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Q 1. If {tex} \left( \frac { 1 + i } { 1 - i } \right) ^ { x } = 1 , {/tex} then
{tex} x = 2 n + 1 , {/tex} where {tex} n {/tex} is any positive integer
{tex} x = 4 n , {/tex} where {tex} n {/tex} is any positive integer
{tex} x = 2 n , {/tex} where {tex} n {/tex} is any positive integer
{tex} x = 4 n + 1 , {/tex} where {tex} n {/tex} is any positive integer
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Q 2. Let {tex} z _ { 1 } {/tex} and {tex} z _ { 2 } {/tex} be the roots of the equation {tex} z ^ { 2 } + a z + b = 0 , z {/tex} being complex number. Further, assume that the origin, {tex} z _ { 1 } {/tex} and {tex} z _ { 2 } {/tex} form an equilateral triangle, then
{tex} a ^ { 2 } = 4 b {/tex}
{tex} a ^ { 2 } = b {/tex}
{tex} a ^ { 2 } = 2 b {/tex}
{tex} a ^ { 2 } = 3 b {/tex}
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Q 3. If {tex} z , \omega {/tex} are two non-zero complex numbers such that {tex} | z \omega | = 1 {/tex} and {tex} \arg ( z ) - \arg ( \omega ) = \pi / 2 , {/tex} then {tex} \overline { z } \omega {/tex} is equal to
{tex}-1{/tex}
{tex}1{/tex}
{tex} -i {/tex}
{tex} i {/tex}
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Q 4. If {tex} \left| z ^ { 2 } - 1 \right| = | z | ^ { 2 } + 1 , {/tex} then {tex} z {/tex} lies on
the real axis
the imaginary axis
a circle
an ellipse
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Q 5. If the cube roots of unity are 1 ,{tex}\omega{/tex}, {tex} \omega ^ { 2 } , {/tex} then the roots of the equation {tex} ( x - 1 ) ^ { 3 } + 8 = 0 {/tex} are
{tex} - 1,1 + 2 \omega , 1 + 2 \omega ^ { 2 } {/tex}
{tex} - 1,1 - 2 \omega , 1 - 2 \omega ^ { 2 } {/tex}
{tex} - 1 , - 1 , - 1 {/tex}
{tex} - 1 , - 1 + 2 \omega, - 1 , - 2 \omega ^ { 2 } {/tex}
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Q 6. If {tex} z _ { 1 } {/tex} and {tex} z _ { 2 } {/tex} are two non-zero complex numbers such that {tex} \left| z _ { 1 } + z _ { 2 } \right| = \left| z _ { 1 } \right| + \left| z _ { 2 } \right| , {/tex} then {tex} \arg \left( z _ { 1 } \right) - \arg \left( z _ { 2 } \right) {/tex} is equal to
{tex} - \frac { \pi } { 2 } {/tex}
{tex}0{/tex}
{tex} - \pi {/tex}
{tex} \frac { \pi } { 2 } {/tex}
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Q 7. If {tex} \omega = \frac { z } { z - \frac { 1 } { 3 } i } {/tex} and {tex} | \omega | = 1 , {/tex} then {tex} z {/tex} lies on
a parabola
a straight line
a circle
an ellipse
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Q 8. If {tex} z ^ { 2 } + z + 1 = 0 , {/tex} where {tex} z {/tex} is a complex number, then the value of {tex} \left( z + \frac { 1 } { z } \right) ^ { 2 } + \left( z ^ { 2 } + \frac { 1 } { z ^ { 2 } } \right) ^ { 2 } + \left( z ^ { 3 } + \frac { 1 } { z ^ { 3 } } \right) ^ { 2 } + \cdots + \left( z ^ { 6 } + \frac { 1 } { z ^ { 6 } } \right) ^ { 2 } {/tex} is
18
54
6
12
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Q 9. If {tex} | z + 4 | \leq 3 , {/tex} then the maximum value of {tex} | z + 1 | {/tex} is
0
4
10
6
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Q 10. The conjugate of a complex number is {tex} \frac { 1 } { i - 1 } {/tex} . Then that complex number is
{tex} \frac { - 1 } { i - 1 } {/tex}
{tex} \frac { 1 } { i + 1 } {/tex}
{tex} \frac { - 1 } { i + 1 } {/tex}
{tex} \frac { 1 } { i - 1 } {/tex}
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Q 11. If {tex} \left| z - \frac { 4 } { z } \right| = 2 , {/tex} then the maximum value of {tex} | z | {/tex} is equal to
{tex} \sqrt { 3 } + 1 {/tex}
{tex} \sqrt { 5 } + 1 {/tex}
{tex}2{/tex}
{tex} 2 + \sqrt { 2 } {/tex}
