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(A) The given quadratic equation is $64x^2 + 5Nx + 1 = 0$. For the equation to have no real roots,the discriminant $D$ must be less than $0$. $D = (5N

Vedclass · Accepted Answer

$27$

Vedclass · Answer

(A) The given quadratic equation is $64x^2 + 5Nx + 1 = 0$. For the equation to have no real roots,the discriminant $D$ must be less than $0$. $D = (5N)^2 - 4(64)(1) $25N^2 - 256 $N^2 Since $N$ is the number of tosses,$N$ must be a positive integer,so $N \in \{1, 2, 3\}$. The probability of getting a head is $P(H) = \frac{1}{4}$ and the probability of getting a tail is $P(T) = \frac{3}{4}$. The probability that the first head appears at the $N$-th toss is given by $P(N) = (\frac{3}{4})^{N-1} 	imes \frac{1}{4}$. For $N=1$: $P(1) = \frac{1}{4}$. For $N=2$: $P(2) = \frac{3}{4} 	imes \frac{1}{4} = \frac{3}{16}$. For $N=3$: $P(3) = (\frac{3}{4})^2 	imes \frac{1}{4} = \frac{9}{64}$. The total probability is $P(N \in \{1, 2, 3\}) = \frac{1}{4} + \frac{3}{16} + \frac{9}{64} = \frac{16 + 12 + 9}{64} = \frac{37}{64}$. Here,$p = 37$ and $q = 64$. Thus,$q - p = 64 - 37 = 27$.

List-$A$	List-$B$
$(I)$ $A, B$ are mutually exclusive events	$(i)$ $P(A \cap B) = P(B) - P(\bar{A})$
$(II)$ $A, B$ are independent events	$(ii)$ $P(A) \leq P(B)$
$(III)$ $A \cap B = A$	$(iii)$ $P(\frac{\bar{A}}{B}) = 1 - P(A)$
$(IV)$ $A \cup B = S$	$(iv)$ $P(A \cup B) = P(A) + P(B)$
	$(v)$ $P(A) + P(B) = 2$

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