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 business maths pair of straight lines condition for perpendicular and coincident lines and bisectors of angles pair of straight lines through origin analytical geometry

The equation $\displaystyle ax^{3}-9yx^{2}-y^{2}x+4y^{3}=0 $ represents three straight lines. If two of the lines are perpendicular to each other, then the value of $a$ is:

  1. 5

  2. -5

  3. 4

  4. -4
Reveal answer Fill a bubble to check yourself
A,D Correct answer
Explanation

The given equation is $ax^3 - 9yx^2 -y^2x + 4y^3 = 0$, which represents three straight lines.


We can see that all the lines passes from origin $(0,0)$

Let's assume the lines are given by equation $y = mx$

Putting $y = mx$ in the given equation of three straight lines, we get,

$\Rightarrow ax^3- 9(mx)(x^2) - (mx)^2x + 4(mx)^3 = 0$

$\Rightarrow (a - 9m - m^2 + 4m^3) x^3 = 0$

$\Rightarrow 4m^3 -m^2 -9m +a = 0$ ....$(1)$

This equation in $m$ has three roots, $m _1$, $m _2$ and $m _3$, which are three slopes of three lines respectively.

If the two lines are perpendicular then let's assume $m _1.m _2 = -1$,

Product of roots in equation $(1)$ is $m _1.m _2.m _3 = \dfrac{-a}{4}$

Hence $m _3 = \dfrac{a}{4}$

also from equation $(1)$, $m _1 + m _2 + m _3 = \dfrac{1}{4}$

$\Rightarrow m _1m _2 + m _3(m _1 + m _2) = \dfrac{-9}{4}$

$\Rightarrow m _1 + m _2 = \dfrac{1-a}{4}$

$\Rightarrow a(1-a) = -20$

By Solving the above equation we get $a = 5, -4$

Multiple choice business maths pair of straight lines condition for perpendicular and coincident lines and bisectors of angles pair of straight lines through origin analytical geometry

Equation $\displaystyle ax^{3}-9yx^{2}-y^{2}x+4y^{3}=0$ represents three straight lines. If two of the lines are perpendicular to each other then the value of a is

  1. 5

  2. -5

  3. 4

  4. -4
Reveal answer Fill a bubble to check yourself
A,D Correct answer
Explanation

The given equation is $ax^3 - 9yx^2 -y^2x + 4y^3 = 0$, which represents three straight lines.


We can see that all the lines passes from origin $(0,0)$

Let's assume the lines are given by equation $y = mx$

Putting $y = mx$ in the given equation of three straight lines, we get,

$\Rightarrow ax^3- 9(mx)(x^2) - (mx)^2x + 4(mx)^3 = 0$

$\Rightarrow (a - 9m - m^2 + 4m^3) x^3 = 0$

$\Rightarrow 4m^3 -m^2 -9m +a = 0$ ....$(1)$

This equation in $m$ has three roots, $m _1$, $m _2$ and $m _3$, which are three slopes of three lines respectively.

If the two lines are perpendicular then let's assume $m _1.m _2 = -1$,

Product of roots in equation $(1)$ is $m _1.m _2.m _3 = \dfrac{-a}{4}$

Hence $m _3 = \dfrac{a}{4}$

also from equation $(1)$, $m _1 + m _2 + m _3 = \dfrac{1}{4}$

$\Rightarrow m _1m _2 + m _3(m _1 + m _2) = \dfrac{-9}{4}$

$\Rightarrow m _1 + m _2 = \dfrac{1-a}{4}$

$\Rightarrow a(1-a) = -20$

By Solving the above equation we get $a = 5, -4$

Multiple choice business maths pair of straight lines condition for perpendicular and coincident lines and bisectors of angles pair of straight lines through origin analytical geometry

The pair of lines represented by $3ax^{2}+5xy+\left ( a^{2}-2 \right )y^{2}= 0$ and $\perp $ to each other for

  1. two values of $a$
  2. for all $a$
  3. for one value of $a$
  4. for no values of $a$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

 Using fact: Pair of lines $\displaystyle Ax^{2}+2hxy+By^{2}=0$ are 


$\displaystyle \perp $ to each other if $\displaystyle A+B=0$ 

$\displaystyle \Rightarrow 3a+a^{2}-2=0 $ $\displaystyle \Rightarrow a^{2}+3a-2=0 $ $\displaystyle\Rightarrow $ There exist two value of a as $\displaystyle D> 0$

Multiple choice business maths pair of straight lines condition for perpendicular and coincident lines and bisectors of angles pair of straight lines through origin analytical geometry

Two of the lines represented by $x^{3}-6x^{2}y+3xy^{2}+dy^{3}=0$ are perpendicular for

  1. all real values of $d$
  2. two real values of $d$
  3. three real values of $d$
  4. no real value of $d$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let $m _{ 1 },{ m } _{ 2 },{ m } _{ 3 }$ be the slopes of the three lines represnted by the given equation such that ${ m } _{ 1 }{ m } _{ 2 }=-1$


