Mathematics · Quantitative Aptitude

Algebraic Equations and Expressions

142 Questions

Algebraic equations and expressions test mathematical skills in solving for unknown variables. Questions involve linear equations, ratios, and evaluating complex expressions. This topic forms a core component of quantitative aptitude sections in various competitive exams.

Linear equationsEvaluating expressionsRatio proportionsExponential equationsHypergeometric functions

Algebraic Equations and Expressions Questions

Multiple choice maths powers and exponents scientific notation use of exponents power of 10

If $ { 9 }^{ x-1 }={ 3 }^{ 2x-1 }-486 $,then the value of x is:

  1. $\dfrac{7}{2}$

  2. 4

  3. 1

  4. 0
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$9^{x-1}=3^{2x-1}-486$
$3^{2x-1}=9^{x-1}t486$
$\dfrac{3^{2x}}{3}=3^{2x-2}+486$
$3^{2x}=\dfrac{3(3^{2x})}{3^{z}}t(486)3$
$3^{2q}=3^{2x-1}+1458$
$2177=729+1458$
$3^{7}=3^{6}+1458$
$2x=7$
$x=\dfrac{7}{2}$
Multiple choice maths powers and exponents scientific notation use of exponents power of 10

If $8^x = 16^{x-1}$, find $x $.

  1. $\dfrac{1}{8}$
  2. $\dfrac{1}{2}$
  3. $2$
  4. $4$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Given, $8^x=(16)^{x-1}$

Left hand side, $8^x=(2^3)^x=2^{3x}$
Right hand side $16^{x-1}=(2^4)^{x-1}=2^{4(x-1)}=2^{(4x-4)}$
Equating both sides, we get
$2^{3x}=2^{{4x-4}}$
As bases are equal, then powers must be equal, so
$\Rightarrow 3x=4x-4$
$ \Rightarrow x=4$

Multiple choice maths powers and exponents scientific notation use of exponents power of 10

If $4^{2x + 2} = 64$, then calculate the value of $x $.

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

Given is $4^{2x+2}=64$

LHS: 
$\Rightarrow 4^{ 2x+2 }=(2^{ 2 })^{ 2x+2 }\ \Rightarrow { 2 }^{ 4x+4 }$
RHS: 
$\Rightarrow 64=2^6$

Now, LHS $=$ RHS
$\Rightarrow { 2 }^{ 4x+4 }={ 2 }^{ 6 }$
As bases are equal, so powers must be equal,
$\Rightarrow 6=4x+4$
$\Rightarrow 4x=2\ \Rightarrow x=\dfrac { 1 }{ 2 } $

Multiple choice maths powers and exponents scientific notation use of exponents power of 10

If $64^{x} = 4^{x^{2} - 4}$, then find the value of $x$.

  1. $x = 4$ or $x = -1$
  2. $x = -4$ or $x = 1$
  3. $x = 10$
  4. $x = \sqrt {20}$
  5. $x = 3$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
  • ${ 64 }^{ x }={ 4 }^{ 3x }={ 4 }^{ { x }^{ 2 }-4 }$ , by equating powers , we get,
  • ${ x }^{ 2 }-4 = 3x$ , which implies ${ x }^{ 2 }-3x-4 = 0$
  • $\Rightarrow x^2-4x+x-4=0$
  • $\Rightarrow x(x-4)+1(x-4)= 0 $
  • $\Rightarrow (x-4)(x+1)=0$
  • The roots are $x=4,-1$
Multiple choice maths powers and exponents scientific notation use of exponents power of 10

If $2^{3x - 2} = 16$, then calculate the value of $x $.

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

Given, ${ 2 }^{ 3x-2 }=16$
$\Rightarrow { 2 }^{ 3x-2 }=16={ 2 }^{ 4 }$

As bases are equal, there powers must be equal.
$\therefore 3x-2=4$
$\therefore 3x=6$
$\therefore x=2$

Multiple choice reciprocal equations theory of equations maths

Find $x$,  $2^{x^2}:2^{2x}=8:1$

  1. $3,-1$
  2. $3,1$
  3. $-3,-1$
  4. $-3,1$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given, $2^{x^2}:2^{2x}=8:1$

$\Rightarrow \dfrac {2{x^2}}{2^{2x}}=\dfrac {8}{1}$
$\Rightarrow 2^{x^2}=8.2^{2x}$
$\Rightarrow 2^{x^2}=2^3.2^{2x}$
$\Rightarrow 2^{x^2}=2^{2x+3}$
$\Rightarrow x^2=2x+3$ ....As bases are equal, powers must be equal
$\Rightarrow x^2-2x-3=0$
$\Rightarrow (x-3)(x+1)=0$
$\therefore x=3,-1$

Multiple choice maths complex numbers and linear inequations identities of complex numbers powers of imaginary unit i algebra of complex numbers

If $(i^{413})(i^x)=1$, then determine the one possible value of x.

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

${ i }^{ 413 }{ i }^{ x }=1$

$\Rightarrow \quad { i }^{ 413 }=1$
now, $\left( 413+x \right) $ must be a multiple of 4 becouse ${ i }^{ 4 }=1$
$\therefore \quad \left( 413+3 \right) $ is divisible by $4$
                                     hence $x=3$

Multiple choice maths ratio, proportion and unitary method more on proportion terms related to proportion proportion

If 150 is the third proportional to 6 and x; find the value of x.

  1. 30

  2. 60

  3. 63

  4. 34
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

If a : b :: b : c, then we say that a, b, c are in continued proportion, and

c is the third proportional of a and b.





Here, $ {b}^{2} = ac $ 





Given  the third proportional of $ 6, x $ is 150 

$ => {x}^{2} = 6 \times 150 = 6 \times 6 \times 25 $

$ => x = \sqrt { 6 \times 6 \times 25 } $

$ x = 6 \times 5 = 30 $


Multiple choice maths squares and square roots finding the square of a number finding square of a number patterns in square numbers

If $\sqrt{\left(12+\sqrt{12+\sqrt{12+....}}\right)}=x$, then the value of x is ____________.

  1. $3$
  2. $4$
  3. $6$
  4. Greater than $6$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\sqrt{12+\sqrt{12+\sqrt{12+....}}} = x$.........................(1)

$\sqrt{12+x}=x$
$(12+x)=x^2$
$x^2-x-12=0$
$(x-4)(x+3)=0$
$ x=4$ or $x=-3$
since x is a positive number (eq 1)
$x=4$