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 decimal fractions rounding decimals non-terminating recurring decimals in rational numbers rounding off decimals rounding of decimals

If $x$ and $y$ are positive real number, then which of the following is correct?

  1. $x > y \Rightarrow -x > -y $
  2. $x > y \Rightarrow -x < -y $
  3. $x > y \Rightarrow \dfrac{1}{x} > \dfrac{1}{y} $
  4. $x > y \Rightarrow \dfrac{1}{x} < \dfrac{-1}{y} $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

If $x$ and $y$ are positive number and $x>y$, then 

$\Rightarrow -x<-y$ 
Also, $x>y$ $\Rightarrow \dfrac{1}{x}<\dfrac{1}{y}$
Hence, B is correct option.

Multiple choice maths decimal fractions rounding decimals non-terminating recurring decimals in rational numbers rounding off decimals rounding of decimals

If $a, b, c$ are distinct $+ve$ real numbers and ${ a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }=1$ then $ab + bc + ca$ is 

  1. less then $1$
  2. equal to $1$
  3. greater then $1$

  4. any real no.
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

${a}^{2} + {b}^{2} + {c}^{2} = 1 \quad \left( \text{Given} \right)$


${\left( a +  b + c \right)}^{2} > 0$

${a}^{2} + {b}^{2} + {c}^{2} + 2 \left( ab + bc + ca \right) > 0$

$1 + 2 \left( ab + bc + ca \right) > 0$

$2 \left( ab + bc + ca \right) > -1$

$\Rightarrow ab + bc + ca >-\dfrac 12$

Multiple choice statistics linear correlation coefficient of correlation correlation coefficient correlation coefficients

The coefficient of correlation is always between

  1. $0\ and \ 1$
  2. $-1\ and \ 1$
  3. $-\infty \ and \ \infty$
  4. $-10\ and \ 10$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\Rightarrow$  Correlation coefficients are expressed as values between $-1$ and $+1$. 

$\Rightarrow$  A coefficient of $+1$ indicates a perfect positive correlation: A change in the value of one variable will predict a change in the same direction in the second variable. 
$\Rightarrow$  A coefficient of $-1$ indicates a perfect negative correlation: A change in the value of one variable predicts a change in the opposite direction in the second variable. 

Multiple choice mathematics and statistics binary operations properties of binary operations discrete mathematics sets and relations

If * is defined on the set R of all real numbers by $a*b=\sqrt{a^2+b^2}$, find the identity element in R with respect to *.

  1. 0

  2. 1

  3. 2

  4. 3
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let e be the identity element in R with respect to . Then,

$a*e=a=e*a$ for all $a\in R$
$a*e=a$ and $e*a=a$ for all $a\in R$
$\sqrt{a^2+e^2}=a$ and $\sqrt{e^2+a^2}=a$ for all $a\in R$
$a^2+e^2=a^2$ and $e^2+a^2=a^2$ for all $a\in R$
$e=0$
Hence, 0 is the identity element in R with respect to $$.

Multiple choice taylor's and maclaurin's series applications of differential calculus maths

$\ln{(1+x)}< x-\cfrac{{x}^{2}}{2}+\cfrac{{x}^{3}}{3}$ for $x> 0$

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The Maclaurin series for ln(1+x) is x - x^2/2 + x^3/3 - x^4/4 + ... for -1 < x <= 1. For x > 0, the next term is -x^4/4, which is negative. Therefore, ln(1+x) is strictly less than the sum of the first three terms.

Multiple choice maths indices negative indices law of indices laws of indices

If the exponent of a negative integer is odd, then the result is a .......... integer.

  1. positive

  2. negative

  3. zero

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

  If the exponent of a negative integer is odd, then the result is a negative integer.

$\Rightarrow$  $(-2)^{3}=(-2)\times (-2)\times (-2)=-8$
$\Rightarrow$  Here, $-2$ is negative base and $3$ is odd power.
$\Rightarrow$  Then result  $-8$ is a negative integer.

Multiple choice maths indices negative indices law of indices laws of indices

If the exponent of a negative integer is even then the result is a ............ integer.

