Mathematical Reasoning and Aptitude Questions and Answers

Q: While entering into the train Anil was ahead of Shyam, who was ahead of Kiran, Anil was behind Shweta. Which of the following shows the correct order of entry into the train ?

The order of entry is Shweta - Anil - Shyam - Kiran.Take the clues one at a time. Anil was ahead of Shyam, and Shyam was ahead of Kiran, which gives Anil, Shyam, Kiran. Anil was behind Shweta, so Shweta comes before Anil. Putting the two together gives Shweta, Anil, Shyam, Kiran.The first option puts Shweta last, although she was ahead of Anil. The second puts Shyam ahead of Anil, reversing the first clue. The fourth puts Anil ahead of Shweta, contradicting the last clue.

Q: If LONQ is converted into JMLO then by the same principles what could be the correct alternative for the GDJM ?

The code for GDJM is EBHK.Compare LONQ with JMLO letter by letter: L to J, O to M, N to L and Q to O. Each letter moves two places back in the alphabet. Apply the same rule to GDJM:G - 2 = E, D - 2 = B, J - 2 = H, M - 2 = K, giving EBHK.ECIL moves the last three letters back by only one place. EHIK and ECHK each get the second letter wrong (D moved two places back is B, not H or C), and EHIK also gets the third letter wrong.

Q: Consider a road map of a city having 5 approach roads from other places, 10 junctions and no internal road has a dead-end. Every junction except one is associated with 4 roads. If n number of roads are associated with the remaining junction then which of the following values is impossible for n ?

The impossible value is 6.Count road-ends at junctions. Each internal road has both ends at junctions, and each of the 5 approach roads has one end at a junction (the other end leads out of the city). So the total number of road-ends at junctions is \(2E + 5\), where E is the number of internal roads, and that is an odd number.\[ \begin{aligned} 9 \times 4 + n &= 2E + 5 \\ 36 + n &= \text{odd} \end{aligned} \]So \(n\) must be odd. 3, 5 and 7 are possible, but 6 is even and cannot occur. This is the handshake (degree-sum) argument from graph theory.

Q: An honest person somehow guaranteed to two different persons X and Y individually to handover complete amount available in his unique purse. In which way/situation he will fulfill his assurance given to X and Y at one and the same time ?

He fulfils both promises if there is no money in his purse.He promised each of X and Y, separately, the complete amount in his purse. Giving the whole amount to one leaves nothing for the other, so both promises can only be kept together if the complete amount is zero: handing over 'everything' to each then means handing over nothing to both, and both promises are honoured.Paying each half breaks both promises, since neither gets the complete amount. Paying nothing when the purse holds money breaks them too, and an even number of notes only allows halving, which still fails.

Q: Immediately after 12 o'clock at what time duration both the minute hand and the hour hand of a watch exactly overlap ?

They overlap after 1 hour five minutes and before one hour six minutes.The minute hand gains \(5.5^\circ\) per minute on the hour hand. After 12 o'clock it must gain a full \(360^\circ\) to overlap again:\[ \begin{aligned} t &= \frac{360}{5.5} = \frac{720}{11}\ \text{min} \\ &= 65\tfrac{5}{11}\ \text{min} \\ &\approx 1\ \text{h}\ 5.45\ \text{min} \end{aligned} \]That is between 1:05 and 1:06. At exactly one hour the hour hand is at 1 while the minute hand is at 12, and at 1:05 the hour hand has moved on past the 1, so the exact one hour and one hour five minutes options are both too early.

Q: As the price of a book is reduced by 20%, then its consumption is increased by 25%. How much percentage difference is there in the previous income ?

There is 0 percent difference.Income (spending) is price times quantity. Taking the old price and quantity as 100 each:\[ \begin{aligned} \text{new} &= 0.80 \times 1.25 \\ &= 1.00 \end{aligned} \]The 20% fall in price is exactly offset by the 25% rise in consumption, so the income is unchanged. The other options come from adding or subtracting the percentages directly (25 - 20 = 5 is not even offered) or from quoting one of the two changes.

Q: The percentage profit earned by selling an item for Rs. 2120 is same as the percentage loss incurred by selling the same item for Rs. 1480. At what price should the item be sold to get 20% profit ?

The item should be sold for 2160.Equal percentage profit and loss mean the cost price lies exactly midway between the two selling prices:\[ \begin{aligned} 2120 - C &= C - 1480 \\ C &= \frac{2120 + 1480}{2} = 1800 \\ 1.2 \times 1800 &= 2160 \end{aligned} \]Check: 2120 is a profit of 320, and 1480 a loss of 320, each 17.8% of 1800. 2200 and 2050 do not correspond to 20% of 1800, and 1900 is only a 5.6% profit.

Q: The number of eight-bit computer words formed by 0s and 1s with exactly three 1s and five 0s is :

There are 56 such words.Choose which 3 of the 8 bit positions hold the 1s; the remaining 5 are 0s:\[ \begin{aligned} \binom{8}{3} &= \frac{8 \times 7 \times 6}{3 \times 2 \times 1} \\ &= 56 \end{aligned} \]Choosing the positions of the five 0s gives the same answer, \(\binom{8}{5} = 56\). The value 15 is \(\binom{6}{2}\), and 30 and 65 do not arise from choosing positions at all.

