Quadratic Equations Test

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Quadratic Equations Test - 27

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Weekly Quiz Competition

Question 1
1 / -0

Find all values of parameter $$a$$ for which the quadratic equation $$ \left( a+1 \right) { x }^{ 2 }+2\left( a+1 \right) x+a-2=0 $$ has two distinct roots.

A
$$a \in (-1,\infty)$$

B
$$a \in (-\infty,-1)$$

C
$$a \in (-1,1)$$

D
$$a \in (-\infty,\infty)$$

Solution

Discriminant of the given quadratic equation is,
$$D = 4\left( a+1 \right) ^{ 2

}-4\left( a+1 \right) \left( a-2 \right) =4\left( a+1 \right) \left(

a+1-a+2 \right) =12\left( a+1 \right) $$
Now for two distinct roots,
$$D>0\Rightarrow a+1>0$$
$$\Rightarrow a \in (-1,\infty)$$.
Question 2
1 / -0

Find the discriminant for the given quadratic equation:
$$x^2\,+\,x\,+\,1\,=\,0$$

A
$$-3$$

B
$$-5$$

C
$$-7$$

D
$$-9$$

Solution

Given equation is,
$$x^2 + x + 1 = 0 $$
We know, $$D = b^2 - 4ac$$
$$a=1,b=1,c=1$$
$$\therefore D = (1)^2 - 4(1)(1)$$
$$\therefore D = -3$$
Question 3
1 / -0

If $$D$$ is the discriminant of $$x^2\,+\,4x\,+\,1\,=\,0$$, then the value of $$D^2$$, is

A
$$100$$

B
$$12$$

C
$$144$$

D
$$10$$

Solution

$$x^2 + 4x + 1 = 0 $$
$$D = b^2 - 4ac$$
$$D = (4)^2 - 4(1)(1)$$
$$D = 16 - 4$$
$$D = 12 $$
$$D^2 = {12}^2 = 144$$
Question 4
1 / -0

Find the discriminant for the given quadratic equation:
$$4x^2\,-\,kx\,+\,2\,=\,0$$

A
$$k^2\,-\,21$$

B
$$k^2\,-\,24$$

C
$$k^2\,-\,28$$

D
$$k^2\,-\,32$$

Solution

Given equation is,
$$4x^2 - kx + 2 = 0 $$
We know, $$D = b^2 - 4ac $$
$$a=4, b=-k, c=2$$
$$D = (-k)^2 - 4(4)(2)$$
$$\therefore D = k^2 - 32$$
Question 5
1 / -0

Solve the following equations.
$$x^4\,-\,3x^2\,+\,2\,=\,0$$, roots are

A
$$x\,=\,\pm\,\sqrt3,\,\pm\,1$$

B
$$x\,=\,\pm\,\sqrt2,\,\pm\,1$$

C
$$x\,=\,\pm\,\sqrt2,\,\pm\,3$$

D
none of the above

Solution

Given equation is $$x^4 - 3x^2 +2 = 0$$
$$x^4 - 2x^2 -x^2 + 2 =0$$
$$(x^2 - 1)(x^2 - 2) = 0$$
$$\therefore x^2 = 2 \Rightarrow x = \pm \sqrt{2}$$

$$\therefore x^2 = 1 \Rightarrow x = \pm 1$$
Question 6
1 / -0

Find the roots of following quadratic equation
$$x^2\,+\,3x\,-\,2\,=\,0$$

A
$$\displaystyle\,x\,=\,\frac{-3\,\pm\,\sqrt{15}}{2}$$

B
$$\displaystyle\,x\,=\,\frac{-3\,\pm\,\sqrt{17}}{2}$$

C
$$\displaystyle\,x\,=\,\frac{-3\,\pm\,\sqrt{19}}{2}$$

D
$$\displaystyle\,x\,=\,\frac{-3\,\pm\,\sqrt{21}}{2}$$

Solution

Gven equation is
$$x^2 + 3x - 2 = 0 $$
Using quadratic formula,
$$a=1, b=3, c=-2$$
$$x = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}$$
$$x = \frac{-3 \pm \sqrt{9 - 4(1)(-2)}}{2\times1}$$
$$\therefore x = \frac{-3 \pm \sqrt{17}}{2}$$
$$\therefore$$ the roots of the given equation are $$\displaystyle \frac {-3\pm \sqrt{17}}{2}$$
Question 7
1 / -0

