Explanation

From 12 January 2024 to 12 January 2025, there are 366 days (2024 is a leap year).
$$366 \equiv 2 \pmod{7}$$

So, 12 January 2025 falls on Sunday.

From 12 January 2025 to 7 June 2025:

• January: 19 days
• February: 28 days
• March: 31 days
• April: 30 days
• May: 31 days
• June: 7 days

Total:$$19+28+31+30+31+7=146 \text{ days}$$ $$146 \equiv 6 \pmod{7}$$

Adding 6 days to Sunday gives Saturday.

Hence, the correct answer is (C) Saturday.

Answer: (B) 5050

Explanation

The sum of the first $n$n natural numbers is given by:
$$\frac{n(n+1)}{2}$$

For $n=100$:$$\frac{100\times101}{2}=50\times101=5050$$

Hence, the correct answer is (B) 5050.

Explanation

The distribution of 12 oranges among 5 baskets is not unique. For example:

• $3,3,2,2,2$ (Option A) is possible.
• $4,2,2,2,2$ (Option C) is also possible.

Since more than one distribution is possible, no single distribution given in the options must occur. Also, Option B is impossible because it uses 6 baskets (3 + 2 + 1), whereas only 5 baskets are available.

Hence, the correct answer is (D) None of the above.

Answer: (D) 180° (as per the given options, but mathematically incorrect)

Explanation

At 9:15:

• The minute hand is at $15 \times 6 = 90^\circ$
  
• The hour hand is at $9 \times 30 + 15 \times 0.5 = 270 + 7.5 = 277.5^\circ$.

Therefore, the angle between the hands is:
$$277.5^\circ – 90^\circ = 187.5^\circ$$

The smaller angle is:
$$360^\circ – 187.5^\circ = 172.5^\circ$$

Thus, the correct angle is 172.5

Answer: (D) Expressible in a recurring decimal.

Answer: (B) 16

Explanation

Observe the first and third numbers in each group:

• $5 \rightarrow 8$ (increase of 3)
• $6 \rightarrow 12$ (increase of 6)
• $7 \rightarrow ?$ (increase should be 9)

The increase follows the pattern:
$$3,\ 6,\ 9$$

Therefore,
$$7 + 9 = 16$$

Hence, the missing number is 16.

Answer: (C) Sum of two irrational numbers is not always an irrational.

Explanation

The sum of two irrational numbers may be rational or irrational.

• Irrational result: $$\sqrt{2}+\sqrt{3}$$
  is irrational.
• Rational result:
  $$\sqrt{2}+(-\sqrt{2})=0,$$
  which is rational.

Therefore, the sum of two irrational numbers is not always irrational.

Hence, the correct answer is (C).

Answer: (B) 4, 6

Explanation

Let the initial numbers of books be:

• Rabi = $2x$
• Madhu = $3x$

After Madhu purchases 6 books:
$$\frac{2x}{3x+6}=\frac{1}{3}$$$$6x=3x+6$$$$3x=6 \Rightarrow x=2$$

Therefore, the initial numbers of books were:

• Rabi = $2\times2=4$
• Madhu = $3\times2=6$

Thus, the correct pair is (4, 6).

Answer: (C) Rs. 138

Explanation

Cost Price (CP) = ₹115 per kg

Profit = 20% of ₹115:
$$\frac{20}{100}\times115=23$$

Selling Price (SP):
$$115+23=138$$

Hence, the required selling price is ₹138 per kg. Therefore, the correct answer is (C) Rs. 138.

Answer: (C) 54

Explanation

Let the tens digit be $x$x and the units digit be $y$y.

Given:
$$x+y=9$$

The difference between the original and the interchanged numbers is:
$$(10x+y)-(10y+x)=9$$$$9(x-y)=9 \Rightarrow x-y=1$$

Solving the two equations:$$x+y=9,\qquad x-y=1$$

gives:$$x=5,\quad y=4$$

Hence, the required number is 54.

Answer: (B) 55, 143

Explanation

Let the two numbers be $5x$ and $13x$.

Since 5 and 13 are coprime, their HCF is $x$. Given that the HCF is 11,
$$x=11$$

Therefore, the numbers are:
$$5\times11=55,\qquad 13\times11=143$$

Hence, the correct answer is (B) 55, 143.

Answer: (C) In 16 years

Explanation

This is a simple interest question.

For the amount to become double, the interest must equal the principal, i.e., 100%.

Using:
$$T=\frac{100}{R}$$

where $R=6\frac{1}{4}\%=6.25\%$,
$$T=\frac{100}{6.25}=16\text{ years}$$

Hence, the correct answer is (C) In 16 years.

Answer: (B) Rs. 1,500

Explanation

Amal’s profit:$$69-46=23$$

So, the profit ratio is:$$23:46=1:2$$

Profit is proportional to Capital × Time.
$$500\times9 : x\times6 = 1:2$$ $$4500:6x=1:2$$$$9000=6x$$ $$x=1500$$

Thus, Bimal’s investment = Rs. 1,500.

