JEE Advanced · Mathematics · 4. P&C

Consider the $6 \times 6$ square in the figure. Let $A_{1}, A_{2}, \dots, A_{49}$ be the points of intersections (dots in the picture) in some order. We say that $A_{i}$ and $A_{j}$ are friends if they are adjacent along a row or along a column. Assume that each point $A_{i}$ has an equal chance of being chosen.

(There are two questions based on $PARAGRAPH " II "$ , the question given below is one of them)
Two distinct points are chosen randomly out of the points $A_{1}, A_{2}, \dots \dots, A_{49}$ . Let $p$ be the probability that they are friends. Then the value of $7 p$ is

A 0.1
B 0.54
C 0.5
D 0.45

Verified Solution

Answer & Solution

Correct Answer

(C) 0.5

Step-by-step Solution

Detailed explanation

Given,
We have to choose two distinct points randomly,
Now number of ways of selecting

2

adjacent dots in row will be

= 6

{as if you pick any dot then you have pick second dot from remaining

6

dots, so total ways will be

1 \times {}^{6}C_{1} = 6

}
Similarly, number of ways of selecting

2

adjacent dots in column

= 6

∴

Number of ways of selecting

2

adjacent dots from the given diagram we will

= 6 \times 7 + 6 \times 7 = 84

So, probability is given by,

p = \frac{84}{{}^{49}C_{2}} = \frac{84 \times 2}{49 \times 48}

\Rightarrow 7 p = \frac{7 \times 84 \times 2}{49 \times 48} = \frac{1}{2} = 0.5

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.

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$(A)$	In a triangle $∆ X Y Z$ let $a, b$ and $c$ be the lengths of the angles $X, Y$ and $Z$ , respectively. If $2 (a^{2} - b^{2}) = c^{2}$ and $λ = \frac{\sin (X - Y)}{sinZ}$ then possible values of $n$ for which $\cos (n π λ) = 0$ is are	$(P)$	$1$
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JEE Advanced 2015 Hard

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