Q: The following is a list of various committees with their members :X1 = {a, b, c, d}, X2 = {x, b, d, y}, X3 = {x, y, a}, X4 = {y, b}, X5 = {x, a, b}, X6 = {y, a, b}, X7 = {x, y, a, b}How many ways can we select representatives of the committees so that no two committees have same representative ?

There are 0 ways.Each of the 7 committees needs a different representative drawn from its own members. But all the committees together have only 6 distinct members: a, b, c, d, x and y.By the pigeonhole principle, 7 committees cannot each be given a different person from a pool of 6, so no such selection exists. In terms of Hall's marriage theorem, the union of all 7 sets is smaller than 7, so a system of distinct representatives is impossible.A selection would be possible only if every group of k committees together had at least k distinct members, which fails here for all seven.

Q: In a rectangular standing arrangement of persons in four rows, each row having four persons. If jth person in the ith row is asked to move to the place of ith person of jth row for all i = 1, 2, 3, 4 and j = 1, 2, 3, 4, then how many persons have changed their original position ?

12 persons change their position.The move sends the person at row i, place j to row j, place i, like transposing a 4 × 4 matrix. A person stays where they are only when i = j, that is, on the diagonal: 4 persons.\[ 16 - 4 = 12 \]Every off-diagonal person swaps with their mirror-image partner, so 12 move. 16 would mean nobody stays, and 4 counts only those who do not move.

21.

While entering into the train Anil was ahead of Shyam, who was ahead of Kiran, Anil was behind Shweta. Which of the following shows the correct order of entry into the train ?

A.Anil - Shyam - Kiran - Shweta
B.Shweta - Shyam - Anil - Kiran
C.Shweta - Anil - Shyam - Kiran
D.Anil - Shweta - Shyam - Kiran

Exam Year(s)

MH SET - 2020, SET Exam - 2020,

Answer: C

The order of entry is Shweta - Anil - Shyam - Kiran.

Take the clues one at a time. Anil was ahead of Shyam, and Shyam was ahead of Kiran, which gives Anil, Shyam, Kiran. Anil was behind Shweta, so Shweta comes before Anil. Putting the two together gives Shweta, Anil, Shyam, Kiran.

The first option puts Shweta last, although she was ahead of Anil. The second puts Shyam ahead of Anil, reversing the first clue. The fourth puts Anil ahead of Shweta, contradicting the last clue.

Discuss in Forum

22.

If LONQ is converted into JMLO then by the same principles what could be the correct alternative for the GDJM ?

A.EBHK
B.ECIL
C.EHIK
D.ECHK

Exam Year(s)

MH SET - 2020, SET Exam - 2020,

Answer: A

The code for GDJM is EBHK.

Compare LONQ with JMLO letter by letter: L to J, O to M, N to L and Q to O. Each letter moves two places back in the alphabet. Apply the same rule to GDJM:

G - 2 = E, D - 2 = B, J - 2 = H, M - 2 = K, giving EBHK.

ECIL moves the last three letters back by only one place. EHIK and ECHK each get the second letter wrong (D moved two places back is B, not H or C), and EHIK also gets the third letter wrong.

Discuss in Forum

23.

Consider a road map of a city having 5 approach roads from other places, 10 junctions and no internal road has a dead-end. Every junction except one is associated with 4 roads. If n number of roads are associated with the remaining junction then which of the following values is impossible for n ?

Exam Year(s)

MH SET - 2021, SET Exam - 2021,

Answer: C

The impossible value is 6.

Count road-ends at junctions. Each internal road has both ends at junctions, and each of the 5 approach roads has one end at a junction (the other end leads out of the city). So the total number of road-ends at junctions is \(2E + 5\), where E is the number of internal roads, and that is an odd number.

\[ \begin{aligned} 9 \times 4 + n &= 2E + 5 \\ 36 + n &= \text{odd} \end{aligned} \]

So \(n\) must be odd. 3, 5 and 7 are possible, but 6 is even and cannot occur. This is the handshake (degree-sum) argument from graph theory.

Discuss in Forum

24.

An honest person somehow guaranteed to two different persons X and Y individually to handover complete amount available in his unique purse. In which way/situation he will fulfill his assurance given to X and Y at one and the same time ?

A.If there is no money in his purse
B.By paying each one of them half of the money available in the purse
C.By paying nothing to anyone of them
D.In the purse the number of notes/coins of same amount is even

Exam Year(s)

MH SET - 2021, SET Exam - 2021,

Answer: A

He fulfils both promises if there is no money in his purse.

He promised each of X and Y, separately, the complete amount in his purse. Giving the whole amount to one leaves nothing for the other, so both promises can only be kept together if the complete amount is zero: handing over 'everything' to each then means handing over nothing to both, and both promises are honoured.

Paying each half breaks both promises, since neither gets the complete amount. Paying nothing when the purse holds money breaks them too, and an even number of notes only allows halving, which still fails.

