SSC CGL Tier 2 Quant Previous Year Paper (Held On: 29 Jan 2022)

The correct answer is 14.3% Key Points

Let the Cost Price (CP) of one article be CP and the Selling Price (SP) of one article be SP. The problem states that the Selling Price of 7 articles is equal to the Cost Price of 8 articles. This gives us the equation: 7 × SP = 8 × CP. To find the relationship between SP and CP, we rearrange the equation: SP / CP = 8 / 7. From this ratio, we can assume the Cost Price (CP) is 7 units and the Selling Price (SP) is 8 units. The Profit is calculated as: Profit = SP − CP = 8 − 7 = 1 unit. The Profit Percentage formula is: (Profit / Cost Price) × 100. Substituting the values: Profit Percentage = (1 / 7) × 100. Performing the division: 100 ÷ 7 = 14.2857...%. Rounding the value to one decimal place as requested, we get 14.3%. Thus, Option 2 is the correct choice.

Additional Information

Cost Price (CP): The price at which an article is purchased. Selling Price (SP): The price at which an article is sold. Profit: If SP > CP, the difference is called profit. Profit = SP − CP. Loss: If CP > SP, the difference is called loss. Loss = CP − SP. The Profit Percentage or Loss Percentage is always calculated on the Cost Price (CP) unless mentioned otherwise in the question. Fractions to percentages: 1/7 ≈ 14.28%, 1/8 = 12.5%, 1/9 ≈ 11.11%. Memorizing these common conversions is helpful for quantitative aptitude exams.

Shortcut Trick

Divide the equation 2x² + 5x + 1 = 0 by 2x to find x + 1/(2x) = −5/2.

Use the direct relationship (x − 1/2x)² = (x + 1/2x)² − 2.

Calculation: (−5/2)² − 2 = 25/4 − 2 = 17/4.

Taking square root: x − 1/2x = √17 / 2.

∴ The correct answer is √17 / 2.

Alternate Method

Given: 2x² + 5x + 1 = 0

Formula Used: (a − b)² = (a + b)² − 4ab

⇒ 2x² + 1 = −5x

⇒ (2x² + 1) / 2x = −5x / 2x

⇒ x + 1/(2x) = −5/2

Now, to find x − 1/(2x), square both sides of the identity:

⇒ (x − 1/(2x))² = (x + 1/(2x))² − 4 × x × (1/2x)

⇒ (x − 1/(2x))² = (−5/2)² − 2

⇒ (x − 1/(2x))² = 25/4 − 2

⇒ (x − 1/(2x))² = (25 − 8) / 4 = 17 / 4

⇒ x − 1/(2x) = ±√(17 / 4)

⇒ x − 1/(2x) = √17 / 2 (taking positive value)

∴ The correct answer is √17 / 2.

Additional Information

Algebraic Identity for Reciprocals

For any value k, if x + 1/x = k, then (x − 1/x)² = k² − 4.

Quadratic Formula

For ax² + bx + c = 0, the roots are x = (−b ± √(b² − 4ac)) / 2a.

Square of a Sum and Difference

(a + b)² − (a − b)² = 4ab is a fundamental identity used to convert between sums and differences.

Shortcut Trick

Profit ∝ Capital × Time ⇒ Capital ∝ Profit / Time

Time (Months): A = 12, B = (12 − 4) = 8, C = (12 − 6) = 6

Ratio of Capitals = (P_A/T_A) : (P_B/T_B) : (P_C/T_C) = 1/12 : 4/8 : 5/6 = 1/12 : 1/2 : 5/6

Multiply by 12 → Ratio = 1 : 6 : 10

Since 1 unit = ₹54,000, Sum of B and C = (6 + 10) = 16 units = 16 × 54,000 = ₹8,64,000

∴ The correct answer is ₹8,64,000.

Alternate Method

Given:

Capital of A (C_A) = ₹54,000

Investment duration for A (T_A) = 12 months

Investment duration for B (T_B) = 12 − 4 = 8 months

Investment duration for C (T_C) = 12 − 6 = 6 months

Profit Ratio (P_A : P_B : P_C) = 1 : 4 : 5

Formula:

Profit = Capital × Time

Calculations:

⇒ P_A / P_B = (C_A × T_A) / (C_B × T_B)

⇒ 1 / 4 = (54,000 × 12) / (C_B × 8)

⇒ C_B = (54,000 × 12 × 4) / 8 = 54,000 × 6 = ₹3,24,000

⇒ P_A / P_C = (C_A × T_A) / (C_C × T_C)

⇒ 1 / 5 = (54,000 × 12) / (C_C × 6)

⇒ C_C = (54,000 × 12 × 5) / 6 = 54,000 × 10 = ₹5,40,000

⇒ Sum of capitals of B and C = C_B + C_C

⇒ Sum = ₹3,24,000 + ₹5,40,000 = ₹8,64,000

∴ The correct answer is ₹8,64,000.

Additional Information

Basic Partnership Rule

When multiple partners invest for the same time, the profit is divided in the ratio of their capitals: P1:P2 = C1:C2.

Compound Partnership Rule

When investments are for different time periods, Profit Ratio = (C1 × T1) : (C2 × T2) : (C3 × T3).

Calculating Capital Ratio

If profit ratio and time ratio are known, the capital ratio is found by: C1:C2:C3 = (P1/T1) : (P2/T2) : (P3/T3).

Shortcut Trick

Volume = (2/3) × π × r³ = 19404

⇒ r³ = 19404 × (21/44) = 441 × 21 ⇒ r = 21 cm

TSA of solid hemisphere = 3 × π × r²

⇒ TSA = 3 × (22/7) × 21 × 21 = 66 × 63 = 4158 cm²

∴ The correct answer is 4158 cm².

Alternate Method

Given:

Volume of solid hemisphere = 19,404 cm³

π = 22/7

Formula Used:

Volume of hemisphere = (2/3)πr³

Total surface area (TSA) of solid hemisphere = 3πr²

Calculations:

⇒ (2/3) × (22/7) × r³ = 19404

⇒ (44/21) × r³ = 19404

⇒ r³ = (19404 × 21) ÷ 44

⇒ r³ = 441 × 21

⇒ r³ = 21 × 21 × 21

⇒ r = 21 cm

Now, calculate Total Surface Area (TSA):

⇒ TSA = 3 × π × r²

⇒ TSA = 3 × (22/7) × 21 × 21

⇒ TSA = 3 × 22 × 3 × 21

⇒ TSA = 66 × 63

⇒ TSA = 4158 cm²

∴ The correct answer is 4158 cm².

Additional Information

Curved Surface Area of Hemisphere

The curved surface area of a hemisphere (excluding the flat base) is given by 2πr².

Volume and Surface Area of a Sphere

For a full sphere, Volume = (4/3)πr³ and Total Surface Area = 4πr².

Relationship between Sphere and Hemisphere

A solid hemisphere has a flat circular base, which is why its TSA (3πr²) is more than half of a sphere's surface area (2πr²).

Shortcut Trick

Divide the base equation 2x − y = 2 by 2 to get x − y/2 = 1.

Determine the product of the terms: x × (y/2) = (xy) ÷ 2 = (3/2) ÷ 2 = 3/4.

Apply the identity a³ − b³ = (a − b)³ + 3ab(a − b) with a = x and b = y/2.

Value = 1³ + 3 × (3/4) × 1 = 1 + 9/4 = 13/4.

∴ The correct answer is 13/4.

Alternate Method

Given: 2x − y = 2 and xy = 3/2

Formula Used: (a − b)³ = a³ − b³ − 3ab(a − b)

⇒ 2x − y = 2

⇒ x − y/2 = 1 (Dividing both sides by 2)

⇒ (x − y/2)³ = 1³ (Cubing both sides)

⇒ x³ − (y/2)³ − 3 × x × (y/2) × (x − y/2) = 1

⇒ x³ − y³/8 − 3 × (xy/2) × (1) = 1

⇒ x³ − y³/8 − 3 × (3/4) = 1

⇒ x³ − y³/8 = 1 + 9/4

⇒ x³ − y³/8 = 13/4

∴ The correct answer is 13/4.

Additional Information

Difference of Cubes Identity

The algebraic identity for the difference of two cubes is a³ − b³ = (a − b)(a² + ab + b²) or a³ − b³ = (a − b)³ + 3ab(a − b).

Sum of Cubes Identity

Similarly, for the sum of cubes: a³ + b³ = (a + b)(a² − ab + b²) or a³ + b³ = (a + b)³ − 3ab(a + b).

Square of Difference

The relation between the square and product is given by (a − b)² = a² + b² − 2ab.

Given:

$0.4\overline 6 + 0.7\overline {23} - 0.3\overline 9 \times 0.\overline 7$

Formula used:

Recurring conversion: $0.ab\overline{c} = \frac{abc - ab}{900}$

Calculation:

$0.4\overline 6 = \frac{46 - 4}{90} = \frac{42}{90}$

$0.7\overline {23} = \frac{723 - 7}{990} = \frac{716}{990}$

$0.3\overline 9 \times 0.\overline 7 = \frac{36}{90} \times \frac{7}{9} = \frac{28}{90}$

⇒ $\frac{42}{90} + \frac{716}{990} - \frac{28}{90}$

⇒ (\frac{42}{90} - \frac{28}{90}) + \frac{716}{990}\)

⇒ $\frac{14}{90} + \frac{716}{990}$

⇒ $\frac{154}{990} + \frac{716}{990}$

⇒ $\frac{870}{990} = \frac{87}{99} = 0.\overline{87}$

∴ The correct answer is option (4).

Shortcut Trick

Direct Geometry property: In ΔABC, if bisectors of ∠B and ∠C meet at P, then ∠BPC − ∠PCA = ∠A + (∠B / 2).

Given: ∠A = 66° and ∠B = 50°

Substitute values: 66° + (50° / 2)

⇒ 66° + 25° = 91°

∴ The correct answer is 91°.

Alternate Method

Given: ∠A = 66° and ∠B = 50° in ΔABC.

Formula Used: ∠A + ∠B + ∠C = 180° | ∠BPC = 90° + (∠A / 2)

⇒ ∠C = 180° − (∠A + ∠B)

⇒ ∠C = 180° − (66° + 50°) = 180° − 116° = 64°

⇒ ∠PCA = ∠C / 2 = 64° / 2 = 32°

⇒ ∠BPC = 90° + (∠A / 2) = 90° + (66° / 2) = 123°

⇒ Required Difference = ∠BPC − ∠PCA = 123° − 32° = 91°

∴ The correct answer is 91°.

Additional Information

Incenter Angle Property

The angle formed at the incenter (intersection of internal bisectors) is always 90° + ∠Vertex/2.

Ex-center Angle Property

The angle formed at the ex-center (intersection of external bisectors) is always 90° − ∠Vertex/2.

Internal Angle Bisector

An angle bisector divides the interior angle into two equal parts and the incenter is equidistant from all sides.

Angle Sum Property

The sum of all interior angles in any triangle is exactly 180°.

Shortcut Trick

In a right pyramid with an equilateral triangle base, Height² = (Slant Edge)² − (Circumradius of base)².

Circumradius (R) = side / √3 = 20 / √3 cm.

Height² = 30² − (20/√3)² = 900 − 400/3 = 2300/3.

Height = √(2300 / 3) = 10√(23/3) cm.

∴ The correct answer is 10√(23/3) cm.

Alternate Method

Given:

Side of equilateral base (a) = 20 cm

Slant edge (e) = 30 cm

Formula Used:

Vertical Height (h) = √[e² − R²], where R is the circumradius of the base.

For an equilateral triangle, Circumradius (R) = a / √3.

Calculations:

⇒ Circumradius (R) = 20 / √3 cm

⇒ h² = e² − R²

⇒ h² = 30² − (20 / √3)²

⇒ h² = 900 − 400 / 3

⇒ h² = (2700 − 400) / 3

⇒ h² = 2300 / 3

⇒ h = √(2300 / 3) = 10√(23/3) cm

∴ The correct answer is 10√(23/3) cm.

Additional Information

Volume of a Pyramid

The volume of any pyramid is given by 1/3 × Base Area × Vertical Height.

