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:

$[(63÷9)\times \{\frac{14}{7}+\frac{19}{1}\times (7-6)\}]$

Calculation:

⇒ $[(63÷9)\times \{\frac{14}{7}+\frac{19}{1}\times (7-6)\}]$

⇒ 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 ̇....

$0.\overline{7}=0.77777\dots$

$0.0\overline{7}=0.077777\dots$

$0.\overline{07}=0.070707\dots$

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:

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

$\Rightarrow ?=(243{)}^{\frac{2}{5}}$

$\Rightarrow ?=(3\times 3\times 3\times 3\times 3{)}^{2⁄5}$

$\Rightarrow ?=(3^5{)}^{2⁄5}$

⇒ ? = 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