We have $\displaystyle m _{ 1 }{ m } _{ 2 }{ m } _{ 3 }=-\frac { 1 }{ d } $ so that 

$\displaystyle { m } _{ 3 }=\frac { 1 }{ d } $

Since $y={ m } _{ 3 }x\Rightarrow x=dy$ satisfies the given equation, we get

${ d }^{ 3 }-6{ d }^{ 2 }+3d+d=0\Rightarrow d\left( { d }^{ 2 }-6d+4 \right) =0$

If $d=0,$ the given equation represents the line $x=0$ and $x^2-6xy+3y^2=0$ which are not perpendicular

$\therefore d\neq 0$ and $\displaystyle d^2-6d+4=0\Rightarrow d=\frac { 6\pm \sqrt { 36-16 }  }{ 2 } =3\pm \sqrt { 5 } $

which gives two real values of $d$

Multiple choice business maths pair of straight lines condition for perpendicular and coincident lines and bisectors of angles pair of straight lines through origin analytical geometry

The pair of lines represented by $3ax^{2}+5xy+(a^{2}-2)y^{2}=0$ are perpendicular to each other for 

  1. two values of $a$
  2. for all values of $a$
  3. for one value of $a$
  4. for no value of $a$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$3ax^2+5xy+(a^2-2)y^2=0$   are perpendicular to each other


$\therefore$ coefficient of $x^2+ $coefficient of $y^2=0$

$\Rightarrow 3a+a^2-2=0$

$\Rightarrow a^2+3a-2=0$


$\Rightarrow a=\dfrac{-3\pm\sqrt{9+8}}{2}$

so $a=\dfrac{-3+\sqrt{17}}{2}$ and $a=\dfrac{-3-\sqrt{17}}{2}$

Multiple choice absolute value real numbers (rational and irrational numbers) real numbers basic algebra maths

The value of x on simplifying $x -2 |x|=-3$ is

  1. -1 or 3

  2. 1 or -3

  3. -1 or -3

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

$x-2|x| = -3$

There two cases possible either $x$ is positive or$x$ is negative
We have to solve both cases
Firstly taking x as positive
$x-2x=-3$
$x=3$
Now taking $x$ as negative
$x-2(-x)=-3$
$3x=-3$
$x=-1$
So we get two values of $x$ that is -1 and 3
 So correct answer will be option A

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

Find the value of $x$ if  $a$ and $ b$  are in direct proportion

  $a$ $2$ $3$ $4$ $5$
  $b$  $14$ $21$ $x$ $35$
  1. $16$
  2. $25$
  3. $27$
  4. $28$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$a$ and $b$ are in direct proportion
$\therefore \dfrac {a}{b} = k = \dfrac {2}{14} = \dfrac {3}{21} = \dfrac {1}{7}$
$\therefore \dfrac {4}{x} = \dfrac {1}{7} \Rightarrow x = 28$.

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

The correct dosage of adult over-the-counter medicine a child can receive is given by a formula by Clark. The child's weight, in pounds, is divided by $150$, and the result is multi pounds lied by the adult dose of the medicine. A mother need to give her daughter acetaminophen, which has an adult dose of $ 1000$ milligrams. She does not know her daughter's exact weight, but she knows the weight is  and between $75 $ and $90 $pounds. Find the range of correct dosage, d, in milligrams of acetaminophen the daughter could receive.

  1. $50$
  2. $500$
  3. $1000$
  4. $1600$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Clark's formula is (weight/150) * adult dose. For 75 lbs: (75/150) * 1000 = 500 mg. For 90 lbs: (90/150) * 1000 = 600 mg. The range is 500 to 600 mg, making 500 the only viable option provided.

Multiple choice maths brackets order operations and algebra using brackets in algebraic expressions order of operations

What is the value of $((x^3-2)\div2^2)\times 4+16$?

  1. $x^3+14$
  2. $x^3-14$
  3. $-x^3+14$
  4. $x^3+16$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$((x^3-2)\div2^2)\times 4+16$
We need to follow BODMAS rule.
=> Brackets (parts of a calculation inside brackets always come first).
=> Orders (numbers involving powers or square roots).
=> Division.
=> Multiplication.
=> Addition.
=> Subtraction.
$=$ $((x^3-2)\div2^2)\times 4+16$
$=$ $\dfrac{x^3-2}{4}\times 4+16$
$=$ $x^3+14$

Multiple choice maths brackets order operations and algebra using brackets in algebraic expressions order of operations

Find the value of the expression using BODMAS rule: $4-x^2\div x +(4\times-(\dfrac{2x^3}{x^2}))-3^2$.