Slant Height vs Slant Edge

Slant height (l) is the height of the triangular face, while slant edge (e) is the length of the edge connecting the vertex to a base corner. Relation: l² = h² + r² (where r = in-radius).

Lateral Surface Area

Lateral Surface Area of a right pyramid = 1/2 × Perimeter of base × Slant height (l).

Equilateral Triangle Radii

For side a, Circumradius R = a/√3 and In-radius r = a/(2√3).

Shortcut Trick

Break each number into its prime components (3, 5, 7, 11) immediately.

847 → 7¹ × 11² | 385 → 5¹ × 7¹ × 11¹

675 → 3³ × 5² | 3025 → 5² × 11²

Total powers: a = 3, b = (1+2+2) = 5, c = (1+1) = 2, d = (2+1+2) = 5.

Calculation: ab − cd = (3 × 5) − (2 × 5) = 15 − 10 = 5.

∴ The correct answer is 5.

Alternate Method

Given: 847 × 385 × 675 × 3025 = 3^a × 5^b × 7^c × 11^d

Step 1: Prime factorization of each number.

⇒ 847 = 7 × 121 = 7¹ × 11²

⇒ 385 = 5 × 77 = 5¹ × 7¹ × 11¹

⇒ 675 = 25 × 27 = 5² × 3³

⇒ 3025 = 25 × 121 = 5² × 11²

Step 2: Combine all factors by adding their exponents.

⇒ (7¹ × 11²) × (5¹ × 7¹ × 11¹) × (3³ × 5²) × (5² × 11²)

⇒ 3³ × 5^(1+2+2) × 7⁽¹⁺¹⁾ × 11^(2+1+2)

⇒ 3³ × 5⁵ × 7² × 11⁵

Step 3: Comparing with 3^a × 5^b × 7^c × 11^d.

⇒ a = 3, b = 5, c = 2, d = 5.

Step 4: Find the value of ab − cd.

⇒ ab − cd = (3 × 5) − (2 × 5)

⇒ ab − cd = 15 − 10 = 5.

∴ The correct answer is 5.

Additional Information

Prime Factorization Method

To find exponents in a product, express every number as a product of prime bases. For any number N = p₁^x × p₂^y, where p are primes.

Laws of Exponents

When multiplying terms with the same base, add the exponents: x^m × xⁿ = x^m+n.

Divisibility Rule for 11

A number is divisible by 11 if the difference between the sum of digits at odd positions and even positions is 0 or a multiple of 11 (e.g., 847 → (8+7)−4 = 11).

Shortcut Trick

Net discount % for successive discounts of x% and y% = x + y − (x × y / 100)%

Total discount % = 10 + 5 − (10 × 5 / 100) = 15 − 0.5 = 14.5%

Selling Price = 100% − 14.5% = 85.5% of Marked Price

Selling Price = 60,000 × 0.855 = Rs. 51,300

∴ The correct answer is Rs. 51,300.

Alternate Method

Given:

Marked Price (MP) = Rs. 60,000

First discount (Online) = 10%

Second discount (Credit Card) = 5%

Formula Used:

Final Price = MP × (1 − d₁/100) × (1 − d₂/100)

Calculations:

⇒ Price after 10% online discount = 60,000 − (10 / 100 × 60,000)

⇒ Price after 10% online discount = 60,000 − 6,000 = Rs. 54,000

⇒ Price after 5% credit card discount = 54,000 − (5 / 100 × 54,000)

⇒ Price after 5% credit card discount = 54,000 − 2,700

⇒ Final Amount = Rs. 51,300

∴ The correct answer is Rs. 51,300.

Additional Information

Successive Discounts Formula

For two successive discounts of a% and b%, the equivalent single discount is given by [a + b − (ab/100)]%.

Fraction Method for Discounts

Discounts can be solved by multiplying factors: 10% discount = 9/10 and 5% discount = 19/20. Final SP = MP × (9/10) × (19/20).

Marked Price vs Selling Price

Marked Price is the price printed on the label, while Selling Price is the actual price paid after deducting all applicable discounts.

Shortcut Trick

Substitute θ = 30° in the given expression: 0° < 30° < 90°.

a/b = 1 + 2 tan²30° + 2 sin30° sec²30° = 1 + 2(1/√3)² + 2(1/2)(2/√3)²

a/b = 1 + 2/3 + 4/3 = 1 + 6/3 = 3 ⇒ Let a = 3, b = 1.

Required value = (a + b) ÷ (a − b) = (3 + 1) ÷ (3 − 1) = 4 ÷ 2 = 2.

From the options, cosec 30° = 2, which matches the required value.

∴ The correct answer is cosecθ.

Alternate Method

Given: 1 + 2tan²θ + 2sinθ sec²θ = a/b

Formula Used: sec²θ = 1 + tan²θ; cos²θ = 1 − sin²θ

⇒ a/b = 1 + 2(sin²θ ÷ cos²θ) + 2sinθ ÷ cos²θ

⇒ a/b = (cos²θ + 2sin²θ + 2sinθ) ÷ cos²θ

⇒ a/b = (1 − sin²θ + 2sin²θ + 2sinθ) ÷ (1 − sin²θ)

⇒ a/b = (1 + sin²θ + 2sinθ) ÷ (1 − sin²θ)

⇒ a/b = (1 + sinθ)² ÷ [(1 + sinθ)(1 − sinθ)]

⇒ a/b = (1 + sinθ) ÷ (1 − sinθ)

⇒ Applying Componendo & Dividendo rule: (a + b) ÷ (a − b) = [(1 + sinθ) + (1 − sinθ)] ÷ [(1 + sinθ) − (1 − sinθ)]

⇒ (a + b) ÷ (a − b) = 2 ÷ 2sinθ = 1/sinθ = cosecθ

∴ The correct answer is cosecθ.

Additional Information

Componendo and Dividendo Rule

If x/y = p/q, then (x + y) ÷ (x − y) = (p + q) ÷ (p − q). This is a useful shortcut for solving ratio-based trigonometric equations.

Fundamental Identities

sin²θ + cos²θ = 1 | 1 + tan²θ = sec²θ | 1 + cot²θ = cosec²θ.

Reciprocal Ratios

cosecθ = 1 ÷ sinθ | secθ = 1 ÷ cosθ | cotθ = 1 ÷ tanθ.

Shortcut Trick

Convert the inner surd √(3 + √5) to √((6 + 2√5) ÷ 2) = (√5 + 1) ÷ √2.

The denominator 4√2 − (√5 + 1) ÷ √2 simplifies to (8 − √5 − 1) ÷ √2 = (7 − √5) ÷ √2.

Original expression becomes (22√2 × √2) ÷ (7 − √5) = 44 ÷ (7 − √5).

Rationalizing 44 ÷ (7 − √5) gives 44(7 + √5) ÷ (49 − 5) = 7 + √5.

Comparing with a + √5b, we get a = 7 and b = 1. Ratio (a × b) : (a + b) = 7 : 8.

∴ The correct answer is 7 : 8.

Alternate Method

Given:

Expression: 22√2 ÷ (4√2 − √(3 + √5)) = a + √5b

Step 1: Simplify the square root √(3 + √5)

⇒ √(3 + √5) = √((6 + 2√5) ÷ 2)

⇒ √((√5 + 1)² ÷ 2) = (√5 + 1) ÷ √2

Step 2: Substitute this value back into the denominator

⇒ Denominator = 4√2 − [(√5 + 1) ÷ √2]

⇒ Denominator = (4√2 × √2 − (√5 + 1)) ÷ √2

⇒ Denominator = (8 − √5 − 1) ÷ √2 = (7 − √5) ÷ √2

Step 3: Simplify the entire fraction

⇒ Expression = 22√2 ÷ [(7 − √5) ÷ √2]

⇒ Expression = (22√2 × √2) ÷ (7 − √5) = 44 ÷ (7 − √5)

Step 4: Rationalize by multiplying by conjugate (7 + √5)

⇒ [44 × (7 + √5)] ÷ [(7 − √5) × (7 + √5)]

⇒ [44(7 + √5)] ÷ [49 − 5] = [44(7 + √5)] ÷ 44

⇒ Expression = 7 + √5

Step 5: Compare and find the ratio

⇒ Comparing 7 + 1√5 with a + b√5 gives a = 7 and b = 1

⇒ (a × b) : (a + b) = (7 × 1) : (7 + 1) = 7 : 8

∴ The correct answer is 7 : 8.

Additional Information

Square root of a Binomial Surd

For any surd of the form √(x + 2√y), find two numbers m and n such that m + n = x and m × n = y; then the square root is √m + √n.

Rationalizing Factor

The rationalizing factor of (a − √b) is its conjugate (a + √b), utilizing the identity (x − y)(x + y) = x² − y² to eliminate roots.

Ratio Properties

A ratio x : y represents the relative size of two quantities and can be expressed as a fraction x/y for calculation purposes.

Shortcut Trick

Total distance = Average Speed × Total Time

Average Speed = (2 × S₁ × S₂) / (S₁ + S₂)

S₁ = 5.5 km/h, S₂ = 4 km/h, Time = 4.75 hours

Average Speed = (2 × 5.5 × 4) / (5.5 + 4) = 44 / 9.5 = 88 / 19 km/h

Total Distance = (88 / 19) × 4.75 = (88 / 19) × (19 / 4) = 22 km

∴ The correct answer is 22 km.

Alternate Method

Given:

Speed while going (S₁) = 5½ km/h = 11/2 km/h = 5.5 km/h

Speed while returning (S₂) = 4 km/h

Total time taken (T) = 4¾ hours = 19/4 hours = 4.75 hours

Calculation:

Let the distance between the student’s home and school be D km.

⇒ Time taken to go = Distance ÷ Speed = D / 5.5 = 2D/11 hours

⇒ Time taken to return = Distance ÷ Speed = D / 4 hours

⇒ Total Time = Time to go + Time to return

⇒ 2D/11 + D/4 = 19/4

⇒ (8D + 11D) / 44 = 19/4

⇒ 19D / 44 = 19/4

⇒ D / 44 = 1/4

⇒ D = 44 / 4 = 11 km

⇒ Total distance covered = One-way distance × 2

⇒ Total distance = 11 × 2 = 22 km

∴ The correct answer is 22 km.

Additional Information

Average Speed for Equal Distance

When an object covers equal distances at different speeds S₁ and S₂, the average speed is (2 × S₁ × S₂) / (S₁ + S₂).

Distance Formula for Total Time

One-way distance D = [ (S₁ × S₂) / (S₁ + S₂) ] × Total Time taken for the round trip.

Speed-Time Relation

For a constant distance, speed is inversely proportional to time (S ∝ 1/T). Therefore, S₁/S₂ = T₂/T₁.

Calculation:

Total number of students passed = 900

The students passed in the examination from institutes D = 28%

The number of students who passed the examination from institute D = (900 × 28/100)

⇒ 252

Total number of students appeared = 1200

The students appeared in the examination from institute A = 72° 

⇒ (72/360 × 1200)

⇒ 240

Now,

The value of x = (252 – 240) = 12

∴ The value of x lies between 11 and 14

Shortcut Trick

In Compound Interest, the amounts at equal time intervals follow a constant multiplier (GP).

Multiplier for 2 years (6 − 4) = 17,820 ÷ 11,880 = 1.5

Since 4 years = 2 years × 2, the multiplier for 4 years is (1.5)² = 2.25

Principal (P) = Amount after 4 years ÷ 2.25 = 11,880 ÷ 2.25 = ₹5,280

Required value = Half of P = 5,280 ÷ 2 = ₹2,640

∴ The correct answer is ₹2640.

Alternate Method

Given: Amount after 4 years (A₄) = ₹11,880 and Amount after 6 years (A₆) = ₹17,820

Formula used: A = P(1 + r/100)ⁿ

⇒ A₄ = P(1 + r/100)⁴ = 11,880 --- (1)

⇒ A₆ = P(1 + r/100)⁶ = 17,820 --- (2)

⇒ Dividing (2) by (1): (1 + r/100)⁶⁻⁴ = 17,820 ÷ 11,880

⇒ (1 + r/100)² = 1.5

⇒ Substituting (1 + r/100)² in equation (1):

⇒ P × [(1 + r/100)²]² = 11,880

⇒ P × (1.5)² = 11,880

⇒ P × 2.25 = 11,880

⇒ P = 11,880 ÷ 2.25 = 11,880 ÷ (9/4)

⇒ P = (11,880 × 4) ÷ 9 = 1,320 × 4 = ₹5,280

⇒ Required value = P ÷ 2 = 5,280 ÷ 2 = ₹2,640

∴ The correct answer is ₹2640.

