Mathematics · Quantitative Aptitude

Algebra and Arithmetic

333 Questions

Algebra and arithmetic questions cover fundamental mathematical operations, inequalities, and binomial products. They assess core quantitative reasoning skills required for various aptitude tests. Solving these problems strengthens the understanding of number systems and algebraic identities.

Binomial productsLinear inequalitiesLeast common multipleQuadratic equationsInteger properties

Algebra and Arithmetic Questions

Multiple choice sum to infinite terms of a gp sequence, progression and series maths

 If  $0<\phi < \pi /2,$   and
 $x= \sum _{n=0}^{\infty} \cos ^{2n} \phi$, $ y=\sum _{n=0}^{\infty } \sin ^{2n} \phi$                     
and $z=\sum _{n=0}^{\infty} \cos ^{2n} \phi \sin ^{2n} \phi $ 
then

  1. xyz $=$xz+y
  2. xyz$=$xy+z
  3. xyz$=$x+y+z
  4. xy$=$yz+z
Reveal answer Fill a bubble to check yourself
B,C Correct answer
Explanation

The series x converges to a value $\dfrac{1}{1-{\cos^2 \phi}}\;=\;{cosec^2 \phi}$

Similarly y converges to  a value ${sec^2 \phi} $

In a similar fashion.. z converges to $\dfrac{{cosec^2 \phi}{sec^2 \phi}}{{cosec^2 \phi}{sec^2 \phi} - 1}$
 $ z = \dfrac{xy}{xy - 1} $

$xy + z = xyz&gt;$; Option B.
And, also
$x + y = xy$
$\therefore xyz = x+ y+ z&gt;$; Option C.

Multiple choice sum to infinite terms of a gp sequence, progression and series maths

If $a=\sum _{ n=0 } ^{\infty  }{x^n } ,b=\sum _{n=0  }^{ \infty  }{ y^n } , c=\sum _{n=0  }^{ \infty  }{ (xy)^n } $ where $|x| ,| y| < 1$ ; then

  1. $abc = a + b + c$
  2. $ab + bc = ac + b$
  3. $ac + bc = ab + c$
  4. $ab + ac = bc + a$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Clearly every summation is infinite series,
$a=\cfrac{1}{1-x}, b = \cfrac{1}{1-y}$ and $c=\cfrac{1}{1-xy}$
or $x=1-\cfrac{1}{a}, y=1-\cfrac{1}{b}$
Simplifying above equation we get, $ac+bc=ab+c$

Multiple choice sum to infinite terms of a gp sequence, progression and series maths

If $|x| > 1$, then
$\left(1-\dfrac{1}{x}\right)+\left(1-\dfrac{1}{x}\right)^2+\left(1-\dfrac{1}{x}\right)^3+.....=$

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

$S=\left( 1-\cfrac { 1 }{ x }  \right) +{ \left( 1-\cfrac { 1 }{ x }  \right)  }^{ 2 }+{ \left( 1-\cfrac { 1 }{ x }  \right)  }^{ 3 }+........$
This is a G.P whose first term, $a=1-\cfrac { 1 }{ x } $ and common ration,  $r=1-\cfrac { 1 }{ x } $
${ S } _{ \infty  }=\cfrac { a }{ 1-r } $
$=\cfrac { 1-\cfrac { 1 }{ x }  }{ 1-\left( 1-\cfrac { 1 }{ x }  \right)  } $
$=\cfrac { { \left( x-1 \right)  }/{ x } }{ { \left[ x-\left( x-1 \right)  \right]  }/{ x } } $
$=\cfrac { x-1 }{ x-x+1 } $
$=x-1$

Multiple choice sum to infinite terms of a gp sequence, progression and series maths

For $0 < \phi < \pi/2$ if $x=\sum _{n=0}^{\infty }\cos ^{2n} \phi, y=\sum _{n=0}^{\infty }\sin ^{2n} \phi, z=\sum _{n=0}^{\infty }\cos ^{2n} \phi \sin^{2n}\phi$, then 

  1. $xyz=xz+y$
  2. $xyz=xy+z$
  3. $xyz=x+y+z$
  4. $xyz=yz+x$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given $\displaystyle x=\sum _{ n=0 }^{ \infty  }{ { \cos }^{ 2n }\phi ,\quad y=\sum _{ n=0 }^{ \infty  }{ { \sin }^{ 2n }\phi  }  } $ and $\displaystyle z=\sum _{ n=0 }^{ \infty  }{ { \cos }^{ 2n }\phi  } { \sin }^{ 2n }\phi $


\since $\displaystyle 0<\phi <\frac { \pi  }{ 2 } $, so each series is geometric series with common ratio $\displaystyle r<1$. Therefore, the series are convergent.


