JEE Advanced · Mathematics · 32. Probability

Let $\omega$ be a complex cube root of unity with $\omega \neq 1$. A fair die is thrown three times. If $r_1, r_2$ and $r_3$ are the numbers obtained on the die, then the probability that $\omega^{r_1}+\omega^{r_2}+\omega^{r_3}=0$ is

A
$\frac{1}{18}$
B
$\frac{1}{9}$
C
$\frac{2}{9}$
D
$\frac{1}{36}$

Verified Solution

Answer & Solution

Correct Answer

(C)
$\frac{2}{9}$

Step-by-step Solution

Detailed explanation

Sample space A dice is thrown thrice, $n(s)=6 \times 6 \times 6$.
Favorable events $\omega^{r_1}+\omega^{r_2}+\omega^{r_3}=0$ ie, $\left(r_1, r_2, r_3\right)$ are ordered 3-triples which can take values, $\left.\begin{array}{llll}(1,2,3), & (1,5,3), & (4,2,3), & (4,5,3) \\ (1,2,6), & (1,5,6), & (4,2,6), & (4,5,6)\end{array}\right\}$ ie, 8 ordered pairs and each can be arranged in 3 ! ways $=6$
\[
\begin{array}{ll}
\therefore & n(E)=8 \times 6 \\
\Rightarrow & P(E)=\frac{8 \times 6}{6 \times 6 \times 6}=\frac{2}{9}
\end{array}
\]

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Let $\mathbb{R}$ denote the set of all real numbers. For a real number $x$, let $[x]$ denote the greatest integer less than or equal to $x$. Let $n$ denote a natural number.
Match each entry in List-I to the correct entry in List-II and choose the correct option.

	LIST - I		LIST - II
(P)	The minimum value of $n$ for which the function $f(x)=\left[\frac{10 x^3-45 x^2+60 x+35}{n}\right]$ is continuous on the interval $[1,2]$,is	(1)	8
(Q)	The minimum value of $n$ for which $g(x)=\left(2 n^2-13 n-15\right)\left(x^3+3 x\right),x \in \mathbb{R}$,is an increasing function on $\mathbb{R}$,is	(2)	9
(R)	The smallest natural number $n$ which is greater than 5,such that $x=3$ is a point of local minima of $h(x)=\left(x^2-9\right)^{\mathrm{n}}\left(x^2+2 x+3\right)$,is	(3)	5
(S)	Number of $x_0 \in \mathbb{R}$ such that $l(x)=\sum_{k=0}^4\left(\sin \|x-k\|+\cos \left\|x-k+\frac{1}{2}\right\|\right),x \in \mathbb{R}$,is NOT differentiable at $x_0$,is	(4)	6
		(5)	10

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Match List I with List II and select the correct answer using the codes given below the lists:

	$\text{List I}$		$\text{List II}$
a.	Boltzmann constant	p.	$\left[M L^2 T^{-1}\right]$
b.	Coefficient of viscosity	q.	$\left[M L^{-1} T^{-1}\right]$
c.	Planck constant	r.	$\left[M L T^{-3} K^{-1}\right]$
d.	Thermal conductivity	s.	$\left[M L^2 T^{-2} K^{-1}\right]$

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Let \(\omega\) be a complex cube root of unity with \(\omega \neq 1\). A fair die is thrown three times. If \(r_1, r_2\) and \(r_3\) are the numbers obtained on the die, then the probability that \(\omega^{r_1}+\omega^{r_2}+\omega^{r_3}=0\) is

Answer & Solution

Step-by-step Solution

See the Complete Solution

	LIST - I		LIST - II
(P)	The minimum value of \(n\) for which the function \(f(x)=\left[\frac{10 x^3-45 x^2+60 x+35}{n}\right]\) is continuous on the interval \([1,2]\),is	(1)	8
(Q)	The minimum value of \(n\) for which \(g(x)=\left(2 n^2-13 n-15\right)\left(x^3+3 x\right),x \in \mathbb{R}\),is an increasing function on \(\mathbb{R}\),is	(2)	9
(R)	The smallest natural number \(n\) which is greater than 5,such that \(x=3\) is a point of local minima of \(h(x)=\left(x^2-9\right)^{\mathrm{n}}\left(x^2+2 x+3\right)\),is	(3)	5
(S)	Number of \(x_0 \in \mathbb{R}\) such that \(l(x)=\sum_{k=0}^4\left(\sin \|x-k\|+\cos \left\|x-k+\frac{1}{2}\right\|\right),x \in \mathbb{R}\),is NOT differentiable at \(x_0\),is	(4)	6
		(5)	10

	\(\text{List I}\)		\(\text{List II}\)
a.	Boltzmann constant	p.	\(\left[M L^2 T^{-1}\right]\)
b.	Coefficient of viscosity	q.	\(\left[M L^{-1} T^{-1}\right]\)
c.	Planck constant	r.	\(\left[M L T^{-3} K^{-1}\right]\)
d.	Thermal conductivity	s.	\(\left[M L^2 T^{-2} K^{-1}\right]\)