Additional Information

Compound Interest GP Property

In compound interest, the amounts at regular time intervals (t, 2t, 3t...) always form a Geometric Progression (GP) where the common ratio is (1 + r/100)^t.

Effective Rate for Compounding

If interest is compounded half-yearly, the effective rate for a year becomes r_eff = r + r + (r×r)/100, where the period rate is half of the annual rate.

CI and SI Difference

For 2 years, the difference between CI and SI on a principal P at rate r% is given by Diff = P(r/100)².

Calculation:

The total production of fertilisers by country Y in 2017 and 2019 and by country X in 2016 = (265 + 285 + 220) = 770

The total production of fertilisers by country Z in 2016, 2018 and 2020 = (320 + 360 + 420) = 1100

Required percentage = (770/1100 × 100)

⇒ (770/11)

⇒ 70%

∴ The required percentage is 70%

Shortcut Trick Persons (55 to <75 kg) = 40 + 50 + 30 + 40 = 160 Persons (80 to <100 kg) = 45 + 25 + 15 + 5 = 90 Required % = [(160 − 90) ÷ 90] × 100 = 77.77...% ≈ 77.8% ∴ The correct answer is 77.8%.

Alternate Method Data extraction from the histogram: Number of persons in weight range 55 − 75 kg: ⇒ (55 − 60) + (60 − 65) + (65 − 70) + (70 − 75) ⇒ 40 + 50 + 30 + 40 = 160 Number of persons in weight range 80 − 100 kg: ⇒ (80 − 85) + (85 − 90) + (90 − 95) + (95 − 100) ⇒ 45 + 25 + 15 + 5 = 90 Calculation of how much 160 is more than 90: ⇒ Difference = 160 − 90 = 70 ⇒ Percentage more = (Difference ÷ Base value) × 100 ⇒ (70 ÷ 90) × 100 = 77.77...% ⇒ Correct to one decimal place = 77.8% ∴ The correct answer is 77.8%.

Additional Information Percentage Change Formula Percentage more or less is calculated as [(New Value − Base Value) ÷ Base Value] × 100. DI Interpretation - Histogram In a histogram, the area or height of the bars represents the frequency of the continuous data intervals. Always sum the frequencies of adjacent bars for a range. DI Percentage Increase/Decrease To compare two groups in DI, identify the base value (the value after the word 'than') and use it as the denominator.

Shortcut Trick

Given ratio of Slant height (l) : Radius (r) = 29 : 20.

Using Pythagorean triplet (20, 21, 29), the height (h) corresponds to 21 parts.

Volume part = 1/3 × r² × h = 1/3 × 20² × 21 = 2800 units³.

Comparing with actual volume: x³ = 4838.4 / 2800 = 1.728 ⇒ x = 1.2.

Radius (r) = 20 × 1.2 = 24 cm.

∴ The correct answer is 24 cm.

Alternate Method

Given:

Ratio of l : r = 29 : 20

Volume of cone (V) = 4838.4 π cm³

Formula:

Volume of cone = (1/3) π r² h

Relation between l, r, and h: l² = r² + h²

Let common ratio be x. Then r = 20x and l = 29x.

⇒ h = √(l² − r²) = √((29x)² − (20x)²)

⇒ h = √(841x² − 400x²) = √(441x²) = 21x

⇒ Volume = (1/3) × π × (20x)² × (21x) = 4838.4 π

⇒ (1/3) × 400x² × 21x = 4838.4

⇒ 400 × 7x³ = 4838.4

⇒ 2800x³ = 4838.4

⇒ x³ = 4838.4 ÷ 2800 = 1.728

⇒ x = ³√1.728 = 1.2

⇒ Radius (r) = 20x = 20 × 1.2 = 24 cm

∴ The correct answer is 24 cm.

Additional Information

Curved Surface Area of Cone

The lateral surface area or CSA of a right circular cone is given by the formula CSA = πrl, where r is radius and l is slant height.

Total Surface Area of Cone

The TSA includes the base area and the curved surface area, calculated as TSA = πr(r + l).

Pythagorean Relation in Cones

In any right circular cone, the height (h), radius (r), and slant height (l) always satisfy the right-angled triangle property l² = r² + h².

Shortcut Trick

Given x² − √7x + 1 = 0, dividing by x gives x + 1/x = √7.

Let k = x + 1/x = √7. Then, x² + 1/x² = k² − 2 = (√7)² − 2 = 5.

Also, x³ + 1/x³ = k³ − 3k = (√7)³ − 3√7 = 4√7.

Value of x⁵ + 1/x⁵ = (x² + 1/x²)(x³ + 1/x³) − (x + 1/x) = (5 × 4√7) − √7 = 19√7.

∴ The correct answer is 19√7.

Alternate Method

Given: x² − √7x + 1 = 0

Formula Used: x⁵ + 1/x⁵ = (x² + 1/x²)(x³ + 1/x³) − (x + 1/x)

⇒ x² − √7x + 1 = 0

⇒ x² + 1 = √7x

⇒ x + 1/x = √7

⇒ Squaring both sides: (x + 1/x)² = (√7)²

⇒ x² + 1/x² + 2 = 7 ⇒ x² + 1/x² = 5

⇒ Cubing both sides: (x + 1/x)³ = (√7)³

⇒ x³ + 1/x³ + 3(x + 1/x) = 7√7

⇒ x³ + 1/x³ + 3√7 = 7√7 ⇒ x³ + 1/x³ = 4√7

⇒ Now, x⁵ + 1/x⁵ = (5 × 4√7) − √7

⇒ x⁵ + 1/x⁵ = 20√7 − √7 = 19√7

∴ The correct answer is 19√7.

Additional Information

Algebraic Identities for x + 1/x

If x + 1/x = k, then x² + 1/x² = k² − 2 and x³ + 1/x³ = k³ − 3k.

Higher Order Powers

x⁴ + 1/x⁴ = (x² + 1/x²)² − 2. For odd powers like 5 or 7, multiply consecutive powers and subtract the base term: x⁷ + 1/x⁷ = (x³ + 1/x³)(x⁴ + 1/x⁴) − (x + 1/x).

Negative Identities

If x − 1/x = k, then x² + 1/x² = k² + 2 and x³ − 1/x³ = k³ + 3k.

Shortcut Trick

In a cyclic quadrilateral ABCD where BC is the diameter, the angle at the vertex A is related to ∠DBC by the direct formula:

∠BAD = 90° + ∠DBC

Given ∠DBC = 29°

⇒ ∠BAD = 90° + 29° = 119°

∴ The correct answer is 119°.

Alternate Method

Given:

ABCD is a cyclic quadrilateral.

BC is the diameter of the circle.

∠DBC = 29°.

Formula Used:

1. Angle subtended by a diameter at any point on the circumference is 90° (Angle in a semicircle).
2. The sum of opposite angles in a cyclic quadrilateral is 180°.

Calculations:

⇒ Since BC is the diameter, the angle subtended by it at point D is 90°.

⇒ ∴ ∠BDC = 90°

⇒ In ΔBCD, the sum of all internal angles is 180°.

⇒ ∠DBC + ∠BDC + ∠BCD = 180°

⇒ 29° + 90° + ∠BCD = 180°

⇒ 119° + ∠BCD = 180°

⇒ ∠BCD = 180° − 119° = 61°

⇒ In cyclic quadrilateral ABCD, opposite angles ∠BAD and ∠BCD sum to 180°.

⇒ ∠BAD + ∠BCD = 180°

⇒ ∠BAD + 61° = 180°

⇒ ∠BAD = 180° − 61° = 119°

∴ The correct answer is 119°.

Additional Information

Angle in a Semicircle

Any angle subtended by the diameter at the circumference of the circle is always a right angle (90°).

Cyclic Quadrilateral Property

For any quadrilateral whose vertices lie on a circle, the sum of opposite internal angles is exactly 180°.

Exterior Angle of a Cyclic Quadrilateral

An exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

Angle Sum Property

The sum of all interior angles of any triangle is 180°.

Shortcut Trick

66 2/3% = 2/3. Let Savings = 3 units, then Expenses = 3 + 2 = 5 units.

Total Income = 3 + 5 = 8 units.

Increase in Income = 8 × 44% = 3.52 units.

Increase in Expenses = 5 × 60% = 3.00 units.

Increase in Savings = 3.52 − 3.00 = 0.52 units.

Given 0.52 units = ₹1,040 → 1 unit = 2,000. Initial Expenditure = 5 × 2,000 = ₹10,000.

∴ The correct answer is ₹10,000.

Alternate Method

Given:

Expense = Savings + (66 2/3% of Savings)

Increase in Income = 44%

Increase in Expenses = 60%

Increase in Savings = ₹1,040

Formula:

Income = Expenditure + Savings

⇒ Let Savings = 300x

⇒ Expenses = 300x + (2/3 × 300x) = 500x

⇒ Initial Income = 300x + 500x = 800x

⇒ New Income = 800x × 144/100 = 1152x

⇒ New Expenses = 500x × 160/100 = 800x

⇒ New Savings = 1152x − 800x = 352x

⇒ Increase in Savings = 352x − 300x = 52x

⇒ 52x = 1040

⇒ x = 1040 ÷ 52 = 20

⇒ Initial Expenditure = 500x = 500 × 20 = 10,000

∴ The correct answer is ₹10,000.

Additional Information

Percentage to Fraction Conversion

Common fractions like 66 2/3% = 2/3, 33 1/3% = 1/3, and 16 2/3% = 1/6 are essential for fast calculations in exams.

Income-Expenditure Relationship

Income = Expenditure + Savings. Any change in total income must be balanced by the sum of changes in expenditure and savings (ΔI = ΔE + ΔS).

Percentage Change Rule

New Value = Old Value × (100 + Change%)/100. This is the basic multiplier used to find values after an increase or decrease.

Shortcut Trick

Total Work = LCM(12.5, 25) = 25 units. Efficiencies: A = 2, B = 1. Total = 3 units/hr.

Normal time = 25 ÷ 3 hours = 8 hours 20 mins. Time with leak = 8h 20m + 1h 40m = 10 hours.

New efficiency (A + B − Leak) = 25 ÷ 10 = 2.5. Thus, Leak Efficiency = 3 − 2.5 = 0.5 units/hr.

Time to empty 70% capacity = (25 × 0.7) ÷ 0.5 = 17.5 ÷ 0.5 = 35 hours.

∴ The correct answer is 35 hours.

Alternate Method

Given:

Time taken by Pipe A = 12.5 hours = 25/2 hours

Time taken by Pipe B = 25 hours

Extra time due to leak = 1 hour 40 minutes = 1 + 40/60 = 5/3 hours

Calculations:

⇒ Total efficiency of Pipes A and B = 2 + 1 = 3 units/hr

⇒ Normal time taken to fill the cistern = 25 ÷ 3 hours = 8 hours 20 minutes

⇒ Total time taken with leakage = 8 hours 20 minutes + 1 hour 40 minutes = 10 hours

⇒ Combined efficiency of (A + B − Leak) = Total Work ÷ Total Time = 25 ÷ 10 = 2.5 units/hr

⇒ Efficiency of the Leak = (Efficiency of A + B) − (Combined Efficiency with Leak)

⇒ Efficiency of the Leak = 3 − 2.5 = 0.5 units/hr

⇒ Work to be done by leak = 70% of 25 = 0.7 × 25 = 17.5 units

⇒ Time taken by leak alone = 17.5 ÷ 0.5 = 35 hours

∴ The correct answer is 35 hours.

Additional Information

Negative Efficiency

In pipes and cisterns, a leak or an outlet pipe is treated as a component with negative efficiency because it removes work (water) from the system.

