Complex Numbers and Quadratic Equations Test -5

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Complex Numbers and Quadratic Equations Test -5

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Weekly Quiz Competition

Question 1
1 / -0

Nature of the roots of the quadratic equation $$2x^{2}-2\sqrt{6}x+3=0$$ is:

A
Real, irrational, unequal

B
Real, rational, equal

C
Real, rational, unequal

D
Complex

Solution

(B) $$b^{2}-4ac=(-2\sqrt{6})^{2}-4(2)(3)$$
$$=24-24$$
$$=0$$
$$\therefore $$ Real, rational, equal
Question 2
1 / -0

Find the value of $$x$$ of the equation $${ \left( 1-i \right) }^{ x }={ 2 }^{ x }$$

A
$$1$$

B
$$2$$

C
$$0$$

D
none of these

Solution

Then $${ \left( 1-i \right) }^{ x }={ 2 }^{ x }$$

$$\Rightarrow |{(1-i)}|^{x} =|2|^{x}$$

$$\Rightarrow { \left( \sqrt {1+1}\right)}^{x}={2}^{x}$$

$$\Rightarrow { \left( \sqrt { 2 } \right) }^{ x }={ 2 }^{ x }$$

$$\Rightarrow \dfrac{x}{2}=x$$

$$\Rightarrow 2x=x\Rightarrow x=0$$
Hence, option C is correct.
Question 3
1 / -0

If the discriminant of a quadratic equation is negative, then its roots are:

A
unequal

B
equal

C
inverse

D
imaginary

Solution

If the discriminant $$ b^2 - 4ac $$ of a quadratic equation is negative, then its roots are imaginary.
Question 4
1 / -0

For what value of 'm', the equation $$(3m+1)x^{2}+2(m+1)x+m=0$$ have equal root ?

A
1 or $$-\frac{1}{2}$$

B
2 or 4

C
4

D
3

Solution

(A) Roots are equal if $$D=0$$
or $$b^{2}-4ac=0$$
$$\Rightarrow 4(m+1)^{2}-4m(3m+1)=0$$
$$\Rightarrow 2m^{2}-m-1=0$$
$$ \therefore m = 1, -1/2$$
Question 5
1 / -0

The roots of $$4x^{2}-2x+8=0$$ are:

A
Real and equal

B
Rational and not equal

C
Irrational

D
Not real

Solution

Given equation is $$4x^2-2x+8=0$$
We know value of discriminant $$D=b^{2}-4ac$$
Here $$a=4, b=-2, c=8$$
Therefore, $$D=(-2)^{2}-4(4)(8)$$
$$=4-128$$
$$=-124< 0$$
Hence, roots are not real.
Question 6
1 / -0

Evaluate :
$$\sqrt{-25} + 3 \sqrt{-4} +2 \sqrt{-9}$$

A
$$-17i$$

B
$$5i$$

C
$$17i$$

D
$$6i$$

Solution

$$\sqrt{-25} + 3 \sqrt{-4} +2 \sqrt{-9} $$

$$=5\sqrt{-1}+6\sqrt{-1}+6\sqrt{-1}$$ we know, $$\sqrt{-1}=i$$

$$= 5i + 6i + 6i = 17i$$
Question 7
1 / -0

Determine the values of $$p$$ for which the quadratic equation $$2x^2 + px + 8 = 0$$ has equal roots.

A
$$p=\pm 64$$

B
$$p=\pm 8$$

C
$$p=\pm 4$$

D
$$p=\pm 16$$

Solution

In $$ax^2+bx+c=0,$$
Discriminant $$D$$ $$=b^2-4ac$$,
D$$=0$$, for the roots to be real and equal

In $$2x^{2}+px+8=0$$
Thus, we have $$a=2 ,b=p$$ and $$c=8$$
Then $$D$$ $$=b^2-4ac=0$$

$$\therefore p^{2}-4\times 2\times 8=0\Rightarrow p^{2}-64=0$$

$$\Rightarrow p=\pm 8$$.
Question 8
1 / -0

$$\displaystyle \frac{\displaystyle i^{4n + 3} + (-i)^{8n - 3}}{\displaystyle(i)^{12 n- 1} - i^{2 - 16 n}}, n \varepsilon N$$ is equal to

A
1 + i

B
2i

C
-2i

D
-1 - i

Solution

Given expression
$$= \displaystyle \frac{\displaystyle i^3 + (-1) (i)^{-3}}{(-i)^{-1} - (i)^2} = \frac{\displaystyle -i - i}{\displaystyle i + 1}$$
$$=\displaystyle \frac{\displaystyle -2i}{1 + i} \times \frac{1 - i}{1 - i} = \frac{\displaystyle -2i - 2}{\displaystyle 1 + 1} = - 1 - i$$
Question 9
1 / -0

1+$$i^2 + i^4 + i^6 + ........+ i^{2n}$$ is

A
Positive

B
Negative

C
Zero

D
Cannot be determined

Solution

$$1+i^2+i^4+........+i^{2n}$$

$$=1-1+1-1+1-1......$$

Hence it can b positive or negative depending on the value of $$2n$$

Hence It cannot be determined from the given data.
Question 10
1 / -0

If $$i^2 = - 1$$, then the value of $$\displaystyle \sum^{200}_{n = 1} i^n $$ is

A
50

B
-50

C
0

D
100

Solution

Given, $$i^2=-1$$
$$\displaystyle \sum_{n = 1}^{200} i^n = i + i^2 + i^3 + ......... + i^{200} = \displaystyle \frac{i (1 - i^{200})}{1 - i} $$ (since G. P.)
$$= \displaystyle \frac{\displaystyle i (1 - 1)}{\displaystyle 1 - i} = 0$$

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Complex Numbers and Quadratic Equations Test -5

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Question 1

Real, irrational, unequal

Real, rational, equal

Real, rational, unequal

Complex

Solution

Question 2

$$1$$

$$2$$

$$0$$

none of these

Solution

Question 3

unequal

equal

inverse

imaginary

Solution

Question 4

1 or $$-\frac{1}{2}$$

2 or 4

4

3

Solution

Question 5

Real and equal

Rational and not equal

Irrational

Not real

Solution

Question 6

$$-17i$$

$$5i$$

$$17i$$

$$6i$$

Solution

Question 7

$$p=\pm 64$$

$$p=\pm 8$$

$$p=\pm 4$$

$$p=\pm 16$$

Solution

Question 8

1 + i

2i

-2i

-1 - i

Solution

Question 9

Positive

Negative

Zero

Cannot be determined

Solution

Question 10

50

-50

0

100

Solution

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