Quantitative Aptitude

Number System and Simplification

585 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 square root square root of perfect square finding square root of a number square root of a perfect square

Simplify the following: 
$\displaystyle \frac{\sqrt{24}}{8}+\frac{\sqrt{54}}{9}$ is equal to $\displaystyle \frac{7\sqrt{6}}{12}$
If true then enter $1$ and if false then enter $0$

  1. $1$
  2. $0$

  3. Cannot be determined, incomplete information

  4. None of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$ =\dfrac { 9\sqrt { 2\times 2\times 2\times 3 } +8\sqrt { 2\times 3\times 3\times 3 }  }{ 72 } $
$ = \dfrac { 18\sqrt { 6 } +24\sqrt { 6 }  }{ 72 } $
$ = \dfrac { 42\sqrt { 6 }  }{ 72 } $
$ = \dfrac { 7\sqrt { 6 }  }{ 12 }  $

Multiple choice maths square root square root of perfect square finding square root of a number square root of a perfect square

Evaluate: $\displaystyle\sqrt{\left (5\, +\, 2\frac{21}{25}\right )\, \times\, \frac{0.169}{1.6}}$ $\times 100$

  1. $91$
  2. $81$
  3. $21$
  4. $54$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\sqrt{\left (5+2\dfrac{21}{25}\right )\times \dfrac{0.169}{1.6}}\times 100$

$=\sqrt{\left (5+\dfrac{71}{25}\right )\times \dfrac{169\times 10}{16\times 1000}}\times 100$

$=\sqrt{\left (\dfrac{125+71}{25}\right )\times \dfrac{169}{16\times 100}}\times 100$

$=\sqrt{\left (\dfrac{196}{25}\right )\times \dfrac{169}{16\times 100}}\times 100$

$=\dfrac{14}{5}\times \dfrac{13}{4\times 10}\times 100$

$=\dfrac{91}{100}\times 100$

$=91$

Multiple choice maths square root square root of perfect square finding square root of a number square root of a perfect square

Evaluate  $\sqrt {\displaystyle \frac { 25 }{ 81 } -\displaystyle\frac { 1 }{ 9 }  } $

  1. $\displaystyle \frac { 16}{ 81 }$
  2. $\displaystyle \frac { 25}{ 81 }$
  3. $\displaystyle \frac { 4}{ 9}$
  4. $\displaystyle \frac { 2}{ 3}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\sqrt {\displaystyle \frac { 25 }{ 81 } -\displaystyle\frac { 1 }{ 9 }  } =\sqrt { \displaystyle\frac { 25-9 }{ 81 }  } =\sqrt {\displaystyle \frac { 16 }{ 81 }  } =\displaystyle\frac { \sqrt { 16 }  }{ \sqrt { 81 }  } =\displaystyle\frac { 4 }{ 9 } $

Multiple choice maths square root square root of perfect square finding square root of a number square root of a perfect square

The value of $\sqrt{1\displaystyle\frac{1}{2}-\begin{bmatrix}1\displaystyle\frac{1}{2}-1\displaystyle\frac{1}{2}+\begin{pmatrix}1\displaystyle\frac{1}{2}-1\displaystyle\frac{1}{2}-1\displaystyle\frac{1}{4}\end{pmatrix}\end{bmatrix}}$ is

  1. $\displaystyle\frac{1}{2}$
  2. $\displaystyle\frac{1}{4}$
  3. $\displaystyle\frac{1}{16}$
  4. $1\displaystyle\frac{1}{5}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\sqrt{1\displaystyle\frac{1}{2}-\begin{bmatrix}1\displaystyle\frac{1}{2}-1\displaystyle\frac{1}{2}+\begin{pmatrix}1\displaystyle\frac{1}{2}-1\displaystyle\frac{1}{2}-1\displaystyle\frac{1}{4}\end{pmatrix}\end{bmatrix}}$
$\Rightarrow \sqrt{\frac{3}{2}-\left [ \frac{3}{2}-\frac{3}{2}+\left ( \frac{3}{2}-\frac{3}{2}-\frac{5}{4} \right ) \right ]}$
$\Rightarrow \sqrt{\frac{3}{2}-\left [ 0+\left ( \frac{6-6-5}{4} \right ) \right ]}$
$\Rightarrow \sqrt{\frac{3}{2}-\frac{5}{4}}$
$\Rightarrow \sqrt{\frac{6 -5}{4}}$
$\Rightarrow \sqrt{\frac{1}{4}}$
$\Rightarrow \frac{1}{2}$