Work-Time Relationship

Efficiency = Work ÷ Time. When work is constant, efficiency is inversely proportional to the time taken to complete the task.

Net Rate Formula

If pipe A fills in x hours and leak L empties in y hours, the net rate of filling is (1/x − 1/y) per hour.

Shortcut Trick

Total Ratio parts (A+D) + (B+C) = 8 + 5 = 13 units.

13 units = 46800 → 1 unit = 3600.

Combined shares: (A+D) = 28800 and (B+C) = 18000.

B : C = 5 : 4 → 9 units = 18000 → B = 10000, C = 8000.

Given A = 18400 → D = 28800 − 18400 = 10400.

x − y = (A − B) − |C − D| = (18400 − 10000) − (10400 − 8000) = 8400 − 2400 = 6000.

∴ The correct answer is ₹6000.

Alternate Method

Given:

Total sum = ₹46,800

(A + D) : (B + C) = 8 : 5

B : C = 5 : 4

A = ₹18,400

Calculations:

⇒ Share of (A + D) = [8 ÷ (8 + 5)] × 46800 = (8/13) × 46800 = 8 × 3600 = ₹28,800

⇒ Share of (B + C) = [5 ÷ (8 + 5)] × 46800 = (5/13) × 46800 = 5 × 3600 = ₹18,000

⇒ Share of B = [5 ÷ (5 + 4)] × 18000 = (5/9) × 18000 = 5 × 2000 = ₹10,000

⇒ Share of C = [4 ÷ (5 + 4)] × 18000 = (4/9) × 18000 = 4 × 2000 = ₹8,000

⇒ Share of D = (A + D) − A = 28800 − 18400 = ₹10,400

⇒ x = Difference between A and B = |18400 − 10000| = ₹8,400

⇒ y = Difference between C and D = |8000 − 10400| = ₹2,400

⇒ Value of (x − y) = 8400 − 2400 = ₹6,000

∴ The correct answer is ₹6000.

Additional Information

Ratio Distribution

When a quantity Q is divided in ratio a:b, the parts are [a/(a+b)] × Q and [b/(a+b)] × Q.

Difference and Modulus

The difference between two values p and q is usually taken as the absolute value |p − q|, which is always non-negative.

Compound Ratios

To solve for multiple variables, ensure they are linked by a common term or common total sum parts.

Shortcut Trick

Let the distances AB and BC be 3 units and 5 units respectively.

Average Speed = Total Distance ÷ Total Time

40 = (3 + 5) ÷ (3/x + 5/50)

40 = 8 ÷ (3/x + 1/10) ⇒ 1/5 = 3/x + 1/10

3/x = 1/5 − 1/10 = 1/10 ⇒ x = 30 km/h.

Ratio = (30 − 10) : (30 + 1) = 20 : 31.

∴ The correct answer is 20 : 31.

Alternate Method

Given:

Distance Ratio AB : BC = 3 : 5

Speed A to B = x km/h, Speed B to C = 50 km/h

Average Speed = 40 km/h

Formula: Average Speed = (Total Distance) ÷ (Total Time)

⇒ Total Distance = 3d + 5d = 8d

⇒ Time (AB) = 3d / x

⇒ Time (BC) = 5d / 50 = d / 10

⇒ 40 = 8d ÷ (3d/x + d/10)

⇒ 40 = 8 ÷ (3/x + 1/10)

⇒ 3/x + 1/10 = 8 / 40

⇒ 3/x + 1/10 = 1/5

⇒ 3/x = 1/5 − 1/10

⇒ 3/x = 1/10

⇒ x = 30

Now, evaluate (x − 10) : (x + 1):

⇒ (30 − 10) : (30 + 1)

⇒ 20 : 31

∴ The correct answer is 20 : 31.

Additional Information

Average Speed for Equal Distances

If a person covers two equal distances at speeds v₁ and v₂, the average speed is (2 × v₁ × v₂) ÷ (v₁ + v₂).

Weighted Average Speed

For different distances d₁, d₂,... and speeds v₁, v₂,..., Average Speed = (Σd_i) ÷ (Σd_i/v_i).

Speed-Time Relationship

For a constant distance, speed is inversely proportional to time (S₁/S₂ = T₂/T₁).

Shortcut Trick

Since the sum of parts in Alloy A (5 + 2 = 7) and Alloy B (3 + 4 = 7) are equal, we can multiply the metal parts directly by the mixing weights.

Quantity of x in C = (5 × 4) + (3 × 5) = 20 + 15 = 35 units.

Total quantity of C = (7 × 4) + (7 × 5) = 28 + 35 = 63 units.

Percentage of x = (35 ÷ 63) × 100 = (5 ÷ 9) × 100 = 55⁵/₉%.

∴ The correct answer is 55⁵/₉%.

Alternate Method

Given:

Ratio of x : y in A = 5 : 2

Ratio of x : y in B = 3 : 4

Mixing ratio of A : B = 4 : 5

Let the total quantity of Alloy C be 9 units (since mixing ratio is 4 : 5).

Quantity of A taken = 4 units

Quantity of B taken = 5 units

⇒ Amount of x in 4 units of A = 4 × (5 ÷ (5 + 2)) = 4 × (5 ÷ 7) = 20 ÷ 7

⇒ Amount of x in 5 units of B = 5 × (3 ÷ (3 + 4)) = 5 × (3 ÷ 7) = 15 ÷ 7

⇒ Total amount of x in Alloy C = (20 ÷ 7) + (15 ÷ 7) = 35 ÷ 7 = 5 units

⇒ Total quantity of Alloy C = 4 + 5 = 9 units

⇒ Percentage of x in Alloy C = (Total x ÷ Total C) × 100

⇒ Percentage of x = (5 ÷ 9) × 100 = 55.55...% = 55⁵/₉%

∴ The correct answer is 55⁵/₉%.

Additional Information

Weighted Average Principle

When two mixtures are combined, the final concentration C_avg = (n₁c₁ + n₂c₂) ÷ (n₁ + n₂), where n is quantity and c is concentration.

Alligation Rule

The ratio of quantities mixed is inversely proportional to the difference between their individual concentrations and the average concentration.

Fraction to Percentage Conversion

1/9 ≈ 11.11% | 5/9 ≈ 5 × 11.11% = 55.55% = 55⁵/₉%.

Shortcut Trick

Ratio of Volume (V) to Curved Surface Area (CSA) for a cylinder is always r / 2.

Given: V = 132 cm³ and CSA = 88 cm².

r / 2 = 132 / 88 ⇒ r / 2 = 3 / 2 ⇒ r = 3 cm.

Using CSA = 2 × π × r × h ⇒ 88 = 2 × 22/7 × 3 × h.

h = (88 × 7) / (44 × 3) = 14 / 3 = 4 2/3 cm.

∴ The correct answer is 4 2/3 cm.

Alternate Method

Given:

Curved Surface Area (CSA) = 88 cm²

Volume (V) = 132 cm³

Value of π = 22/7

Formula Used:

Curved Surface Area of Cylinder = 2πrh

Volume of Cylinder = πr²h

Calculations:

⇒ Volume / CSA = (πr²h) / (2πrh) = 132 / 88

⇒ r / 2 = 3 / 2

⇒ r = 3 cm

⇒ CSA = 2 × (22/7) × 3 × h = 88

⇒ (132 / 7) × h = 88

⇒ h = (88 × 7) / 132

⇒ h = (2 × 7) / 3

⇒ h = 14 / 3 = 4 2/3 cm

∴ The correct answer is 4 2/3 cm.

Additional Information

Total Surface Area of Cylinder

The total surface area includes the curved surface and the two circular bases: TSA = 2πrh + 2πr² = 2πr(h + r).

Volume of a Cone

If a cone has the same radius and height as a cylinder, its volume is exactly one-third of the cylinder: V = (1/3)πr²h.

Ratio Property

For any cylinder, the ratio of its volume to its curved surface area is always r/2, independent of the height h.

Shortcut Trick

Total additional marks obtained = 35 + 11 + 4 = 50

Net increase in average marks = 76 − 71 = 5

Number of papers = Total extra marks ÷ Increase in average

Number of papers = 50 ÷ 5 = 10

∴ The correct answer is 10 papers.

Alternate Method

Given:

Initial average = 71 marks

New average = 76 marks

Increase in Science = 35 | Increase in History = 11 | Increase in Computer Science = 4

Formula:

Average = Total Marks ÷ Number of Papers

Calculations:

⇒ Let the total number of papers be n.

⇒ Total initial marks = 71 × n = 71n

⇒ Total extra marks = 35 + 11 + 4 = 50

⇒ New total marks = 71n + 50

⇒ According to the problem, New Average = 76

⇒ New total marks = 76 × n = 76n

⇒ Equating both: 76n = 71n + 50

⇒ 76n − 71n = 50

⇒ 5n = 50

⇒ n = 10

∴ The correct answer is 10 papers.

Additional Information

General Average Formula

Average = (Sum of all observations) ÷ (Number of observations). To find the sum, use: Sum = Average × Count.

Net Deviation Concept

If the total sum increases by S and the count of items remains constant, the change in average is ΔA = S ÷ n.

Balanced Addition Rule

If every observation in a set is increased or decreased by a constant value k, the average of the set also increases or decreases by k.

Shortcut Trick

Number 75462A97B6 is divisible by 72, so it must be divisible by 8 and 9.

For divisibility by 8, last three digits 7B6 must be divisible by 8 → B = 3 (since 736 ÷ 8 = 92).

For divisibility by 9, sum of digits (7+5+4+6+2+A+9+7+B+6 = 46+A+B) must be divisible by 9.

Substitute B = 3 ⇒ 46 + A + 3 = 49 + A → A = 5 (since 54 is divisible by 9).

Value = √(8 × 5 − 4 × 3) = √(40 − 12) = √28.

∴ The correct answer is √28.

Alternate Method

Given number: 75462A97B6 is divisible by 72.

Factors of 72 are 8 × 9 (where 8 and 9 are co-prime).

Step 1: Divisibility by 8

The last three digits of the number are 7B6.

⇒ 7B6 ÷ 8 should leave remainder 0.

⇒ If B = 3, 736 ÷ 8 = 92 (Possible)

⇒ If B = 7, 776 ÷ 8 = 97 (Possible)

Step 2: Divisibility by 9

Sum of digits = 7 + 5 + 4 + 6 + 2 + A + 9 + 7 + B + 6 = 46 + A + B.

Case 1: If B = 3

⇒ Sum = 46 + A + 3 = 49 + A

⇒ For (49 + A) to be divisible by 9, A must be 5 (since 54 is the next multiple).

Case 2: If B = 7

⇒ Sum = 46 + A + 7 = 53 + A

⇒ For (53 + A) to be divisible by 9, A must be 1 (since 54 is the next multiple).

Step 3: Finding the value of √(8A − 4B)

For Case 1 (A=5, B=3):

⇒ Value = √(8 × 5 − 4 × 3)

⇒ Value = √(40 − 12) = √28.

For Case 2 (A=1, B=7):

⇒ Value = √(8 × 1 − 4 × 7) = √(8 − 28) = √(−20) (Not a real number).

Comparing with options, √28 is correct.

∴ The correct answer is √28.

Additional Information

Divisibility Rule of 8

A number is divisible by 8 if its last three digits are divisible by 8 or are zeros (000).

Divisibility Rule of 9

A number is divisible by 9 if the sum of all its individual digits is a multiple of 9.

Divisibility of Composite Numbers

To check divisibility by a composite number like 72, 88, or 99, the number must be divisible by its co-prime factors (e.g., 72 = 8 × 9, 88 = 8 × 11).

Shortcut Trick

In a trapezium ABCD where AB ∥ DC, the distance between the midpoints of the diagonals AC and BD is given by the formula:

Length of EF = ½ × |AB − CD|

Given AB = 18 cm and CD = 6 cm.

EF = ½ × (18 − 6) = ½ × 12 = 6 cm.

∴ The correct answer is 6 cm.

Alternate Method

Given:

Quadrilateral ABCD is a trapezium with AB ∥ DC.

Length of parallel side AB = 18 cm.

