{6}{5}, \frac{9}{6}\)

Formula used:

Reciprocal of the sum of reciprocals of the given numbers:

Calculation:

Reciprocals: 5/2, 4/5, 8/6, 5/6, 6/9

Sum of reciprocals = 5/2 + 4/5 + 8/6 + 5/6 + 6/9

⇒ Sum = 5/2 + 4/5 + 4/3 + 5/6 + 2/3

⇒ Sum = 75/30 + 24/30 + 40/30 + 25/30 + 20/30

⇒ Sum = (75 + 24 + 40 + 25 + 20)/30

⇒ Sum = 184/30 = 92/15

So, reciprocal of sum = 1/(92/15) = 15/92

∴ The correct answer is option (2).

Given:

\(\left[ (63 \div 9) × \left\{ \frac{14}{7} + \frac{19}{1} × (7 - 6) \right\} \right]\)

Calculation:

⇒ \(\left[ (63 \div 9) × \left\{ \frac{14}{7} + \frac{19}{1} × (7 - 6) \right\} \right]\)

⇒ 7 × {2 + 19 × 1}

⇒ 7 × {2 + 19}

⇒ 7 × 21

⇒ 147

∴ The correct answer is option (2).

Given:

Fractions: \(\frac{3}{2}, \frac{1}{5}, \frac{4}{3}, \frac{5}{4}\)

Formula used:

The difference between the largest and smallest fraction is determined by finding their values and subtracting the smallest from the largest.

Calculation:

Convert fractions to decimals for comparison:

\(\frac{3}{2} = 1.5\)

\(\frac{1}{5} = 0.2\)

\(\frac{4}{3} \approx 1.333\)

\(\frac{5}{4} = 1.25\)

Smallest fraction = \(\frac{1}{5} = 0.2\)

Largest fraction = \(\frac{3}{2} = 1.5\)

⇒ Difference = Largest - Smallest

⇒ Difference = \((1.5 - 0.2)\)

⇒ Difference = \(\frac{13}{10}\)

∴ The correct answer is option (3).

Given:

Calculations:

⇒

⇒

⇒

⇒

⇒

⇒ 4

∴ The correct answer is option (4).

Concebt used

a.b̅ = a.bbbbbb

a.0b̅ = a.0bbbb

Calculation

0.7 = 0.700000 ̇....

\(0.\bar7 = 0.77777 \ldots\)

\(0.0\bar7 = 0.077777 \ldots\)

\(0.\overline {07} = 0.070707 \ldots\)

Solution:

\(12\frac{1}{2} + 12\frac{1}{3} + 12\frac{1}{6}\)

= 25/2 + 37/3 + 73/6

= (75 + 74 + 73)/6

= 222/6

= 37

Shortcut Trick

\(12\frac{1}{2} + 12\frac{1}{3} + 12\frac{1}{6}\)

= 12 + 12 + 12 + (1/2 + 1/3 + 1/6)

= 36 + 1 = 37

Solution:

We have to follow the BODMAS rule

Calculation:

\(? = \sqrt[5]{{{{\left( {243} \right)}^2}}}\)

\(⇒ ? = (243)^{\frac{2}{5}}\)

\(⇒ ? = (3 × 3 × 3 × 3 × 3)^{2⁄5}\)

\(⇒ ? = (3^5)^{2⁄5}\)

⇒ ? = 3²

Formula used:

(a + b)² = a² + b² + 2ab

Calculation:

Given expression is:

\(\sqrt {8\; + \;2\sqrt {15} \;} \)

⇒  \(\sqrt {5\; + \;3\; + \;2\times \sqrt 5 \times \sqrt 3 \;} \)

⇒  \(\sqrt {{{(\sqrt 5 )}^2}\; + \;{{\left( {\sqrt 3 } \right)}^2}\; + \;2 \times \sqrt 5 \times \sqrt 3 \;} \)

⇒  \(\sqrt {{{\left( {\;\sqrt 5 \; + \;\sqrt 3 \;} \right)}^2}\;} \)

⇒  \(\sqrt 5 + \sqrt 3 \)

\(\sqrt {{{\left( {0.65} \right)}^2} - {{\left( {0.16} \right)}^2}} \)

Since,

a2 - b2 = (a - b) ( a + b)

\(\begin{array}{l} \Rightarrow \sqrt {\left( {0.65 + 0.16} \right)\left( {0.65 - 0.16} \right)} \\ \Rightarrow \sqrt {\left( {0.81} \right)\left( {0.49} \right)} \\ \Rightarrow \sqrt {\left( {0.9} \right)\left( {0.9} \right) \times \left( {0.7} \right)\left( {0.7} \right)} \end{array}\)

⇒ 0.9 × 0.7 = 0.63

Concept:

We can find √x using the factorisation method.

Calculation:

√[(10 + √25) (12 - √49)]

⇒ √[(10 + 5)(12 – 7)]

⇒ √(15 × 5)

⇒ √(3 × 5 × 5)

⇒ 5√3

Given,

2³ × 3⁴ × 1080 ÷ 15 = 6^x

⇒ 2³ × 3⁴ × 72 = 6^x

⇒ 2³ × 3⁴ × (2 × 6²) = 6^x

⇒ 2⁴ × 3⁴ × 6² = 6^x

⇒ (2 × 3)⁴ × 6² = 6^x           [∵ x^m × y^m = (xy)^m]

⇒ 6⁴ × 6² = 6^x

⇒ 6^{(4 + 2)} = 6^x

⇒ x = 6

Given:

√3n = 729

Formulas used:

(x^a)^b = x^ab

If x^a = x^b then a = b 

Calculation:

√3n = 729

⇒ √3n = (3²)³

⇒ (3n)^1/2 = (32)3

⇒ (3n)1/2 = 3⁶

⇒ n/2 = 6 

∴  n = 12 

Given expression,

(81.84 + 118.16) ÷ 5³ = 1.2 × 2 + ?

⇒ 200 ÷ 5³ = 1.2 × 2 + ?

⇒ 200 ÷ 125 = 1.2 × 2 +?

⇒ 1.6 = 2.4 + ?

⇒ ? = -0.8