Tag: maths

Questions Related to maths

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

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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.

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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.

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

reciprocal equations theory of equations maths

  1. Reciprocal equation

  2. Radical equation

  3. Exponential equation

  4. Linear equation

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Correct Option: A
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.

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

reciprocal equations theory of equations maths

  1. $(1,2,3,4)$

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

  3. $\left(1,\dfrac{1}{3},3,1\right)$

  4. $\left(0,1,1,\dfrac{1}{2}\right)$

Show answer
Correct Option: B
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.

The domain of reciprocal equation is :

reciprocal equations theory of equations maths

  1. $R$

  2. $R-{0}$

  3. $Q$

  4. $R^+$

Show answer
Correct Option: B
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.

The range of reciprocal equation is:

reciprocal equations theory of equations maths

  1. $R$

  2. $R-{0}$

  3. $R^+$

  4. $Q$

Show answer
Correct Option: B
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 $R-0$