Quadratic Equations Test

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Quadratic Equations Test - 23

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Weekly Quiz Competition

Question 1
1 / -0

The number of triples $$(x, y, z)$$ of real numbers satisfying the equation
$$x^{4} + y^{4} + z^{4} + 1 = 4xyz$$ is

A
$$0$$

B
$$4$$

C
$$8$$

D
More than $$8$$

Solution

Applying $$ AM \geq GM $$
We get $$ \dfrac{ x^{4} + y^{4} + z^{4} + 1}{4} \geq \sqrt[4]{x^4y^4z^4} $$
$$ \Longrightarrow x^{4} + y^{4} + z^{4} + 1 \geq 4xyz$$
$$ \textrm {Thus, min value of } x^{4} + y^{4} + z^{4} + 1 \textrm { must be } 4xyz$$
Clearly our function is increasing for x,y,z>0 and decreasing for x,y,z<0.
Thus min value must occur for integral values like 1,0,-1.
For 0, it doesn't satisfy the equation $$x^{4} + y^{4} + z^{4} + 1 = 4xyz$$
But triplets of 1 and -1 do.
Thus, we can check that the triplets $$(1,-1,-1); (-1,1,-1); (-1,-1,1) and (1,1,1)$$ satisfy the equation.
We don't need to look for further values as we won't get min value of $$x^{4} + y^{4} + z^{4} + 1$$ by them.
Thus, these are the only solutions.
Question 2
1 / -0

Which of the following methods is/are guaranteed to solve any Quadratic Equation?

A
Factorisation Method

B
Completing Square Method

C
Standard Quadratic Formula

D
Both option b and option c

Solution

We have learnt that the Factorisation Method might not be helpful in solving different quadratic equations. But the Completing Square Method and the Standard Quadratic Formula are guaranteed to solve any given Quadratic Equation. Hence both options $$b$$ and $$c$$ are correct. Thus option $$d$$ is the correct answer.
Question 3
1 / -0

Identify which of the following equations is a Quadratic Equation.

A
$$5x + 8 = 0$$

B
$$-2x^2+x=21$$

C
$$ax + by + c = 0$$

D
None of these.

Solution

Only option b is an equation where there is only one variable x and whose degree is 2. Thus it is a quadratic equation.Hence option b is the correct answer.
Question 4
1 / -0

Which of the following is a Quadratic Equation?

A
$$5x + 8 = 0$$

B
$$6x^2 + 7x =19$$

C
$$ x + 1$$

D
None of these

Solution

A Quadratic Equation is an equation with one variable and degree $$2$$. In options a, and c we have equations in one variable i.e. x . but the degree of the variable $$x$$ is $$1$$. While option b has one variable and the degree is $$2$$, hence $$b$$ is the correct option.
Question 5
1 / -0

The standard quadratic formula to solve the quadratic equation $$ax^2+bx+c=0$$ is given by.

A
$$x=\dfrac{b\pm \sqrt{b^2+4ac}}{2a}$$

B
$$x=\dfrac{\sqrt{b^2-4ac}}{2a}$$

C
$$x=\dfrac{-b\pm \sqrt{b^2+4ac}}{2a}$$

D
$$x=\dfrac{-b\pm \sqrt{b^2- 4ac}}{2a}$$

Solution

The standard quadratic formula to solve the quadratic equation $$ax^2+bx+c=0$$ is given by $$x=\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$$
Option $$d$$ is the correct answer.
Question 6
1 / -0

The Completing Square Method converts the quadratic equation $$ax^2+bx+c=0$$ into which of the following forms?

A
$$(px+q)(rx+s)=0$$

B
$$px+q=0$$

C
$$(x+p)^2=q$$

D
All of the above.

