Mathematics · Quantitative Aptitude

Algebra and Arithmetic

406 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 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 business maths matrix properties of matrix multiplication properties of multiplication of matrix multiplication of matrices

If $A^k=0$ for some value of $k$ and $B=1+A+A^2+...+A^{k-1},$ then $B^{-1}$ equal

  1. $I-A$
  2. $I+A$
  3. $I-A^{k-1}$

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

Let $\displaystyle B=I+A+{ A }^{ 2 }+{ A }^{ 3 }+...+{ A }^{ k-1 }$

$\displaystyle \therefore B\left( I-A \right) =\left( I+A+{ A }^{ 2 }+...+{ A }^{ k-1 } \right) \left( I-A \right) $
$\displaystyle =I-A+A-{ A }^{ 2 }+{ A }^{ 2 }+...+{ A }^{ k-1 }-{ A }^{ k }=I={ A }^{ k }=I\quad \quad \quad \left( \because { A }^{ k }=O \right) $
Hence, $\displaystyle { \left( I-A \right)  }^{ -1 }=I+A+{ A }^{ 2 }+...+{ A }^{ k }-1=B$
$\displaystyle \Rightarrow { B }^{ -1 }=I-A$

Multiple choice business maths matrix properties of matrix multiplication properties of multiplication of matrix multiplication of matrices

If $D=diag({d} _{1}, {d} _{2}, {d} _{3}........{d} _{n})$, where ${d} _{1}\ne 0$ for all $i=1, 2,.....n$, then ${D}^{-1}$ is equal to

  1. $D$
  2. ${I} _{n}$
  3. diag $({d} _{1}^{-1}, {d} _{2}^{-1}, ........{d} _{n}^{-1})$

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

For a diagonal matrix D = diag(d1, d2, ..., dn), the inverse is D^-1 = diag(1/d1, 1/d2, ..., 1/dn), provided all di are non-zero.

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.$

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

If ${ y }^{ 2 }={ x }^{ 2 }-x+1$ and $\quad { I } _{ n }=\int { \cfrac { { x }^{ n } }{ y }  } dx$ and $A{ I } _{ 3 }+B{ I } _{ 2 }+C{ I } _{ 1 }={ x }^{ 2 }y$ then ordered triplet $A,B,C$ is

  1. $\quad \left( \cfrac { 1 }{ 2 } ,-\cfrac { 1 }{ 2 } ,1 \right) $
  2. $\left( 3,1,0 \right) $
  3. $\left( 1,-1,2 \right) $
  4. $\left( 3,-\cfrac { 5 }{ 2 } ,2 \right) $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

We have given


${y^2} = {x^2} - x + 1$

${I _n} =\displaystyle \int {\dfrac{{{x^n}}}{y}dx} $
And,

$A{I _3} + B{I _2} + C{I _1} = {x^2}y$

Order triplet $A,\, B,\,C$ $ = \left( {\dfrac{1}{2},\,\dfrac{{ - 1}}{2},1} \right)$

Hence, the option $(A)$ is correct.

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

If a, b are constants then, $Var(a+bX)$ is

  1. $Var(a)+Var(X)$
  2. $Var(a)-Var(X)$
  3. $b^2Var(X)$

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

$Var(a+bX)=Var(a)+Var(bX)$


$\implies Var(a+bX)=0+Var(bX)$ (Since, $Var(c)=0$)

$\implies Var(a+bX)=b^2Var(X)$ (Since, $Var(aX)=a^2Var(X)$)

Multiple choice business economics and quantitative methods measures of dispersion and skewness shortcut method to find variance and standard deviation variance and standard deviation measures of dispersion

${X}^2$ test is equal to____________.

  1. ${\sum _{i = 1}^{n}} {A{x}^1} = A{x}^1 + A{x}^2 + ... + A{x}^n$
  2. $V = (r - 1) (e - 1)$
  3. $\dfrac {{\Sigma}(O - E)^2}{E}$
  4. $ r = \dfrac {{\Sigma} _{xy}}{\sqrt{{\Sigma}{{x}^2}{{y}^2}}}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The correct formula for the estimation of Chi-Square test is mentioned in Option C, where O represents the frequency observed while E represents the frequency expected.