Quadratic Equations Test

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Quadratic Equations Test - 20

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Weekly Quiz Competition

Question 1
1 / -0

Check whether the following is a quadratic equation.
$$(x - 3) (2x + 1) = x (x + 5)$$

A
Yes

B
No

C
Ambiguous

D
Data insufficient

Solution

Given, $$(x-3)(2x+1)=(x)(x+5)$$
$$\Rightarrow 2x^{2}+x-6x-3=x^{2}+5x$$
$$\Rightarrow x^{2}-10x-3=0$$
Using quadratic formula, we get
$$x=\dfrac{10\pm\sqrt{100-12}}{2}$$
$$=\dfrac{10\pm7\sqrt{2}}{2}$$
Question 2
1 / -0

Find the value of the polynomial $$5x-4x^2+3$$ at $$x=0$$

A
$$5$$

B
$$2$$

C
$$3$$

D
$$1$$

Solution

Let,
$$P(x)=5x-4x^2+3$$

putting $$x=0$$

$$P(0)=5(0)-4(0)^2+3$$

$$=0-0+3$$

$$=3$$
Question 3
1 / -0

Is the following equation a quadratic equation?
$$\displaystyle \frac{3x}{4} - \frac{5x^2}{8} = \frac{7}{8}$$

A
Yes

B
No

C
Ambiguous

D
Data insufficient

Solution

Given, $$\dfrac{3x}{4}-\dfrac{5x^{2}}{8}=\dfrac{7}{8}$$
Hence, $$6x-5x^{2}=7$$
$$\Rightarrow 5x^{2}-6x+7=0$$
$$\Rightarrow ax^{2}+bx+c=0$$
Hence, it is a quadratic equation.
Question 4
1 / -0

Is the following equation a quadratic equation?
$$16x^2 - 3 = (2x + 5) (5x - 3)$$

A
Yes

B
No

C
Ambiguous

D
Data insufficient

Solution

$$16x^2 - 3 = (2x + 5) (5x - 3)$$
$$\Rightarrow 16x^2 - 3=10x^2-6x+25x-15$$
$$\Rightarrow 16x^2-10x^2+6x-25x+15-3=0$$
$$\Rightarrow 6x^2-19x+12=0$$
Comparing it with the standard form of quadratic equation $$x^2+bx+c=0, a\neq 0$$.
Here $$a=6$$
Thus, it is a quadratic equation.
Question 5
1 / -0

$$\displaystyle a^2x^2 - 3abx + 2b^2 $$= 0 then $$x = \cfrac{a}{b}\ and\ x = \cfrac{b}{a}$$ are the roots of the equation.

A
True

B
False

C
Ambiguos

D
Can't say

Solution

$$a^{2}x^{2}-3abx+2b^{2}=0$$
Therefore
$$2b^{2}.a^{2}=2b^{2}a^{2}$$
$$=2ab.ab$$
Hence
$$a^{2}x^{2}-abx-2abx+2b^{2}=0$$
$$ax(ax-b)-2b(ax-b)=0$$
$$(ax-2b)(ax-b)=0$$
Thus, the roots are
$$\dfrac{2b}{a},\dfrac{b}{a}$$
Hence, the given statement is false.
Question 6
1 / -0

The condition for a general quadratic equation such that its both roots are equal, is

A
$$ b^{2}+4ac= 0$$

B
$$ b^{2}-4ac= 0$$

C
$$ b^{2}+ac= 0$$

D
$$ b^{2}+2ac= 0$$

Solution

General quadratic equation is $$ax^2+bx+c=0$$
For both roots to be equal discriminant of this quadratic should be zero $$\Rightarrow b^2-4ac=0$$
Question 7
1 / -0

