Quantitative Aptitude PYQs

Shortcut Trick

If each number in a set is decreased by a constant value k, the average of the set also decreases by the same value k.

Given Old Average = 45 and Decrease per number = 3.

New Average = 45 − 3 = 42.

∴ The correct answer is 42.

Alternate Method

Given:

Total numbers = 8

Initial Average = 45

Formula Used:

Average = Sum of observations ÷ Number of observations

Calculations:

⇒ Initial Sum = Average × Count = 45 × 8 = 360

⇒ Total Reduction = 8 × 3 = 24

⇒ New Sum = 360 − 24 = 336

⇒ New Average = 336 ÷ 8

⇒ New Average = 42

∴ The correct answer is 42.

Additional Information

Addition Rule

If k is added to every observation, the new average is Old Average + k.

Multiplication Rule

If every observation is multiplied by a constant n, the new average becomes Old Average × n.

Division Rule

If every observation is divided by a non-zero constant m, the new average is Old Average ÷ m.

Shortcut Trick

Percentage difference = 45% − 20% = 25%

The question states that this 25% difference is equal to 30.

Using the Unitary method:

25% → 30

100% → 30 × (100 ÷ 25) = 30 × 4 = 120

∴ The correct answer is 120.

Alternate Method

Given: 45% of a number exceeds 20% of the same number by 30.

Formula Used: Percentage of a number = (Percent ÷ 100) × Number

⇒ Let the number be x. ⇒ (45% of x) − (20% of x) = 30 ⇒ (45/100) × x − (20/100) × x = 30 ⇒ (45 − 20)/100 × x = 30 ⇒ (25/100) × x = 30 ⇒ (1/4) × x = 30 ⇒ x = 30 × 4 ⇒ x = 120

∴ The correct answer is 120.

Additional Information

Percentage to Fraction Conversion

To solve quickly, remember common fractions: 25% = 1/4, 20% = 1/5, 50% = 1/2, and 75% = 3/4.

Base Value Property

When comparing two percentages of the same number, you can directly subtract or add the percentage values before applying them to the base.

Percentage Change Formula

Percentage change = (Difference ÷ Original Value) × 100. This is used to find how much one value is greater or smaller than another in percentage terms.

Shortcut Trick

Let the smaller number be x. Then the larger number is 54 − x.

Given condition: 3 × x − 2 × (54 − x) = 6

⇒ 3x − 108 + 2x = 6 ⇒ 5x = 114 ⇒ x = 22.8.

Larger number = 54 − 22.8 = 31.2.

Difference = 31.2 − 22.8 = 8.4.

∴ The correct answer is 8.4.

Alternate Method

Given: Sum of two numbers = 54. Let smaller number = x and larger number = y.

Formulas: (i) x + y = 54 | (ii) 3x − 2y = 6

⇒ From (i), y = 54 − x.

⇒ Substitute y in (ii): 3x − 2(54 − x) = 6

⇒ 3x − 108 + 2x = 6

⇒ 5x = 114 ⇒ x = 22.8

⇒ y = 54 − 22.8 = 31.2

⇒ Difference = y − x = 31.2 − 22.8 = 8.4

∴ The correct answer is 8.4.

Additional Information

Linear Equations in Two Variables

Equations of the form ax + by = c can be solved using the Elimination Method or Substitution Method to find unique values for the variables.

Sum and Difference Relation

If the Sum (S) and Difference (D) of two numbers are known, the Larger number = (S + D) ÷ 2 and the Smaller number = (S − D) ÷ 2.

Algebraic Modeling

Key phrases like exceeds or more than indicate subtraction (Larger − Smaller = Difference) when setting up equations.

Shortcut Trick

In a right-angled triangle with angles 30°−60°−90°, the ratio of sides is 1 : √3 : 2.

Here, the side opposite to 60° (Height of wall) corresponds to √3 parts.

The hypotenuse (Ladder length) corresponds to 2 parts = 8 m.

⇒ 1 part = 8 ÷ 2 = 4 m.

⇒ Height (opposite to 60°) = 4 × √3 = 4√3 m.

