Quantitative Aptitude PYQs

Shortcut Trick

Total Cost Price (CP) of 6 notebooks = ₹30.

Total Selling Price (SP) of 6 notebooks = ₹36.

Multiplier for SP = CP × (100 + P%) ÷ 100.

Profit% = (SP − CP) ÷ CP × 100 = (36 − 30) ÷ 30 × 100.

Profit% = 6 ÷ 30 × 100 = 1 ÷ 5 × 100 = 20%.

∴ The correct answer is 20%.

Alternate Method

Given: CP of 6 notebooks = ₹30; SP of 6 notebooks = ₹36.

Formulas: Profit = SP − CP | Profit% = (Profit ÷ CP) × 100

⇒ Cost Price (CP) of 1 notebook = 30 ÷ 6 = ₹5.

⇒ Selling Price (SP) of 1 notebook = 36 ÷ 6 = ₹6.

⇒ Profit per notebook = SP − CP = 6 − 5 = ₹1.

⇒ Profit Percentage = (1 ÷ 5) × 100 = 20%.

∴ The correct answer is 20%.

Additional Information

Profit and Loss Basics

Profit occurs when Selling Price is greater than Cost Price. It is always calculated on the Cost Price unless specified otherwise.

Unitary Method

To find profit percentage, you can compare the total CP and total SP for the same quantity of items, or find the price per single unit.

Shortcut Trick

Total amount = ₹8,516.

Total deduction = 88 + 31 + 90 = ₹209.

Remaining amount = 8516 − 209 = ₹8,307.

Ratio after deduction = 18 : 17 : 4 (Total parts = 18 + 17 + 4 = 39).

Value of (B − C) ratio difference = (17 − 4) = 13 units.

Difference in original shares = (Difference in ratio parts × Value per unit) + (B deduction − C deduction) = (13 × 8307 ÷ 39) + (31 − 90) = 2769 − 59 = ₹2,710.

∴ The correct answer is ₹2,710.

Alternate Method

Given: Total = ₹8,516 | Deductions: A = ₹88, B = ₹31, C = ₹90 | Ratio (A−88):(B−31):(C−90) = 18:17:4.

Formula: Share = (Ratio Part ÷ Total Parts) × Remaining Sum + Deduction Amount

⇒ Sum after deduction = 8516 − (88 + 31 + 90) = 8516 − 209 = 8307

⇒ Total ratio parts = 18 + 17 + 4 = 39 units

⇒ 39 units = 8307

⇒ 1 unit = 8307 ÷ 39 = 213

⇒ B's share = (17 × 213) + 31 = 3621 + 31 = 3652

⇒ C's share = (4 × 213) + 90 = 852 + 90 = 942

⇒ Difference = 3652 − 942 = 2710

∴ The correct answer is ₹2,710.

Additional Information

Ratio and Proportion Basics

If a sum S is divided in the ratio a:b:c, the shares are (a/Σr)×S, (b/Σr)×S, and (c/Σr)×S respectively.

Effective Sum Adjustment

When fixed amounts are added or subtracted from shares, always adjust the Total Sum first to align with the given ratio.

Compound Ratio

The compound ratio of a:b and c:d is (a×c):(b×d). This is useful in Partnership and Mixture problems.

Shortcut Trick

Teacher's Age = New Average + (Old Count × Increase in Average)

Given: Old Average = 25 years | New Average = 26 years

Old Count = 43 students | Increase = 26 − 25 = 1 year

⇒ Teacher's Age = 26 + (43 × 1) = 26 + 43 = 69 years

∴ The correct answer is 69 years.

Alternate Method

Given: Number of students = 43, Average age = 25 years, Average with teacher included = 26 years.

Formula Used: Total Sum = Average × Number of Observations

⇒ Total age of 43 students = 43 × 25 = 1075 years

⇒ Total persons (including teacher) = 43 + 1 = 44

⇒ Total age of 44 persons = 44 × 26 = 1144 years

⇒ Teacher's age = (Total age of 44 persons) − (Total age of 43 students)

⇒ Teacher's age = 1144 − 1075 = 69 years

∴ The correct answer is 69 years.

Additional Information

General Inclusion Formula

Value of new entry = (New Number of items × New Average) − (Old Number of items × Old Average).

Net Deviation Method

Sum of deviations from the average is always zero. If the average increases, the new value must be greater than the old average by (Total New Count × Increase).

