JEE Main 2023 (25 January Shift 2)

LEVELJEE Main

Animated Solution for Mathematics - Probability: Let $N$ be the sum of the numbers appeared when two fair dice are rolled and let the probability that $N - 2, 3 N, N + 2$ are in geometric progression be $\frac{k}{48}$ . Then the value of $k$ is

Q: Wait, why do we reject $N = -1$? It is a valid root of the quadratic equation!

That is a brilliant observation! Mathematically, $N = -1$ is indeed a root of the equation $N^2 - 3N - 4 = 0$. However, in physics and probability, we must respect the physical constraints of the system. Since $N$ is the sum of two dice, the minimum value is $1+1=2$. A sum of $-1$ is physically impossible in our sample space, so we must discard it.

Select Answer:

Visualized Solution

Defining the Variable $N$

Let $N$ be the sum of numbers on two fair dice.
Condition: $N - 2$ , $3 N$ , $N + 2$ are in Geometric Progression (G.P.).
Probability of this event is given as $\frac{k}{48}$ .

The Geometric Progression Condition

For three numbers $a$ , $b$ , $c$ to be in G.P., the condition is $b^{2} = a c$ .

Substituting the Terms

Substitute $a = N - 2$ , $b = 3 N$ , and $c = N + 2$ into $b^{2} = a c$ .
Equation: $(3 N)^{2} = (N - 2) (N + 2)$

Simplifying the Equation

Simplify the left-hand side: $(3 N)^{2} = 3 N$
Simplify the right-hand side using $(a - b) (a + b) = a^{2} - b^{2}$ .
$(N - 2) (N + 2) = N^{2} - 4$
Resulting equation: $3 N = N^{2} - 4$

Forming the Quadratic Equation

Rearrange the terms to form a standard quadratic equation.
Move $3 N$ to the right side: $N^{2} - 3 N - 4 = 0$

Solving for $N$

Factorize the quadratic: $N^{2} - 4 N + N - 4 = 0$
$(N - 4) (N + 1) = 0$
Possible values: $N = 4$ or $N = - 1$

Validating the Value of $N$

Recall that $N$ is the sum of the numbers on two dice.
The minimum sum is $1 + 1 = 2$ , and the maximum is $6 + 6 = 12$ .
Therefore, $2 \leq N \leq 12$ .
Reject $N = - 1$ . The only valid solution is $N = 4$ .

Analyzing the Sample Space

Total number of outcomes when rolling two fair dice: $n (S) = 6 \times 6 = 36$ .
Each outcome is an ordered pair $(x, y)$ where $1 \leq x, y \leq 6$ .

Identifying Favorable Outcomes

We need the outcomes where the sum $N = 4$ .
Let $A$ be the favorable event: $A = {(1, 3), (2, 2), (3, 1)}$
Number of favorable outcomes: $n (A) = 3$ .

Calculating the Probability

Probability of event $A$ : $P (A) = \frac{n ( A )}{n ( S )}$
Substitute the values: $P (A) = \frac{3}{36}$
Simplify the fraction: $P (A) = \frac{1}{12}$

Finding the Value of $k$

The problem states the probability is $\frac{k}{48}$ .
Equate our calculated probability to the given expression: $\frac{1}{12} = \frac{k}{48}$
Solve for $k$ : $k = \frac{48}{12} = 4$
Final Answer: $k = 4$

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The Sigma Insight: Classical Definition of Probability

Solution Diagram

Analyzing the Geometric Constraint

For any three terms

a

b

, and

c

to be in a Geometric Progression (G.P.), the middle term must satisfy the condition:

b^{2} = a c

In this problem, the terms are

N - 2

3 N

, and

N + 2

. Substituting these into our "Golden Key" condition, we obtain:

(3 N)^{2} = (N - 2) (N + 2)

Solving the Algebraic Equation

The left side simplifies to

3 N

, while the right side follows the difference of squares identity, resulting in

N^{2} - 4

. This yields the following quadratic equation:

N^{2} - 3 N - 4 = 0

Factoring the quadratic expression, we get:

(N - 4) (N + 1) = 0

This provides two potential solutions:

N = 4

N = - 1

Interpreting Physical Reality

Here lies the "JEE Trap." Since

N

represents the sum of two dice, the minimum possible sum is

1 + 1 = 2

and the maximum is

6 + 6 = 12

Because

N = - 1

falls outside the valid range

[2, 12]

, it is physically impossible and must be discarded. We proceed strictly with

N = 4

Calculating Probability

The total sample space for rolling two dice is

6 \times 6 = 36

. We identify the favorable outcomes

(x, y)

such that

x + y = 4

(1, 3), (2, 2), (3, 1)

There are exactly

3

such outcomes. The probability

P

is therefore:

P = \frac{3}{36} = \frac{1}{12}

Final Calculation

We are given that this probability is equal to

\frac{k}{48}

. Equating the two expressions:

\frac{1}{12} = \frac{k}{48}

Solving for

k

, we find:

$k = 4$

Common Traps & FAQs

Wait, why do we reject

N = - 1

? It is a valid root of the quadratic equation!

That is a brilliant observation! Mathematically,

N = - 1

is indeed a root of the equation

N^{2} - 3 N - 4 = 0

. However, in physics and probability, we must respect the physical constraints of the system. Since

N

is the sum of two dice, the minimum value is

1 + 1 = 2

. A sum of

- 1

is physically impossible in our sample space, so we must discard it.

Why is the total sample space

36

Think of the two dice as independent events. The first die has

6

faces, and the second has

6

faces. By the fundamental counting principle, the total number of outcomes is

6 \times 6 = 36

. Every outcome is an ordered pair

(x, y)

where

1 \leq x, y \leq 6

How do I know which outcomes sum to

4

Systematic listing is your best friend here! Start with the smallest possible value for the first die. If the first die is

1

, the second must be

3

. If the first is

2

, the second must be

2

. If the first is

3

, the second must be

1

. If the first is

4

, the second would have to be

0

, which is impossible. So, we have exactly

3

favorable outcomes.

Animated Solution for Mathematics - Probability: Let $N$ be the sum of the numbers appeared when two fair dice are rolled and let the probability that $N - 2, 3 N, N + 2$ are in geometric progression be $\frac{k}{48}$ . Then the value of $k$ is

Select Answer:

Visualized Solution

Defining the Variable $N$

The Geometric Progression Condition

Substituting the Terms

Simplifying the Equation

Forming the Quadratic Equation

Solving for $N$

Validating the Value of $N$

Analyzing the Sample Space

Identifying Favorable Outcomes

Calculating the Probability

Finding the Value of $k$

The Sigma Insight: Classical Definition of Probability

Analyzing the Geometric Constraint

Solving the Algebraic Equation

Interpreting Physical Reality

Calculating Probability

Final Calculation

Similar Questions

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Select Answer:

Visualized Solution

Defining the Variable .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } N

The Geometric Progression Condition

Substituting the Terms

Simplifying the Equation

Forming the Quadratic Equation

Solving for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } N

Validating the Value of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } N

Analyzing the Sample Space

Identifying Favorable Outcomes

Calculating the Probability

Finding the Value of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } k

The Sigma Insight: Classical Definition of Probability

Analyzing the Geometric Constraint

Solving the Algebraic Equation

Interpreting Physical Reality

Calculating Probability

Final Calculation

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