Tag: reciprocal equations

Questions Related to reciprocal equations

The equation $3x^4-5x^3+3x^2-4x+5=0$ is of the type

reciprocal equations theory of equations maths

  1. Quadratic

  2. Linear

  3. Reiprocal

  4. None of the above

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Correct Option: D
Explanation:

Given equation is $3x^4-5x^3+3x^2-4x+5=0$ .... $(i)$

The maximum power of $x$ in this equation is $ 4$, so this is $4th$ degree equation  
So, $(i)$ is neither linear nor quadratic equation.
In reciprocal equation ($ ax^4 +bx^3 +cx^2 +dx +e = 0 $)  the multiplication of the roots i.e ($\dfrac{e}{a}$) should be  $1$

In the given equation $(i)$, $\dfrac{e}{a}  = \dfrac{5}{3}$  
So the multiplication of roots is not equal to one
Therefore, equation $(i)$ is not a reciprocal equation.

Hence, option D is correct.

$x+\dfrac{1}{x}=2, x^{1680}+\dfrac{1}{x^{1680}}$

reciprocal equations theory of equations maths

  1. $1$

  2. $-1$

  3. $2$

  4. $-2$

Show answer
Correct Option: C
Explanation:

We have,

$x+\dfrac{1}{x}=2$
We know that if $x+\dfrac{1}{x}=2$, then
$x^n+\dfrac{1}{x^n}=2$
Therefore,
$x^{1680}+\dfrac{1}{x^{1680}}=2$

Simplify the reciprocal equation $\dfrac{3}{12}=\dfrac{3}{2x}$

reciprocal equations theory of equations maths

  1. $0$

  2. $3$

  3. $6$

  4. $1$

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Correct Option: C
Explanation:

Given equation is $\dfrac {3}{12}=\dfrac {3}{2x}$

$\Rightarrow 6x=36$
$\Rightarrow x=\dfrac {36}{6}$
$\Rightarrow x=6$
Therefore, the reciprocal of the given function is $6$.

Simplify $\sqrt { 1+{ \left( \cfrac { { x }^{ 4 } }{ -2{ x }^{ 2 } }  \right)  }^{ 2 } } $

reciprocal equations theory of equations maths

  1. $\cfrac { { x }^{ 4 }+1 }{ 2{ x }^{ 2 } } $

  2. $\cfrac{\sqrt{{x}^{2}+1}}{2}$

  3. $\cfrac{{x}^{4}+2{x}^{2}-1}{2{x}^{2}}$

  4. $\cfrac { { x }^{ 4 }-1 }{ 2{ x }^{ 4 } } $

Show answer
Correct Option: B
Explanation:

$\sqrt{1+(\dfrac{x^4}{-2x^2})^2}$


$\Rightarrow \sqrt{1+\dfrac{x^8}{4x^4}}$

$\Rightarrow \sqrt{\dfrac{4x^4+x^8}{4x^4}}$

$\Rightarrow \dfrac{\sqrt{4x^4+x^8}}{2x^2}$

$\Rightarrow\cfrac{\sqrt{{x}^{2}+1}}{2}$

The equation $2x^4-9x^3+14x^2-9x+2=0$ is of the type

reciprocal equations theory of equations maths

  1. Quadratic equation

  2. Linear equation

  3. Reciprocal Equation

  4. None

Show answer
Correct Option: C
Explanation:

Given equation is $2x^4-9x^3+14x^2-9x+2=0$ .... $(i)$

The maximum power of $x$ in this equation is $ 4$, so this is $4th$ degree equation  
So, it is neither linear nor quadratic equation.
In reciprocal equation $ ax^4 +bx^3 +cx^2 +dx +e = 0 $, the multiplication of the roots i.e ($\dfrac{e}{a}$) should be  $1$

In the given equation $(i)$, $\dfrac{e}{a}  = \dfrac{2}{2}$  
So the multiplication of roots is equal to one
Therefore, equation $(i)$ is a reciprocal equation.

Hence, option C is correct.

What is a reciprocal equation?

reciprocal equations theory of equations maths

  1. It involves reciprocal of the given variable.

  2. It involves square of the given variable.

  3. It involves squareroot of the given variable.

  4. It involves square and reciprocal of the given variable.

Show answer
Correct Option: A
Explanation:

Reciprocal is said to be divide $1$ by a number. Reciprocal equation involves reciprocal of the given number.

Determine the root of the equation: $\dfrac{9}{x}-\dfrac{7}{x}=1$

reciprocal equations theory of equations maths

  1. $x=2$

  2. $x=-2$

  3. $x=1$

  4. None of these

Show answer
Correct Option: A
Explanation:

Given reciprocal equation can be written as

$\dfrac{9}{x}=\dfrac{7+x}{x}$
Cancelling out the denominator on both side, we get
$9=7+x$
$\Rightarrow x=2$
Hence, option A is correct.

Which of the following is not a reciprocal function?

reciprocal equations theory of equations maths

  1. $f(x)=\dfrac{1}{x}$

  2. $f(x)={x}^{-1}$

  3. $f(x)=x$

  4. None of the above

Show answer
Correct Option: C
Explanation:

This is the reciprocal function:

$f(x)=\dfrac {1}{x}$
Its domain is the real numbers, except $0$, because $\dfrac {1}{0}$ is undefined.
The reciprocal function can also be written as an exponent.
$f(x)=x^{-1}$

$\cfrac { \left( 2x-1 \right) { \left( x-1 \right)  }^{ 4 }{ \left( x-2 \right)  }^{ 4 } }{ (x-2){ \left( x-4 \right)  }^{ 4 } } \le 0$

reciprocal equations theory of equations maths

  1. $(\dfrac{1}{2},2)$

  2. $R$

  3. $\phi$

  4. $(1/3,2)$

Show answer
Correct Option: A
Explanation:

$\dfrac{(2{x}-1)(x-1)^{4}(x-2)^{4}}{(x-2)(x-4)^{4}} \le 0\implies x\neq 2$


$(2{x}-1)(x-1)^{4}(x-2)^{3}\le 0$


$(x-\dfrac{1}{2})(x-2)^{3}\le 0$

$x\in \bigg(\dfrac{1}{2},2\bigg)$

If $b$ is a root of a reciprocal equation, $f(x)=0$, then another root of $f(x)=0$ is:

reciprocal equations theory of equations maths

  1. $\dfrac{-1}{b}$

  2. $\dfrac{1}{b^2}$

  3. $\sqrt b$

  4. $\dfrac{1}{b}$

Show answer
Correct Option: D
Explanation:

A reciprocal equation is an equation whose roots can be divided into pairs of numbers, each the reciprocal of the other. 

(equivalently) an equation which is unchanged if the variable $x$ is replaced by its reciprocal $\dfrac{1}{x}$ is reciprocal equation.
Since one root is $b$, then the other root is $\dfrac{1}{b}$

Hence, option D is correct.