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

Given:

The value of \(\frac{{{{\left( {0.96} \right)}^3} - {{\left( {0.1} \right)}^3}}}{{{{\left( {0.96} \right)}^2} + 0.096 + {{\left( {0.1} \right)}^2}}}{\rm{ }}\)

Formula Used:

\(\frac{a^3 - b^3}{a^2 + ab + b^2} = a - b\)

Calculation:

Here, a = 0.96 and b = 0.1 .

Using the formula:

\(\frac{{(0.96)^3 - (0.1)^3}}{{(0.96)^2 + (0.96 \times 0.1) + (0.1)^2}}\)

⇒ \(\frac{{(0.96)^3 - (0.1)^3}}{{(0.96)^2 + 0.096 + (0.1)^2}}\)

⇒ 0.96 - 0.1

⇒ 0.86

The correct answer is option 3.

Given:

42^0.66 = x, 42^0.65 = y, and x^z = y⁴

Calculation:

Express x and y in terms of 42:

x = 42^0.66

y = 42^0.65

Substitute x and y into the given equation x^z = y⁴:

(42^0.66)^z = (42^0.65)⁴

Simplify the exponents:

42^0.66z = 42^2.6

Equate the powers of 42:

0.66z = 2.6

Solve for z:

z = 2.6 / 0.66

z ≈ 3.94

The value of z is approximately 3.94.

Given:

\(\rm 5\frac{1}{2}+3+2\frac{2}{3}+7\frac{1}{2}+6\frac{1}{3}\)

Formula used:

Basic arithmetic operations

Calculation:

\(\rm 5\frac{1}{2}+3+2\frac{2}{3}+7\frac{1}{2}+6\frac{1}{3}\)

⇒11/2 + 3 + 8/3 + 15/2 + 19/3

⇒ (11 × 3 + 3 × 6 + 8 × 2 + 15 × 3 + 19 × 2) / 6

⇒ (33 + 18 + 16 + 45 + 38) / 6

⇒ 150 / 6 = 25

∴ The correct answer is option (3).

Given:

Expression: [(91 ÷ 7) × {(64 ÷ 4) + (17 ÷ 6) × (8 - 2)}]

Formula Used:

Order of operations (BODMAS): Solve inside brackets first, then division/multiplication, and finally addition/subtraction.

Calculation:

⇒ [(91 ÷ 7) × {(64 ÷ 4) + (17 ÷ 6) × (8 - 2)}]

⇒ (13) × {(16) + (17 ÷ 6) × (6)}

⇒ (13) × {16 + (17 × 6 ÷ 6)}

⇒ (13) × {16 + 17}

⇒ 13 × 33

⇒ 429

∴ The value of [(91 ÷ 7) × {(64 ÷ 4) + (17 ÷ 6) × (8 - 2)}] is 429.

Given :

\(\dfrac{27×3^n-3×3^{n+1}}{81×3^{n+1}-9×3^{n+2}}\)

Formula used:

a^m × aⁿ = a^{m + n}

am ÷ an = am - n

Calculation :

\(\dfrac{27×3^n-3×3^{n+1}}{81×3^{n+1}-9×3^{n+2}}\)

⇒ \(\rm \dfrac{3^3×3^n-3^1×3^{n+1}}{3^4× 3^{n+1}-3^2×3^{n+2}}\) 

⇒ \(\rm \dfrac{3^{3+n}-3^{n+2}}{3^{n+5}-3^{n+4}}\)

⇒ \(\rm \dfrac{3^{n+2}\times(3-1)}{3^{n+4}\times(3-1)}\) 

⇒ \(\rm \dfrac{3^{n+2}}{3^2× 3^{n+2}}\) = \(\dfrac{1}{9}\)

∴ The answer is \(\dfrac{1}{9}\) .

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