  1. Positive

  2. Negative

  3. 0

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

  If the exponent of a negative integer is even, then the result is a $positive$ integer.

$\Rightarrow$  $(-2)^{4}=(-2)\times (-2)\times (-2)\times(-2)=16$
$\Rightarrow$  Here, $-2$ is negative base and $4$ is even power.
$\Rightarrow$  Then result  $16$ is a positive integer.

Multiple choice maths indices negative indices law of indices laws of indices

Which of the following has an exponent with negative index?

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

$\dfrac {1}{3^{4}} = 3^{-4}$

$\therefore \dfrac {1}{3^{4}}$ has a negative index.
So, option $C$ is correct.

Multiple choice maths indices negative indices law of indices laws of indices

Which of the following has a negative index?

  1. ${ x }^{ 5 }$
  2. $\dfrac { 1 }{ x } $
  3. ${ (x) }^{ \tfrac { 3 }{ 2 } }$
  4. ${ (-x) }^{ \tfrac { 3 }{ 2 } }$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\dfrac{1}{x} = x^{-1}$. Index of $x$ is $(-1)$, which is negative.


Hence, option $B$ is correct

Multiple choice maths indices negative indices law of indices laws of indices

Which of the following has a negative index?

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

$\dfrac{1}{x^3}=x^{-3}$ . Index of $x$ is  $(-3)$, which is negative.


Hence, option $D$ is correct.

Multiple choice maths indices negative indices law of indices laws of indices

Which of the following has a negative index?

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

$\dfrac { 1 }{ { x }^{ 2 } } $ = ${ x }^{ -2 }$
Thus ${ x }^{ -2 }$ has a negative index
Therefore option $A$ is the correct answer.

Multiple choice maths indices negative indices law of indices laws of indices

Which of the following has a negative index?

  1. ${ (x) }^{ \tfrac{ 1 }{4 } }$
  2. ${ (x) }^{ \tfrac { 2 }{ 3 } }$
  3. $\dfrac { 1 }{ { x }^{ 2 } } $
  4. ${ (-x) }^{\tfrac { 2 }{ 3 } }$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
Option A has positive index i.e. $\dfrac {1}{4}$.
Option B also has positive index i.e., $\dfrac {2}{3}$
In option C, $\dfrac{1}{x^2} = x^{-2}$.  
Index of $x$ is $(-2)$, which is negative.
Hence, option C is correct

Multiple choice maths real number fundemental theorem of arithmetic real numbers on number line fundamental theorem of arithmetic

If any positive' even integer is of the form 4q or 4q + 2, then q belongs to:

  1. whole number

  2. rational number

  3. real number

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

Let a be any positive even integer and b=$4$.Then by division algorithm,
$a=4q+r$ for some integer $q \ge 0$ and $r=0,1,2,3$
So,
$a=4q$ or, 
$4q+1$,
$4q+2$
$4q+3$
Because $0 \ge r \ge 4$
Now,
$4q$i.e $2(2q)$ is an even number
$\therefore$ $4q+1$ is an odd number
$4q+2$ i.e. $2(2q+1)$ is an even number
$\therefore (4q+2)+1=4q+3$ is an odd number
Thus, We can say that any even integer can be written as in the form of $4q, 4q+2$ where $q$ is the whole number

Multiple choice economics consumption and investment functions determinants of consumption function and savings function production, consumption, saving and economic units ex ante and ex post

The value of _____ can never be negative, while can have a value equal to one. 

  1. APS, APC

  2. MPG, APS

  3. APC, APS

  4. MPS, APC
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

APC refers to Average Propensity to Consume which defines the amount of consumption in every 1 rupee of income for all level of income which can never be negative because consumption is never negative owing to the very existence of life and basic needs to satisfy it while it can be more than one as long as consumption is more national income, i.e. before the break-even point, APC > 1. 

Marginal Propensity to save refers to the percentage change in savings for every one rupee of change in the income. It is the ratio between the change in income and its corresponding change in savings. It can never be negative as it is a change in the value when the value of income changes while it can be equal to one if the overall change in the value of income is used in savings.