Now, $\displaystyle x=\frac { 1 }{ 1-{ \cos }^{ 2 }\phi  } $


$\displaystyle =\frac { 1 }{ { \sin }^{ 2 }\phi  } $      $(\because S _{ \infty  }=\dfrac { a }{ 1-r } )$


$\displaystyle y=\frac { 1 }{ 1-{ \sin }^{ 2 }\phi  } $      $(\because S _{ \infty  }=\dfrac { a }{ 1-r } )$


$\displaystyle =\frac { 1 }{ { \cos }^{ 2 }\phi  } $


$\displaystyle z=\frac { 1 }{ 1-{ \sin }^{ 2 }{ \phi \cos }^{ 2 }\phi  } $ 

   

$(\because S _{ \infty  }=\dfrac { a }{ 1-r } )$


Consider, $\displaystyle xyz=\frac { 1 }{ { \sin }^{ 2 }\phi { \cos }^{ 2 }\phi (1-{ \sin }^{ 2 }{ \phi \cos }^{ 2 }\phi ) } $      $(1)$


Also, $\displaystyle =\frac { 1 }{ { \sin }^{ 2 }\phi { \cos }^{ 2 }\phi  } +\frac { 1 }{ 1-{ \sin }^{ 2 }{ \phi \cos }^{ 2 }\phi  } $


$\displaystyle xy+z=\frac { 1-{ \sin }^{ 2 }{ \phi \cos }^{ 2 }\phi +{ \sin }^{ 2 }\phi { \cos }^{ 2 }\phi  }{ { \sin }^{ 2 }\phi { \cos }^{ 2 }\phi (1-{ \sin }^{ 2 }{ \phi \cos }^{ 2 }\phi ) } $


$\displaystyle =\frac { 1 }{ { \sin }^{ 2 }\phi { \cos }^{ 2 }\phi (1-{ \sin }^{ 2 }{ \phi \cos }^{ 2 }\phi ) } $

 

$\displaystyle =xyz$      $[From (1)]$

 

Multiple choice maths concept of directed numbers and number line subtraction of directed numbers subtraction of integers subtraction of integers on number line

Addition of integer and its additive inverse is

  1. Zero

  2. double the number

  3. half the number

  4. none
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Additive inverse of an integer is a number which is of opposite sign and when added to the integer, gives zero.

Thus, Addition of integer and its additive inverse is Zero.

Multiple choice maths concept of directed numbers and number line subtraction of directed numbers subtraction of integers subtraction of integers on number line

$0$ is the __________ identify for whole numbers, whereas $1$ is the ___________ identify for whole numbers.

  1. Additive, Multiplicative

  2. Multiplicative, Additive

  3. Commutative, Additive

  4. Positive, Not-defined
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

When $0$ is added to any given whole number, the resultant number is the given whole number itself. This shows that $0$ is additive identity for whole number.

When $1$ is multiplied to any whole number, the resultant number is the number itself.
Here $0$ and $1$ are not making any change to the number.
Hence option A is correct.

Multiple choice maths geometric sequences sum of terms of g.p sum of n terms of an gp summing geometric series

Given $A=2^{65}$ and $B=(2^{64}+2^{63}+2^{62}+....+2^0)$

  1. B is $2^{64}$ larger than A

  2. A and B are equal
  3. B is larger than A by $1$
  4. A is larger than B by $1$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

B is in G.P. with $a=2^0, r=2, n=65$
$\therefore S _n=\frac {a(r^n-1)}{r-1}=\frac {2^0(2^{65}-1)}{2-1}$
$\therefore B=2^{65}-1$
$\Rightarrow B=A-1$
$\therefore$ A is larger than B by $1$.

Multiple choice maths geometric sequences sum of terms of g.p sum of n terms of an gp summing geometric series

If $1+a+a^{2}+a^{3}+.........+a^{n}=(1+a)(1+a^2)(1+a^4)$ then $n$ is given by 

  1. $3$
  2. $5$
  3. $7$
  4. $9$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given, 

$1+a+a^{2}+a^{3}+.........+a^{n}=(1+a)(1+a^2)(1+a^4)$
or, $1+a+a^{2}+a^{3}+.........+a^{n}=(1+a+a^2+a^3)(1+a^4)$
or, $1+a+a^{2}+a^{3}+.........+a^{n}=(1+a+a^2+a^3+a^4+a^5+a^6+a^7)$
Comparing both sides we get, $n=7$.

Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

If A= {0, 1} and B ={1, 0}, then what is A x B equal to ?

  1. {(0, 1), (1, 0)}

  2. {(0, 0), (1, 1)}

  3. {(0, 1), (1, 0), (1, I)}

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

$\left{ { 0,1 } \right} \times \left{ 1,0 \right} ={ \left{ (0,1),(0,0),(1,1),(1,0) \right}  }$

$\left{ { 0,1 } \right} \times \left{ 0,1 \right} ={ \left{ (0,0),(0,1),(1,0),(1,1) \right}  }$

So, $A\times B=A\times A$
Hence, D is correct.

Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

Given $(a - 2, b + 3) = (6, 8)$, are equal ordered pair. Find the value of $a$ and $b$.

  1. $a = 8$ and $b = 5$
  2. $a = 8$ and $b = 3$
  3. $a = 5$ and $b = 5$
  4. $a = 8$ and $b = 6$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

By equality of ordered pairs, we have
$(a - 2, b + 3) = (6, 8)$
On equating we get
$a - 2 = 6$
$a = 8$
$b + 3 = 8$
$b = 5$
So, the value of$ a = 8 $ and $ b = 5.$