Multiple choice maths square root square root of perfect square finding square root of a number square root of a perfect square

If $\sqrt{1\, +\, \displaystyle \frac{27}{169}}\, =\, 1\, +\, \displaystyle \frac{x}{13}$, then the value of $x$ is

  1. 1

  2. 14

  3. Cannot be determined

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\sqrt{1\, +\, \displaystyle \cfrac{27}{169}}\, =\, \displaystyle \cfrac{196}{169}\, =\, \displaystyle \cfrac{14}{13}$
$\Rightarrow 1\, \displaystyle \cfrac{1}{13}\, =\, 1\, +\, \displaystyle \cfrac{1}{13}$
$\therefore\, x\, =\, 1$

Multiple choice maths square root square root of perfect square finding square root of a number square root of a perfect square

If $x=5+2\sqrt { 6 } $, then $\sqrt{ x }+\dfrac{1}{\sqrt { x }} $ is ?

  1. $2\sqrt{ 2 } $
  2. $2\sqrt { 3 } $
  3. $\sqrt { 3 } +\sqrt { 2 } $
  4. $\sqrt { 3 } -\sqrt { 2 } $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let $A=\sqrt { x } +\cfrac { 1 }{ \sqrt { x }  } $
$\Rightarrow { A }^{ 2 }=x+\cfrac { 1 }{ x } -2=\left( 5+2\sqrt { 6 }  \right) +\cfrac { 1 }{ 5+2\sqrt { 6 }  } -2$
$=5+2\sqrt { 6 } +\cfrac { 5-2\sqrt { 6 }  }{ 25-24 } -2=8$ $\text{[Rationalising the denominator]}$
$=5+2\sqrt { 6 } + { 5-2\sqrt { 6 }  } -2=8$
$ \Rightarrow A^2=8 $
$ \Rightarrow A=2\sqrt { 2 } $

Multiple choice maths square root square root of perfect square finding square root of a number square root of a perfect square

For what value of $\displaystyle x+\frac { 1 }{ 4 } \sqrt { x } +{ a }^{ 2 }$ will be perfect square -

  1. $\displaystyle \pm { 1 }/{ 18 }$
  2. $\displaystyle \pm { 1 }/{ 8 }$
  3. $\displaystyle \pm { 1 }/{ 5 }$
  4. $\displaystyle { 1 }/4$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

If $\displaystyle x+\frac { 1 }{ 4 } \sqrt { x } +{ a }^{ 2 }$ is a perfect square
then $\displaystyle \frac { 1 }{ 4 } \sqrt { x } =2\times \sqrt { x } \times \left( \pm a \right) $
$\displaystyle \therefore \quad a=\pm \frac { 1 }{ 8 } $

Multiple choice physics measurements and experimentation measuring distance of celestial bodies unconventional units of measurements units of mass

1 AU is equal to

  1. $\displaystyle 1.5\times { 10 }^{ 11 }m$
  2. $\displaystyle 1.5\times { 10 }^{ 10 }m$
  3. $\displaystyle 1.5\times { 10 }^{ 9 }m$
  4. $\displaystyle 1.5\times { 10 }^{ -11 }m$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
One astronomical unit is defined as the distance between the sun and the earth.
$ 1 \ AU = 1.5\times 10^{11} \ m$
Multiple choice maths remainder and factor theorems linear and synthetic method of division factorising expressions using factor theorem division algorithm for polynomials

If $ \displaystyle 2x^{3}+4x^{2}+2ax+b $  is exactly divisible by $ \displaystyle x^{2}-1 $  Then the value of $a$ and $b$ respectively will be 

  1. $1,2$
  2. $-1,4$
  3. $1,-2$
  4. $-1,-4$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Since $\displaystyle f\left ( x \right )=2x^{3}+4x^{2}+2ax+b$ is exactly divisible
by $x\displaystyle ^{2}-1=\left ( x-1 \right )\left ( x+1 \right )$
$\displaystyle \therefore f\left ( 1 \right )=0$ and $\displaystyle f\left ( -1 \right )=0$
These give
    $2+4+2a+b=0$
or $2a+b+6=0$          .....(i)
and $-2+4-2a+b=0$
or $2a-b-2=0$        ....(ii)
Solving equations (i) and (ii) we get 
$a=-1, b=-4$