Answer: (D) 10 seconds

Explanation

To completely cross the platform, the train must cover:
$$80 + 120 = 200 \text{ m}$$

Convert the speed:
$$72 \text{ km/h} = 72 \times \frac{5}{18} = 20 \text{ m/s}$$

Time taken:$$\frac{200}{20}=10 \text{ seconds}$$

Hence, the correct answer is (D) 10 seconds.

Answer: (C) 11

Explanation

If the number of rows is equal to the number of saplings in each row, then the total number of saplings required is a perfect square.

Out of 127 saplings, 6 are left unused, so the number of saplings actually planted is:
$$127 – 6 = 121$$

Since,
$$121 = 11^2,$$

there are 11 rows with 11 saplings in each row.

Hence, the correct answer is (C) 11.

Answer: (C) 24=2.12+0

Explanation

According to the Division Algorithm,
$$\boxed{\text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder}}$$

where the remainder must satisfy:
$$0 \leq \text{Remainder} < \text{Divisor}$$

Here, the divisor is 2, so the remainder can only be 0 or 1.

Checking the options:

• (A) Remainder = 2 (x) (not less than 2)
• (B) Remainder = 4 (x)
• (C) 24=2.12+0 ✅ (remainder is valid)
• (D) Remainder = 6 (x)

Hence, the correct answer is (C) 24=2.12+0

Answer: (A) 7

Explanation

Let the three continued proportional numbers be:
$$\frac{a}{b}=\frac{b}{c}$$

So they can be written as:
$$\frac{x}{r},\; x,\; xr$$

Their product is:
$$\frac{x}{r}\times x\times xr=x^3=343$$$$x=\sqrt[3]{343}=7$$

The mean (median) proportional between the first and third numbers is the middle term, which is 7.

Hence, the correct answer is (A) 7.

Answer: (D) None of the above

Explanation

Maximum Retail Price (MRP) = ₹550

After allowing a 20% discount, the selling price is:
$$550 \times \frac{80}{100}=₹440$$

This selling price is 15% more than the cost price.
$$\text{Cost Price}=\frac{440\times100}{115} =\frac{44000}{115} \approx ₹382.61$$

Hence, the purchase price is approximately ₹382.61, which is not given in the options.

Therefore, the correct answer is (D) None of the above.

Answer: (D) 337

Explanation

The required number minus 1 must be divisible by 12, 16, and 28.

Find the LCM:

• $12 = 2^2 \times 3$
• $16 = 2^4$
• $28 = 2^2 \times 7$

$$\text{LCM} = 2^4 \times 3 \times 7 = 336$$

Therefore,
$$\text{Required number} = 336 + 1 = 337$$

Hence, the correct answer is (D) 337.

Answer: (D) None of the above

Explanation

Profit is shared in the ratio of Capital × Time.

• Mita’s investment:
  $$700 \times 4 + 1200 \times 8 = 2800 + 9600 = 12400$$
• Arun’s investment:
  $$900 \times 4 + 600 \times 8 = 3600 + 4800 = 8400$$

Therefore, the profit ratio is:
$$12400:8400 = 31:21$$

Total profit = ₹1720.

• Mita’s share:
  $$\frac{31}{52}\times1720 \approx ₹1025.38$$
• Arun’s share:
  $$\frac{21}{52}\times1720 \approx ₹694.62$$

These values do not match any of the given options.

Hence, the correct answer is (D) None of the above.

Answer: (A) At 160 minutes

Explanation

In the first 40 minutes $\left(\frac{2}{3}\text{ hour}\right)$:

• Faster train covers:
  $$60 \times \frac{2}{3}=40\text{ km}$$
• Slower train covers:
  $$30 \times \frac{2}{3}=20\text{ km}$$

Total distance covered = $40+20=60$.

So, the remaining distance between them is:
$$120-60=60\text{ km}$$

The faster train stops, and the slower train continues at 30 km/hr.

Time to cover the remaining 60 km:

$$\frac{60}{30}=2\text{ hours}=120\text{ minutes}$$

Total time from the start:
$$40+120=160\text{ minutes}$$

Thus, the correct mathematical answer is 160 minutes.

Answer: (B) Rs. 45 and Rs. 55

Explanation

The ratio of A : B is:
$$9:11$$

Total parts:
$$9+11=20$$

Value of one part:
$$\frac{100}{20}=5$$

Therefore,

• A’s share $=9\times5=₹45$
• B’s share $=11\times5=₹55$

Hence, the correct answer is (B) Rs. 45 and Rs. 55.

Answer: (D) In 18 days

Explanation

A’s one-day work:
$$\frac{1}{9}$$

B’s one-day work:
$$\frac{1}{6}$$

A, B, and C together complete the work in 3 days:
$$\frac{1}{3}$$

Therefore, C’s one-day work is:
$$\frac{1}{3}-\frac{1}{9}-\frac{1}{6} =\frac{6-2-3}{18} =\frac{1}{18}$$

Hence, C alone will complete the work in 18 days.

