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Graduate Aptitude Test in Engineering

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General Aptitude

1

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

$${1 \over 9}$$

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)

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

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

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

$${1 \over {30}}$$

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

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

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

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

$${1 \over 12}$$

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)

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

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)

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

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