Quantitative Aptitude

Number System and Simplification

548 Questions

Number system and simplification questions assess mathematical proficiency in handling fractions, decimals, and complex algebraic expressions. Test takers must simplify surds, calculate reciprocals, and solve intricate number pattern equations. This foundational quantitative aptitude topic is critical for achieving high scores in SSC and banking exams.

Fraction simplificationDecimal operationsSurds and indicesPercentage calculationsComplex number algebra

Number System and Simplification Questions

Multiple choice maths 5-digit numbers expanded form introduction to numbers and number systems numbers in general form

In the formula $T = 2\pi \sqrt{\dfrac{L}{g}}, \pi$ and $g$ are constants. If we solve the formula for $L$

  1. $\dfrac{Tg}{2\pi}$
  2. $\dfrac{Tg^2}{2\pi}$
  3. $\dfrac{T^2}{4\pi^2g}$
  4. $\dfrac{T^2}{4\pi g^2}$
  5. $\dfrac{gT^2}{4\pi^2}$
Reveal answer Fill a bubble to check yourself
E Correct answer
Explanation

Given $T = 2\pi \sqrt{\dfrac{L}{g}}$
Now square it on both sides

$\Rightarrow {T}^{2} =4{\pi}^{2}\dfrac{L}{g}$ 
$\Rightarrow L=\dfrac{g{T}^{2}}{4{\pi}^{2}}$

Multiple choice finding nth roots of a complex number n th root of unity demoivre's theorem complex numbers maths

The value of the expression 

$1 \cdot (2 - \omega) (2 - \omega^2) + 2\cdot (3 - \omega) (3 - \omega^2) +$ _____$+ (n - 1)(n - \omega)(n - \omega^2)$

  1. $\dfrac{1}{4}n(n - 1)(n^2 + 3n + 4)$
  2. $n(n - 1)(n^2 + 3n + 4)$
  3. $\dfrac{1}{4}n(n - 1)(n^2 + 3n - 4)$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Here, $a _n= (n-1)(n-\omega )(n-\omega ^2)$

$\sum a _n= \sum (n-1)(n-\omega )(n-\omega ^2)$

$ = \sum (n^2-\omega n-n+\omega )(n-\omega ^2)$

$=\sum (n^3-\omega n^2 n^2+\omega ^3 n +\omega ^2n-\omega ^3)$

$= \sum (n^3-(\omega +\omega ^3)n^2-n^2(\omega +\omega ^2)n+n-1) \,\,\,\,\, [\because \omega ^3 =1]$

$= \sum (n^3-(-1)n^2-n^2+(-1)n+n-1) \,\,\,\,\, [\because 1+\omega +\omega ^2=0]$

$= \sum (n^3+n^2-n^2-n+n-1)$

$=\sum (n^3-1)$

$=\sum n^3-\sum 1$

$= \left [ \dfrac {n(n+1)}{2} \right ]^2-n$

$=\dfrac {n^2(n+1)^2}{4}-n$

$=\dfrac {n^4+n^2+2n^3-4n}{4}$

$=\dfrac {n}{4}(n^3+2n^2+n-4)$

$=\dfrac {n(n-1)(n^2+3n+4)}{4}$

$\therefore 1 \cdot (2 - \omega) (2 - \omega^2) + 2\cdot (3 - \omega) (3 - \omega^2) +$ _____$+ (n - 1)(n - \omega)(n - \omega^2) = \dfrac {1}{4}n (n-1)(n^2+3n+4)$
Multiple choice maths decimal numbers adding and subtracting decimals addition and subtraction of decimals operations on decimals

The simplified value of $\displaystyle\frac{10.24 \div 1.6}{20 - 19.8}$ is

  1. $1.6$
  2. $3.2$
  3. $16$
  4. $32$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\displaystyle\frac { 10.24 }{ 1.6 } = \displaystyle\frac { 102.4 }{ 16 } = 6.4$
$\therefore \displaystyle\frac { 10.24\div 1.6 }{ 20-19.8 } = \displaystyle\frac { 6.4 }{ 0.2 } = \displaystyle\frac { 64 }{ 2 } = 32$

Multiple choice maths decimal numbers adding and subtracting decimals addition and subtraction of decimals operations on decimals

Simplify :$\displaystyle \left ( 0.\overline{1} \right )^{2}\left { 1-9\left ( 0.\overline{16} \right )^{2} \right }$

  1. $\displaystyle \frac{1}{162}$
  2. $\displaystyle \frac{1}{108}$
  3. $\displaystyle \frac{7696}{10{6}}$
  4. $\displaystyle \frac{1}{106}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\displaystyle \left ( 0.\overline{1} \right )^{2}\left { 1-9\left ( 0.\overline{16} \right )^{2} \right }$


= $\displaystyle \left ( \frac{1}{9} \right )^{2}\left { 1-9\left ( \frac{16-1}{90} \right )^{2} \right }=\frac{1}{81}\left { 1-9\times\left ( \frac{15}{90} \right )^{2}  \right }$

= $\displaystyle \frac{1}{81}\left { 1-9\times \frac{1}{36} \right }=\frac{1}{81}\times \frac{3}{4}=\frac{1}{108}$

Multiple choice maths decimal numbers adding and subtracting decimals addition and subtraction of decimals operations on decimals

$\displaystyle (0.34\overline{67}+0.13\overline{33})$ is equal to 

  1. $\displaystyle 0.\overline{48}$

  2. 0.4803
  3. $\displaystyle 0.48\overline{01}$
  4. $\displaystyle 0.4\overline{8}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\displaystyle 0.34\overline{67}+0.13\overline{33}=\frac{3467-34}{9900}+\frac{1333-13}{9900}$


= $\displaystyle \frac{3433}{9900}+\frac{1320}{9900}=\frac{4753}{9900}$

= $\displaystyle \frac{4801-48}{9900}=0.4801 $

Multiple choice maths decimal numbers adding and subtracting decimals addition and subtraction of decimals operations on decimals

The value of $\displaystyle 0.\overline{2}+0.\overline{3}+0.\overline{4}+0.\overline{9}+0.\overline{39}$ is 

  1. $\displaystyle 0.\overline{57}$
  2. $\displaystyle 1\frac{20}{33}$
  3. $\displaystyle 2\frac{1}{3}$
  4. $\displaystyle 2\frac{13}{33}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$\displaystyle 0.\overline{2}+0.\overline{3}+0.\overline{4}+0.\overline{9}+0.\overline{39}$


= $\displaystyle \frac{2}{9}+\frac{3}{9}+\frac{4}{9}+\frac{9}{9}+\frac{39}{99}$

= $\displaystyle \frac{22+33+44+99+39}{99}$

= $\displaystyle \frac{237}{99}=2\frac{13}{33}$

Multiple choice maths decimal numbers adding and subtracting decimals addition and subtraction of decimals operations on decimals

Simplify $\displaystyle :0.\overline{4}+0.\overline{61}+0.\overline{11}-0.\overline{36}$

  1. $\displaystyle 0.\overline{83}$
  2. $\displaystyle 0.\overline{87}$
  3. $\displaystyle 0.\overline{80}$
  4. $\displaystyle 0.\overline{85}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\displaystyle :0.\overline{4}+0.\overline{61}+0.\overline{11}-0.\overline{36}=\frac{4}{9}+\frac{61}{99}+\frac{11}{99}-\frac{36}{99}=\frac{4}{9}+\frac{72}{99}-\frac{36}{99}$


$\displaystyle=\frac{4}{9}+\frac{36}{99}=\frac{44}{99}+\frac{36}{99}=\frac{80}{99}=0.\overline{80}$