Length of parallel side CD = 6 cm.

E is the midpoint of diagonal AC and F is the midpoint of diagonal BD.

Formula Used:

For a trapezium with parallel sides a and b, the segment joining the midpoints of the diagonals is parallel to the bases and has length |a − b| / 2.

Calculations:

⇒ Parallel sides are AB = 18 cm and CD = 6 cm.

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

⇒ EF = ½ × (AB − CD)

⇒ EF = ½ × (18 − 6)

⇒ EF = ½ × 12

⇒ EF = 6 cm.

∴ The correct answer is 6 cm.

Additional Information

Median of a Trapezium

The line segment joining the midpoints of the non-parallel sides (legs) is called the median. Its length is the average of the parallel sides: Median = (AB + CD) / 2.

Area of a Trapezium

The area of a trapezium is calculated as Area = ½ × (sum of parallel sides) × height, where height is the perpendicular distance between the parallel sides.

Isosceles Trapezium Property

In an isosceles trapezium, the non-parallel sides are equal in length, the base angles are equal, and the diagonals are equal (AC = BD).

Cyclic Quadrilateral

An isosceles trapezium is always a cyclic quadrilateral, meaning all its vertices lie on a single circle.

Shortcut Trick

Exterior angle of ▵ADP ⇒ ∠ADC = ∠APD + ∠DAP = 25° + 39° = 64°

Same segment angles equal ⇒ ∠ABC = ∠ADC = 64°

AD is diameter ⇒ ∠ABD = 90°

∠CBD = ∠ABD − ∠ABC = 90° − 64° = 26°

∴ The correct answer is 26°.

Alternate Method

Given:

AB and CD are two chords of a circle with centre O and AD is a diameter.

AB and CD meet at an external point P.

∠APD = 25°

∠DAP = 39°

Concept Used:

Exterior Angle Theorem: An exterior angle of a triangle is equal to the sum of the two opposite interior angles.

Angle in a semicircle = 90°

Angles in the same segment are equal.

Calculation:

⇒ In △ADP

⇒ ∠ADC = ∠DPA + ∠DAP

⇒ ∠ADC = 25° + 39°

⇒ ∠ADC = 64°

⇒ Angles in the same segment are equal

⇒ ∠ADC = ∠ABC = 64°

⇒ AD is a diameter

⇒ ∠ABD = 90°

⇒ ∠ABC + ∠CBD = 90°

⇒ 64° + ∠CBD = 90°

⇒ ∠CBD = 26°

∴ ∠CBD = 26°.

Additional Information

Angle in a Semicircle

The angle subtended by a diameter at any point on the circumference of a circle is always a right angle (90°).

Angles in the Same Segment

Angles subtended by the same arc at any point on the remaining part of the circle are equal (∠CBD = ∠CAD).

Cyclic Quadrilateral Properties

The sum of opposite angles of a cyclic quadrilateral is 180°, and an exterior angle is equal to the interior opposite angle.

Shortcut Trick

Last Year Ratio A ÷ B = 4 ÷ 3

Present Year Ratio = (4 × 4/3) ÷ (3 × 6/5) = 16/3 ÷ 18/5 = 80 ÷ 54 = 40 ÷ 27

Total units (Present Year) = 40 + 27 = 67 units = Rs. 24.12 lakhs

1 unit = 24.12 ÷ 67 = 0.36 lakhs

Sum = A(Last Year) + B(Present Year) = (A_Pres × 3/4) + B_Pres = (40 × 0.36 × 3/4) + (27 × 0.36) = 10.8 + 9.72 = 20.52 lakhs

∴ The correct answer is Rs. 20.52 lakhs.

Alternate Method

Given: Last Year Income Ratio A ÷ B = 4 ÷ 3

Individual Growth Ratios: A(L:P) = 3 ÷ 4 and B(L:P) = 5 ÷ 6

Total Present Year Income = Rs. 24.12 lakhs

⇒ Let last year incomes of A and B be 4x and 3x.

⇒ Present income of A = 4x × (4/3) = 16x/3

⇒ Present income of B = 3x × (6/5) = 18x/5

⇒ Total Present Income = (16x/3) + (18x/5) = 24.12

⇒ (80x + 54x) / 15 = 24.12

⇒ 134x = 24.12 × 15 = 361.8

⇒ x = 361.8 ÷ 134 = 2.7

⇒ Income of A (Last Year) = 4x = 4 × 2.7 = 10.8 lakhs

⇒ Income of B (Present Year) = 18x/5 = 3.6 × 2.7 = 9.72 lakhs

⇒ Required Sum = 10.8 + 9.72 = 20.52 lakhs

∴ The correct answer is Rs. 20.52 lakhs.

Additional Information

Compound Ratio

When two or more ratios are multiplied together (a/b × c/d), the resulting ratio (ac/bd) is called the compound ratio.

Income-Expenditure-Saving Rule

Income = Expenditure + Savings. If ratios for any two are given, the third can be calculated using common variables.

Scaling Ratio Parts

To equate different ratios sharing a common term, multiply the parts by a constant such that the common term reaches its LCM.

Shortcut Trick

In ΔABC, if O is the intersection of angle bisectors, it is the incenter.

Property: ∠BOC = 90° + (∠A / 2)

Given ∠BOC = 108° ⇒ 108° = 90° + (∠A / 2) ⇒ ∠A / 2 = 18°

Since AO is the bisector of ∠A, ∠BAO = ∠A / 2 = 18°.

∴ The correct answer is 18°.

Alternate Method

Given: In ΔABC, O is the intersection of bisectors of ∠A and ∠B. ∠BOC = 108°.

⇒ In any triangle, the intersection of internal angle bisectors is called the Incenter.

⇒ The angle subtended by a side at the incenter: ∠BOC = 90° + (∠A / 2)

⇒ 108° = 90° + (∠A / 2)

⇒ ∠A / 2 = 108° − 90°

⇒ ∠A / 2 = 18°

⇒ ∠A = 36°

⇒ Since AO is the bisector of ∠A, ∠BAO = ∠OAC = ∠A / 2

⇒ ∠BAO = 18°

∴ The correct answer is 18°.

Additional Information

Incenter Property

The incenter O is the point where all three internal angle bisectors meet. The angle ∠AOB = 90° + ∠C/2 and ∠AOC = 90° + ∠B/2.

Excenter Angle

The angle formed by the intersection of two external angle bisectors (Excenter E) is given by ∠BEC = 90° − ∠A/2.

Orthocenter and Circumcenter

At the orthocenter (H), ∠BHC = 180° − ∠A. At the circumcenter (C′), ∠BC′C = 2 × ∠A.

Shortcut Trick

Profit Ratio (A : B : C) = (Capital × Time) ratio

Profit Ratio = (2 × 4) : (3 × 2) : (5 × 3) = 8 : 6 : 15

Difference between A and B = 8 − 6 = 2 units

2 units = ₹ 1,86,000 ⇒ 1 unit = ₹ 93,000

C's share = 15 units = 15 × 93,000 = ₹ 13,95,000

∴ The correct answer is ₹13,95,000.

Alternate Method

Given:

Ratio of Capitals (A : B : C) = 2 : 3 : 5

Ratio of Time (A : B : C) = 4 : 2 : 3

Difference between Profit of A and B = ₹ 1,86,000

Formula Used:

Profit = Capital × Time Duration

Calculations:

⇒ Let the capitals be 2k, 3k, and 5k and the time periods be 4t, 2t, and 3t.

⇒ Profit of A = 2k × 4t = 8kt

⇒ Profit of B = 3k × 2t = 6kt

⇒ Profit of C = 5k × 3t = 15kt

⇒ Ratio of Profits A : B : C = 8kt : 6kt : 15kt = 8 : 6 : 15

⇒ Difference between Profit A and B = 8 − 6 = 2 units

⇒ 2 units = 1,86,000

⇒ 1 unit = 1,86,000 ÷ 2 = 93,000

⇒ Share of C = 15 units = 15 × 93,000

⇒ Share of C = ₹ 13,95,000

∴ The correct answer is ₹13,95,000.

Additional Information

Compound Partnership

When partners invest different amounts for different periods of time, the profit is distributed in the ratio of the products of their capital and time: P₁ : P₂ = (C₁ × T₁) : (C₂ × T₂).

Finding Capital or Time Ratio

If the profit ratio and one other ratio are known, the unknown can be found using: Capital Ratio = (Profit Ratio ÷ Time Ratio) and Time Ratio = (Profit Ratio ÷ Capital Ratio).

Working vs Sleeping Partner

A working partner manages the business and may receive a fixed salary or commission from the total profit before the remaining profit is divided according to the investment ratio.

Shortcut Trick

Simplify given equations to standard linear form ax + by = c.

Equation 1: 4x + y/3 = 8/3 → 12x + y = 8

Equation 2: x/2 + 3y/4 = −5/2 → 2x + 3y = −10

Solve quickly: Multiply Eq 1 by 3 and subtract: (36x + 3y) − (2x + 3y) = 24 − (−10) ⇒ 34x = 34 ⇒ x = 1, y = −4.

Check options with P(1, −4): Option (2) ⇒ 3(1) − (−4) − 7 = 3 + 4 − 7 = 0.

∴ The correct answer is 3x − y − 7 = 0.

Alternate Method

Given:

Equation 1: 4x + y/3 = 8/3

Equation 2: x/2 + 3y/4 + 5/2 = 0

Formula Used:

Substitution or Elimination method for linear equations in two variables.

Calculations:

⇒ From Equation 1: 12x + y = 8 ⇒ y = 8 − 12x ... (i)

⇒ From Equation 2: (2x + 3y + 10) / 4 = 0 ⇒ 2x + 3y = −10 ... (ii)

⇒ Substitute (i) into (ii): 2x + 3(8 − 12x) = −10

⇒ 2x + 24 − 36x = −10

⇒ −34x = −10 − 24 ⇒ −34x = −34 ⇒ x = 1

⇒ Substitute x = 1 in (i): y = 8 − 12(1) = −4

⇒ Intersection Point P = (1, −4)

⇒ Test Option (2): 3x − y − 7 = 3(1) − (−4) − 7 = 3 + 4 − 7 = 0

∴ The correct answer is 3x − y − 7 = 0.

Additional Information

Nature of Roots for Linear Equations

For a system a₁x + b₁y = c₁ and a₂x + b₂y = c₂, if a₁/a₂ ≠ b₁/b₂, the lines intersect at exactly one point (Consistent system).

Parallel and Coincident Lines

If a₁/a₂ = b₁/b₂ ≠ c₁/c₂, lines are parallel (No solution). If a₁/a₂ = b₁/b₂ = c₁/c₂, lines are coincident (Infinite solutions).

Point-Line Relationship

A point (x₀, y₀) lies on a graph of an equation if substituting the coordinates into the equation satisfies the equality f(x₀, y₀) = 0.

Shortcut Trick

Part filled = 4 units in 6 minutes.

Total units = 13 units.

Remaining units to be filled = 13 − 4 = 9 units.

Required time = (6 ÷ 4) × 9 = 1.5 × 9 = 13.5 minutes.

13.5 minutes = 13 minutes 30 seconds.

∴ The correct answer is 13 minutes 30 seconds.

Alternate Method

Given: 4/13 part of the bucket is filled in 6 minutes.

Part of the bucket remaining = 1 − 4/13 = 9/13.

⇒ Time taken to fill 4/13 part = 6 minutes

⇒ Time taken to fill 1/13 part = 6 ÷ 4 = 1.5 minutes

⇒ Time taken to fill 9/13 part = 9 × 1.5 = 13.5 minutes

⇒ To convert 0.5 minutes to seconds: 0.5 × 60 = 30 seconds

⇒ Total time required = 13 minutes 30 seconds

∴ The correct answer is 13 minutes 30 seconds.

Additional Information

Unitary Method for Fractions

If a/b part of a work is completed in T time, then the whole work (1 unit) is completed in T × (b/a) time.

Fractional Remainder

The remaining part of any quantity is calculated by subtracting the given fraction from 1. Remaining = 1 − (Filled Fraction).

Time Conversion

To convert decimal minutes into seconds, multiply the decimal portion by 60. Example: 0.25 min = 0.25 × 60 = 15 seconds.