Therefore, the correct answer is (D) In 18 days.

Answer: (D) Composite

Explanation

A number is composite if it has more than two factors.

Since,
$$171 = 3 \times 57 = 3 \times 3 \times 19 = 9 \times 19,$$

171 has factors other than 1 and itself. Therefore, it is not prime.

Hence, the correct answer is (D) Composite.

Answer: (D) 11

Explanation

The number of times $\frac{7}{11}$117​ is contained in $7$7 is:
$$7 \div \frac{7}{11} =7\times\frac{11}{7} =11$$

Therefore, $\frac{7}{11}$ occurs 11 times in 7.

Hence, the correct answer is (D) 11.

Answer: (B) 35

Explanation

Let the numbers be $3x$ and $4x$.

Since 3 and 4 are coprime,
$$\text{LCM} = 3 \times 4 \times x = 12x$$

Given,
$$12x = 60 \Rightarrow x = 5$$

Therefore, the numbers are:
$$3 \times 5 = 15,\qquad 4 \times 5 = 20$$

Their sum is:
$$15 + 20 = 35$$

Hence, the correct answer is (B) 35.

Answer: (C) 50 cm

Explanation

Let the side of the resulting square be $x$ cm.

Then,

• Original length $= x+4$
• Original width $= x-3$

Since the square has the same area as the original rectangle,
$$(x+4)(x-3)=x^2$$$$x^2+x-12=x^2$$$$x=12$$

Therefore,

• Length $=12+4=16$
• Width $=12-3=9$

Perimeter of the original rectangle:
$$2(16+9)=2\times25=50\text{ cm}$$

Hence, the correct answer is (C) 50 cm.

Answer: (B) $\frac{2}{3}$

Explanation

Simplify the denominator:
$$1+\frac{x}{1-x} =\frac{1-x+x}{1-x} =\frac{1}{1-x}$$

Hence,
$$\frac{1}{1+\frac{x}{1-x}} =1-x$$

So the equation becomes:$$\frac{2x}{1+(1-x)}=1$$$$\frac{2x}{2-x}=1$$$$2x=2-x$$$$3x=2$$$$\boxed{x=\frac{2}{3}}$$

Hence, the correct answer is (B) $\frac{2}{3}$.

​.

Answer: (D) 4 : 3

Explanation

Given,
$$5\%\text{ of }A + 4\%\text{ of }B =\frac{2}{3}\left(6\%\text{ of }A + 8\%\text{ of }B\right)$$

Substituting percentages:
$$\frac{5A}{100}+\frac{4B}{100} =\frac{2}{3}\left(\frac{6A}{100}+\frac{8B}{100}\right)$$

Multiply both sides by 300:
$$15A+12B=12A+16B$$$$3A=4B$$$$\boxed{A:B=4:3}$$

Hence, the correct answer is (D) 4 : 3.

Answer: (A) 1,00,000

Explanation

Let the population in 1999 be $P$.

Since the population increases by 5% per year, after two years:
$$P \times (1.05)^2 = 110250$$$$P=\frac{110250}{1.1025}=100000$$

Therefore, the population in 1999 was 1,00,000.

Hence, the correct answer is (A) 1,00,000.

Answer: (D) 285

Explanation

A 30-day month beginning with a Sunday has 5 Sundays (1st, 8th, 15th, 22nd, and 29th) and 25 other days.

Total visitors:
$$5 \times 510 + 25 \times 240 =2550+6000 =8550$$

Average visitors per day:
$$\frac{8550}{30}=285$$

Hence, the correct answer is (D) 285.

Answer: (A) 288

Explanation

Let the required number be $x$.

According to the question,$$\frac{5x}{6}-\frac{5x}{16}=150$$

Taking LCM $48$48,$$\frac{40x-15x}{48}=150$$$$\frac{25x}{48}=150$$$$25x=7200$$$$x=\frac{7200}{25}=288$$

Hence, the correct answer is (A) 288.

Answer: (B) 20%

Explanation

Let the original price be ₹100 per unit and consumption be 100 units.

After a 25% increase, the new price becomes:
$$₹100 \rightarrow ₹125$$

To keep the total expenditure unchanged,
$$\text{New consumption}=\frac{100}{125}\times100=80\%$$

Hence, the reduction in consumption is:
$$100\%-80\%=20\%$$

Therefore, the correct answer is (B) 20%.

Answer: (D) $46 \dfrac{46}{49}\%$

Explanation

Runs scored through boundaries:

• 4 fours $= 4 \times 4 = 16$
• 6 sixes $= 6 \times 6 = 36$

Total runs from boundaries:
$$16+36=52$$

Runs scored by running:
$$98-52=46$$

Percentage of runs made by running:
$$\frac{46}{98}\times100 =\frac{23}{49}\times100 =46\frac{46}{49}\%$$

Hence, the correct answer is (D) $46 \dfrac{46}{49}\%$

Answer: (D) 55 kmph

Explanation

The train covers its own length while crossing the man.