Average of Groups

Combined Average = (n₁x₁ + n₂x₂) ÷ (n₁ + n₂), where n is the count and x is the average of each group.

Shortcut Trick

Apply the inverse proportionality rule: Total Work (Man-Days) remains constant.

M₁ × D₁ = M₂ × D₂

1209 × 20 = (1209 + 403) × D₂

1209 × 20 = 1612 × D₂

D₂ = (1209 × 20) ÷ 1612 = 15 days.

∴ The correct answer is 15 days.

Alternate Method

Given: Initial men (M₁) = 1209, Initial days (D₁) = 20, Men added = 403.

Formula Used: M₁D₁ = M₂D₂

⇒ Total food capacity = 1209 × 20 units

⇒ New number of men (M₂) = 1209 + 403 = 1612

⇒ Let the new number of days be D₂.

⇒ 1209 × 20 = 1612 × D₂

⇒ D₂ = (1209 × 20) ÷ 1612

⇒ D₂ = 24180 ÷ 1612

⇒ D₂ = 15

∴ The correct answer is 15 days.

Additional Information

Chain Rule (Inverse Proportion)

When the number of men increases, the number of days for which provisions last decreases proportionally. M₁D₁ = M₂D₂.

Total Work Concept

The product of men and time gives the total volume of work or resources (like food units). Constant Resource = Men × Days.

Man-Hours Formula

If hours per day are also given, use: (M₁ × D₁ × H₁) ÷ W₁ = (M₂ × D₂ × H₂) ÷ W₂.

Shortcut Trick

Use the multiplying factor method for successive changes.

Net multiplier = (1 − 0.45) × (1 − 0.15) × (1 − 0.50) × (1 + 0.70)

Net multiplier = 0.55 × 0.85 × 0.50 × 1.70 = 0.397375

Decrease % = (1 − 0.397375) × 100 = 60.2625%

∴ The correct answer is Decreases by 60.26%.

Alternate Method

Given:

Initial price of fuel = 100 units

Successive changes: 45% decrease, 15% decrease, 50% decrease, and 70% increase.

Calculations:

⇒ Price after 1^st month = 100 × (55/100) = 55

⇒ Price after 2^nd month = 55 × (85/100) = 46.75

⇒ Price after 3^rd month = 46.75 × (50/100) = 23.375

⇒ Price after 4^th month = 23.375 × (170/100) = 39.7375

⇒ Net Change = Final Price − Original Price = 39.7375 − 100 = −60.2625

⇒ Percentage Change = (−60.2625 / 100) × 100 = −60.2625%

∴ The correct answer is Decreases by 60.26%.

Additional Information

Successive Percentage Change

For two changes x% and y%, the net change is given by the formula: [x + y + (xy/100)]%.

Base Value Change

Percentage increase/decrease is always calculated on the preceding value in a successive series, not the original 100 throughout.

Multiplying Factor

An x% decrease corresponds to a multiplier of (1 − x/100), while an x% increase corresponds to (1 + x/100).

Shortcut Trick

Distribute the outer values into the brackets directly to expand the expression.

⇒ 4 × (6/4)x² − 4 × 25x + 4 × 12 − 6x² − 30x + 78

⇒ 6x² − 100x + 48 − 6x² − 30x + 78

⇒ (6 − 6)x² − (100 + 30)x + (48 + 78)

⇒ −130x + 126

∴ The correct answer is −130x + 126.

Alternate Method

Given:

Expression: 4 × ((6/4)x² − 25x + 12) − 6 × (x² + 5x − 13)

Concept Used:

Use the distributive property of multiplication over addition and subtraction: a(b ± c) = ab ± ac

Calculation:

⇒ (4 × (6/4)x²) − (4 × 25x) + (4 × 12) − (6 × x²) − (6 × 5x) − (6 × −13)

⇒ 6x² − 100x + 48 − 6x² − 30x + 78

Grouping the like terms together (the x² terms, x terms, and constant terms):

⇒ (6x² − 6x²) + (−100x − 30x) + (48 + 78)

⇒ 0 − 130x + 126

⇒ −130x + 126

∴ The correct answer is −130x + 126.

Additional Information

BODMAS Rule

It defines the correct sequence to solve mathematical expressions: Brackets, Order, Division, Multiplication, Addition, and Subtraction.

