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Mathematics : Complex Numbers and Quadratic Equations

Multiple Choice Questions

21.

The values of p for which the difference between the roots of the equation x2 + px + 8 = 0 is 2, are

C.

Given equation is x2 + px + B = 0. Let  be the roots of this equation, then

22.

The number of non-zero integral solutions of the equation  = 2x

• infinite

• 1

• 2

• None of these

B.

1

Therefore, the number of non-zero integral solutions is one.

23.

If the difference between the roots of the equation x2 + ax + 1 = 0 is less than 5 , then the set of possible values of a is

•  (−3, 3)

• (−3, ∞)

• (3, ∞)

• (−∞, −3)

A.

(−3, 3)

D.

(−∞, −3)

x2 + ax + 1 = 0
α + β = −a αβ = 1

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24.

If , then the greatest and the least value of  are

• 6, - 6

• 6, 0

• 7, 2

• 0, - 1

B.

6, 0

We have,

Hence, greatest and least values of  are 6 and 0 respectively

25.

Let be the roots of x2 + x + 1 = 0, then the equation whose roots are  is

• x2 - x + 1 = 0

• x2 + x - 1 = 0

• x2 + x + 1 = 0

• x60 + x30 + 1 = 0

C.

x2 + x + 1 = 0

Given equation is x2 + x + 1 = 0. Since,   are the roots of the equation.

26.

The complex numbers sin(x) + i cos(2x) and cos(x) - i sin(2x) are conjugate to each other for

• x = $\mathrm{n\pi }$

• x = 0

• No value of x

D.

No value of x

Given, complex number sin(x) + i cos(2x) and cos(x) - i sin(2x) are conjugate to each other, if sin(x) = cos(x) and cos(2)x = sin(2x)

There existes no value of x common in Eqs. (i) and (ii).

Therefore, there is no value of x for which the given complex numbers are conjugate.

27.

The region of the complex plane for which  = 1, [Re (a) $\ne$ 0] is

• x - axis

• y - axis

• the straight line x = a

• None of the above

B.

y - axis

Given,  = 1

28.

If w ( )is a cube root of unity and (1 + w2)n = (1 + w4)n, then the least positive value of n is

• 2

• 3

• 5

• 6

B.

3

We have, (1 + w2)n = (1 + w4)n

29.

For any two complex numbers z1 and z2 and any real numbers a and b,  is equal to

• None of these

B.

30.

The locus of the points z which satisfy the condition

• a straight line

• a circle

• a parabola

• None of the above

B.

a circle