Relative speed:
$$\frac{100}{6}=\frac{50}{3}\text{ m/s}$$

Convert the man’s speed:
$$5\text{ km/h}=\frac{25}{18}\text{ m/s}$$

Since they move in opposite directions,
$$\text{Train’s speed} =\frac{50}{3}-\frac{25}{18} =\frac{300-25}{18} =\frac{275}{18}\text{ m/s}$$

Converting to km/h:
$$\frac{275}{18}\times\frac{18}{5}=55\text{ km/h}$$

Hence, the correct answer is (D) 55 kmph.

Answer: (C) Rs. 4000

Explanation

Let the annual incomes of A and B be $5x$ and $4x$, and their annual expenditures be $3y$ and $2y$.

Since each saves ₹1600,
$$5x-3y=1600$$$$4x-2y=1600$$

Multiply the second equation by 3:
$$12x-6y=4800$$

Multiply the first equation by 2:
$$10x-6y=3200$$

Subtracting,
$$2x=1600 \Rightarrow x=800$$

Therefore, A’s annual income is:
$$5x=5\times800=₹4000$$

Hence, the correct answer is (C) Rs. 4000.

Answer: (A) 37.5%

Explanation

Let the Marked Price (MP) be ₹100.

• Cost Price (CP) = 64% of MP = ₹64.
• After a 12% discount, Selling Price (SP) = ₹88.

Profit:
$$88-64=24$$

Profit percentage:
$$\frac{24}{64}\times100=37.5\%$$

Hence, the correct answer is (A) 37.5%.

Answer: (B) 0.262

Explanation

Let
$$a=0.362,\qquad b=0.1$$

Since,
$$0.0362=0.362\times0.1=ab,$$

the denominator becomes:
$$a^2+ab+b^2$$

Using the identity,
$$a^3-b^3=(a-b)(a^2+ab+b^2),$$

the expression simplifies to:
$$\frac{a^3-b^3}{a^2+ab+b^2}=a-b$$$$=0.362-0.100=0.262$$

Hence, the correct answer is (B) 0.262.

Answer: (C) 60 days

Explanation

Let A’s and B’s one-day work be $a$a and $b$b, respectively.

Since A and B together complete the work in 30 days,
$$a+b=\frac{1}{30}$$

A works alone for 16 days and B alone for 44 days to complete the work:
$$16a+44b=1$$

Substitute $a=\frac{1}{30}-b$:
$$16\left(\frac{1}{30}-b\right)+44b=1$$$$\frac{8}{15}+28b=1$$
$$28b=\frac{7}{15}$$
$$b=\frac{1}{60}$$

Therefore, B alone can complete the work in 60 days.

Hence, the correct answer is (C) 60 days.

Answer: (C) 15 years

Explanation

If the sum doubles in 5 years, then:
$$(1+r)^5=2$$

To become 8 times the principal:
$$8=2^3$$

So the amount must double three times:

• After 5 years → $2P$
• After 10 years → $4P$
• After 15 years → $8P$

Hence, the required time is 15 years.

Therefore, the correct answer is (C) 15 years.

Answer: (C) 3 kmph

Explanation

Let the speed of the stream be $x$x kmph.

• Downstream speed $=10+x$
• Upstream speed $=10-x$

Since the times are equal,
$$\frac{26}{10+x}=\frac{14}{10-x}$$

Cross-multiplying,
$$26(10-x)=14(10+x)$$
$$260-26x=140+14x$$
$$120=40x$$
$$x=3$$

Therefore, the speed of the stream is 3 kmph.

Hence, the correct answer is (C) 3 kmph.

Answer: (D) 12 : 15 : 10

Explanation

For the same distance,
$$\text{Time}=\frac{\text{Distance}}{\text{Speed}}$$

Thus, time is inversely proportional to speed.

Given the speed ratio:
$$5:4:6$$

The corresponding time ratio is:
$$\frac{1}{5}:\frac{1}{4}:\frac{1}{6}$$

Taking the LCM of $5, 4,$5,4, and $6$6, which is 60:
$$\frac{60}{5}:\frac{60}{4}:\frac{60}{6} =12:15:10$$

This ratio is equivalent to:
$$\boxed{12:15:10}$$

Hence, the correct answer is (D) 12 : 15 : 10.

Answer: (A) Rs. 1080

Explanation

The rate of interest is:$$16\dfrac{2}{3}\%=\frac{1}{6}$$

For 3 years,
$$\text{CI}=P\left[\left(1+\frac16\right)^3-1\right] =P\left[\left(\frac76\right)^3-1\right]$$
$$1270=P\left(\frac{343-216}{216}\right) =P\cdot\frac{127}{216}$$
$$P=\frac{1270\times216}{127}=2160$$

Now, the simple interest for 3 years is:
$$\text{SI}=P\times\frac16\times3 =2160\times\frac12 =1080$$

Hence, the correct answer is (A) Rs. 1080.

