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

If $x+\cfrac{1}{x}=4$, then ${x}^{4}+\cfrac{1}{{x}^{4}}$ is equal to

  1. $196$
  2. $194$
  3. $192$
  4. $190$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Given,

$x+\dfrac{1}{x}=4$

squaring on both sides, we get,

$x^2+\dfrac{1}{x^2}+2=16$

$x^2+\dfrac{1}{x^2}=16-2=14$

squaring on both sides, we get,

$x^4+\dfrac{1}{x^4}+2=196$

$x^4+\dfrac{1}{x^4}=196-2=194$
Multiple choice maths rational numbers proof of irrationality of numbers proofs of irrationality introduction to irrational numbers

Find x if $\dfrac{\sqrt{3x+1}+\sqrt{3x-6}}{\sqrt{3x+1}-\sqrt{3x-6}}=7$.

  1. $2$
  2. $5$
  3. $3$
  4. $7$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$\dfrac { \sqrt { 3x+1 } +\sqrt { 3x-6 }  }{ \sqrt { 3x+1 } -\sqrt { 3x-6 }  } =7$

Rotational give :-

$\dfrac { \left( 3x+1 \right) +\left( 3x-6 \right) +2\sqrt { \left( 3x+1 \right) \left( 3x-6 \right)  }  }{ \left( 3x+1 \right) -\left( 3x-6 \right)  } =7$

$\Rightarrow 6x-5+2\sqrt { \left( 3x+1 \right) \left( 3x-6 \right)  } =49$

$\Rightarrow 2\sqrt { \left( 3x+1 \right) \left( 3x-6 \right)  } =-6x+54$

$\Rightarrow \sqrt { \left( 3x+1 \right) \left( 3x-6 \right)  } =-3x+27$
which gives, $x=5$
Multiple choice sum to infinite terms of a gp sequence, progression and series maths

The value of x that satisfies the relation 
$x=1-x+{ x }^{ 2 }-{ x }^{ 3 }+{ x }^{ 4 }-{ x }^{ 5 }+........\infty $ 

  1. $2cos{ 3 }6^{ \circ }$
  2. $2cos144^{ \circ }$
  3. $2sin18^{ \circ }$

  4. none
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The series $1-x+x^2-....$ form $GP$ with $a=1 ,r=-x$

Sum of infinte GP is $x=\dfrac{a}{1-r}\x=\dfrac{1}{1+x}\x+x^2=1\x^2+x-1=0$
By quadratic formulae 
$x=\dfrac{-1\pm\sqrt{1+4}}2\2\dfrac{-1\pm\sqrt5}{4}\2\cos 36^{\circ}$

Multiple choice maths geometric sequences sum of terms of g.p sum of n terms of an gp summing geometric series

The value of $x$ that satisfies the relation $x=1-x+x^{2}-x^{3}+x^{4}-x^{5}+.\infty$ if $|x|<1$ 

  1. $\dfrac{-1\pm\sqrt5}{2}$
  2. $\dfrac{-1\pm3i}{2}$
  3. $0$
  4. $none$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Here the first term is $a$ 
Common ratio is given by $-x$
The sum of infinite series is of an GP is given by $\dfrac{a}{1-r}\\x=dfrac{1}{1+x}\\x^2+x=1\\x^2+x-1=0$
Using quadratic formulae $\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$
Here $a=1b=1c=-1$
$\implies x=\dfrac{-1\pm\sqrt{1+4}}{2}\\x=\dfrac{-1\pm\sqrt{5}}{2}$
Multiple choice maths parts and whole multiplication of a fraction multiplication of a fractions multiplication of fraction finding the whole when a fraction is given

Find $x$ if $\left (\cfrac {1}{2}\times \cfrac {1}{3}\right )\times \cfrac {1}{4}= x \times \left (\cfrac {1}{3}\times \cfrac {1}{4}\right )$.

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

The given question shows the associative property of multiplication i.e. 

$\left( a\times b \right) \times c=a\times \left( b\times c \right) $
$\left (\cfrac {1}{2}\times \cfrac {1}{3}\right )\times \cfrac {1}{4}= x \times \left (\cfrac {1}{3}\times \cfrac {1}{4}\right )$
Therefore, $x=\dfrac{1}{2}$
Hence, the correct answer is option C.

Multiple choice maths equation reducing simple equations to simpler form solving linear equations solution of a linear equation in one variable

If $a\neq 0$ and $\dfrac{5}{x}=\dfrac{5+a}{x+a}$, what is the value of $x$?

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

Given:

$a$ $\neq$ $0$ and $\dfrac {5}{x}$ $=$ $\dfrac {5 \space + \space a}{x \space + \space a}$
To find the value of $x$,
$\Rightarrow \dfrac {5}{x}$ $=$ $\dfrac {5 \space + \space a}{x \space + \space a}$
$\Rightarrow 5$ $\times$ $(x$ $+$ $a)$ $=$ $x$ $\times$ $(5$ $+$ $a)$
$\Rightarrow 5x$ $+$ $5a$ $=$ $5x$ $+$ $xa$
Get the co-efficients of $'x'$ on one-side,
$\Rightarrow 5x$ $+$ $xa$ $-$ $5x$ $=$ $5a$
$\Rightarrow xa$ $=$ $5a$
As $(a$ $\neq$ $0)$,    $x$ $=$ $5$
Therefore, the value of $'x'$ is $'5'$.

Multiple choice maths equation reducing simple equations to simpler form solving linear equations solution of a linear equation in one variable

If $t = x+2$, find the value of x .If $2t-7 +\dfrac{3(t-1)}{2}=3$

  1. $\frac{23}{7}$
  2. $\frac{3}{7}$
  3. $\frac{9}{7}$
  4. $\frac{37}{7}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$(2t-7)+(\frac{3t-3}{2})=3\\(\frac{4t-14+3t-3}{2})=3\\7t-17=6\\\therefore 7t=6+17\\t=(\frac{23}{7})\\then x+2=(\frac{23}{7})\\\therefore x=(\frac{23}{7})-2\\=(\frac{23-14}{7})\\=(\frac{9}{7})$

Multiple choice maths equation reducing simple equations to simpler form solving linear equations solution of a linear equation in one variable

Find the value of $a$, if $x = 0.5$ is a solution of equation $ax^{2}\, +\, (a\, -\, 1)\,
x\, +\, 3\, =\, a$.

  1. $24$
  2. $15$
  3. $10$
  4. $8$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Given equation is $ax^2+(a-1)x+3=a$
The equation can be written as
$ax^2+(a-1)x+3-a=0$
Since $x=0.5$ is a solution of the equation, 
$a(0.5)^2 + (a-1)(0.5) + 3 - a = 0$
$\Rightarrow 0.25a + 0.5a - 0.5 + 3 - a = 0$
$\Rightarrow   (0.25 + 0.5 -1)a - 0.5+3 = 0$
$\Rightarrow  -0.25a + 2.5 = 0$
$\Rightarrow  0.25a = 2.5$
$\Rightarrow  a = \dfrac{2.5}{0.25}$
$\Rightarrow a = 10$