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 hcf-lcm introduction to multiples multiples lcm

If A, B and C are three numbers such that L.C.M. of A and B is B and the L.C.M. of B and C is C then the L.C.M. of A, B and C is

  1. A

  2. B

  3. C
  4. $\displaystyle \frac{A+B+C}{3}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

LCM of A and B is B it means that B is multiple of A. LCM of B and C is C it means C is multiple of B or we can say that C is multiple of A also.

So LCM of A,B ,C is C so correct answer is option C

Multiple choice range and mean deviation statistics and probability maths coefficient of variance variance and standard deviation

If $n> 1, x> -1, x\neq 0$, then the statement $\left ( 1+x \right )^{n}> 1+nx$ is true for

  1. $ \;n\;\epsilon \;N$
  2. $\forall \;n\;> 1$
  3. $x> -1 \;and\; x\neq 0$

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

$P(1)$ is not true 


For $n=2,P\left( 2 \right) :{ \left( 1+x \right)  }^{ 2 }>1+2x$ is true if $x\neq 0$

Let $P(k):{ \left( 1+x \right)  }^{ k }>1+kx$ be two 

$\therefore{ \left( 1+x \right)  }^{ k+1 }=\left( 1+x \right) { \left( 1+x \right)  }^{ k}>\left( 1+x \right) \left( 1+kx \right)> 1+\left( k+1 \right) x+k{ x }^{ 2}>1+\left(k+1\right) x$

$\left( \because k{ x }^{ 2 }>0 \right) $
$\therefore$ By PMI
Given statement is true for every $n\in N$.

Multiple choice maths vectors and transformations vectors from a geometric viewpoint basic concepts of vector introduction to vectors

If $a +2b +3c = 0$, then $a \times b + b\times c + c\times a = ka\times b,$
Where $k$ is equal to ?

  1. $0$
  2. $1$
  3. $2$
  4. $3$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given,

$\begin{array}{l} a+2b+3c=0 \ a=-\left( { 2b+3c } \right)  \ a\times a=-\left( { 2\left( { b\times a } \right) +3\left( { c\times a } \right)  } \right)  \ \Rightarrow 0=-\left( { 2b\times a+3\left( { c\times a } \right)  } \right) ..........\left( i \right)  \end{array}$
Again,
$\begin{array}{l} 2b=-\left( { a+3c } \right)  \ 2b\times b=-\left( { a\times b+3c\times b } \right)  \ \Rightarrow 0=-\left( { a\times b+3c\times b } \right) ..........\left( { ii } \right)  \end{array}$
equation (i) and (ii)
$\begin{array}{l} -\left( { 2b\times a+3c\times a } \right) +\left( { a\times b } \right) +3c\times b=0 \ \Rightarrow 3a\times b-3c\times a-3b\times c=0 \end{array}$
We know that,
$a \times b =  - b \times a$
$a \times b = c \times a + b \times c...............\left( {iii} \right)$
Now,$ATQ;$
$a \times b + b \times c + c \times a = K\,\,\,a \times b$
$1 = 2a \times b = k\,a \times b$         {from equation (iii)}
$\therefore k=2$
Option $C$ is correct answer.

Multiple choice maths vectors:planes in three dimensions cartesian equation of plane general form of the equation of a plane lines in space

Let a,b,c be any real numbers.Suppose that there are real numbers x,y,z not all zero such that $x=cy+bz , y=az+cx$ and $z=bx+ay$, then ${a^2} + {b^2} + {c^2} + 2abc $ is equal to

  1. 2

  2. -1

  3. 0

  4. 1
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
$a, b, c$ real numbers

$x, y, z$ real numbers not all zero

$x=cy+bz\rightarrow x-cy-bz=0 --- (1)$

$y=az+cx\rightarrow cx+y-az=0---(2)$

$z=bx+cy\rightarrow -bx-ay+z=0 --- (3)$

The system of equation have trivial solution then

$\begin{vmatrix} 1 & -c & -b \\ c & -1 & a \\ b & a & -1 \end{vmatrix}=0$

$\Rightarrow 1(1-a^{2})+c(-c-ab)-b(a+b)=0$

$\Rightarrow a^{2}+b^{2}+c^{2}+2abc=1$

$D$ is correct
Multiple choice biology respiration and energy transfer respiratory quotient-basic respiration-plants respiratory quotient

The value of R.Q. at the compensation point is

  1. Infinity.

  2. Two.

  3. > 1.

