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 power and exponent power of powers laws of exponents and powers law of indices

The value of $\cfrac { { 2 }^{ 2n-2 } }{ { 2 }^{ n(n-1) } }-\cfrac { { 8 }^{ n-1 } }{ { 2 }^{ (n-1)(n+1) } } $ will be

  1. $2$
  2. $0$
  3. $\dfrac {1}{2}$
  4. $\dfrac {1}{4}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Given,

$\dfrac{2^{2n-2}}{2^{n(n-1)}} - \dfrac{8^{n-1}}{2^{(n-1)(n+1)}}$

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

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

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

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

$=\dfrac{2^2}{2^n}-\dfrac{2^3}{2^n \times 2^1}$

$=\dfrac{2^2}{2^n}-\dfrac{2^2}{2^n}$

$=0$
Multiple choice maths ellipse normal to an ellipse tangent and normal to an ellipse two dimensional analytical geometry-ii

If $y=mx+7\sqrt{3}$ is normal to $\dfrac{x^2}{18}-\dfrac{y^2}{24}=1$ then the value of m can be?

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

$7\sqrt{3}=\dfrac{42m}{\sqrt{24-18m^2}}\Rightarrow \sqrt{3}=\dfrac{\sqrt{6}m}{\sqrt{4-3m^2}}\Rightarrow 4-3m^2=2m^2$
$m=\dfrac{2}{\sqrt{5}}$.

Multiple choice physics units and measurement: error analysis significant figures significant figures and rounding of digits units and measurements

In the final answer of the expression  $\dfrac { ( 29.2 - 20.2 ) \left( 1.79 \times 10 ^ { 5 } \right) } { 1.37 }.$  The number of significant figures is

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

29.2 - 20.2 = 9.0 (two sig figs). 9.0 * 1.79 = 16.11. 16.11 / 1.37 = 11.759. The result should be limited by the precision of the subtraction (two sig figs), but the options suggest three.

Multiple choice de moivre’s theorem and its applications demoivre's theorem complex numbers maths

The value of $\displaystyle { \left( \frac { 1+i }{ \sqrt { 2 }  }  \right)  }^{ 8 }+{ \left( \frac { 1-i }{ \sqrt { 2 }  }  \right)  }^{ 8 }$ is equal to

  1. $4$
  2. $6$
  3. $8$
  4. $2$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

We have $\displaystyle { \left( \frac { 1+i }{ \sqrt { 2 }  }  \right)  }^{ 8 }+{ \left( \frac { 1-i }{ \sqrt { 2 }  }  \right)  }^{ 8 }$


$\displaystyle={ \left[ \cos { \frac { \pi  }{ 4 }  } +i\sin { \frac { \pi  }{ 4 }  }  \right]  }^{ 8 }+{ \left[ \cos { \frac { \pi  }{ 4 }  } -i\sin { \frac { \pi  }{ 4 }  }  \right]  }^{ 8 }$


$=\cos { 2\pi  } +i\sin { 2\pi  } +\cos { 2\pi  } -i\sin { 2\pi  } $      [by de-moivre's theorem]

$=2\cos { 2\pi  } =2\left( 1 \right) =2$  

Multiple choice maths operations adding and subtracting numbers using place value addition & subtraction mental additions and subtractions

If P : Q : R = 6 : 5 : 4 and $\displaystyle P^{2}+Q^{2}+R^{2}=192500$ then find $\displaystyle \frac{(P+Q-R)}{2}$

  1. 175

  2. 165

  3. 185

  4. 200
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$
:Q:R\quad =\quad 6:5:4\ Let\quad P\quad \quad =\quad 6x\ Q=\quad 5x\ R\quad =\quad 4x\ { P }^{ 2 }+{ Q }^{ 2 }{ +\quad R }^{ 2 }\quad =\quad 192500\ { (6x) }^{ 2 }+{ (5x) }^{ 2 }+(4x)^{ 2 }\quad =\quad 192500\ 77{ x }^{ 2 }\quad =\quad 192500\ { x }^{ 2 }\quad =\quad 2500\ x\quad =\quad 50\ \ \frac { P+Q-R }{ 2 } \quad =\quad \frac { 6x+5x-4x }{ 2 } \quad =\quad \frac { 7x }{ 2 } \quad =\quad \frac { 7\times 50 }{ 2 } \quad =\quad 175
$

Multiple choice maths operations adding and subtracting numbers using place value addition & subtraction mental additions and subtractions

Two rows of numbers are given The resultant number in each row is to be worked out separately based on the following rules and the question below the rows of numbers is to be answered The operations of numbers progress from left to right
Rules
I. If an odd number is followed by a two-digit even number then they are to be added.
II. If an odd number is followed by a two-digit odd number then the second number is to be subtracted from the first number
III. If an even number is followed by a number which is a perfect square of a number then the second number is to be divided by the first number
IV. If an even number is followed by a two-digit even number then the first number is to be multiplied by the second number
                    8  16  16  14
                   13  11  12  144
What is the difference between the resultant of the first set of numbers and the second set of numbers ?