Distributive Property

The distributive property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products: a(b + c) = ab + ac.

Like Terms

Terms whose variables and their exponents are exactly the same are called like terms.

Only like terms can be added or subtracted from each other to simplify algebraic expressions.

Shortcut Trick

The fastest way to check divisibility is by looking at the last digit.

The given number is 386937355, which ends in the digit 5.

According to the divisibility rule of 5, any number ending in 0 or 5 is exactly divisible by 5.

Since the last digit is 5, the entire number is directly divisible by 5.

∴ The correct answer is 5.

Alternate Method

Given:

The number to be divided is 386937355.

Method Used:

We will apply the divisibility rules for each given option to find the correct divisor.

Calculations:

⇒ Option 1 (Divisibility by 3): A number is divisible by 3 if the sum of its digits is a multiple of 3.

⇒ Sum of digits = 3 + 8 + 6 + 9 + 3 + 7 + 3 + 5 + 5 = 49.

⇒ Since 49 is not divisible by 3, the given number is not divisible by 3.

⇒ Option 2 (Divisibility by 15): A number must be divisible by both 3 and 5 to be divisible by 15.

⇒ Since the number fails the divisibility test for 3, it is not divisible by 15.

⇒ Option 4 (Divisibility by 14): A number must be even to be divisible by an even number like 14.

⇒ The given number ends in 5, making it an odd number. Hence, it is not divisible by 14.

⇒ Option 3 (Divisibility by 5): A number is divisible by 5 if its last digit is either 0 or 5.

⇒ The last digit of 386937355 is 5, which satisfies the condition.

∴ The correct answer is 5.

Additional Information

Divisibility Rule of 3 & 9

A number is divisible by 3 or 9 if the sum of all its digits is divisible by 3 or 9 respectively.

Divisibility Rule of 4 & 8

A number is divisible by 4 if its last two digits form a number divisible by 4. It is divisible by 8 if its last three digits form a number divisible by 8.

Divisibility Rule of 11

A number is divisible by 11 if the difference between the sum of digits at odd places and the sum of digits at even places is 0 or a multiple of 11.

Divisibility Rule of Composite Numbers

To check divisibility by a composite number (like 12 or 15), check divisibility by its co-prime factors (e.g., 3 and 4 for 12; 3 and 5 for 15).

Shortcut Trick

M's profit = (6705 + 745) ÷ 2 = ₹3,725 and N's profit = 6705 − 3725 = ₹2,980.

Profit ratio of M : N = 3725 : 2980 = 5 : 4.

Investment ratio M : N = (Profit M ÷ Time M) : (Profit N ÷ Time N) = (5 ÷ 7) : (4 ÷ 6) = 15 : 14.

Difference in investment = 15 − 14 = 1 unit → ₹79,000.

Capital contributed by M = 15 units = 15 × 79000 = ₹11,85,000.

∴ The correct answer is ₹11,85,000.

Alternate Method

Given:

Total profit = ₹6,705

M's profit − N's profit = ₹745

Time period of M = 7 months, Time period of N = 6 months.

Partner M
Capital: x + 79,000
Time: 7 months

Partner N
Capital: x
Time: 6 months

Profit Ratio (M : N) = 5 : 4

Formula Used:

Profit Ratio = Ratio of (Investment × Time)

Calculations:

⇒ M's profit = (6705 + 745) ÷ 2 = ₹3,725

⇒ N's profit = 6705 − 3725 = ₹2,980

⇒ Profit Ratio (M : N) = 3725 : 2980 = 5 : 4

⇒ Let N's capital be x, then M's capital = x + 79000

⇒ ((x + 79000) × 7) ÷ (x × 6) = 5 ÷ 4

⇒ (x + 79000) ÷ x = (5 × 6) ÷ (4 × 7)

⇒ (x + 79000) ÷ x = 30 ÷ 28 = 15 ÷ 14

⇒ 15x = 14(x + 79000)

⇒ 15x = 14x + 1106000

⇒ x = ₹11,06,000

⇒ M's Capital = x + 79000 = 1106000 + 79000 = ₹11,85,000

∴ The correct answer is ₹11,85,000.

Additional Information

Partnership Profit Ratio

The profit earned in a partnership is distributed in the ratio of the product of capital invested and the time period of investment. Profit Ratio = C₁×T₁ : C₂×T₂.