Answer:(B) $6\dfrac{3}{4}$

Explanation

Evaluate each expression:
$$\frac{6}{7/8}=6\times\frac{8}{7}=\frac{48}{7}$$

and
$$\frac{6/7}{8}=\frac{6}{7}\times\frac{1}{8}=\frac{6}{56}=\frac{3}{28}$$

Now find the difference:
$$\frac{48}{7}-\frac{3}{28} =\frac{192-3}{28} =\frac{189}{28} =\frac{27}{4} =6\frac{3}{4}$$

Hence, the correct answer is (B) $6\dfrac{3}{4}$

Answer: (D) 25%

Explanation

Let the third number be 100.

• First number = $100 – 40\% = 60$
• Second number = $100 – 55\% = 45$

The second number is less than the first by:
$$60-45=15$$

Percentage by which the second is less than the first:

$$\frac{15}{60}\times100=25\%$$

Hence, the correct answer is (D) 25%.

Answer: (C) Rs. 6600

Explanation

Let the total capital be ₹x.

Investments:

• $\frac{x}{3}$​ at 7%
• $\frac{x}{4}$​ at 8%
• Remaining:

$$x-\frac{x}{3}-\frac{x}{4} =x\left(1-\frac13-\frac14\right) =\frac{5x}{12}$$

at 10%.

Total annual interest:

$$\frac{x}{3}\times\frac{7}{100} +\frac{x}{4}\times\frac{8}{100} +\frac{5x}{12}\times\frac{10}{100} =561$$

Taking LCM $=300$=300,
$$\frac{7x+6x+12.5x}{300}=561$$
$$\frac{25.5x}{300}=561$$
$$17x=561\times200=112200$$
$$x=\frac{112200}{17}=6600$$

Hence, the total capital invested is ₹6600.

Therefore, the correct answer is (C) Rs. 6600.

Answer: (D) Rs. 1200

Explanation

Let the cost price be ₹x.

• Profit when sold for ₹1060:

$$1060-x$$

• Loss when sold for ₹950:

$$x-950$$

Given that the profit is 20% more than the loss:
$$1060-x=\frac{6}{5}(x-950)$$
$$5(1060-x)=6(x-950)$$
$$5300-5x=6x-5700$$
$$11000=11x$$

$$x=1000$$

Thus, the cost price is ₹1000.

To earn 20% profit,$$\text{Selling Price}=1000\times\frac{120}{100}=₹1200$$

Hence, the correct answer is (D) Rs. 1200.

Answer: (A) 25 years

Explanation

Let Ajay’s age at the time of his marriage be $x$x years.

Present age:$$x+6=\frac{5}{4}x$$$$4x+24=5x$$$$x=24$$

So, Ajay’s present age is:
$$24+6=30\text{ years}$$

His sister was 5 years younger than him at the time of marriage.

Sister’s age at that time:
$$24-5=19\text{ years}$$

Therefore, her present age is:
$$19+6=25\text{ years}$$

Hence, the correct answer is (A) 25 years.

Answer: (C) 11 A.M.

Explanation

From 8 A.M. to 9 A.M., the first train travels alone.

Distance covered by the first train:
$$60 \times 1 = 60 \text{ km}$$

Remaining distance between the trains at 9 A.M.:
$$330 – 60 = 270 \text{ km}$$

Now both trains move towards each other.

Relative speed:
$$60 + 75 = 135 \text{ kmph}$$

Time taken to meet:
$$\frac{270}{135} = 2 \text{ hours}$$

Therefore, they meet 2 hours after 9 A.M., i.e.,
$$\boxed{\text{11:00 A.M.}}$$

Hence, the correct answer is (C) 11 A.M.

Answer: (C) Rs. 2,00,000

Explanation

Let the value of the machine 2 years ago be $P$P.

Since the machine depreciates by 10% per year, its present value is:
$$P \times (0.9)^2 = 162000$$$$P \times 0.81 = 162000$$$$P=\frac{162000}{0.81}=200000$$

Therefore, the value of the machine 2 years ago was ₹2,00,000.

Hence, the correct answer is (C) Rs. 2,00,000.

Answer: (D) 1260

Explanation

If 75% of the students are boys, then 25% are girls.

Given:

• Girls = 420 = 25% of the total students.

Total students:$$\frac{420 \times 100}{25}=1680$$

Number of boys:$$75\% \text{ of } 1680=\frac{75}{100}\times1680=1260$$

Hence, the correct answer is (D) 1260.

Answer: (B) $33\frac{1}{3}\%$

Explanation

Let B’s salary be ₹100.

Then A’s salary is 25% less:$$100-25=75$$

Now, the percentage by which B’s salary is more than A’s salary is:
$$\frac{100-75}{75}\times100 =\frac{25}{75}\times100 =33\frac{1}{3}\%$$

Hence, the correct answer is (B) $33\frac{1}{3}\%$

Answer: (B) Rs. 800

Explanation

Let the Cost Price (CP) be ₹x.