  1. 106

  2. 118

  3. 210

  4. 222
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice maths average arithmetic mean of ap introduction to averages means

If $A _1,A _2$ be two arithmetic means between $\dfrac{1}{3}$ and $\dfrac{1}{24}$, then their value are 

  1. $\dfrac{7}{72},\dfrac{5}{36}$
  2. $\dfrac{17}{72},\dfrac{5}{36}$
  3. $\dfrac{7}{36},\dfrac{5}{72}$
  4. $\dfrac{5}{72},\dfrac{17}{72}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For two means A1, A2 between 1/3 and 1/24, the common difference d = (1/24 - 1/3) / (2 + 1) = (-7/24) / 3 = -7/72. A1 = 1/3 - 7/72 = 17/72. A2 = 17/72 - 7/72 = 10/72 = 5/36.

Multiple choice maths average arithmetic mean of ap introduction to averages means

Let $(1-2x+3x^{2})^{10}=a _{0}+a _{1}x+a _{2}x^{2}+....+a _{n}x^{n},a _{n}\neq 0$, then the arithmetic mean of $a _{0},a _{1},a _{2},....a _{n}$ is

  1. $\dfrac{1024}{11}$
  2. $\dfrac{512}{7}$
  3. $\dfrac{512}{11}$
  4. $\dfrac{1024}{21}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The arithmetic mean of coefficients a_0, a_1, ..., a_n is the sum of coefficients divided by the number of terms, which is (1/21) * sum. By substituting x = 1 and x = -1 into the polynomial expansion, we find the sum of all coefficients and the alternating sum, allowing us to find the total sum of coefficients and compute the mean as 1024/21.

Multiple choice maths average arithmetic mean of ap introduction to averages means

The arithmetic mean of 1, 2, 3, ..., n, is

  1. $\displaystyle \frac{n-1}{2}$
  2. $\displaystyle \frac{n+1}{2}$
  3. $\displaystyle \frac{n}{2}$
  4. $\displaystyle \frac{n}{2}+1$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\Rightarrow$   We have the sequence, $1,2,3.......n$

$\Rightarrow$   This is an AP, with the initial term $a=1$ and the common difference $d=1$.
$\therefore$   The sum of $n$ terms of an AP is given by,
$\Rightarrow$  $S _n=\dfrac{n}{2}[2a+(n-1)d]$

$\Rightarrow$  $S _n=\dfrac{n}{2}[2\times 1+(n-1)\times 1]$

$\Rightarrow$  $S _n=\dfrac{n}{2}[2+(n-1)]$

$\Rightarrow$  $S _n=\dfrac{n}{2}[n+1]$
$\rightarrow$   Arithmetic mean of $n$ numbers $a _1,a _2,a _3,a _4,... a _n$ is given by the formula
$\Rightarrow$  $Arithmetic\,mean=\dfrac{a _1+a _2+a _3+a _4+...+a _n}{n}$

$\Rightarrow$  $Arithmetic\,mean=\dfrac{S _n}{n}$

$\Rightarrow$  $Arithmetic\, mean=\dfrac{\dfrac{n}{2}[n+1]}{n}$

$\therefore$     $Arithmetic\, mean=\dfrac{n+1}{2}$

Multiple choice maths fraction lowest form of a fraction simplest ratio lowest form of fractions

If $2a-5b = 0$ then find the value of $\displaystyle \frac{a+b}{a-b}$.

  1. $\displaystyle \frac{7}{2}$
  2. $\displaystyle \frac{7}{3}$
  3. $\displaystyle \frac{3}{2}$
  4. $\displaystyle \frac{7}{5}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given, $ 2a-5b = 0 $

$=>2a = 5b $

$ => \dfrac {a}{b} = \dfrac {5}{2} $

Applying componendo and dividendo,

Now, $ \dfrac {a+b}{a-b} = \dfrac{5+2}{5-2} =

\dfrac {7}{3} $

Multiple choice maths fraction lowest form of a fraction simplest ratio lowest form of fractions

If $2a-5b = 0$ then find the value of  $\displaystyle \frac{a-b}{b}$

  1. $\displaystyle \frac{7}{2}$
  2. $\displaystyle \frac{3}{2}$
  3. $\displaystyle \frac{5}{2}$
  4. $\displaystyle \frac{7}{5}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given, $ 2a-5b = 0 $
$=> 2a = 5b $
$ => \dfrac {a}{b} = \dfrac {5}{2} $
 
Now, $ \dfrac {a-b}{b} = \dfrac {a}{b} - 1 = \dfrac {5}{2} - 1 = \dfrac{5-2}{2} = \dfrac {3}{2} $

Multiple choice maths fraction lowest form of a fraction simplest ratio lowest form of fractions

If $2a-5b = 0$ then find the value of  $\displaystyle \frac{a+b}{b}$

  1. $\displaystyle \frac{7}{2}$
  2. $\displaystyle \frac{7}{5}$
  3. $\displaystyle \frac{2}{7}$
  4. $\displaystyle \frac{7}{6}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given, $ 2a-5b = 0 $
$=>. 2a = 5b $
$ => \frac {a}{b} = \frac {5}{2} $
 
Now, $ \frac {a+b}{b} = \frac {a}{b} + 1 = \frac {5}{2} + 1 = \frac{5+2}{2} = \frac {7}{2} $