Active vs Sleeping Partner

An active partner receives a fixed percentage or salary from the total profit for managing the business, and the remaining profit is divided according to capital ratios.

Mid-way Join or Withdraw

If a partner adds or withdraws capital mid-way, their total effective capital is the sum of (capital × duration) for each respective period within the total time.

Shortcut Trick

Solve the inner brackets mentally: (5 − 4) = 1

Multiply: 2 × 1 = 2

Add to 20: 20 + 2 = 22

Divide the final sum: 682 ÷ 22 = 31

∴ The correct answer is 31.

Alternate Method

Given:

[682 ÷ {20 + 2 × (5 − 4)}]

Method Used:

BODMAS Rule: First solve Brackets (), then {}, then []. Next apply Of, Division, Multiplication, Addition, and Subtraction respectively.

Calculation:

⇒ [682 ÷ {20 + 2 × (5 − 4)}]

⇒ [682 ÷ {20 + 2 × 1}]

⇒ [682 ÷ {20 + 2}]

⇒ [682 ÷ 22]

⇒ 31

∴ The correct answer is 31.

Additional Information

BODMAS Rule

It dictates the order of mathematical operations: Brackets, Orders (powers/roots), Division, Multiplication, Addition, and Subtraction. Brackets are solved in the order: () round, {} curly, [] square.

VBODMAS Extension

The V stands for Vinculum (bar over an expression). Any expression under a Vinculum is evaluated first before applying the standard BODMAS sequence.

Important Property

When simplifying equations, always perform multiplication and division from left to right, followed by addition and subtraction from left to right to ensure an accurate result.

Shortcut Trick

Shopkeeper gives 16 books but charges for 15 with an additional 4% discount on Marked Price.

Effective SP / MP ratio = (15 × 0.96) / 16 = 14.4 / 16 = 90 / 100 ⇒ 10% effective discount.

Using direct formula: CP / MP = (100 − D%) / (100 + P%)

⇒ CP / MP = (100 − 10) / (100 + 35) = 90 / 135 = 2 / 3

⇒ Percentage by which MP is higher than CP = [(3 − 2) / 2] × 100% = 50%.

∴ The correct answer is 50%.

Alternate Method

CP100MP150Paid SP144Eff. SP135+50%-4%× 15/16Overall Gain = +35%1 Free book on 15 means customer gets 16 books but pays for 15.Hence, Effective SP per book = Paid SP × (15 / 16)

Given:

Discount given on Marked Price = 4%

Offer = 1 book free on the purchase of every 15 books (Customer pays for 15 but receives 16)

Overall Profit Gained = 35%

Calculations:

⇒ Let the Cost Price (CP) of 1 book be ₹100.

⇒ Total Cost Price for 16 books given to the customer = 16 × 100 = ₹1600.

⇒ The seller gains a 35% profit on the total cost incurred.

⇒ Total Selling Price (SP) for 16 books = 1600 × (100 + 35) / 100 = ₹2160.

⇒ Since the customer pays for only 15 books, the discounted SP of 1 book = 2160 ÷ 15 = ₹144.

⇒ This SP was arrived at after allowing a 4% discount on the Marked Price (MP).

⇒ So, 96% of MP = 144

⇒ MP = 144 × (100 / 96) = ₹150.

⇒ Difference between Marked Price and Cost Price = 150 − 100 = ₹50.

⇒ Percentage by which MP is higher than CP = (50 / 100) × 100% = 50%.

∴ The correct answer is 50%.

Additional Information

Buy X Get Y Free Offer

When a shopkeeper offers 'Buy X Get Y Free', the customer gets a total of (X + Y) items while paying for only X items. The effective discount percentage due to this scheme is [Y / (X + Y)] × 100%.

Direct Relation between CP and MP

A very useful shortcut formula to directly find the ratio of Cost Price to Marked Price when Profit (P%) and Discount (D%) are known is: CP / MP = (100 − D%) / (100 + P%).

Markup Percentage

Markup is the amount added to the Cost Price to arrive at the Marked Price. It is always calculated over the Cost Price: Markup % = [(MP − CP) / CP] × 100%. Alternatively, Discount is always calculated over the Marked Price.

Shortcut Trick

Total Cost Price (CP) for 83 mirrors = ₹83.

Number of mirrors available for sale = 83 − 22 = 61 mirrors.

Total Selling Price (SP) = 61 × 3.60 = ₹219.60.

