Mathematics · Quantitative Aptitude

Algebraic Equations and Expressions

152 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 composition of ratios types of ratios ratio and proportions ratio and proportion maths

The value of $x$ is ____ if $(3x + 1) : (5x - 4)$ is the duplicate ratio of $5 : 6$

  1. $2$
  2. $4$
  3. $8$
  4. $6$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$(3x + 1) : (5x - 4)$ is the duplicate ratio of $5 : 6$.
Also, the duplicate ratio of $5 : 6$ is $5^{2} : 6^{2} = 25 : 36$
$\therefore \dfrac {3x + 1}{5x - 4} = \dfrac {25}{36}$
$\therefore 108 x + 36 = 125x - 100$
$\therefore 125x - 108x = 36 + 100$
$\therefore 17x = 136$
$\therefore x = \dfrac {136}{17} = 8$

Multiple choice composition of ratios types of ratios ratio and proportions ratio and proportion maths

If $(4x + 3) : (9x + 10)$ is the triplicate ratio of $3 : 4$, then the value of x is ___

  1. $6$
  2. $12$
  3. $5$
  4. $4$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The triplicate ratio of $3 : 4$ is $3^{3} : 4^{3} = 27 : 64$
$\therefore \dfrac {4x + 3}{9x + 10} = \dfrac {27}{64}$
$\Rightarrow 256x + 192 = 243x + 270$
$\Rightarrow 256x - 243x = 270 - 192$
$\Rightarrow 13x = 78$
$\Rightarrow x = \dfrac {78}{13} = 6$
$\Rightarrow x = 6$

Multiple choice composition of ratios types of ratios ratio and proportions ratio and proportion maths

The value of $x$ is ____ if $(x - 4) : (x + 2)$ is the triplicate ratio of $1 : 2$

  1. $\dfrac {34}{7}$
  2. $\dfrac {7}{34}$
  3. $\dfrac {30}{7}$
  4. $\dfrac {7}{30}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The triplicate ratio of $1 : 2$ is $1^{3} : 2^{3} = 1 : 8$
$\therefore \dfrac {x - 4}{x + 2} = \dfrac {1}{8}\Rightarrow 8x - 32 = x + 2$
$\Rightarrow 8x - x = 2 + 32$
$\Rightarrow 7x = 34$
$\Rightarrow x = \dfrac {34}{7}$

Multiple choice

22x-1 = (16/√8)6

  1. 7

  2. 0

  3. 8

  4. 0.25
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Simplify the right side: (16 / sqrt(8))^6 = (16 / (2*sqrt(2)))^6 = (8 / sqrt(2))^6 = (4*sqrt(2))^6. This simplifies to (2^2 * 2^0.5)^6 = (2^2.5)^6 = 2^15. Setting 2^(2x-1) = 2^15, we get 2x-1 = 15, so 2x = 16, x = 8.

Multiple choice

3x+1 = (9/√3)

  1. 2.5

  2. 0.5

  3. 3.5

  4. -0.5
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

3^(x+1) = 9 / sqrt(3) = 3^2 / 3^0.5 = 3^(2 - 0.5) = 3^1.5. Thus, x + 1 = 1.5, which means x = 0.5.

Multiple choice

√x

  1. 1/x

  2. X1/2
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The square root of x is equivalent to x raised to the power of 1/2.

Multiple choice maths remainder and factor theorems linear and synthetic method of division factorising expressions using factor theorem division algorithm for polynomials

$p(x)=(x^2-10x-24)$ , when divided by $x+2$ and $x\neq -2$ gives the quotient $Q$. Find $Q$.

  1. $x -22$
  2. $x-12$
  3. $x+12$
  4. $x+22$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
 Quotient  $x - 12$
 $x+2$   $x^2-10x-24$  $x^2+2x$$- $   $-$
         $ -12x-24$        $ -12x-24$       $+$        $+$------------------------------                 $ 0$

Using division algorithm method we get the value of $Q = x - 12$.

Multiple choice maths power and exponent power of powers laws of exponents and powers law of indices

If $(\sqrt{2})^x + (\sqrt{3})^x = (\sqrt{13})^{\frac{x}{2}}$, then the value of $x$ is ___.

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

$(\sqrt2)^x+(\sqrt3)^x=(\sqrt{13})^{\frac{x}{2}}$

$\Rightarrow 2^{\frac{x}{2}}+3^{\frac{x}{2}}=13^{\frac{x}{4}}$

$x$ should be the multiple of $4$.
If we put $x=4$
L.H.S$: 2^2+3^2=13$
and R.H.S $: 13^{\frac{4}{4}}=13$
$\therefore x=4$

Multiple choice maths power and exponent power of powers laws of exponents and powers law of indices

Find $x:[3+\left { 2+(1+x^{2}) \right }^{2}]^{2}=144$

  1. $1$
  2. $0$
  3. $5$
  4. $6$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Given,

$\left [ 3+\left\{2+\left(1+x^2\right)\right\}^2 \right ]^2=144$

taking square root on both sides, we get,

$\left [ 3+\left\{2+\left(1+x^2\right)\right\}^2 \right ]=12$

$\left\{2+\left(1+x^2\right)\right\}^2=12-3=9$

again taking square root on both sides, we get,

$2+(1+x^2)=3$

$1+x^2=3-2=1$

$x^2=1-1=0$

$\therefore x=0$
Multiple choice maths power and exponent power of powers laws of exponents and powers law of indices

Find the sum of all values of $x$, so that $16^{\left(x^{2}+3x-1\right)}=8^{\left(x^{2}+3x+2\right)}$.

  1. $0$
  2. $3$
  3. $-3$
  4. $-5$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given, $16^{(x^2 + 3x -1)} = 8^{(x^2 + 3x + 2)}$


As, 16 = $2^4 and \ \ \ 8 = 2^3$


$2^{4(x^2 + 3x -1)} = 2^{3(x^2 + 3x + 2)}$

So, we can write 

${4(x^2 + 3x -1)} = {3(x^2 + 3x + 2)}$

${4x^2 + 12x - 4} = {3x^2 + 9x + 6}$

${4x^2 + 12x - 4} - {(3x^2 + 9x + 6)} = 0$
$4x^2 + 12x - 4 - 3x^2 - 9x - 6 = 0$

$x^2 + 3x - 10  = 0$

$x^2 + 5x - 2x - 10  = 0$

$x(x + 5) - 2(x + 5)  = 0$

$(x - 2)(x + 5)  = 0$

So, $x = 2, - 5$

Sum of values of $x = 2 + (-5) = -3$

Multiple choice maths operations adding and subtracting numbers using place value addition & subtraction mental additions and subtractions

A student was asked to multiply a certain number by 3 and to add 2 to it, but he divided the number by 3 and subtracted 2 from it and got the number 1. The correct answer is

  1. 9

  2. 25

  3. 29

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

Let the certain number be x
As per problem,
$\Rightarrow \frac { x }{ 3 } -2=1$
 $\Rightarrow \frac { x-6 }{ 3 } =1$
 $\Rightarrow x-6=3$
 $\Rightarrow x=9$
The correct answer will be $9\times 3+2=27+2=29$