Shortcut Trick

Selling Price 1 (SP₁) at 10.5% discount = 100% − 10.5% = 89.5%

Selling Price 2 (SP₂) at 6.5% discount = 100% − 6.5% = 93.5%

Given: 89.5% → ₹716

1% → 716 ÷ 89.5 = 8

93.5% → 93.5 × 8 = ₹748

∴ The correct answer is ₹748.

Alternate Method

Given:

Initial Selling Price (SP₁) = ₹716

Initial Discount (D₁) = 10.5%

New Discount (D₂) = 6.5%

Formula Used:

Marked Price (MP) = SP ÷ (1 − Discount%)

New SP = MP × (1 − New Discount%)

Calculations:

⇒ SP₁ = MP × (100 − 10.5) / 100

⇒ 716 = MP × 89.5 / 100

⇒ MP = (716 × 100) / 89.5

⇒ MP = 8 × 100 = ₹800

⇒ New Selling Price (SP₂) = 800 × (100 − 6.5) / 100

⇒ SP₂ = 800 × (93.5 / 100)

⇒ SP₂ = 8 × 93.5

⇒ SP₂ = ₹748

∴ The correct answer is ₹748.

Additional Information

Selling Price and Discount Relation

Selling Price (SP) = Marked Price (MP) − Discount. When comparing two scenarios for the same article, SP₁ / (100 − D₁) = SP₂ / (100 − D₂).

Marked Price and Cost Price Relationship

The ratio of Marked Price to Cost Price can be found using the formula: MP / CP = (100 + Profit%) / (100 − Discount%).

Successive Discounts

If two successive discounts of a% and b% are given, the effective single discount is (a + b − ab/100)%.

Shortcut Trick

Check if it is a right-angled triangle: 12² + 35² = 144 + 1225 = 1369 = 37².

Since AB² + BC² = AC², ΔABC is right-angled at B.

In a right-angled triangle, the median to the hypotenuse is half of the hypotenuse: Median BM = AC ÷ 2.

The centroid G divides the median BM in the ratio 2 : 1, so BG = (2 ÷ 3) × BM.

Direct Formula: BG = Hypotenuse ÷ 3 = 37 ÷ 3 = 12.33 cm.

∴ The correct answer is 12.3 cm.

Alternate Method

Given: AB = 35 cm, BC = 12 cm, AC = 37 cm, G is the centroid.

Formula: Length of median to hypotenuse (BM) = AC ÷ 2; BG = (2 ÷ 3) × BM.

⇒ Verify right triangle: 35² + 12² = 1225 + 144 = 1369.

⇒ √1369 = 37, which is the third side AC. So, ∠ABC = 90°.

⇒ Median BM to hypotenuse AC = AC ÷ 2 = 37 ÷ 2 = 18.5 cm.

⇒ Centroid G divides median BM in ratio 2 : 1 starting from vertex B.

⇒ BG = (2 ÷ 3) × BM = (2 ÷ 3) × 18.5.

⇒ BG = 37 ÷ 3 = 12.333... cm.

⇒ Rounding to one decimal place, BG ≈ 12.3 cm.

∴ The correct answer is 12.3 cm.

Additional Information

Centroid Property

The centroid (G) of a triangle divides each median in the ratio 2 : 1 from the vertex to the base.

Median to Hypotenuse

In any right-angled triangle, the length of the median drawn to the hypotenuse is exactly half the length of the hypotenuse.

Circumradius of Right Triangle

The circumcenter of a right triangle is the midpoint of the hypotenuse, and the circumradius (R) = Hypotenuse ÷ 2.

Calculation:

The ratio of the total production of fertilisers by country X in 2017 and country Y in 2020  = (236 + 334) = 570

The production of fertilisers by country Z in 2019 = 380

Now,

The ratio of the total production of fertilisers by country X in 2017 and country Y in 2020 to the production of fertilisers by country Z in 2019 = (570 : 380)

⇒ 3 : 2

∴ The required ratio is 3 : 2

Given:

x = $\frac{5\frac{3}{4} - \frac{3}{7} \times 15\frac{3}{4} + 2\frac{2}{35} \div 1\frac{11}{25}}{\frac{3}{4} \div 5\frac{1}{4} + 5\frac{3}{5} \div 3\frac{4}{15}}$

Result = $\frac{7}{13}$ when y is added to x.

Formula used:

BODMAS Rule (Bracket, Of, Division, Multiplication, Addition, Subtraction)

Calculation:

⇒ Numerator = $\frac{23}{4} - \frac{3}{7} \times \frac{63}{4} + \frac{72}{35} \div \frac{36}{25}$

⇒ Numerator = $\frac{23}{4} - \frac{27}{4} + (\frac{72}{35} \times \frac{25}{36})$

⇒ Numerator = $-1 + \frac{10}{7} = \frac{3}{7}$

⇒ Denominator = $\frac{3}{4} \times \frac{4}{21} + \frac{28}{5} \times \frac{15}{49}$

⇒ Denominator = $\frac{1}{7} + \frac{12}{7} = \frac{13}{7}$

⇒ x = $\frac{3/7}{13/7} = \frac{3}{13}$

⇒ x + y = $\frac{7}{13}$

⇒ $\frac{3}{13}$ + y = $\frac{7}{13}$

⇒ y = $\frac{7}{13} - \frac{3}{13}$

⇒ y = $\frac{4}{13}$

∴ The correct answer is option 2

Calculation:

Total number of students appeared = 1200

Numbers of students appeared in the examination from institutes A = 72° 

⇒ (72/360 × 1200) 

⇒ 240

Numbers of students appeared in the examination from institutes B = 54° 

⇒ (54/360 × 1200) 

⇒ 180

Numbers of students appeared in the examination from institutes C = 90° 

⇒ (90/360 × 1200) 

⇒ 300

Numbers of students appeared in the examination from institutes D = 105°

⇒ (105/360 × 1200) 

⇒ 350

Numbers of students appeared in the examination from institutes E = 39°

⇒ (39/360 × 1200) 

⇒ 130

Now,

Total number of students passed = 900

Numbers of students passed in the examination from institutes A = 15%

⇒ (900 × 15/100)

⇒ 135

Numbers of students passed in the examination from institutes B = 18%

⇒ (900 × 18/100)

⇒ 162

Numbers of students passed in the examination from institutes C = 30%

⇒ (900 × 30/100)

⇒ 270

Numbers of students passed in the examination from institutes D = 28%

⇒ (900 × 28/100)

⇒ 252

Numbers of students passed in the examination from institutes E = 9%

⇒ (900 × 9/100)

⇒ 81

The percentage of students of institutes A passed to the students who appeared from that institute = (135/240 × 100)

⇒ 56.25%

The percentage of students of institutes B passed to the students who appeared from that institute = (162/180 × 100)

⇒ 90%

The percentage of students of institutes C passed to the students who appeared from that institute = (270/300 × 100)

⇒ 90%

The percentage of students of institutes D passed to the students who appeared from that institute = (252/350 × 100)

⇒ 72%

The percentage of students of institutes E passed to the students who appeared from that institute = (81/130 × 100)

⇒ 62.30%

Now,

We can see that the percentage of B and C is same which is also the highest = 90%

So, the second highest percentage of students who passed to the  students who appeared is from institute D = 72%

∴ The required answer is Institute D

Shortcut Trick

Direct formula for speed: Distance = [Product of speeds ÷ Difference in speeds] × Difference in time.

240 = [v × (v − 8) ÷ 8] × 1

v × (v − 8) = 1920 ⇒ By factorizing: 48 × 40 = 1920 ⇒ v = 48 km/h.

Time for 480 km = Distance ÷ Original Speed = 480 ÷ 48 = 10 hours.

∴ The correct answer is 10 hours.

Alternate Method

Given:

Distance₁ = 240 km, Distance₂ = 480 km, Time difference = 1 hour, Speed reduction = 8 km/h.

Formula Used:

Time = Distance ÷ Speed

Let the original speed be v km/h.

⇒ 240/(v − 8) − 240/v = 1

⇒ 240 [ (v − (v − 8)) ÷ (v(v − 8)) ] = 1

⇒ 240 × [ 8 ÷ (v² − 8v) ] = 1

⇒ 1920 = v² − 8v

⇒ v² − 8v − 1920 = 0

⇒ v² − 48v + 40v − 1920 = 0

⇒ (v − 48)(v + 40) = 0

⇒ v = 48 km/h (speed cannot be negative)

⇒ Required time for 480 km = 480 ÷ 48

⇒ Time = 10 hours.

∴ The correct answer is 10 hours.

Additional Information

Speed and Time Relationship

When distance is constant, speed is inversely proportional to time (s ∝ 1/t). If speed decreases by a factor, time increases by the inverse of that factor.

Average Speed for Constant Distance

If a journey is covered at speed 'a' and returned at speed 'b', the average speed is given by [2 × a × b ÷ (a + b)].

Speed-Time Variance Formula

For a fixed distance D, if speed changes from v₁ to v₂ resulting in time change Δt, then D = [v₁ × v₂ ÷ |v₁ − v₂|] × Δt.

Shortcut Trick

Numerator = 4(1/√3)² + (1/2)²(1/√2)² + (sec²48° − tan²48°) = 4/3 + 1/8 + 1 = 59/24.

Denominator = sin(37° + 53°) + tan 18° cot 18° = sin 90° + 1 = 1 + 1 = 2.

Value = (59/24) ÷ 2 = 59/48.

∴ The correct answer is 59/48.

Alternate Method

Given: [4 tan²30° + sin²30° cos²45° + sec²48° − cot²42°] / [cos 37° sin 53° + sin 37° cos 53° + tan 18° tan 72°]

Formulas Used:

1. sec²θ − tan²θ = 1
2. sin(A + B) = sin A cos B + cos A sin B
3. tan θ × tan(90° − θ) = 1

Calculations:

⇒ Numerator Part 1: 4 × (1/√3)² = 4/3

⇒ Numerator Part 2: (1/2)² × (1/√2)² = 1/4 × 1/2 = 1/8

⇒ Numerator Part 3: sec²48° − cot²42° = sec²48° − tan²(90° − 42°) = sec²48° − tan²48° = 1

⇒ Total Numerator = 4/3 + 1/8 + 1 = (32 + 3 + 24) / 24 = 59/24

⇒ Denominator Part 1: cos 37° sin 53° + sin 37° cos 53° = sin(37° + 53°) = sin 90° = 1

⇒ Denominator Part 2: tan 18° tan 72° = tan 18° cot(90° − 72°) = tan 18° cot 18° = 1

⇒ Total Denominator = 1 + 1 = 2

⇒ Final Value = (59/24) / 2 = 59/48

∴ The correct answer is 59/48.

Additional Information

Complementary Angle Relations

sin(90° − θ) = cosθ, tan(90° − θ) = cotθ, and sec(90° − θ) = cosecθ.

Square Identities

sin²θ + cos²θ = 1, sec²θ − tan²θ = 1, and cosec²θ − cot²θ = 1.

Compound Angle Formulas

sin(A ± B) = sin A cos B ± cos A sin B; cos(A ± B) = cos A cos B ∓ sin A sin B.

Given:

x + y = 65

$\sqrt{xy}$ = 26

Formula used:

$\frac{1}{x} + \frac{1}{y} = \frac{x + y}{xy}$

Calculations:

⇒ $\sqrt{xy}$ = 26

⇒ xy = 26²

⇒ xy = 676

⇒ $\frac{1}{x} + \frac{1}{y} = \frac{65}{676}$

⇒ $\frac{1}{x} + \frac{1}{y} = \frac{5 \times 13}{52 \times 13}$

⇒ $\frac{1}{x} + \frac{1}{y} = \frac{5}{52}$

∴ The correct answer is option (1).

Shortcut Trick

Radius (r) = 21/2 = 10.5 cm

Total Surface Area = 2πr(r + h) ⇒ 4092 = 2 × 22/7 × 10.5 × (10.5 + h)

4092 = 66 × (10.5 + h) ⇒ 62 = 10.5 + h ⇒ h = 51.5 cm

50% Volume = 1/2 × πr²h = 1/2 × 22/7 × 10.5 × 10.5 × 51.5

50% Volume = 11 × 1.5 × 10.5 × 51.5 = 173.25 × 51.5 = 8922.375 ≈ 8922 cm³

∴ The correct answer is 8922 cm³.