  1. $-9x-12$
  2. $-9x-4x^2$
  3. $-9x-5$
  4. $-4x-5$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$4-x^2\div x +(4\times-(\dfrac{2x^3}{x^2}))-3^2$
We need to follow BODMAS rule.
=> Brackets (parts of a calculation inside brackets always come first).
=> Orders (numbers involving powers or square roots).
=> Division.
=> Multiplication.
=> Addition.
=> Subtraction.
$=$ $4-x^2\div x +(4\times-(\dfrac{2x^3}{x^2}))-3^2$
$=$ $4-x+(4\times -2x)-9$
$=$ $4-x-8x-9$
$=$ $-9x-5$

Multiple choice maths brackets order operations and algebra using brackets in algebraic expressions order of operations

Find the value of $5x[2x(x^2+x^3)-x^3]-4x^2\div x^2-12x$.

  1. $15x^{12}-12x-4$
  2. $15x^4-12x-4$
  3. $15x^4+12x-4$
  4. $5x^4-12x-4$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$5x[2x(x^2+x^3)-x^3]-4x^2\div x^2-12x$
We need to follow BODMAS rule.
=> Brackets (parts of a calculation inside brackets always come first).
=> Orders (numbers involving powers or square roots).
=> Division.
=> Multiplication.
=> Addition.
=> Subtraction.
$=$ $5x[2x(x^2+x^3)-x^3]-4x^2\div x^2-12x$
$=$ $5x[2x^3+2x^4-x^3]-\dfrac{4x^2}{x^2}-12x$
$=$ $5x[x^3+2x^4]-4-12x$
$=$ $5x^4+10x^4-12x-4$
$=$ $15x^4-12x-4$

Multiple choice properties of proportion ratio and proportions ratio and proportion maths

If $x=3+\sqrt 8$ then the value of $x^2+\frac {1}{x^2}$ is

  1. 30

  2. 32

  3. 34

  4. 36
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$x=3+\sqrt 8\Rightarrow \frac {1}{x}=\frac {1}{3+\sqrt 8}\times \frac {3-\sqrt 8}{3-\sqrt 8}=\frac {3-\sqrt 8}{9-8}$
$=3-\sqrt 8$
$x^2+\frac {1}{x^2}=\left (x+\frac {1}{x}\right )^2-2=36-2=34$

Multiple choice properties of proportion ratio and proportions ratio and proportion maths

Here 'x' in the following is : $\dfrac{\sqrt{a+x}+\sqrt{a-x}}{\sqrt{a+x}-\sqrt{a-x}}=b$

  1. $\dfrac{2ab}{(b^2+1)}$
  2. $\dfrac{2ab}{a+b}$
  3. $\dfrac{a+b}{2ab}$
  4. $\dfrac{b^2+1}{2ab}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\dfrac{\sqrt{a+x}+\sqrt{a-x}}{\sqrt{a+x}-{\sqrt{a-x}}}=\dfrac{b}{1}$

$\Rightarrow \dfrac{\sqrt{a+x}}{\sqrt{a-x}}=\dfrac{b+1}{b-1}$

$\Rightarrow \dfrac{a+x}{a-x}=\dfrac{(b+1)^2}{(b-1)^2}$

$\Rightarrow \dfrac{(a+x)+(a-x)}{(a+x)-(a-x)}=\dfrac{(b+1)^2+(b-1)^2}{(b+1)^2-(b-1)^2}$

$\Rightarrow \dfrac{2a}{2x}=\dfrac{2(b^2+1)}{4b}$

$\Rightarrow x=\dfrac{2ab}{b^2+1}$

Multiple choice properties of proportion ratio and proportions ratio and proportion maths

If $x=\cfrac { 2\sqrt { 5 }  }{ \sqrt { 3 } +\sqrt { 5 }  } $, then what is the value of $\cfrac { x+\sqrt { 5 }  }{ x-\sqrt { 5 }  } +\cfrac { x+\sqrt { 3 }  }{ x-\sqrt { 3 }  } $

  1. $\sqrt {5}$
  2. $\sqrt {3}$
  3. $\sqrt {15}$
  4. $2$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$x=\cfrac { 2\sqrt { 5 }  }{ \sqrt { 3 } +\sqrt { 5 }  } \Rightarrow \cfrac { x }{ \sqrt { 3 }  } =\cfrac { 2\sqrt { 5 }  }{ \sqrt { 3 } +\sqrt { 5 }  } $
and $\cfrac { x }{ \sqrt { 5 }  } =\cfrac { 2\sqrt { 3 }  }{ \sqrt { 3 } +\sqrt { 5 }  } $
Applying components and dividendo, we get
$\cfrac { x+\sqrt { 5 }  }{ x-\sqrt { 5 }  } =-\left( 7+2\sqrt { 15 }  \right) $
and
$\cfrac { x+\sqrt { 3 }  }{ x-\sqrt { 3 }  } =9+2\sqrt { 15 } $
$\Rightarrow \cfrac { x+\sqrt { 5 }  }{ x-\sqrt { 5 }  } +\cfrac { x+\sqrt { 3 }  }{ x-\sqrt { 3 }  } =2$

Multiple choice computer and ms office mathematical methods for economics economics

A statement of relationship between two quantities is called a/an ____________.

  1. formula

  2. relationship

  3. equation

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

An equation is a mathematical statement that asserts the equality of two expressions, typically linked by an equals sign.