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Q 12. The number of complex number z such that {tex} | z - 1 | = | z + 1 | = | z - i | {/tex} equals
{tex} \infty {/tex}
0
1
2
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Q 13. If {tex} \alpha {/tex} and {tex} \beta {/tex} are the roots of the equation {tex} x ^ { 2 } - x + 1 = 0 {/tex} , then {tex} \alpha ^ { 2009 } + \beta ^ { 2009 } = ? {/tex}
2
-2
-1
1
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Q 14. If {tex} \omega ( \neq 1 ) {/tex} is a cube root of unity, and {tex} ( 1 + \omega ) ^ { 7 } = A + B \omega . {/tex} Then {tex} ( A , B ) {/tex} equals
{tex} ( 1,1 ) {/tex}
{tex} ( 1,0 ) {/tex}
{tex} ( - 1,1 ) {/tex}
{tex} ( 0,1 ) {/tex}
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Q 15. Let {tex} \alpha , \beta {/tex} be real and {tex} z {/tex} be a complex number. If {tex} z ^ { 2 } + \alpha z + \beta = 0 {/tex} has two distinct roots on the line {tex} R z = 1 {/tex} , then it is necessary that
{tex} \beta \in ( - 1,0 ) {/tex}
{tex} | \beta | = 1 {/tex}
{tex} \beta \in ( 1 , \infty ) {/tex}
{tex} \beta \in ( 0,1 ) {/tex}
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Q 16. If {tex} z {/tex} is a complex number of unit modulus and argument {tex} \theta {/tex} , then {tex} \arg \left( \frac { 1 + z } { 1 + \overline { z } } \right) {/tex} equals
{tex}0{/tex}
{tex} \frac { \pi } { 2 } - \theta {/tex}
{tex}\theta{/tex}
{tex} \pi - \theta {/tex}
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Q 17. If , but
, then the equation whose roots are
and
is
None of these
,
⇒
. α, β are roots of
. Therefore
,
∵ ⇒
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Q 18. If one root of the equation is the square of the other, then
Let α and be the roots then
,
Now ⇒
⇒
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Q 19. If the roots of the equation x2 - 2ax + a2 + a - 3 = 0 are less than 3 then
a < 2
2 ≤ a ≤ 3
3 < a ≤ 4
a > 4
Since roots are less than a real number, roots must be real
⇒ 4a2 - 4(a2 + a - 3) ≥ 0 ⇒ a ≤ 3. ... (1)
Let f(x) = x2 - 2ax + a2 + a - 3. Since 3 lie outside the roots,
f(3) > 0 ⇒ a < 2 or a > 3. . . . .(2)
From (1), (2) we have a < 2
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Q 20. Sum of the real roots of the equation x2 + 5|x| + 6 = 0
Equals to 5
Equals to 10
Equals to -5
Does not exit
Since x2, 5|x| and 6 are positive so x2 + 5|x| + 6 = 0 does not have any real root. Therefore sum does not exist
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Q 21. If = 7 then x =
log7(3/4)
log7(4/3)
x = log(3/4) 7 = =
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Q 22. The value of 'a' for which the sum of the squares of the roots of the equation x2 - (a - 2)x - a - 1 = 0 assumes the least value is :
0
1
2
3
Here α+β=a-2,αβ=-(a-1), Now,α2+β2=(α+β)2- 2αβ
= (a- 2)2 + 2 (a +1) = a2 - 2a + 6 = (a-1)2 + 5
So α2 + β2 will be maximum at a = 1.
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Q 23. If a, b, c are in G.P. then the equation ax2 + 2bx + c = 0 and dx2 + 2ex + f = 0 have a common root if d/a , e/b, f/c are in :
AP
GP
HP
None of these
Here b = (As a, b, c → GP)
So, ax2 + 2 + c = 0 ⇒ (x
+
)2 = 0
or x = -/
, Now, substituting in eqn (2);
dc/a - 2e/
+ f = 0 ⇒ d/a - 2e/
+ f/c = 0
⇒ d/a -2e/b + f/c = 0
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Q 24. If the roots of the equation =
are such that α + β = 0, then the value of λ is-
c
(λ + 1) (x2 - bx) = (λ - 1) (ax - c)
⇒ λx2 + (-bλ - b - aλ + a) x + cλ - c = 0
Since α + β = 0, we have = 0
⇒ -bλ - b -aλ + a = 0 ⇒ λ = .
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Q 25. The equation x + = 1 +
has-
No real root
One real root
Two equal roots
Infinitely
We have x + = 1 +
⇒ x = 1 provided x = 1 ≠ 0 i.e., x ≠ 1.
The given equation has no solution.