Alternate Method

Given:

Total Surface Area (TSA) = 4092 cm²

Diameter (d) = 21 cm ⇒ Radius (r) = 10.5 cm

Formula Used:

TSA of Cylinder = 2πr(r + h)

Volume of Cylinder = πr²h

Calculations:

⇒ 4092 = 2 × (22/7) × 10.5 × (10.5 + h)

⇒ 4092 = 44 × 1.5 × (10.5 + h)

⇒ 4092 = 66 × (10.5 + h)

⇒ 10.5 + h = 4092 ÷ 66

⇒ 10.5 + h = 62

⇒ h = 62 − 10.5 = 51.5 cm

⇒ Volume of Cylinder = πr²h = (22/7) × 10.5 × 10.5 × 51.5

⇒ Volume = 22 × 1.5 × 10.5 × 51.5

⇒ Volume = 33 × 10.5 × 51.5 = 17844.75 cm³

⇒ 50% Volume = 17844.75 ÷ 2 = 8922.375 cm³

⇒ Nearest integer = 8922 cm³

∴ The correct answer is 8922 cm³.

Additional Information

Curved Surface Area (CSA)

The lateral or curved surface area of a cylinder is given by the formula 2πrh.

Hollow Cylinder

The volume of a hollow cylinder with external radius R and internal radius r is π(R² − r²)h.

Volume of Cone relationship

A cone with the same base radius and height as a cylinder has 1/3 the volume of that cylinder: V_cone = (1/3)πr²h.

Shortcut Trick

Compare the ratio change: 8 years ago (5 : 4) → Present (6 : 5).

Difference for A = 6 − 5 = 1 unit; Difference for B = 5 − 4 = 1 unit.

Since the age gap (8 years) is same for both, 1 unit = 8 years.

Present total ratio sum = 6 + 5 = 11 units.

Present sum = 11 × 8 = 88 years.

Sum after 7 years = Present Sum + (2 × 7) = 88 + 14 = 102 years.

∴ The correct answer is 102 years.

Alternate Method

Given: Ratio 8 years ago = 5 : 4; Present ratio = 6 : 5; Future time = 7 years.

Formula Used: Present Age = (Ratio part × Value of 1 unit); Sum of ages = A + B.

⇒ Let common multiplier for present ages be x. Present age of A = 6x, B = 5x.

⇒ Ages 8 years ago: A = 6x − 8, B = 5x − 8.

⇒ (6x − 8) / (5x − 8) = 5 / 4

⇒ 4 × (6x − 8) = 5 × (5x − 8)

⇒ 24x − 32 = 25x − 40

⇒ x = 40 − 32 = 8

⇒ Present Age of A = 6 × 8 = 48 years; Present Age of B = 5 × 8 = 40 years.

⇒ Age of A after 7 years = 48 + 7 = 55 years.

⇒ Age of B after 7 years = 40 + 7 = 47 years.

⇒ Sum of ages = 55 + 47 = 102 years.

∴ The correct answer is 102 years.

Additional Information

Constant Age Difference

The difference between the ages of two people always remains constant throughout their lives. If A is 5 years older than B today, A will always be 5 years older than B.

Ratio Balancing Trick

If the unit differences in ratios are not equal (e.g., 6−5 ≠ 5−4), multiply the first ratio by the difference of the second ratio parts, and vice versa, to balance the units.

Future and Past Sums

To find the sum of ages of N people after T years, add (N × T) to the present sum. Conversely, subtract (N × T) to find the sum T years ago.

Shortcut Trick

Let ∠QPS = x, then ∠PSR = 2x.

∠SPR = ∠QPR − ∠QPS = 78° − x.

In ΔPSR, sum of angles: 2x + (78° − x) + 44° = 180°.

x + 122° = 180° ⇒ x = 58°.

∠PSQ = 180° − ∠PSR = 180° − 116° = 64°.

∴ The correct answer is 64°.

Alternate Method

Given: ∠QPR = 78°, ∠PRS = 44°, ∠QPS = 1/2 ∠PSR.

Formula Used: Angle Sum Property of a triangle (Δ = 180°) and Linear Pair property.

⇒ Let ∠QPS = x. According to the question, ∠PSR = 2x.

⇒ Since ∠QPR = 78°, then ∠SPR = ∠QPR − ∠QPS = 78° − x.

⇒ In ΔPSR, the sum of internal angles is 180°.

⇒ ∠PSR + ∠SPR + ∠PRS = 180°.

⇒ 2x + (78° − x) + 44° = 180°.

⇒ x + 122° = 180°.

⇒ x = 180° − 122° = 58°.

⇒ The value of ∠PSR = 2 × 58° = 116°.

⇒ Since QSR is a straight line, ∠PSQ + ∠PSR = 180° (Linear Pair).

⇒ ∠PSQ = 180° − 116° = 64°.

∴ The correct answer is 64°.

Additional Information

Angle Sum Property

The sum of all interior angles of any triangle is always equal to 180°.

Exterior Angle Theorem

The exterior angle of a triangle is equal to the sum of the two opposite interior angles (∠PSR = ∠PQS + ∠QPS).

Linear Pair of Angles

When two angles are adjacent and their non-common sides form a straight line, their sum is 180°.

Shortcut Trick

Identify Pythagorean triplets: For angle A (5, 12, 13) and for angle B (7, 24, 25).

sin A = 5/13 → sec A = 13/12 and tan A = 5/12.

cot B = 24/7 → cos B = 24/25 and cosec B = 25/7.

Expression = (sec A × cos B) × (cosec B × tan A) = (13/12 × 24/25) × (25/7 × 5/12).

Simplifying: (13 × 2 / 25) × (25 × 5 / (7 × 12)) = (26/25) × (125/84) = 130 / 84 = 65/42.

∴ The correct answer is 65/42.

Alternate Method

Given:

sin A = 5/13

7 cot B = 24 ⇒ cot B = 24/7

Formula Used:

sin θ = P/H, cos θ = B/H, tan θ = P/B, cot θ = B/P, sec θ = H/B, cosec θ = H/P

H² = P² + B²

Calculations:

⇒ For angle A: sin A = 5/13 (P=5, H=13) ⇒ B = √(13² − 5²) = √(169 − 25) = 12.

⇒ sec A = H/B = 13/12 and tan A = P/B = 5/12.

⇒ For angle B: cot B = 24/7 (B=24, P=7) ⇒ H = √(24² + 7²) = √(576 + 49) = 25.

⇒ cos B = B/H = 24/25 and cosec B = H/P = 25/7.

⇒ Required value = (sec A × cos B) × (cosec B × tan A)

⇒ Required value = (13/12 × 24/25) × (25/7 × 5/12)

⇒ Required value = (13/12 × 24/7 × 25/25 × 5/12)

⇒ Required value = (13 × 2 × 1 × 5) / (7 × 12) = 130 / 84

⇒ Required value = 65/42.

∴ The correct answer is 65/42.

Additional Information

Pythagorean Triplets

Common triplets used in exams include (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), and (9, 40, 41).

Reciprocal Relations

sec θ = 1/cos θ, cosec θ = 1/sin θ, and tan θ = sin θ/cos θ.

Fundamental Identity

sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ.

Shortcut Trick 33 1/3% = 1/3 of the value. Selling at 1/3 of original SP results in 1/3 loss ⇒ 1/3 SP = (1 − 1/3) CP = 2/3 CP. ∴ Original SP = 2 CP. New SP = 80% of Original SP = 0.8 × 2 CP = 1.6 CP. Profit % = (1.6 − 1) × 100 = 60%. ∴ The correct answer is 60%. Alternate Method Given: Original Selling Price = SP New Selling Price1 = 33 1/3% of SP → Loss = 33 1/3% New Selling Price2 = 80% of SP

Formula Used: CP = SP × 100 / (100 − Loss%) Profit % = [(SP − CP) / CP] × 100 Calculations: ⇒ Let the original selling price (SP) be ₹300. ⇒ New SP1 = 33 1/3% of 300 = 1/3 × 300 = ₹100. ⇒ Loss% = 33 1/3% = 100/3 %. ⇒ CP = 100 × 100 / (100 − 100/3) = 100 × 100 / (200/3) = 100 × 3/2 = ₹150. ⇒ New SP2 = 80% of original SP = 0.8 × 300 = ₹240. ⇒ Profit = New SP2 − CP = 240 − 150 = ₹90. ⇒ Profit % = (90 / 150) × 100 = (3/5) × 100 = 60%. ∴ The correct answer is 60%. Additional Information Relationship between SP and CP Profit occurs when SP > CP, Profit % = [(SP − CP)/CP] × 100. Loss occurs when SP < CP, Loss % = [(CP − SP)/CP] × 100. Fraction to Percentage Conversion 33 1/3% = 1/3 | 66 2/3% = 2/3 | 20% = 1/5 | 80% = 4/5. Using fractions simplifies calculation steps in competitive exams. Successive Percentage Changes If a value changes by x% and then by y%, the net change is [x + y + (xy/100)]%.

Shortcut Trick

Polygon A: Number of sides n_A = (Sum of angles ÷ 180°) + 2 = (1260 ÷ 180) + 2 = 7 + 2 = 9.

Polygon B: Exterior angle = 180° − Interior angle = 180° − 128 4/7° = 51 3/7° = 360/7°.

Number of sides n_B = 360° ÷ Exterior angle = 360 ÷ (360/7) = 7.

Total number of sides = n_A + n_B = 9 + 7 = 16.

∴ The correct answer is 16.

Alternate Method

Given:

Sum of interior angles of A = 1260°

Each interior angle of B = 128 4/7° = 900/7°

Formula Used:

1. Sum of interior angles = (n − 2) × 180°
2. Each interior angle of a regular polygon = 180° − (360° ÷ n)

For Polygon A:

⇒ (n_A − 2) × 180 = 1260

⇒ n_A − 2 = 1260 ÷ 180

⇒ n_A − 2 = 7

⇒ n_A = 9

For Polygon B:

⇒ Each Interior Angle = 128 4/7° = 900/7°

⇒ 180 − (360 ÷ n_B) = 900/7

⇒ 360 ÷ n_B = 180 − 900/7

⇒ 360 ÷ n_B = (1260 − 900) ÷ 7

⇒ 360 ÷ n_B = 360 ÷ 7

⇒ n_B = 7

Sum of number of sides = n_A + n_B = 9 + 7 = 16.

∴ The correct answer is 16.

Additional Information

Exterior Angle Property

For any convex polygon, the sum of all exterior angles is always 360°. For a regular polygon, each exterior angle = 360° ÷ n.

Number of Diagonals

The number of diagonals in a polygon with n sides is given by the formula [n × (n − 3)] ÷ 2.

Angle Relationship

At each vertex of a polygon, the sum of the interior angle and the exterior angle is always 180° (Linear Pair).

Shortcut Trick

Check for Pythagorean triplet: 16² + 30² = 256 + 900 = 1156 = 34². Base is a right triangle.

Perimeter of base = 16 + 30 + 34 = 80 cm.

Lateral Surface Area = Perimeter × Height = 80 × 32 = 2560 cm².

Base Area = 1/2 × 16 × 30 = 240 cm².

Volume = Base Area × Height = 240 × 32 = 7680 cm³.

∴ The correct answer is 2560 cm² and 7680 cm³.

Alternate Method

Given:

Sides of triangular base (a, b, c) = 16 cm, 30 cm, 34 cm.

Height of prism (h) = 32 cm.

Formula Used:

1. Lateral Surface Area (LSA) = Perimeter of base × height
2. Volume = Area of base × height
3. Area of triangle = 1/2 × base × height (if right-angled)

Calculations:

⇒ Identification of Base: 16² + 30² = 256 + 900 = 1156 = 34². The base is a right-angled triangle.

⇒ Base Area = 1/2 × 16 × 30 = 8 × 30 = 240 cm².