Selling at 20% loss:
$$\text{SP}_1=0.8x$$

If sold for ₹200 more, there would be a 5% profit:
$$0.8x+200=1.05x$$$$200=0.25x$$$$x=\frac{200}{0.25}=800$$

Answer: (B) $11\frac{1}{9}\%$

Explanation

Assume the cost price of 1 kg of goods is ₹100.

The dealer charges ₹100 for “1 kg”, but actually gives only 900 g.

Cost of 900 g:
$$100\times\frac{900}{1000}=₹90$$

Profit:
$$100-90=₹10$$

Profit percentage:
$$\frac{10}{90}\times100 =11\frac{1}{9}\%$$

Hence, the correct answer is (B) $11\frac{1}{9}\%$

Answer: (B) 1 : 5

Explanation

Let the cost price of 1 litre of spirit be ₹100.

To earn a 20% profit while selling at the cost price, the total mixture must increase by 20%.

So, for every 5 litres of spirit, add 1 litre of water (which costs nothing).

• Cost of 5 litres of spirit = ₹500
• Add 1 litre of water ⇒ Total mixture = 6 litres
• Selling at the cost price of spirit (₹100 per litre):

$$6 \times 100 = ₹600$$

Profit:
$$600 – 500 = ₹100$$

Profit percentage:
$$\frac{100}{500}\times100 = 20\%$$

Therefore, the required proportion of water : spirit is
$$\boxed{1:5}$$

Hence, the correct answer is (B) 1 : 5.

Answer: (A) 99

Explanation

Here,
$$999\frac{494}{495} =999+\frac{494}{495} =1000-\frac{1}{495}$$

Therefore,
$$\left(1000-\frac{1}{495}\right)\times99 =99000-\frac{99}{495} =99000-\frac{1}{5}$$

Now,
$$\frac{1}{5}+\left(99000-\frac{1}{5}\right)=99000$$

Hence,
$$\frac{1}{1000}\times99000=99$$

Therefore, the correct answer is (A) 99.

Answer: (B) 3.3

Explanation

Let the number be $x$.

The student multiplied it by 3:
$$3x=29.7$$$$x=\frac{29.7}{3}=9.9$$

The correct operation was to divide the number by 3:
$$\frac{x}{3}=\frac{9.9}{3}=3.3$$

Hence, the correct answer is (B) 3.3.

Answer: (C) 15 sec.

Explanation

In 1 minute (60 seconds), the cistern is filled to:
$$\frac{4}{5}$$​

Therefore, the remaining part is:
$$1-\frac{4}{5}=\frac{1}{5}$$

Since $\frac{1}{5}$​ is one-fourth of $\frac{4}{5}$​,
$$\text{Required time} =\frac{\frac{1}{5}}{\frac{4}{5}}\times60 =\frac{1}{4}\times60 =15\text{ seconds}$$

Hence, the correct answer is (C) 15 sec.

Answer: (A) 5

Explanation

X’s one-day work:
$$\frac{1}{20}$$​

Y’s one-day work:
$$\frac{1}{12}$$​

Work done by Y in 9 days:
$$9\times\frac{1}{12}=\frac{3}{4}$$

Remaining work:
$$1-\frac{3}{4}=\frac{1}{4}$$

Time taken by X to complete the remaining work:
$$\frac{\frac{1}{4}}{\frac{1}{20}} =\frac{1}{4}\times20 =5\text{ days}$$

Hence, the correct answer is (A) 5.

Answer: (B) 3 hrs. 12 mins.

Explanation

For the same amount of work,
$$\text{Workers} \times \text{Time} = \text{Constant}$$

Let the required time be $x$x hours.
$$12 \times 4 = 15 \times x$$ $$x=\frac{48}{15}=3.2\text{ hours}$$

Convert $0.2$0.2 hour into minutes:
$$0.2 \times 60 = 12 \text{ minutes}$$

Therefore, the required time is:
$$\boxed{3\text{ hours }12\text{ minutes}}$$

Hence, the correct answer is (B) 3 hrs. 12 mins.

Answer: (B) 134

Explanation

If the number of rows equals the number of trees in each row, then:
$$(\text{Number of trees in a row})^2 = 17956$$

Let the number of trees in each row be $n$.
$$n=\sqrt{17956}$$

Since,
$$134^2=134\times134=17956$$

Therefore,
$$n=134$$

Hence, the number of trees in each row is 134.

Therefore, the correct answer is (B) 134.

Answer: (A) Rs. 6600

Explanation

Let the total capital be ₹x.

Investments:

• $\frac{x}{3}$​ at 7%
• $\frac{x}{4}$​ at 8%
• Remaining:

$$x-\frac{x}{3}-\frac{x}{4} =\frac{5x}{12}$$

at 10%.

