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 Multiple Choice QuestionsMultiple Choice Questions

21.

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

  • ± 2

  • ± 4

  • ± 6

  • ± 8


C.

± 6

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

           α + β = - pand        α . β = 8Given, α - β = 2   α - β2 = α + β2 - 4αβ = 22 p2 - 8 × 4 = 4                p2 = 36                  p = ± 6


22.

The number of non-zero integral solutions of the equation 1 - ix = 2x

  • infinite

  • 1

  • 2

  • None of these


B.

1

Given, 1 - ix = 2x  1 + 1x = 2x               x2 = x                x = 0, 2

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

vertical line straight alpha space minus straight beta vertical line space equals space square root of left parenthesis straight alpha space plus straight beta right parenthesis squared space minus 4 αβ end root
vertical line straight alpha minus straight beta vertical line space equals space square root of straight a squared minus 4 end root
square root of straight a squared minus 4 end root space less than square root of 5
straight a squared space minus space 4 less than 5
straight a to the power of 2 minus end exponent space 9 less than 0
straight a element of space left parenthesis negative 3 comma 3 right parenthesis

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24.

If z + 4  3, then the greatest and the least value of z + 1 are

  • 6, - 6

  • 6, 0

  • 7, 2

  • 0, - 1


B.

6, 0

We have, z + 4  3

  - 3  z + 4  3  - 6  z + 1  0 0 - z + 1  6         0  z + 1  6

Hence, greatest and least values of z + 1 are 6 and 0 respectively


25.

Let α, α2 be the roots of x2 + x + 1 = 0, then the equation whose roots are α31, α62 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, α, α2  are the roots of the equation.

 α + α2 = - 1           ...(i)        α3 = 1               ...(ii)Now, for the equation of roots are α31 and α62α31 + α62 = α311 + α31  α31 + α62 = α30α1 + α30 . α α31 + α62 = α310 . α1 + α310 . α α31 + α62 = α1 + α            using Eq. (ii) α31 + α62 = - 1                    using Eq. (i)Again α31 . α62 = α93                        = α331 = 1  Required equation is,x2 -  α31 + α62 x + α31 . α62 = 0 x2 + x +1 = 0


26.

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

  • x = 

  • x = n + 12π

  • 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)

      tanx = 1  x = π4, 5π4, 9π4, ...          ...(i)and  tan2x = 1  2x = π4, 5π4, 9π4, ...       ...(ii)

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 z - az + a = 1, [Re (a)  0] is

  • x - axis

  • y - axis

  • the straight line x = a

  • None of the above


B.

y - axis

Given, z - az + a = 1

   z - a = z + a  z - a2 = z + a2 z - az - a = z + az + a z - az - a = z + az + a            a = a zz - za - az + aa = zz + za + a z + aaza + za + a z + az = 0         z + za + a = 0                     z + z = 0            a + a = 2Re(a)  0, given 2Re(z) = 0        2x = 0 x = 0 i.e., y-axis


28.

If w ( 1 )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

 - wn = - w2n       wn = 1         n = 3 is least positive value of n.


29.

For any two complex numbers z1 and z2 and any real numbers a and b, az1 - az22 + bz1 + az22 is equal to

  • a2 + b2z1 + z2

  • a2 + b2z12 + z22

  • a2 + b2z12 - z22

  • None of these


B.

a2 + b2z12 + z22

az1 - bz22 + bz1 - az2= a2z12 + b2z22 - 2ab Rez1z2 + b2z12              + a2z22 + 2ab Rez1z2= a2 + b2z12 + z22


30.

The locus of the points z which satisfy the condition argz - 1z + 1 = π3

  • a straight line

  • a circle

  • a parabola

  • None of the above


B.

a circle

Given, argz - 1z + 1 = π3Let                       z = x + iy               z - 1z + 1 = x + iy - 1x + iy + 1                               = x2 + y2 - 1 + 2iyx + 12 + y2       argz - 1z + 1 = tan-12yx2 + y2 - 1 = π3    2yx2 + y2 - 1 = tanπ3 = 3      x2 + y2 - 1 = 23y x2 + y2 - 23y - 1 = 0Which is the equation of circle.