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

If (2a + 3b)(2c - 3d)=(2a - 3b)(2c + 3d)Then

  1. a/b = c/d

  2. a/d = c/b

  3. a/b = d/c

  4. b/a = c/d

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

(2a + 3b)(2c - 3d)=(2a - 3b)(2c + 3d)(2a + 3b) / (2a - 3b) = (2c + 3d) / (2c - 3d)4ac -6ad +6bc -9bd = 4ac + 6ad - 6bc - 9bd12bc = 12 adbc = ada/b = c/dHence the option is correct

Multiple choice mathematics and statistics set language de morgan's law for set theory complement of sets different sets de morgan's law

If $U = \left {x|x\epsilon N, x < 5\right }, A = \left {x|x\epsilon N, x\leq 2\right }$ then $A' =$ __________.

  1. $\left \{1, 2\right \}$
  2. $\left \{1, 2, 3, 4, 5\right \}$
  3. $\left \{3, 4\right \}$
  4. $\left \{3, 4, 5\right \}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$ \cup  = { x|x \in N,x < 5} $

${\rm A} = \left| {x|x \in N,x \leqslant 2} \right|$
then $A' = { 3,4} $
part $C$ is correct answer.

Multiple choice maths four operations whole number operations on the number line whole numbers on number line introduction to multiples and factors

Let $x$ be a real variable, and let $3 < x < 4.$ Which of the following values, $x$ might have?

  1. $-3.1$
  2. $\sqrt{11}$
  3. $\sqrt{\dfrac{10}{11}}$
  4. $4.1$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$x$ can have any values between $3$ and $4$ so it can be rational or irrational. 
option $c$=$0.9$  and options $A$ and $D$ are outside the given range.
Hence, correct answer is option $B=\sqrt11=3.3$.

Multiple choice maths cube and cube root estimation of cube root estimation of cube roots cubes and cube roots

If $x = \sqrt [3]{a + \sqrt {a^{2} - b^{3}}} + \sqrt [3]{a - \sqrt {a^{2} - b^{3}}}$ then $x^{3} + 3bx = $ ____________.

  1. $2a$
  2. $2b$
  3. $3a$
  4. $4a$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Given,
$x = \sqrt [3]{a + \sqrt {a^{2} - b^{3}}} + \sqrt [3]{a - \sqrt {a^{2} - b^{3}}}$.......(1).
Now cubing both sides we get,
$x^3=a+\sqrt{a^2-b^3}+a-\sqrt{a^2-b^3}-3$$\sqrt [3]{a + \sqrt {a^{2} - b^{3}}}  \sqrt [3]{a - \sqrt {a^{2} - b^{3}}}$$( \sqrt [3]{a + \sqrt {a^{2} - b^{3}}} + \sqrt [3]{a - \sqrt {a^{2} - b^{3}}})$
or, $x^3=2a-3bx$ [ Using (1)and $ (\sqrt [3]{a + \sqrt {a^{2} - b^{3}}})(\sqrt [3]{a - \sqrt {a^{2} - b^{3}}})=\sqrt[3]{a^2-(a^2-b^3)}=b$]
or, $x^3+3bx=2a$.
Multiple choice business maths limits and continuity of a function graphs of the form y=ax^2+bx+c some more types of functions functions and their graphs

If $|z-1|+ |z+3| \le 8$, then the range of values of $|z-4|$ is

  1. $(0, 7)$
  2. $(1,8)$
  3. $[1,9]$
  4. $[2,5]$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The equation |z-1| + |z+3| = 8 represents an ellipse with foci at 1 and -3. The distance between foci is 4. The major axis 2a = 8, so a = 4. The center is at -1. The vertices are at -1 +/- 4, i.e., 3 and -5. The range of |z-4| is the distance from 4 to the ellipse. The minimum distance is |3-4| = 1 and the maximum is |-5-4| = 9.

Multiple choice lines in planes applications of determinants inverse of a matrix and linear equations matrix algebra maths

If $\mathrm{a}\neq b\neq \mathrm{c}$ and if $ax+by+\mathrm{c}=0\  bx+cy+\mathrm{a}=0$ and $cx+ay+b=0$ are concurrent, 

then find the value of 
$ 2^{\mathrm{a}^{2}b^{-1}\mathrm{c}^{-1}}2^{b^{2}\mathrm{c}^{-1}\mathrm{a}^{-1}}2^{\mathrm{c}^{2}\mathrm{a}^{-1}b^{-1}}$

  1. 1

  2. 4

  3. 8

  4. 16
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For the lines to be concurrent, the determinant of the coefficients must be zero: |a, b, c; b, c, a; c, a, b| = 0. This expands to -(a^3 + b^3 + c^3 - 3abc) = 0, implying a+b+c=0 or a=b=c. Given a!=b!=c, we must have a+b+c=0. The exponent is (a^2/bc + b^2/ca + c^2/ab) = (a^3+b^3+c^3)/abc. Since a^3+b^3+c^3 = 3abc when a+b+c=0, the exponent is 3. Thus, 2^3 = 8.

Multiple choice maths multiply and divide division trick division division of numbers

Let $Q = \dfrac{x}{y}$ where $x$ and $y$ are real numbers. If both $x$ and $y$ are increased equally then

  1. $Q$ will increase
  2. $Q$ will decrease
  3. $Q$ will remain the same

  4. none of the above
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

If x/y = 1, adding 1 to both gives 2/2 = 1 (no change). If x/y < 1, adding a constant increases the ratio. If x/y > 1, adding a constant decreases the ratio. No single outcome is universal.

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

For rational numbers, $x$ and $y,$ if $x > y,$ then which of the following is always a positive rational number?

  1. $ y - xy$
  2. $ xy-x$
  3. $ y-x$
  4. $ x- y $
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
If $x>y$

$y-xy\rightarrow $ can be both positive and negative.

Example coside $x>1$ & $y>0$

$\left(y-xy\right)<0$

$xy-x\rightarrow $ can be both positive and negative 

$y-x\rightarrow $ always negative

$\boxed {x-y\rightarrow always\ positive\ since\ x>y}$
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 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.