Profit = SP − CP = 219.60 − 83 = ₹136.60.

∴ The correct answer is ₹136.60.

Alternate Method

Given:

Total mirrors purchased = 83

Total Cost Price (CP) = ₹83

Mirrors broken in transit = 22

Selling Price (SP) of each remaining mirror = ₹3.60

Formula Used:

Profit = Total SP − Total CP

Total Mirrors = 83, Broken = 22 → Remaining = 61
SP per remaining mirror = ₹3.60

CP = ₹83
SP = ₹219.60

Profit = ₹136.60

Profit = SP − CP = 219.60 − 83 = ₹136.60

Calculations:

⇒ Number of remaining mirrors = 83 − 22 = 61

⇒ Total Selling Price (SP) = Remaining mirrors × SP per mirror

⇒ Total SP = 61 × 3.60

⇒ Total SP = ₹219.60

⇒ Profit = Total SP − Total CP

⇒ Profit = 219.60 − 83

⇒ Profit = ₹136.60

∴ The correct answer is ₹136.60.

Additional Information

Profit and Loss Percentages

Profit % = (Profit ÷ CP) × 100, where Profit = SP − CP. Loss % = (Loss ÷ CP) × 100, where Loss = CP − SP.

Marked Price and Discount

Discount is always calculated on Marked Price (MP). SP = MP − Discount. Discount % = (Discount ÷ MP) × 100.

Successive Discounts

If two successive discounts of a% and b% are given, then the single equivalent discount % is [a + b − (a × b ÷ 100)]%.

Shortcut Trick

Total population is 362000 and males are 181000, so females = 181000.

Ratio of Males : Females = 1 :

1. Let female literacy be x%.

Using weighted average: (35% × 1 + x% × 1) ÷ 2 = 47%

⇒ 35 + x = 94 ⇒ x = 59%.

∴ The correct answer is 59%.

Alternate Method

Given:

Total population = 362000

Total males = 181000

Formula Used:

Percentage = (Part value ÷ Total value) × 100

Males
35%

Females
x%

Total
47%

Ratio: 1

Ratio: 1

x − 47 = 47 − 35 = 12 ⇒ x = 59%

Calculations:

⇒ Number of females = Total population − Total males

⇒ Number of females = 362000 − 181000 = 181000

⇒ Total literate population = 47% of 362000 = (47 ÷ 100) × 362000 = 170140

⇒ Literate males = 35% of 181000 = (35 ÷ 100) × 181000 = 63350

⇒ Literate females = Total literate − Literate males

⇒ Literate females = 170140 − 63350 = 106790

⇒ Percentage of literate females = (106790 ÷ 181000) × 100

⇒ Percentage of literate females = 59%

∴ The correct answer is 59%.

Additional Information

Weighted Average Method

Used when two or more groups are combined. If group 1 has average A₁ and weight W₁, and group 2 has average A₂ and weight W₂, then the combined average A = (A₁W₁ + A₂W₂) ÷ (W₁ + W₂).

Rule of Alligation

It helps to find the ratio in which two quantities must be mixed to produce a mixture of a given mean value. Ratio = (Dearer value − Mean value) : (Mean value − Cheaper value).

Base of Percentage

When calculating percentage of a specific group, the base should always be the total count of that specific group, not the overall population. Required % = (Specific value ÷ Group total) × 100.

Shortcut Trick

Assume the base average for all 33 matches is 40.

Deviation for the next 10 matches = 16 − 40 = −24 per match.

Total deviation = 10 × −24 = −240.

Change in overall average = −240 ÷ 33 = −7.2727... ≈ −7.27.

New combined average = 40 − 7.27 = 32.73.

∴ The correct answer is 32.73 runs.

Alternate Method

Given:

Average runs for the first 23 matches = 40

Average runs for the next 10 matches = 16

Formula Used:

Combined Average = (Sum of all observations) ÷ (Total number of observations)

23 matches
Average = 40

10 matches
Average = 16

Combined 33 matches | Average = 32.73

Calculations:

⇒ Total runs in the first 23 matches = 23 × 40 = 920

⇒ Total runs in the next 10 matches = 10 × 16 = 160

⇒ Total runs in all 33 matches = 920 + 160 = 1080

⇒ Combined Average = 1080 ÷ 33 = 360 ÷ 11

⇒ Combined Average = 32.7272... ≈ 32.73

∴ The correct answer is 32.73 runs.

