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Given:

tan(3A) = cot(A - 22°), where 3A is an acute angle,

Calculation:

cot(x) = tan(90° - x).

\(\tan(3A) = \tan\left(90^∘ - (A - 22^∘)\right)\)

\(\tan(3A) = \tan\left(90^∘ - A + 22^∘\right)\)

\(\tan(3A) = \tan(112^∘ - A)\)

Since the tangent function is periodic with a period of 180°, we have:

\(3A = 112^∘ - A + n \cdot 180^∘\)

where n is an integer. Since 3A is an acute angle, (i.e. (0^∘  < A < 90^∘).

3A = 112^∘ - A

4A = 112^∘

A = 28^∘

∴ The correct answer is 28∘.

Shortcut Trick

Average of consecutive numbers is always the middle term.

Middle number = Total Sum ÷ Count ⇒ 2457 ÷ 3 = 819.

Since they are multiples of 9, the numbers are (819 − 9), 819, and (819 + 9).

Smallest number = 819 − 9 = 810.

∴ The correct answer is 810.

Alternate Method

Let the three consecutive multiples of 9 be (x − 9), x, and (x + 9).

Given: Sum of these numbers = 2457

⇒ (x − 9) + x + (x + 9) = 2457

⇒ 3x = 2457

⇒ x = 2457 ÷ 3

⇒ x = 819

⇒ Smallest number = x − 9 = 819 − 9 = 810

∴ The correct answer is 810.

Additional Information

Arithmetic Progression (AP)

Consecutive multiples of any number 'n' always form an Arithmetic Progression with common difference 'd' = n.

Middle Term Property

For an odd number of terms in AP, the Average (Sum ÷ Count) is always equal to the middle term of the series.

Multiple Check

A number is a multiple of 9 if the sum of its digits is divisible by 9. For 810, 8 + 1 + 0 = 9, which is divisible by 9.

Shortcut Trick

Identify numbers in the range 400 to 430 that are not divisible by 2, 3, 5, 7, 11, 13, 17, or 19.

Numbers to check: 401, 403, 407, 409, 413, 419, 421, 427, 429.

Eliminate: 403 (13×31), 407 (11×37), 413 (7×59), 427 (7×61), 429 (3×143).

Remaining Prime Numbers: 401, 409, 419, 421.

∴ The correct answer is 4.

Alternate Method

Given: Range of numbers between 400 and 430.

Method: To check if a number n is prime, test divisibility by all primes ≤ √n. Here, √430 ≈ 20.7, so test primes 2, 3, 5, 7, 11, 13, 17, 19.

⇒ 401: Not divisible by 2, 3, 5, 7, 11, 13, 17, 19 → Prime

⇒ 403: 13 × 31 = 403 → Not Prime

⇒ 407: 11 × 37 = 407 → Not Prime

⇒ 409: Not divisible by 2, 3, 5, 7, 11, 13, 17, 19 → Prime

⇒ 419: Not divisible by 2, 3, 5, 7, 11, 13, 17, 19 → Prime

⇒ 421: Not divisible by 2, 3, 5, 7, 11, 13, 17, 19 → Prime

⇒ List of primes: {401, 409, 419, 421}

⇒ Total count = 4

∴ The correct answer is 4.

Additional Information

Definition of Prime Numbers

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Primality Test Rule

To check if N is prime, divide N by all prime numbers less than or equal to √N. If none divide N, then N is prime.

Twin Primes

Pairs of prime numbers that have a difference of 2 (e.g., 419 and 421 are twin primes).

Given

Marked price is 150% of the cost price

Discount given is 20%

Concept:

Profit percentage is calculated as (Selling price - Cost price)/Cost price × 100%

Solution:

Selling price after discount = 150% - 20% of 150% = 150% - 30% = 120%

Profit percentage = (Selling price - Cost price)/Cost price × 100% = (120% - 100%)/100% × 100% = 20%

Therefore, the profit percentage is 20%.

Alternate MethodMP/CP = (100 + profit%)/(100 - loss%)

150/100 = (100 + p)/80 ⇒ P = 120 -100 = 20%

Shortcut Trick

Total Votes = 2,48,000

Valid Votes = 80% of 2,48,000 (since 20% are invalid)

Other Candidate's Votes = 25% of Valid Votes (since first gets 75%)

⇒ Required Votes = 2,48,000 × (80/100) × (25/100)

⇒ Required Votes = 2,48,000 × (4/5) × (1/4) = 2,48,000 ÷ 5 = 49,600

∴ The correct answer is 49600 votes.

Alternate Method

Given:

Total votes cast = 2,48,000

Invalid votes = 20%

First candidate's share of valid votes = 75%

Calculations:

⇒ Number of Valid Votes = 2,48,000 − (20% of 2,48,000)

⇒ Valid Votes = 2,48,000 × 0.80 = 1,98,400

⇒ Share of the Other Candidate = 100% − 75% = 25% of valid votes

⇒ Votes for other candidate = 1,98,400 × (25/100)

⇒ Votes for other candidate = 1,98,400 ÷ 4

⇒ Votes for other candidate = 49,600

∴ The correct answer is 49600 votes.