Discriminant of the equation $$ -3x^2 + 2x -8 = 0$$ is

A
$$- 92$$

B
$$- 29$$

C
$$39$$

D
$$49$$

Solution

The quadratic equation is $$-3x^2 + 2x -8 = 0$$
Comparing it with $$ax^2+bx+c=0$$ we get, $$a=-3, b=2, c=-8$$
Therefore, $$D=b^2-4ac$$
$$2^2-4\times (-3)\times (-8)$$
$$=4-96$$
$$=-92$$
Question 8
1 / -0

Solve for $$y:$$ $${2y^2 +4=9y}$$

A
$$\{2,4\}$$

B
$$\{\dfrac 12,2\}$$

C
$$\{2,2\}$$

D
$$\{\dfrac 12,4\}$$

Solution

Given, $$2y^2+4=9y$$
$$\Rightarrow 2y^2-9y+4=0$$
Comparing it with standard form quadratic equation $$ax^2+bx+c=0$$, we get,
$$a=2, b=-9, c=4$$
Thus required roots are $$y=\displaystyle \cfrac{-b\pm \sqrt{b^2-4ac}}{2a}$$
$$y=\displaystyle \cfrac{9\pm\sqrt{81-32}}{4}=\frac{9\pm 7}{4}=4,\frac{1}{2}$$
Hence, correct choice is option D.
Question 9
1 / -0

The sum of a number and its reciprocal is $$ \displaystyle \frac{125}{22} $$ The number is

A
$$ \displaystyle \frac{2}{11} $$

B
$$ \displaystyle \frac{1}{11} $$

C
$$ \displaystyle \frac{3}{11} $$

D
None of these

Solution

Let the number be x
Now, According to question
$$x+ \dfrac{1}{x}= \dfrac{125}{22}$$
$$\dfrac{x^2 +1}{x}= \dfrac{125}{22}$$
$$22{x^2 +22}= {125}x$$
$$22{x^2 +22}-{125}x=0$$
$$x= \dfrac{125\pm\sqrt{(125)^2-4.22.22} }{44}$$
$$x= \dfrac{125\pm\sqrt{(15625-1936} }{44}$$
$$x= \dfrac{125\pm\sqrt{(13689} }{44}$$
$$x= \dfrac{125\pm117}{44}$$
$$x= \dfrac{125+117}{44}$$, $$x= \dfrac{125-117}{44}$$
$$x= \dfrac{242}{44}$$, $$x= \dfrac{8}{44}$$
Option A is correct
$$x= \dfrac{2}{11}$$
Question 10
1 / -0

Choose the best possible option.
$$\displaystyle { x }^{ 3 }-5x+2{ x }^{ 2 }+1=0$$ is quadratic equation.

A
Yes

B
No

C
Can't predict

D
None

Solution

The degree of equation is 3.
$$\displaystyle \therefore \quad { x }^{ 3 }-5x+2{ x }^{ 2 }+1=0$$ is not quadratic.
It is cubic equation.

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Re-Attempt Weekly Quiz Competition

Quadratic Equations Test - 20

Result

Quadratic Equations Test - 20

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Rank

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SHARING IS CARING

Question 1

Yes

No

Ambiguous

Data insufficient

Solution

Question 2

$$5$$

$$2$$

$$3$$

$$1$$

Solution

Question 3

Yes

No

Ambiguous

Data insufficient

Solution

Question 4

Yes

No

Ambiguous

Data insufficient

Solution

Question 5

True

False

Ambiguos

Can't say

Solution

Question 6

$$ b^{2}+4ac= 0$$

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

$$ b^{2}+ac= 0$$

$$ b^{2}+2ac= 0$$

Solution

Question 7

$$- 92$$

$$- 29$$

$$39$$

$$49$$

Solution

Question 8

$$\{2,4\}$$

$$\{\dfrac 12,2\}$$

$$\{2,2\}$$

$$\{\dfrac 12,4\}$$

Solution

Question 9

$$ \displaystyle \frac{2}{11} $$

$$ \displaystyle \frac{1}{11} $$

$$ \displaystyle \frac{3}{11} $$

None of these

Solution

Question 10

Yes

No

Can't predict

None

Solution

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