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Multiple choice reciprocal equations theory of equations maths

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

  1. Quadratic equation

  2. Linear equation

  3. Reciprocal Equation

  4. None
Reveal answer Fill a bubble to check yourself
C Correct answer
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.

Multiple choice reciprocal equations theory of equations maths

What is a reciprocal equation?

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

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

Multiple choice reciprocal equations theory of equations maths

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

  1. $x=2$
  2. $x=-2$
  3. $x=1$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
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.

Multiple choice reciprocal equations theory of equations maths

Which of the following is not a reciprocal function?

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

  4. None of the above
Reveal answer Fill a bubble to check yourself
C Correct answer
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.
$f(x)=x^{-1}$

Multiple choice reciprocal equations theory of equations maths

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

  1. $(\dfrac{1}{2},2)$
  2. $R$
  3. $\phi$
  4. $(1/3,2)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

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


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


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

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

Multiple choice reciprocal equations theory of equations maths

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

  1. $\dfrac{-1}{b}$
  2. $\dfrac{1}{b^2}$
  3. $\sqrt b$
  4. $\dfrac{1}{b}$
Reveal answer Fill a bubble to check yourself
D Correct answer
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.

Multiple choice reciprocal equations theory of equations maths

A ............ equation is one which remains the same when $x$ is replaced by $\dfrac{1}{x}$.

  1. Reciprocal equation

  2. Radical equation

  3. Exponential equation

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

(Originally) an equation whose roots can be divided into pairs of numbers, each the reciprocal of the other is called Reciprocal equation

(Equivalently) an equation which is unchanged if the variable $x$ is replaced by its reciprocal $\dfrac{1}{x}$ is known as reciprocal equation.
Hence, option A is correct.

Multiple choice reciprocal equations theory of equations maths

The roots of equation $2x^4-9x^3+14x^2-9x+2=0$ are

  1. $(1,2,3,4)$
  2. $\left(1,1,\dfrac{1}{2},2\right)$
  3. $\left(1,\dfrac{1}{3},3,1\right)$
  4. $\left(0,1,1,\dfrac{1}{2}\right)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

We can see that in the giving equation, the multiplication of roots is $1$ 

i.e multiplication of roots $=\dfrac{e}{a}$ in the equation $ax^4 +bx^3 + cx^2 +dx  +e  =0$  
$\Rightarrow $ $\dfrac{e}{a}  = \dfrac{2}{2}  = 1 $
Now, sum of the roots is $\dfrac{-b}{a}  = -(\dfrac{-9}{2}) =  \dfrac{9}{2}$

$\Rightarrow $ it is only possible in option B.
Hence, option B is correct.

Multiple choice reciprocal equations theory of equations maths

The domain of reciprocal equation is :

  1. $R$
  2. $R-\{0\}$
  3. $Q$
  4. $R^+$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

We know that in reciprocal equation, if one value of $x$ is $a$ then the other value of $x$  will be $\dfrac{1}{a}$

But, if $a = 0$ then $\dfrac{1}{a}$ does not exist  

Therefore, $0$ can not be in the domain of a reciprocal equation.
So, domain of reciprocal equation will be set of all real numbers excluding $0$.
Hence, option B is correct.

Multiple choice reciprocal equations theory of equations maths

The range of reciprocal equation is:

  1. $R$
  2. $R-{0}$
  3. $R^+$
  4. $Q$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The range is defined as the set of all output values or the set of all those possible values which appear on Y axis in the graph of given function. 

The domain and range of a reciprocal function are all the real number except for zero. 
This is because reciprocal of $0$ i.e. is undefined.
The range of reciprocal equation is $R-0$