Additional Information

Weighted Average

When dealing with groups of different sizes, the combined average is calculated as A_w = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Net Deviation Method

Instead of multiplying large numbers, assume a base average and calculate the net deviation. True Average = Assumed Average + (Net Deviation ÷ Total Count). This makes calculations significantly faster.

Shortcut Trick

Average of the ratio parts = (3 + 5 + 7) ÷ 3 = 15 ÷ 3 = 5 parts.

Given the average price is ₹5,325, this corresponds directly to the average of the ratio parts.

Value of 1 part = 5,325 ÷ 5 = ₹1,065.

Price of the cheapest item (3 parts) = 3 × 1,065 = ₹3,195.

∴ The correct answer is ₹3,195.

Alternate Method

Given:

Average price of 3 items = ₹5,325

Ratio of their prices = 3 : 5 : 7

Formula Used:

Total value = Average × Number of items

Average (5 parts) = ₹5,325

3 parts

5 parts

7 parts

Cheapest
Middle
Expensive

Cheapest item (3 parts) = 3 × ₹1,065 = ₹3,195

Calculations:

⇒ Let the prices of the three items be 3x, 5x, and 7x.

⇒ Total price of the three items = 3 × 5,325 = ₹15,975

⇒ Sum of the prices = 3x + 5x + 7x = 15x

⇒ 15x = 15,975

⇒ x = 15,975 ÷ 15

⇒ x = 1,065

⇒ Price of the cheapest item = 3x

⇒ Cheapest item = 3 × 1,065 = ₹3,195

∴ The correct answer is ₹3,195.

Additional Information

Ratio and Arithmetic Progression

When terms in a ratio are in an Arithmetic Progression (like 3, 5, 7 with a common difference of 2), their average is always directly equal to the middle term. Here, the average ratio directly equals 5 parts.

Value of One Part

In ratio problems, finding the value of '1 part' simplifies the calculation drastically. It can be found by dividing the total value by the total ratio parts or the average value by the average ratio parts.

Divisibility Check Trick

Since the cheapest item corresponds to 3 parts, the correct answer must be a multiple of 3. You can verify this by checking if the sum of the digits is divisible by 3 (3 + 1 + 9 + 5 = 18, which is divisible by 3).

Shortcut Trick

Let θ = 30°, then sec 30° + tan 30° = 2/√3 + 1/√3 = 3/√3 = √3.

Given s = sec θ + tan θ, so s = √3 ⇒ s² = 3.

Substitute s² = 3 in the given expression: (s² − 1) / (s² + 1).

⇒ (3 − 1) / (3 + 1) = 2 / 4 = 1/2.

Check the given options for θ = 30°: Option 3 gives sin 30° = 1/2, which matches perfectly.

∴ The correct answer is sin θ.

Alternate Method

Given:

sec θ + tan θ = s

Formula Used:

sec² θ − tan² θ = 1

Calculations:

⇒ (sec θ − tan θ)(sec θ + tan θ) = 1

⇒ sec θ − tan θ = 1/s

Subtracting (sec θ − tan θ = 1/s) from (sec θ + tan θ = s):

⇒ 2 tan θ = s − 1/s = (s² − 1) / s

⇒ tan θ = (s² − 1) / 2s

θ
Base = 2s
Perp = s² − 1
Hypotenuse = s² + 1

In a right-angled triangle, if tan θ = Perpendicular / Base = (s² − 1) / 2s:

⇒ Perpendicular = s² − 1

⇒ Base = 2s

⇒ Hypotenuse = √[(s² − 1)² + (2s)²]

⇒ Hypotenuse = √[s⁴ − 2s² + 1 + 4s²]

⇒ Hypotenuse = √[s⁴ + 2s² + 1] = s² + 1

We need to find the value of (s² − 1) / (s² + 1):

⇒ (s² − 1) / (s² + 1) = Perpendicular / Hypotenuse

⇒ Perpendicular / Hypotenuse = sin θ

∴ The correct answer is sin θ.

Additional Information

Fundamental Trigonometric Identities

The three core identities are: sin² θ + cos² θ = 1, sec² θ − tan² θ = 1, and cosec² θ − cot² θ = 1.

Inverse Relation of Secant and Tangent

Since sec² θ − tan² θ = 1, we can write (sec θ − tan θ)(sec θ + tan θ) = 1. Thus, if sec θ + tan θ = x, then sec θ − tan θ = 1/x.