Annual interest:
$$\frac{x}{3}\times\frac{7}{100} +\frac{x}{4}\times\frac{8}{100} +\frac{5x}{12}\times\frac{10}{100} =561$$
$$\frac{7x}{300}+\frac{2x}{100}+\frac{x}{24}=561$$

Taking LCM $=600$,$$\frac{14x+12x+25x}{600}=561$$
$$\frac{51x}{600}=561$$
$$51x=561\times600$$
$$x=\frac{561\times600}{51}=6600$$

Hence, the total capital is ₹6600.

Therefore, the correct answer is (A) Rs. 6600.

Answer: (C) 9:24 A.M.

Explanation

Let the distance between Burdwan and Howrah be LCM(6, 4) = 12 units.

• Speed of the first train:

$$\frac{12}{6}=2 \text{ units/hour}$$

• Speed of the second train:

$$\frac{12}{4}=3 \text{ units/hour}$$

Combined speed:$$2+3=5 \text{ units/hour}$$

Time taken to meet:$$\frac{12}{5}=2.4 \text{ hours}$$

Convert $0.4$0.4 hour into minutes:
$$0.4\times60=24\text{ minutes}$$

They start at 7:00 A.M., so they meet at:
$$7:00+2\text{ hours }24\text{ minutes}=9:24\text{ A.M.}$$

Hence, the correct answer is (C) 9:24 A.M.

Answer: (C) 24 years

Explanation

Let the ages of Vijoy, Ajoy, and Tarun be $V$, $A$, and $T$, respectively.

Since Ajoy is as much younger than Vijoy as he is older than Tarun,
$$V-A=A-T$$$$2A=V+T$$

Given,$$V+T=48$$

So,$$2A=48$$$$A=24$$

Therefore, Ajoy’s age is 24 years.

Hence, the correct answer is (C) 24 years.

Answer: (B) 61.2%

Explanation

Monthly income = ₹25,000

Step 1: Education = 10% of ₹25,000
$$=2500$$

Remaining income:
$$25000-2500=22500$$

Step 2: Housing = 20% of ₹22,500
$$=4500$$

Remaining income:
$$22500-4500=18000$$

Step 3: Savings = 15% of ₹18,000
$$=2700$$

Remaining amount spent on food and clothes:

$$18000-2700=15300$$

Percentage of total income:
$$\frac{15300}{25000}\times100=61.2\%$$

Hence, the correct answer is (B) 61.2%.

Answer: (D) 56.87 km/h

Explanation

Distance covered in each part:

• First part:

$$35 \times 2 = 70 \text{ km}$$

• Second part:

$$60 \times 3.5 = 210 \text{ km}$$

• Third part:

$$70 \times 2.5 = 175 \text{ km}$$

Total distance:
$$70+210+175=455 \text{ km}$$

Total time:
$$2+3.5+2.5=8 \text{ hours}$$

Average speed:
$$\frac{455}{8}=56.875 \text{ km/h}$$$$\approx 56.87 \text{ km/h}$$

Hence, the correct answer is (D) 56.87 km/h.

Answer: (C) Rs. 10

Explanation

Let the original price be ₹x per dozen.

After a 20% reduction, the new price becomes:$$0.8x$$

With ₹50, the buyer gets:

• Before reduction: $\frac{50}{x}$
  
• After reduction: $\frac{50}{0.8x}=\frac{62.5}{x}$

Given that he gets 1 dozen more:
$$\frac{62.5}{x}-\frac{50}{x}=1$$$$\frac{12.5}{x}=1$$$$x=12.5$$

Reduced price:$$0.8\times12.5=10$$

Thus, the reduced price per dozen is ₹10.

Answer: (B) 6

Explanation

Find the perfect squares nearest to 15370.
$$123^2=15129$$$$124^2=15376$$

Since,
$$15370<15376,$$

the least number to be added is:
$$15376-15370=6$$

Hence, the correct answer is (B) 6.

Explanation

Observe that:
$$2^2=(2\times1)^2=4\times1^2,\quad 4^2=(2\times2)^2=4\times2^2,\quad \ldots,\quad 20^2=(2\times10)^2=4\times10^2$$

Therefore,
$$2^2+4^2+\cdots+20^2 =4(1^2+2^2+\cdots+10^2)$$

Given,
$$1^2+2^2+\cdots+10^2=385$$

Hence,$$4\times385=1540$$

Therefore, the correct answer is (B) 1540.

Answer: (A) 3 : 5

Explanation

Given,
$$x:y=2:1$$

Let,
$$x=2k,\qquad y=k$$

Then,
$$x^2=(2k)^2=4k^2,\qquad y^2=k^2$$

So,
$$x^2-y^2=4k^2-k^2=3k^2$$

and$$x^2+y^2=4k^2+k^2=5k^2$$

Therefore,
$$(x^2-y^2):(x^2+y^2)=3k^2:5k^2=3:5$$

Hence, the correct answer is (A) 3 : 5.

Answer: (A) 3 Litre

Explanation

Cost of 12 litres of milk:$$12 \times 10 = ₹120$$

Let x litres of water be added.

Since water is free, the total cost remains ₹120.

The mixture is sold at ₹8 per litre.

