JEE Advanced 2020 · Paper 1 · Mathematics Q5

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MathematicsSingle correct+3 / -1Question 41 of 54

Let $C_1$ and $C_2$ be two biased coins such that the probabilities of getting head in a single toss are $\dfrac23$ and $\dfrac13$ , respectively. Suppose $\alpha$ is the number of heads that appear when $C_1$ is tossed twice, independently, and suppose $\beta$ is the number of heads that appear when $C_2$ is tossed twice, independently. Then the probability that the roots of the quadratic polynomial $x^2 - \alpha x + \beta$ are real and equal, is

\dfrac{40}{81}

\dfrac{20}{81}

\dfrac12

\dfrac14

Answer and worked solution

Official answer: B

Equal roots need $\alpha^2=4\beta$ . With $\alpha,\beta\in\{0,1,2\}$ , the only possibilities are $(\alpha,\beta)=(0,0)$ and $(2,1)$ .

Probabilities.

$P(\alpha=0)=\left(\tfrac13\right)^2=\tfrac19$ and $P(\beta=0)=\left(\tfrac23\right)^2=\tfrac49$ .
$P(\alpha=2)=\left(\tfrac23\right)^2=\tfrac49$ and $P(\beta=1)=2\cdot\tfrac13\cdot\tfrac23=\tfrac49$ .

$P=\frac19\cdot\frac49+\frac49\cdot\frac49=\frac4{81}+\frac{16}{81}=\frac{20}{81}.$
(B)

Topics:Independent events Nature of roots

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JEE Advanced 2020 · Paper 1, Mathematics question 5