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Study Stuff for MCA Examinations - NIMCET
Study Stuff for MCA Examinations - NIMCET
Previous Year Question (PYQs)
Coefficients a, b, c of $ax^2 + bx + c = 0$ are chosen by tossing 3 fair coins.
Head means 1, Tail means 2.
Find the probability that the roots are imaginary
Solution
Since each of $a,b,c$ can be $1$ or $2$:Total possible triples $(a,b,c)$:
$2^3 = 8$.
Roots are imaginary if:
$b^2 - 4ac < 0$
So we check all $8$ cases.
List all possibilities
$(1,1,1)$: $1^2 - 4(1)(1) = 1 - 4 = -3 < 0$ ✔
$(1,1,2)$: $1 - 8 = -7 < 0$ ✔
$(1,2,1)$: $4 - 4 = 0$ ✘ (not imaginary)
$(1,2,2)$: $4 - 8 = -4 < 0$ ✔
$(2,1,1)$: $1 - 8 = -7 < 0$ ✔
$(2,1,2)$: $1 - 16 = -15 < 0$ ✔
$(2,2,1)$: $4 - 8 = -4 < 0$ ✔
$(2,2,2)$: $4 - 16 = -12 < 0$ ✔
Count imaginary root cases
Total imaginary cases = $7$ out of $8$.
So probability:
$\displaystyle P = \frac{7}{8}$
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