⇒ Base Perimeter = 16 + 30 + 34 = 80 cm.

⇒ Lateral Surface Area = Perimeter × Height = 80 × 32 = 2560 cm².

⇒ Volume = Base Area × Height = 240 × 32 = 7680 cm³.

∴ The correct answer is 2560 cm² and 7680 cm³.

Additional Information

Total Surface Area of Prism

Total Surface Area (TSA) = Lateral Surface Area + 2 × Area of Base. In this case, TSA = 2560 + 2(240) = 3040 cm².

Right Angled Triangles

A triangle is right-angled if the sum of the squares of the two shorter sides equals the square of the longest side (Pythagoras Theorem: a² + b² = c²).

General Prism Formula

A right prism is a 3D solid with two identical parallel bases and vertical rectangular sides. Volume is always Base Area × perpendicular height.

Shortcut Trick

 

मध्य-बिंदु प्रमेय का उपयोग करने पर,

ΔBAP में,

MN = $AP\over2$ = $\frac{5}{{2}}$ = 2.5 सेमी

Alternate Method

Given:

In ΔABC, M is the midpoint of AB.

CN is the bisector of ∠C and CN ⟂ NB.

BC = 10 cm

AC = 15 cm

Formula used:

Midpoint Theorem: The line segment joining the midpoints of two sides of a triangle is parallel to the third side and equal to half of it.

Calculations:

Extend BN to P such that it meets AC at P.

Join MN.

In ΔNPC and ΔNBC,

⇒ ∠NPC = ∠NBC = 90°

⇒ BC = PC = 10 cm

⇒ CN is common

⇒ ΔNPC ≅ ΔNBC

⇒ NP = NB

⇒ N is the midpoint of BP

Now,

⇒ AP = AC - PC

⇒ AP = 15 - 10 = 5 cm

In ΔABP,

⇒ M is the midpoint of AB

⇒ N is the midpoint of BP

By Midpoint Theorem,

⇒ MN = AP/2

⇒ MN = 5/2

∴ MN = 2.5 cm.

Additional Information

Midpoint Theorem

The line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is exactly half the length of the third side.

Isosceles Triangle Properties

In an isosceles triangle, the angle bisector of the vertex angle, the median to the base, and the altitude to the base all coincide as the same line segment.

Angle Bisector Theorem

An angle bisector of a triangle divides the opposite side into two segments that are proportional to the other two sides of the triangle.

Shortcut Trick

Let PR = x cm.

Given: PQ = PR + 2 = (x + 2) cm.

And PQ = QR + 5 ⇒ QR = PQ − 5 = (x + 2) − 5 = (x − 3) cm.

Perimeter = PQ + QR + PR = 32 cm.

⇒ (x + 2) + (x − 3) + x = 32 ⇒ 3x − 1 = 32 ⇒ 3x = 33 ⇒ x = 11.

∴ The correct answer is 11 cm.

Alternate Method

Given:

PQ = (QR + 5) cm

PQ = (PR + 2) cm

Formula used:

Perimeter of ΔPQR = PQ + QR + PR

Calculations:

Let QR = x cm.

⇒ PQ = (QR + 5) ⇒ PQ = (x + 5)

Again,

⇒ PQ = (PR + 2)

⇒ (x + 5) = (PR + 2)

⇒ PR = (x + 5 - 2) ⇒ PR = (x + 3)

According to the question,

Perimeter of ΔPQR = 32 cm

⇒ PQ + QR + PR = 32

⇒ (x + 5) + x + (x + 3) = 32

⇒ 3x + 8 = 32

⇒ 3x = 24

⇒ x = 8

Therefore,

⇒ PR = x + 3

⇒ PR = 8 + 3  = 11 cm

∴ The required value of PR is 11 cm.

Additional Information

Tangents from an External Point

Lengths of tangents drawn from an external point to a circle are equal. In this triangle, QT = QS, RS = RU, and PT = PU.

Incircle Property

The distance from a vertex to the points of tangency can be calculated as s − opposite side, where s is the semi-perimeter. For example, PT = PU = s − a.

Incenter

The center of the inscribed circle (I) is the point where the internal angle bisectors of the triangle intersect.

∴ The correct answer is 11 cm.

Shortcut Trick

Combine the percentages since they apply to the same base: 40% + 30% = 70%.

Given that 70% of the number is equal to 70.

Therefore, 1% of the number = 70 ÷ 70 = 1.

The full number (100%) = 1 × 100 = 100.

∴ The correct answer is 100.

Alternate Method

Given:

40% of a number + 30% of the same number = 70

Formula Used:

Value = (Percentage ÷ 100) × Total Number

Calculations:

⇒ Let the required number be x.

⇒ (40/100) × x + (30/100) × x = 70

⇒ (40x + 30x) ÷ 100 = 70

⇒ 70x ÷ 100 = 70

⇒ 70x = 70 × 100

⇒ x = (70 × 100) ÷ 70

⇒ x = 100

∴ The correct answer is 100.

Additional Information

Percentage Addition Rule

Percentages can be added or subtracted directly if they are calculated on the same base value: a% of x + b% of x = (a + b)% of x.

Fraction Equivalents

Converting percentages to fractions simplifies calculations. For example, 40% = 2/5 and 30% = 3/10.

Finding the Whole

If P% of a number is equal to V, then the original number is (V ÷ P) × 100.

Shortcut Trick

Rationalize each term using the rule: 1/(√a − √b) = √a + √b when a − b = 1.

Check the squares: 4² − 15 = 1, 15 − 14 = 1, ..., 10 − 3² = 1, 3² − 8 = 1.

The series becomes: (4 + √15) − (√15 + √14) + (√14 + √13) − ... − (3 + √8).

In this alternating telescoping series, all intermediate terms cancel out.

Value = First part of first term − Last part of last term = 4 − √8 = 4 − 2√2.

∴ The correct answer is 4 − 2√2.

Alternate Method

Given: 1/(4 − √15) − 1/(√15 − √14) + 1/(√14 − √13) − 1/(√13 − √12) + 1/(√12 − √11) − 1/(√11 − √10) + 1/(√10 − 3) − 1/(3 − √8)

Formula: Rationalize the denominator by multiplying by the conjugate: 1/(√a − √b) × (√a + √b)/(√a + √b) = (√a + √b)/(a − b).

⇒ 1/(4 − √15) = (4 + √15)/(16 − 15) = 4 + √15

⇒ 1/(√15 − √14) = (√15 + √14)/(15 − 14) = √15 + √14

⇒ 1/(√10 − 3) = (√10 + 3)/(10 − 9) = √10 + 3

⇒ 1/(3 − √8) = (3 + √8)/(9 − 8) = 3 + √8

⇒ Substituting all rationalized forms into the expression:

⇒ (4 + √15) − (√15 + √14) + (√14 + √13) − (√13 + √12) + (√12 + √11) − (√11 + √10) + (√10 + 3) − (3 + √8)

⇒ 4 + √15 − √15 − √14 + √14 + √13 − √13 − √12 + √12 + √11 − √11 − √10 + √10 + 3 − 3 − √8

⇒ 4 − √8

⇒ 4 − 2√2

∴ The correct answer is 4 − 2√2.

Additional Information

Rationalization of Surds

To rationalize a denominator of the form (√a − √b), multiply both the numerator and the denominator by its conjugate (√a + √b).

Telescoping Series

A telescoping series is a summation where consecutive terms cancel out, leaving only the initial and final parts of the series.

Conjugate Rule

The product of a surd and its conjugate results in a rational number: (x + √y)(x − √y) = x² − y.

Shortcut Trick

Put θ = 45° in the given expression.

Numerator = (1 + √2 × √2)² × (√2 − 1)² × (1 + 1/√2) = 9 × (3 − 2√2) × ((√2 + 1)/√2) = 9(1 − 1/√2).

Denominator = (sin 45° + sec 45°)² + (cos 45° + cosec 45°)² = (1/√2 + √2)² + (1/√2 + √2)² = 4.5 + 4.5 = 9.

Expression = 9(1 − 1/√2) ÷ 9 = 1 − 1/√2, which is equal to 1 − sin 45°.

∴ The correct answer is 1 − sin θ.

Alternate Method

Given: [(1 + sec θ cosec θ)² (sec θ − tan θ)² (1 + sin θ)] ÷ [(sin θ + sec θ)² + (cos θ + cosec θ)²]

Formula used: sec θ = 1/cos θ, cosec θ = 1/sin θ, sin² θ + cos² θ = 1

⇒ Numerator part 2: (sec θ − tan θ)²(1 + sin θ) = ((1 − sin θ)/cos θ)²(1 + sin θ)

⇒ (1 − sin θ)²(1 + sin θ)/cos² θ = (1 − sin θ)(1 − sin² θ)/cos² θ = 1 − sin θ

⇒ Denominator: (sin θ + 1/cos θ)² + (cos θ + 1/sin θ)² = ((sin θ cos θ + 1)/cos θ)² + ((cos θ sin θ + 1)/sin θ)²

⇒ (sin θ cos θ + 1)² [1/cos² θ + 1/sin² θ] = (sin θ cos θ + 1)² [1/(sin² θ cos² θ)]

⇒ Denominator = ( (sin θ cos θ + 1) / (sin θ cos θ) )² = (1 + sec θ cosec θ)²

⇒ Expression = [(1 + sec θ cosec θ)² × (1 − sin θ)] ÷ (1 + sec θ cosec θ)²

⇒ Expression = 1 − sin θ

∴ The correct answer is 1 − sin θ.

Additional Information

Pythagorean Identities

Fundamental relations: sin²θ + cos²θ = 1, sec²θ − tan²θ = 1, and cosec²θ − cot²θ = 1.

Reciprocal Relations

sec θ = 1/cos θ, cosec θ = 1/sin θ, and tan θ = sin θ/cos θ.

Value Substitution Method

To simplify complex trigonometric identities, substitute a standard angle like 45° or 30° to quickly verify options.

Shortcut Trick

Volume of 8 cubes = Volume of Cuboid

8 × a³ = 12 × 54 × 72

a³ = 12 × 54 × 9 ⇒ a³ = 5832 ⇒ a = 18 cm

Lateral Surface Area of 2 cubes = 2 × (4a²) = 8 × 18²

Total LSA = 8 × 324 = 2592 cm²

∴ The correct answer is 2592 cm².

Alternate Method

Given: Dimensions of cuboid = 12 cm × 54 cm × 72 cm, Number of cubes = 8.

Formula: Volume of cuboid = L × B × H; Volume of cube = a³; LSA of cube = 4a².

⇒ Volume of cuboid = 12 × 54 × 72 = 46656 cm³.

⇒ Volume of 8 cubes = 46656 cm³.

⇒ Volume of 1 cube = 46656 ÷ 8 = 5832 cm³.

⇒ a³ = 5832 ⇒ a = √³5832 = 18 cm.

⇒ Lateral Surface Area (LSA) of 1 cube = 4a².

⇒ LSA of 1 cube = 4 × (18)² = 4 × 324 = 1296 cm².

⇒ Sum of LSAs of 2 cubes = 2 × 1296 = 2592 cm².

∴ The correct answer is 2592 cm².

Additional Information

Total Surface Area (TSA) of Cube and Cuboid

TSA of a cube = 6a² and TSA of a cuboid = 2(lb + bh + hl).

Diagonal of Cube and Cuboid

Length of the longest rod (diagonal) in a cube = a√3 and in a cuboid = √(l² + b² + h²).

Lateral Surface Area (LSA)

LSA refers to the area of the vertical faces only. For a cube, it is 4a²; for a cuboid, it is 2h(l + b).

Calculation:

The average production of fertilisers by country Z in 2017, 2018 and 2020 = $(\frac{330 + 360 + 420}{{3}})$

⇒ $\frac{1110}{{3}}$

⇒ 370

The average production of fertilisers by country X in 2018 and 2020 = $(\frac{240 + 260}{{2}})$

⇒ $\frac{500}{{2}}$

⇒ 250

Required percentage = $\frac{(370 - 250)}{{250}} × 100$

⇒ $\frac{120}{{250}} × 100$

⇒ (24 × 2)%

⇒ 48%

∴ The required average production is 48%