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Let α, β be real and z be a complex number. If z2 + αz + β = 0 has two distinct roots on the line Re z = 1, then it is necessary that
β ∈(0, 1)
β ∈(-1, 0)
|β| = 1
β ∈ (1, ∞)
D.
β ∈ (1, ∞)
Let roots be p + iq and p - iq p, q ∈ R
root lie on line Re(z) = 1
⇒ p = 1
product of roots = p2 + q2 = β = 1 + q2
⇒ β∈ (1, ∞), (q ≠ 0, ∵ roots are distinct)
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The coefficient of x7 in the expansion of (1 - x - x2 +x3)6 is :
144
-132
-144
132
C.
-144
A man saves Rs. 200 in each of the first three months of his service. In each of the subsequent months his saving increases by Rs. 40 more than the saving of immediately previous month. His total saving from the start of service will be Rs. 11040 after
18 Months
19 Months
20 Months
21 Months
D.
21 Months
a = Rs. 200
d = Rs. 40
savings in first two months = Rs. 400
remained savings = 200 + 240 + 280 + ..... upto n terms =
n/2[400 + (n -1)40] = 11040 - 400
200n + 20n2 - 20n = 10640
20n2 + 180 n - 10640 = 0
n2 + 9n - 532 = 0
(n + 28) (n - 19) = 0
n = 19
∴ no. of months = 19 + 2 = 21
Consider the following statements
P: Suman is brilliant
Q: Suman is rich
R: Suman is honest. The negation of the statement ì Suman is brilliant and dishonest if and only if Suman is richî can be ex- pressed as
~ P ^ (Q ↔ ~ R)
~ (Q ↔ (P ^ ~R)
~ Q ↔ ~ P ^ R
~ (P ^ ~ R)↔ Q
B.
~ (Q ↔ (P ^ ~R)
Negation of (PΛ~ R) ↔ Q is ~ ↔(PΛ ~ R)↔Q)
It may also be written as ~ (Q ↔ (PΛ ~ R))
If ω(≠1) is a cube root of unity, and (1 + ω)7 = A + Bω.Then (A, B) equals
(0,1)
(1,1)
(1,0)
(-1,1)
B.
(1,1)
(1 + ω)7 = A + Bω
(-ω2)7 = A + Bω
- ω14 = A + Bω
-ω2 = A + Bω
1 + ω = A + Bω
therefore,
(A, B) = (1, 1)
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Mock Test Series