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

During aerobic respiration, $O _2$ is consumed and $CO _2$ is released. The ratio of the volume of $CO _2$ evolved to the volume of $O _2$ consumed in respiration is called the respiratory quotient (RQ) or respiratory ratio. Thus the following expression is used to calculate the value of RQ.
RQ= volume of $CO _2$ evolved/ volume of $O _2$ consumed.
At compensation point there is no net evolution of carbon dioxide. Hence, numerator in above expression is zero, giving the value of RQ as zero.

Multiple choice biology respiration and energy transfer respiratory quotient-basic respiration-plants respiratory quotient
Value of R.Q. in succulents is ____________.

  1. Unity

  2. Infinite

  3. Less than unity

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

Succulents do not evolve carbon dioxide during the night (when their stomata are open) as the same is in carbon fixation. They also change carbohydrates to organic acids which utilise oxygen but do not evolve carbon dioxide.

${2C} _{6}{H} _{12}{O} _{6}+{3O} _{2}\longrightarrow\underset {Malic \,acid}{{3C} _{4}{H} _{6}{O} _{5}}+{3H} _{2}O$
$R.Q.=\frac{Zero\,{CO} _{2}}{{3O} _{2}}=Zero$
So the correct option is D.

Multiple choice maths direct proportion and inverse proportion rule of three types of proportions direct proportion

If $a:b=c:d$ then how many of the following statements are true?

  1. $c(a+b)=a(c+d)$
  2. $d(a-b)=b(c-d)$
  3. $(a^{2}+b^{2})(ac-bd)=(a^{2}-b^{2})(ac+bd)$
  4. $(a^{2}-b^{2})(ad-bc)=(a^{2}+b^{2})(ac-bd)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\dfrac{a}{b}=\dfrac{c}{d}$
$\dfrac{b}{a}=\dfrac{d}{c}$

$\left(1+\dfrac{b}{a}\right)=\left(1+\dfrac{d}{c}\right)$
$\dfrac{(a+b)}{a}=\dfrac{(c+d)}{c}$
$c(a+b)=a(c+d)$

Multiple choice maths direct proportion and inverse proportion rule of three types of proportions direct proportion

If $a : b = 5 : 9$ and $b : c = 4 : 7$ find $a : b : c$

  1. $5:9:\dfrac{63}{4}$
  2. $20:63:36$
  3. $4:36:63$
  4. $20:36:63$
Reveal answer Fill a bubble to check yourself
A,D Correct answer
Explanation

Given, $a : b = 5 : 9 $ and $b : c = 4 : 7$ $=$ $\displaystyle \left ( 4\times

\frac{9}{4} \right ):\left ( 7\times \frac{9}{4} \right

)=9:\frac{63}{4}$
$\displaystyle \Rightarrow a:b:c=5:9:\frac{63}{4}=20:36:63$

Multiple choice maths negative numbers and integers even and odd numbers sum of numbers odd and even numbers

a, b, c are even numbers and x, y, z are odd numbers. Which of the following relationships can't be justified at any cost?
(a) $\dfrac{a\times b}{c} = x\times y$ (b) $\dfrac{a\times b}{x}=yz$ (c) $\dfrac{xy}{z} = ab$

  1. Only B

  2. Only C

  3. All the three

  4. Only B and C
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

In option B a×b is always even & xyz is always odd therefore equality not holds.

In option C ab is always even therefore abz is also even & xy is always odd hence equality not holds.
In option A xy is always odd but (a×b)/c can be odd or even therefore equality can hold in this case.

Multiple choice maths negative numbers and integers even and odd numbers sum of numbers odd and even numbers

If $a$ and $b$ are odd numbers, then which of the following is even?

  1. $a+b$
  2. $a+b+1$
  3. $ab$
  4. $ab+2$

  5. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
We know the following rule :

odd + odd = even,

even + even = even,

odd + even = odd,

even + odd = odd,

odd × odd = odd.

(A) The given expression is

a + b = odd + odd = even.

(B) The given expression is

a + b + 1 = odd + odd + odd = even + odd = odd.

(C) The given expression is

ab = odd × odd = odd.

(D) The given expression is

ab + 2 = odd × odd + 2= odd + even = odd.

Thus, the correct option is (A) a+b.