1

### JEE Main 2021 (Online) 1st September Evening Shift

Two squares are chosen at random on a chessboard (see figure). The probability that they have a side in common is : A
${2 \over 7}$
B
${1 \over 18}$
C
${1 \over 7}$
D
${1 \over 9}$

## Explanation

Total ways of choosing square = ${}^{64}{C_2}$

$= {{64 \times 63} \over {2 \times 1}} = 32 \times 63$

ways of choosing two squares having common side = 2 (7 $\times$ 8) = 112

Required probability $= {{112} \over {32 \times 63}} = {{16} \over {32 \times 9}} = {1 \over {18}}$.

Ans. (b)
2

### JEE Main 2021 (Online) 31st August Evening Shift

Let S = {1, 2, 3, 4, 5, 6}. Then the probability that a randomly chosen onto function g from S to S satisfies g(3) = 2g(1) is :
A
${1 \over {10}}$
B
${1 \over {15}}$
C
${1 \over {5}}$
D
${1 \over {30}}$

## Explanation

g(3) = 2g(1) can be defined in 3 ways

number of onto functions in this condition = 3 $\times$ 4!

Total number of onto functions = 6!

Required probability = ${{3 \times 4!} \over {6!}} = {1 \over {10}}$
3

### JEE Main 2021 (Online) 27th August Evening Shift

Each of the persons A and B independently tosses three fair coins. The probability that both of them get the same number of heads is :
A
${1 \over 8}$
B
${5 \over 8}$
C
${5 \over 16}$
D
1
4

### JEE Main 2021 (Online) 27th August Morning Shift

When a certain biased die is rolled, a particular face occurs with probability ${1 \over 6} - x$ and its opposite face occurs with probability ${1 \over 6} + x$. All other faces occur with probability ${1 \over 6}$. Note that opposite faces sum to 7 in any die. If 0 < x < ${1 \over 6}$, and the probability of obtaining total sum = 7, when such a die is rolled twice, is ${13 \over 96}$, then the value of x is :
A
${1 \over 16}$
B
${1 \over 8}$
C
${1 \over 9}$
D
${1 \over 12}$

## Explanation

Probability of obtaining total sum 7 = probability of getting opposite faces.

Probability of getting opposite faces

$= 2\left[ {\left( {{1 \over 6} - x} \right)\left( {{1 \over 6} + x} \right) + {1 \over 6} \times {1 \over 6} + {1 \over 6} \times {1 \over 6}} \right]$

$\Rightarrow 2\left[ {\left( {{1 \over 6} - x} \right)\left( {{1 \over 6} + x} \right) + {1 \over 6} \times {1 \over 6} + {1 \over 6} \times {1 \over 6}} \right] = {{13} \over {96}}$ (given)

$\Rightarrow$ $x = {1 \over 8}$