Find the discriminant for the given quadratic equation:
$$\sqrt{3}x^2\,+\,2\sqrt{2}x\,-\,2\sqrt{3}\,=\,0$$

A
$$26$$

B
$$32$$

C
$$38$$

D
$$44$$

Solution

Given equation is,
$$\sqrt{3}x^2\,+\,2\sqrt{2}x\,-\,2\sqrt{3}\,=\,0$$
We know, $$D = b^2 - 4ac$$
$$a=\sqrt 3, b=2 \sqrt 2, c=-2\sqrt3$$
$$D = (2\sqrt{2})^2 - 4\times\sqrt{3}\times(-2\sqrt{3})$$
$$D = 8 + 24$$
$$\therefore D = 32$$
Question 8
1 / -0

Find the discriminant for the given equation:
$$3x^2\,+\,2x\,-\,1\,=\,0$$

A
$$11$$

B
$$13$$

C
$$15$$

D
$$16$$

Solution

Given equation is,
$$3x^2 + 2x- 1 = 0$$
We know, $$D = b^2 - 4c$$
$$a=3,b=2,c=-1$$
$$D = 2^2 - 4\times3\times(-1)$$
$$D = 4 + 12$$
$$\therefore D = 16$$
Question 9
1 / -0

Find the discriminant for the given quadratic equation: $$x^2\,+\,4x\,+\,k\,=\,0$$

A
$$6\,-\,4k$$

B
$$16\,-\,4k$$

C
$$16\,-\,3k$$

D
$$13\,-\,2k$$

Solution

Given equation is,
$$x^2 + 4x + k = 0 $$
We know,
$$D = b^2 - 4ac $$
$$a=1, b=4, c=k$$
$$D = (4)^2 - 4(1)(k)$$
$$\therefore D = 16 - 4k$$
Question 10
1 / -0

Find the value of discriminant for the following equation.
$$x^{2}\, +\, 4x\, +\, k\, =\, 0$$

A
$$\Delta\, =\, 16\, + 4k$$

B
$$\Delta\, =\, 16\, -4k$$

C
$$\Delta\, =\, 16\, - k$$

D
$$\Delta\, =\, 16+k$$

Solution

Given equation is:
$$x^2+4x+k=0$$
Discriminant = $$b^2 - 4ac$$
$$a=1, b=4, c=k$$
$$\therefore D= (4)^2 - 4(k)(1)$$
$$= 16 - 4k$$
$$\therefore \Delta =16-4k$$

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Quadratic Equations Test - 27

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Quadratic Equations Test - 27

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Question 1

$$a \in (-1,\infty)$$

$$a \in (-\infty,-1)$$

$$a \in (-1,1)$$

$$a \in (-\infty,\infty)$$

Solution

Question 2

$$-3$$

$$-5$$

$$-7$$

$$-9$$

Solution

Question 3

$$100$$

$$12$$

$$144$$

$$10$$

Solution

Question 4

$$k^2\,-\,21$$

$$k^2\,-\,24$$

$$k^2\,-\,28$$

$$k^2\,-\,32$$

Solution

Question 5

$$x\,=\,\pm\,\sqrt3,\,\pm\,1$$

$$x\,=\,\pm\,\sqrt2,\,\pm\,1$$

$$x\,=\,\pm\,\sqrt2,\,\pm\,3$$

none of the above

Solution

Question 6

$$\displaystyle\,x\,=\,\frac{-3\,\pm\,\sqrt{15}}{2}$$

$$\displaystyle\,x\,=\,\frac{-3\,\pm\,\sqrt{17}}{2}$$

$$\displaystyle\,x\,=\,\frac{-3\,\pm\,\sqrt{19}}{2}$$

$$\displaystyle\,x\,=\,\frac{-3\,\pm\,\sqrt{21}}{2}$$

Solution

Question 7

$$26$$

$$32$$

$$38$$

$$44$$

Solution

Question 8

$$11$$

$$13$$

$$15$$

$$16$$

Solution

Question 9

$$6\,-\,4k$$

$$16\,-\,4k$$

$$16\,-\,3k$$

$$13\,-\,2k$$

Solution

Question 10

$$\Delta\, =\, 16\, + 4k$$

$$\Delta\, =\, 16\, -4k$$

$$\Delta\, =\, 16\, - k$$

$$\Delta\, =\, 16+k$$

Solution

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