Therefore,$$8(12+x)=120$$$$12+x=15$$$$x=3$$

Hence, 3 litres of water should be added.

Therefore, the correct answer is (A) 3 Litre.

Answer: (D) 5

Explanation

Let$$x=\sqrt{20+\sqrt{20+\sqrt{20+\cdots}}}$$

Then,$$x=\sqrt{20+x}$$

Squaring both sides,$$x^2=20+x$$$$x^2-x-20=0$$$$(x-5)(x+4)=0$$

So,
$$x=5 \quad \text{or} \quad x=-4$$

Since a square root cannot be negative,
$$x=5$$

Hence, the correct answer is (D) 5.

Answer: (A) Rs. 2900

Explanation

Convert the capital ratio:
$$\frac12:\frac13:\frac15 =15:10:6$$

Effective capitals (Capital × Time):

• A: For the first 5 months = $15 \times 5 = 75$ After withdrawing $\frac13$​ of his capital, remaining capital:
  $$15-\frac{15}{3}=10$$15−315​=10 For the next 7 months:
  $$10 \times 7 = 70$$
  Total = $75+70=145$
  
• B: $10 \times 12 = 120$
• C: $6 \times 12 = 72$

Therefore, the profit-sharing ratio is:
$$145:120:72$$

Sum of the ratio:
$$145+120+72=337$$

A’s share of the profit:$$6740 \times \frac{145}{337}$$

Since,$$6740 \div 337 = 20,$$$$20 \times 145 = 2900$$

Therefore, A’s share of the profit is ₹2900.

Hence, the correct answer is (A) Rs. 2900.

Answer: (C) 40

Explanation

The student’s marks were recorded as 83 instead of 63.

Increase in total marks:
$$83-63=20$$

This increased the average by:$$\frac{1}{2}$$

Let the number of students be $n$.
$$\frac{20}{n}=\frac{1}{2}$$ $$n=\frac{20\times2}{1}=40$$

Hence, the number of students in the class is 40.

Therefore, the correct answer is (C) 40.

Answer: (A) Rs. 15,000

Explanation

Let the monthly income be ₹x.

He spends in 4 months what he earns in 3 months.

So, monthly expenditure is:
$$\frac{3x}{4}$$

Monthly savings:
$$x-\frac{3x}{4}=\frac{x}{4}$$

Annual savings:
$$12\times\frac{x}{4}=3x$$

Given,
$$3x=45000$$$$x=15000$$

This gives ₹15,000, which is the mathematically correct result.

Answer: (B) 10 hours

Explanation

Let the longer pipe take x hours.

Then the smaller pipe takes x+5 hours.

Together they fill the pool in 6 hours:
$$\frac{1}{x}+\frac{1}{x+5}=\frac{1}{6}$$
$$\frac{2x+5}{x(x+5)}=\frac{1}{6}$$$$6(2x+5)=x(x+5)$$$$12x+30=x^2+5x$$$$x^2-7x-30=0$$$$(x-10)(x+3)=0$$

Ignoring the negative value,$$x=10$$

Therefore, the longer (faster) pipe alone fills the pool in 10 hours.

Hence, the correct answer is (B) 10 hours.

Answer: (C) Rs. 39,000

Explanation

Lottery amount:
$$₹60,000$$

Tax deducted = 35% of ₹60,000:
$$\frac{35}{100}\times60000=₹21,000$$

Amount actually received:
$$60000-21000=₹39,000$$

Hence, the correct answer is (C) Rs. 39,000.

Answer: (A) 7 : 4

Explanation

Given,$$X:Y=4:7$$

Let,$$X=4k,\qquad Y=7k$$

Then,
$$21X+2Y=21(4k)+2(7k)=84k+14k=98k$$

and
$$7X+4Y=7(4k)+4(7k)=28k+28k=56k$$

Therefore,
$$(21X+2Y):(7X+4Y)=98:56=7:4$$

Hence, the correct answer is (A) 7 : 4.

Answer: (B) Rs. 36

Explanation

Let the wages of the two labourers be 4x and 3x.

After the changes:

• First labourer’s wages = $4x – 9$
• Second labourer’s wages = $3x + 9$

The new ratio is the reverse of the original ratio, i.e.,
$$\frac{4x-9}{3x+9}=\frac{3}{4}$$

Cross-multiplying,
$$4(4x-9)=3(3x+9)$$$$16x-36=9x+27$$$$7x=63$$$$x=9$$

Therefore, the first labourer’s wages are:
$$4x=4\times9=36$$

Thus, the correct answer is (B) Rs. 36.

Answer: (A) 99000

Explanation

First simplify the bracket:
$$999+\frac{494}{495} =1000-\frac{1}{495}$$

Now,
$$\left(1000-\frac{1}{495}\right)\times99 =99000-\frac{99}{495} =99000-\frac{1}{5}$$

Adding $\frac{1}{5}$​,$$\frac{1}{5}+99000-\frac{1}{5}=99000$$