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 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 maths negative numbers and integers odd and even numbers operations on integers different types of numbers

If k is a positive integer, then $k(k+1)(k+3)$ is

  1. even only when k is even

  2. even only when k is odd

  3. Always odd

  4. Always even
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Suppose $k$ is even

Then $k(k+1)(k+3)$ is even as $k$ will be divisible by $2$
Now if $k$ is odd.
Then $(k+1)$ and $(k+3)$ are even
$\Rightarrow k(k+1)(k+3)$ is also even.
So $k(k+1)(k+3)$ is always even.
So option $D$ is correct.

Multiple choice mathematical modelling proof by contradiction similar triangles

To prove " If  $x,x\in N$  is even then $x^2$ is even". By direct method, we must start with the assumption:

  1. Let $x ^2 $ not even
  2. Let $x$ not even
  3. Let $x$ be even natural number.
  4. Let $x\notin N$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

To prove an implication by straight forward approach or direct method, we have to take the hypothesis of the statement as an assumption and then reach at the conclusion. 

In the statement, "If $p$ then $q$",  $p$ is hypothesis and $q$  is conclusion.
So, here $C$ is correct

Multiple choice decimal representation of rational numbers rational and irrational numbers maths

If a and b are any two such real numbers that ab $ = 0 $ , then

  1. $a = 0, b \leq 0$
  2. $b = 0, a \leq 0$

  3. a = 0 or b = 0 or both
  4. $a = b$ and $b = 0$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

if both number are real the either a or b or both should be zero.
then only ab will be 0.
if any real number is multiplied by 0 then result will be zero.
So, answer is
C a = 0 or b = 0 or both

Multiple choice business mathematics and statistics moving averages introduction to time series introduction to time series and forecasting time series and forecasting uses of average in day-to-day life

If $A : B =3 : 4$ and $B : C =5 : 6$, then $A : C$ is

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

$A:B=3:4$
 $B:C=5:6$
 $\therefore \frac { A }{ B } =\frac { 3 }{ 4 } & \frac { B }{ C } =\frac { 5 }{ 6 } $
 $\therefore \frac { A }{ B } \times \frac { B }{ C } =\frac { A }{ C } $
$\therefore \frac { 3 }{ 4 } \times \frac { 5 }{ 6 } =A:C$
 $=\frac { 15 }{ 24 } =\frac { 5 }{ 8 } $
A:C=5:8