Discuss in Forum

25.

Immediately after 12 o'clock at what time duration both the minute hand and the hour hand of a watch exactly overlap ?

A.One hour
B.One hour five minutes
C.After 1 hour five minutes and before one hour six minutes
D.After one hour six minutes

Exam Year(s)

MH SET - 2021, SET Exam - 2021,

Answer: C

They overlap after 1 hour five minutes and before one hour six minutes.

The minute hand gains \(5.5^\circ\) per minute on the hour hand. After 12 o'clock it must gain a full \(360^\circ\) to overlap again:

\[ \begin{aligned} t &= \frac{360}{5.5} = \frac{720}{11}\ \text{min} \\ &= 65\tfrac{5}{11}\ \text{min} \\ &\approx 1\ \text{h}\ 5.45\ \text{min} \end{aligned} \]

That is between 1:05 and 1:06. At exactly one hour the hour hand is at 1 while the minute hand is at 12, and at 1:05 the hour hand has moved on past the 1, so the exact one hour and one hour five minutes options are both too early.

Discuss in Forum

26.

As the price of a book is reduced by 20%, then its consumption is increased by 25%. How much percentage difference is there in the previous income ?

A.10
B.20
C.25
D.0

Exam Year(s)

MH SET - 2021, SET Exam - 2021,

Answer: D

There is 0 percent difference.

Income (spending) is price times quantity. Taking the old price and quantity as 100 each:

\[ \begin{aligned} \text{new} &= 0.80 \times 1.25 \\ &= 1.00 \end{aligned} \]

The 20% fall in price is exactly offset by the 25% rise in consumption, so the income is unchanged. The other options come from adding or subtracting the percentages directly (25 - 20 = 5 is not even offered) or from quoting one of the two changes.

Discuss in Forum

27.

The percentage profit earned by selling an item for Rs. 2120 is same as the percentage loss incurred by selling the same item for Rs. 1480. At what price should the item be sold to get 20% profit ?

A.2160
B.2200
C.2050
D.1900

Exam Year(s)

MH SET - 2021, SET Exam - 2021,

Answer: A

The item should be sold for 2160.

Equal percentage profit and loss mean the cost price lies exactly midway between the two selling prices:

\[ \begin{aligned} 2120 - C &= C - 1480 \\ C &= \frac{2120 + 1480}{2} = 1800 \\ 1.2 \times 1800 &= 2160 \end{aligned} \]

Check: 2120 is a profit of 320, and 1480 a loss of 320, each 17.8% of 1800. 2200 and 2050 do not correspond to 20% of 1800, and 1900 is only a 5.6% profit.

Discuss in Forum

28.

The number of eight-bit computer words formed by 0s and 1s with exactly three 1s and five 0s is :

A.15
B.56
C.65
D.30

Exam Year(s)

MH SET - 2023, SET Exam - 2023,

Answer: B

There are 56 such words.

Choose which 3 of the 8 bit positions hold the 1s; the remaining 5 are 0s:

\[ \begin{aligned} \binom{8}{3} &= \frac{8 \times 7 \times 6}{3 \times 2 \times 1} \\ &= 56 \end{aligned} \]

Choosing the positions of the five 0s gives the same answer, \(\binom{8}{5} = 56\). The value 15 is \(\binom{6}{2}\), and 30 and 65 do not arise from choosing positions at all.

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29.

The following is a list of various committees with their members :

X₁ = {a, b, c, d}, X₂ = {x, b, d, y}, X₃ = {x, y, a}, X₄ = {y, b}, X₅ = {x, a, b}, X₆ = {y, a, b}, X₇ = {x, y, a, b}

How many ways can we select representatives of the committees so that no two committees have same representative ?

Exam Year(s)

MH SET - 2023, SET Exam - 2023,

Answer: D

There are 0 ways.

Each of the 7 committees needs a different representative drawn from its own members. But all the committees together have only 6 distinct members: a, b, c, d, x and y.

By the pigeonhole principle, 7 committees cannot each be given a different person from a pool of 6, so no such selection exists. In terms of Hall's marriage theorem, the union of all 7 sets is smaller than 7, so a system of distinct representatives is impossible.

A selection would be possible only if every group of k committees together had at least k distinct members, which fails here for all seven.

Discuss in Forum

30.

In a rectangular standing arrangement of persons in four rows, each row having four persons. If j^th person in the i^th row is asked to move to the place of i^th person of j^th row for all i = 1, 2, 3, 4 and j = 1, 2, 3, 4, then how many persons have changed their original position ?

A.4
B.5
C.12
D.16

Exam Year(s)

MH SET - 2023, SET Exam - 2023,

Answer: C

12 persons change their position.

The move sends the person at row i, place j to row j, place i, like transposing a 4 × 4 matrix. A person stays where they are only when i = j, that is, on the diagonal: 4 persons.

\[ 16 - 4 = 12 \]

Every off-diagonal person swaps with their mirror-image partner, so 12 move. 16 would mean nobody stays, and 4 counts only those who do not move.

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