Additional Information

Valid Votes Concept

In election problems, the winner and loser percentages are always calculated on the Valid Votes unless stated otherwise. Valid Votes = Total Votes − Invalid Votes.

Percentage Multiplier

To find a value after a percentage decrease (like invalid votes), multiply by (100 − x)/100. For 20% invalid, the multiplier is 0.80.

Majority Margin

The Majority or Margin is the difference between the winner's votes and the runner-up's votes. Margin = (Winner % − Loser %) × Valid Votes.

Data:

Average age of a family of 4 members 4 years ago = 16 years.

Concept:

The current average age is calculated by adding the years passed to the old average, and the average age after adding a new member is calculated by dividing the total age by the number of members.

Solution:

Current average age = 16 + 4 = 20 years,

Total age of family now = 20 × 4 = 80 years,

Average age after the baby is born = (80 + 0) / 5 = 16 years.

Therefore, the present average age of the family of 5 members is 16 years.

Shortcut Trick

Total population increase = 8680 − 8000 = 680.

If all 8000 were males, increase = 8% of 8000 = 640.

Excess increase = 680 − 640 = 40.

This 40 represents the extra 4% (12% − 8%) of the females.

4% of females = 40 ⇒ Total females = (40 ÷ 4) × 100 = 1000.

∴ The correct answer is 1000.

Alternate Method

Given:

Initial population = 8000

Final population = 8680

Male growth rate = 8%

Female growth rate = 12%

Formula Used:

Total Increase = (Growth % of Males × Males) + (Growth % of Females × Females)

Calculations:

⇒ Total increase = 8680 − 8000 = 680

⇒ Overall increase percentage = (680 ÷ 8000) × 100 = 8.5%

⇒ Using alligation method:

⇒ Ratio of Males to Females = (12 − 8.5) : (8.5 − 8)

⇒ Ratio = 3.5 : 0.5 = 7 : 1

⇒ Total ratio parts = 7 + 1 = 8 units

⇒ 8 units = 8000

⇒ 1 unit = 1000

⇒ Number of females = 1 unit = 1000

∴ The correct answer is 1000.

Additional Information

Rule of Alligation

It is a shortcut to find the ratio in which two components are mixed to get a mean value: Ratio = (Higher Value − Mean) : (Mean − Lower Value).

Percentage Change

The percentage change is calculated as: (Final Value − Initial Value) ÷ Initial Value × 100.

Weighted Average

The net change in a population can be seen as a weighted average of the changes in its subgroups (males and females).

Given:

Total students in class = 45

There are 20 girls who scored an average of 42 marks a test of 50 marks.

The class average is 36.

Calculations:

Let the average marks scored by boys be a.

According to the question

⇒ Average of class = 36 = (42 × 20 + 25 × a)/45

⇒1620 – 840 = 25a

⇒ a = 780/25 = 31.20

∴ Boys average marks are 31.20.

Let the rate of interest be r

We know that, CA = \(p{\left[ {1; + ;\frac{r}{{100}}} \right]^n}\)

Here, p = Rs.10000, n = 2 years, CA = Rs. 11664

\(\begin{array}{l} 11664; = ;10000{\left[ {1; + ;\frac{r}{{100}}} \right]^2}\ \Rightarrow ;\frac{{11664}}{{10000}}; = ;{\left[ {1; + ;\frac{r}{{100}}} \right]^2} \end{array}\)

Taking square - root of both sides,

\(\begin{array}{l} \left[ {1; + ;\frac{r}{{100}}} \right]; = ;\frac{{108}}{{100}}\ \Rightarrow ;\frac{r}{{100}}; = ;\frac{{108}}{{100}} - 1 \end{array}\)

⇒ r = 8%

Shortcut Trick

Let total work = LCM(20, 40) = 40 units.

Combined efficiency of A, B, and C = 40 ÷ 20 = 2 units/day.

Efficiency of A alone = 40 ÷ 40 = 1 unit/day.

Efficiency of B and C together = 2 − 1 = 1 unit/day.

Part of the work completed by B and C in 1 day = 1 / 40.

∴ The correct answer is 1/40.

Alternate Method

Given:

Time taken by A, B, and C together to finish the work = 20 days

Time taken by A alone to finish the work = 40 days

Formula Used:

Total Work = LCM of given days

Efficiency = Total Work ÷ Time Taken

Calculations:

⇒ Efficiency of (B + C) = Efficiency of (A + B + C) − Efficiency of A

⇒ Efficiency of (B + C) = 2 − 1 = 1 unit/day

⇒ Part of the work B and C together finish in a day = Efficiency of (B + C) ÷ Total Work

⇒ Part of work = 1 ÷ 40 = 1/40

∴ The correct answer is 1/40.

Additional Information

Time & Work Relation

Total Work = Efficiency × Time taken. Time taken and efficiency are inversely proportional to each other for a given amount of work.

Pipes & Cisterns

Works on the same concept as Time & Work, but filling pipes have positive efficiency while emptying pipes have negative efficiency.

Work & Wages

Total wages are always distributed in the ratio of the actual total work done by the individuals, not just their efficiency or time.