Algebraic Pythagorean Triplets

For any real number s > 1, the values 2s, s² − 1, and s² + 1 always form the sides of a right-angled triangle, where s² + 1 represents the hypotenuse.

Shortcut Trick

For any trapezium, the length of the line segment joining the midpoints of its diagonals is directly equal to half the difference of the parallel sides.

The non-parallel sides (AD and BC) are given only to confuse and are not needed for this calculation.

Given parallel sides are AB = 12 cm and DC = 30 cm.

EF = (DC − AB) / 2

EF = (30 − 12) / 2 = 18 / 2 = 9 cm.

∴ The correct answer is 9 cm.

Alternate Method

Given:

A quadrilateral ABCD where AB || DC (which makes it a Trapezium).

Parallel sides: AB = 12 cm, DC = 30 cm.

Non-parallel sides: AD = 45 cm, BC = 40 cm.

E and F are the midpoints of the diagonals AC and BD respectively.

A
B
D
C
E
F

12 cm
30 cm
45 cm
40 cm
EF = ?

Formula Used:

Length of the line segment joining the midpoints of the diagonals of a trapezium = 1/2 × (Difference of parallel sides).

Calculations:

⇒ Parallel sides are AB and DC, so the formula is applied directly to them.

⇒ EF = 1/2 × (DC − AB)

⇒ EF = 1/2 × (30 − 12)

⇒ EF = 1/2 × 18

⇒ EF = 9 cm

∴ The correct answer is 9 cm.

Additional Information

Midsegment of a Trapezium

The line segment joining the midpoints of the non-parallel sides is parallel to the bases, and its length is the exact average of the bases: 1/2 × (AB + DC).

Intersection of Diagonals

The diagonals of a trapezium intersect each other proportionally in the ratio of the parallel sides. If diagonals AC and BD intersect at point O, then AO / OC = BO / OD = AB / DC.

Area of a Trapezium

The area is calculated as: Area = 1/2 × (Sum of parallel sides) × (Perpendicular height between them).

Shortcut Trick

Check if the given sides form a Pythagorean triplet: 28² + 45² = 784 + 2025 = 2809.

Since 53² = 2809, the sides 28 cm, 45 cm, and 53 cm form a right-angled triangle.

For a right-angled triangle, Area = (1 ÷ 2) × Base × Perpendicular.

Area = (1 ÷ 2) × 28 × 45 = 14 × 45 = 630 cm².

∴ The correct answer is 630 cm².

Alternate Method

Given:

Sides of the triangle: a = 28 cm, b = 45 cm, c = 53 cm.

28 cm
45 cm
53 cm
A
B
C

Formula Used:

Heron's Formula: Area = √[s × (s − a) × (s − b) × (s − c)]

Semi-perimeter (s) = (a + b + c) ÷ 2

Calculations:

⇒ s = (28 + 45 + 53) ÷ 2

⇒ s = 126 ÷ 2 = 63 cm

⇒ Area = √[63 × (63 − 28) × (63 − 45) × (63 − 53)]

⇒ Area = √[63 × 35 × 18 × 10]

⇒ Area = √[(9 × 7) × (7 × 5) × (9 × 2) × (5 × 2)]

⇒ Area = √[9² × 7² × 5² × 2²]

⇒ Area = 9 × 7 × 5 × 2

⇒ Area = 630 cm²

∴ The correct answer is 630 cm².

Additional Information

Heron's Formula

Used to find the area of a scalene triangle when all three sides are known: Area = √[s(s − a)(s − b)(s − c)].

Pythagorean Triplets

Sets of three integers satisfying a² + b² = c². Common examples include (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), and (28, 45, 53).

Right-Angled Triangle Properties

The circumradius is half the hypotenuse (R = c ÷ 2). The inradius can be found using the formula r = (a + b − c) ÷ 2.

Shortcut Trick

Directly calculate the time taken for each segment of the journey using Time = Distance ÷ Speed.

Total Time = 198÷66 + 584÷73 + 656÷82 = 3 + 8 + 8 = 19 hours.

Total Distance = 198 + 584 + 656 = 1438 km.

Average Speed = Total Distance ÷ Total Time = 1438 ÷ 19 = 75.684... km/hr.

∴ The correct answer is 75.68 km/hr.

