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Mechanical engineering Questions and solutions set - 19

Mechanical engineering Questions and solutions set - 19
Question:-
Option (A)
(P2 + P + 2I)
Option (B)
(P2 + P + 1)
Option (C)
-(P2 + P + 1)
Option (D)
-(P2 + P + 2I)
Correct Option:
D
Question Solution:
If characteristic equation is
λ3 + λ2 + 2λ + 1 = 0
Then by cayley - hamilton theorem,
P3 + P2 + 2P + I = 0
I = -P3 - P2 - 2P
Multiplying by P-1 on both sides,
P-1 = -P2 - P - 2 I
= -(P2 + P + 2 I)
Question:-
Option (A)
Independence of P and Q implies that probability (P ∩ Q) = 0
Option (B)
Probability (P ∪ Q) ≥ Probability (P) + Probability (Q)
Option (C)
If P and Q are mutually exclusive, then they must be independent
Option (D)
Probability (P ∩ Q) ≤ Probability (P)
Correct Option:
D
Question Solution:
(a)is false since of P & Q are independent
pr(P ∩ Q) = pr(P) * pr(Q)
which need not be zero.
(b)is false since pr(P ∪ Q) = pr(P) + pr(Q) - pr(P ∩ Q)
therefore, pr(P ∪ Q) ≤ pr(P) + pr(Q)
(c)is false since independence and mutually exclusion are unrelated properties.
(d)is true
sinceP ∩ Q P
=> n(P ∩ Q) ≤ n(P)
=> pr(P ∩ Q) ≤ pr(P)
Question:-
Option (A)
Augmented matrix [Pq] must have the same rank as matrix P
Option (B)
Vector q must have only non-zero elements
Option (C)
Matrix P must be singular
Option (D)
Matrix P must be square
Correct Option:
A
Question Solution:
rank [pq] = rank [p] is necessary for existence of at least one solution to Px = q.