Solution

The Completing Square Method involves converting a quadratic equation of the form $$ax^2+bx+c=0$$ into the form $$(x+p)^2=q$$ so that this equation can be solved easily as it now contains only one term of $$x$$ unlike the original quadratic equation which has two terms in $$x$$ ,$$ax^2$$ and bx thus making it difficult to solve the equation. Hence option c is the correct answer.
Question 7
1 / -0

For a quadratic equation in the standard form $$ax^2+bx+c=0$$, the nature of roots is dependent on the value of ?

A
$$b^2+4ac$$

B
$$b^2 \pm 4ac$$

C
$$\sqrt{b^2}-4ac$$

D
$$b^2-4ac$$

Solution

Nature of roots is dependent on the value of determinant and discriminant$$={ b }^{ 2 }-4ac$$
Question 8
1 / -0

If $$px^2 +qx+r=0$$ has equal roots, then $$q^2 =$$

A
$$4p$$

B
$$4pr$$

C
$$8pr$$

D
$$pr$$

Solution

$$px^2 +qx+r=0$$

The standard quadratic equation is $$ax^2+bx+c=0$$

For roots to be equal
$$\begin{array}{l}{b^2} - 4ac = 0\\or,{b^2} = 4ac\\so,{(q)^2} = 4 \times p \times r\\{q^2} = 4pr \end{array}$$
Question 9
1 / -0

If the roots of a quadratic equation $$ax^2+bx+c=0$$ are all real then which one of the following is true?

A
$$b^2-4ac=0$$

B
$$b^2-4ac>0$$

C
$$b^2-4ac\ge 0$$

D
$$b^2-4ac<0$$

Solution

Given, the roots of a quadratic equation $$ax^2+bx+c=0$$........(1) are all real.
Then the discreminant of the equation (1) is $$\ge 0$$.
The discreminant of the equation (1) is $$b^2-4ac$$.
So $$b^2-4ac\ge 0$$.
Question 10
1 / -0

Solve $$(x+5)(x-5)=$$

A
$$x^{2}+25$$

B
$$x^{2}-25$$

C
$$25x^{2}$$

D
$$x^{2}+10x$$

Solution

$$(x+5)(x-5)$$
We know, $$(a+b)(a-b)=a^2-b^2$$
$$(x+5)(x-5)=(x)^2-(5)^2$$
$$(x+5)(x-5)=x^2-25$$

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Re-Attempt Weekly Quiz Competition

Quadratic Equations Test - 23

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Quadratic Equations Test - 23

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SHARING IS CARING

Question 1

$$0$$

$$4$$

$$8$$

More than $$8$$

Solution

Question 2

Factorisation Method

Completing Square Method

Standard Quadratic Formula

Both option b and option c

Solution

Question 3

$$5x + 8 = 0$$

$$-2x^2+x=21$$

$$ax + by + c = 0$$

None of these.

Solution

Question 4

$$5x + 8 = 0$$

$$6x^2 + 7x =19$$

$$ x + 1$$

None of these

Solution

Question 5

$$x=\dfrac{b\pm \sqrt{b^2+4ac}}{2a}$$

$$x=\dfrac{\sqrt{b^2-4ac}}{2a}$$

$$x=\dfrac{-b\pm \sqrt{b^2+4ac}}{2a}$$

$$x=\dfrac{-b\pm \sqrt{b^2- 4ac}}{2a}$$

Solution

Question 6

$$(px+q)(rx+s)=0$$

$$px+q=0$$

$$(x+p)^2=q$$

All of the above.

Solution

Question 7

$$b^2+4ac$$

$$b^2 \pm 4ac$$

$$\sqrt{b^2}-4ac$$

$$b^2-4ac$$

Solution

Question 8

$$4p$$

$$4pr$$

$$8pr$$

$$pr$$

Solution

Question 9

$$b^2-4ac=0$$

$$b^2-4ac>0$$

$$b^2-4ac\ge 0$$

$$b^2-4ac<0$$

Solution

Question 10

$$x^{2}+25$$

$$x^{2}-25$$

$$25x^{2}$$

$$x^{2}+10x$$

Solution

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