Alternate Method

198 km
584 km
656 km

66 km/hr
73 km/hr
82 km/hr

3 hrs
8 hrs
8 hrs

Total Distance = 1438 km

Total Time = 19 hrs

Given:

Distance 1 = 198 km, Speed 1 = 66 km/hr

Distance 2 = 584 km, Speed 2 = 73 km/hr

Distance 3 = 656 km, Speed 3 = 82 km/hr

Formula Used:

Time = Distance ÷ Speed

Average Speed = Total Distance ÷ Total Time

Calculation:

⇒ Time for 1^st part = 198 ÷ 66 = 3 hours

⇒ Time for 2^nd part = 584 ÷ 73 = 8 hours

⇒ Time for 3^rd part = 656 ÷ 82 = 8 hours

⇒ Total Time = 3 + 8 + 8 = 19 hours

⇒ Total Distance = 198 + 584 + 656 = 1438 km

⇒ Average Speed = 1438 ÷ 19

⇒ Average Speed ≈ 75.684... km/hr

∴ The correct answer is 75.68 km/hr.

Additional Information

Average Speed (General Formula)

Average Speed is always calculated by taking the sum of all individual distances traveled and dividing it by the total duration of the journey.

Average Speed for Equal Distances

If two exactly equal distances are covered at speeds 'x' and 'y', the overall average speed simplifies to (2 × x × y) ÷ (x + y).

Average Speed for Equal Times

If multiple distances are covered in identical time periods at different speeds, the average speed becomes the simple arithmetic mean of those speeds.

Relative Speed

When two bodies travel towards each other (opposite directions), relative speed = (x + y). When they travel in the same direction, relative speed = (x − y).

Shortcut Trick

Convert the given decimals to their lowest fractions: 6.3 = 63/10 and 0.07 = 7/100.

LCM of fractions = LCM of numerators ÷ HCF of denominators.

LCM(63, 7) = 63 and HCF(10, 100) = 10.

LCM = 63 ÷ 10 = 6.3.

∴ The correct answer is 6.3.

Alternate Method

Given:

The numbers are 6.3 and 0.07

Formula Used:

LCM of decimals: Make the number of decimal places equal, remove the decimals to find the LCM of the resulting integers, and place the decimal point back in the result at the exact same position.

Calculation:

⇒ Make the decimal places equal by adding a zero: 6.30 and 0.07

⇒ Remove the decimal point to consider them as integers: 630 and 7

⇒ Find the LCM of 630 and 7.

⇒ Since 630 is exactly divisible by 7 (630 = 7 × 90), their LCM is 630.

⇒ Put the decimal point back at two places from the right (as in the equated decimals): 6.30

⇒ Therefore, the LCM of 6.3 and 0.07 is 6.3.

∴ The correct answer is 6.3.

Additional Information

HCF of Decimals

Make the decimal places equal, find the HCF of the resulting integers, and place the decimal point back at the same position in the final answer.

LCM of Fractions

LCM = LCM of Numerators ÷ HCF of Denominators (

Note:

Always convert fractions to their simplest form before applying the formula).

HCF of Fractions

HCF = HCF of Numerators ÷ LCM of Denominators (

Note:

Always convert fractions to their simplest form before applying the formula).

Shortcut Trick

We know that the product of two numbers = HCF × LCM

Second Number = (11 × 330) ÷ 110

Second Number = 11 × 3 = 33

∴ The correct answer is 33.

Alternate Method

Given:

HCF = 11

LCM = 330

First Number = 110

Formula Used:

First Number × Second Number = HCF × LCM

Calculations:

⇒ 110 × Second Number = 11 × 330

⇒ 110 × Second Number = 3630

⇒ Second Number = 3630 ÷ 110

⇒ Second Number = 33

∴ The correct answer is 33.

Additional Information

Product Rule for Two Numbers

For any two positive integers a and b, the product of the numbers equals the product of their HCF and LCM: a × b = HCF(a, b) × LCM(a, b). Note that this rule strictly applies only to two numbers, not three or more.

Co-prime Numbers

Two numbers are considered co-prime if their highest common factor (HCF) is 1. For such numbers, their least common multiple (LCM) is directly the product of the two numbers.

HCF and LCM of Fractions

HCF = (HCF of Numerators) ÷ (LCM of Denominators)

LCM = (LCM of Numerators) ÷ (HCF of Denominators)