∴ The correct answer is 4√3 m.

Alternate Method

Given:

Length of the ladder (Hypotenuse, AC) = 8 m

Angle of elevation with the ground (θ) = 60°

Let the height of the wall reached by the ladder be AB.

Formula Used:

In a right-angled triangle, sin θ = Perpendicular (Height) ÷ Hypotenuse

Calculations:

⇒ sin 60° = AB ÷ AC

⇒ √3 ÷ 2 = AB ÷ 8

⇒ AB = 8 × (√3 ÷ 2)

⇒ AB = 4 × √3

⇒ AB = 4√3 m

∴ The correct answer is 4√3 m.

Additional Information

Trigonometric Ratios

For an angle θ, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, and tan θ = opposite/adjacent.

Standard Trigonometric Values

Key values include sin 30° = 1/2, sin 45° = 1/√2, and sin 60° = √3/2.

Pythagoras Theorem

In any right-angled triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides: a² + b² = c².

Shortcut Trick

Total units in the ratio = 3 + 3 + 4 = 10 units.

Sum of angles in a triangle = 180°.

10 units → 180°

1 unit → 18°

Measure of the largest angle (4 units) = 4 × 18° = 72°.

∴ The correct answer is 72degree.

Alternate Method

Given:

Ratio of three angles = 3 : 3 : 4

Formula Used:

The Angle-Sum Property of a triangle states that the sum of all interior angles is exactly 180°.

Calculations:

⇒ Let the three angles of the triangle be 3x, 3x, and 4x.

⇒ 3x + 3x + 4x = 180°

⇒ 10x = 180°

⇒ x = 180° / 10 = 18°

⇒ Measure of the first angle = 3 × 18° = 54°

⇒ Measure of the second angle = 3 × 18° = 54°

⇒ Measure of the third (largest) angle = 4 × 18° = 72°

∴ The correct answer is 72degree.

Additional Information

Angle-Sum Property

For any polygon with n sides, the sum of interior angles is (n − 2) × 180°.

Isosceles Triangle

When a triangle has two equal angles, the sides opposite those angles are also equal.

Classification by Angles

A triangle where all angles are less than 90° is an acute triangle; if one is exactly 90°, it is a right triangle.

Shortcut Trick

Area of Rectangle = Area of Circle ⇒ 98 × 77 = (22/7) × r²

r² = (98 × 77 × 7) ÷ 22 ⇒ r² = 49 × 11 × 7 × 7 ÷ 11 × 2 ÷ 2 (Simplifying: 98 = 49 × 2 and 77 = 11 × 7)

r² = 49 × 49 ⇒ r = 49 m

Perimeter of Circle = 2 × (22/7) × 49 = 44 × 7 = 308 m

∴ The correct answer is 308 m.

Alternate Method

Given: Length of rectangular field = 98 m, Breadth = 77 m. Area of circle = Area of rectangle.

⇒ Area of Rectangular Field = Length × Breadth

⇒ Area = 98 × 77 = 7546 m²

⇒ Area of Circular Field = πr²

⇒ Since Area of Circle = Area of Rectangle:

⇒ (22/7) × r² = 7546

⇒ r² = (7546 × 7) ÷ 22

⇒ r² = 343 × 7 = 2401

⇒ Radius (r) = √2401 = 49 m

⇒ Perimeter (Circumference) of Circular Field = 2 × π × r

⇒ Perimeter = 2 × (22/7) × 49 = 2 × 22 × 7 = 308 m

∴ The correct answer is 308 m.

Additional Information

Area and Perimeter of a Rectangle

The Area is length × breadth (l × b) and the Perimeter is 2(l + b).

Circle Mensuration Formulas

For a circle of radius r, the Area = πr² and the Circumference (Perimeter) = 2πr.

Relation Between Radius and Area

If the radius is multiplied by a factor k, the Area increases by k² and the Perimeter increases by k.

Shortcut Trick

Total Work = LCM(14, 21, 42) = 42 units.

Efficiencies: P = 3, Q = 2, R = 1. Combined (P+Q+R) = 6.

Work done in 4 days = 6 × 4 = 24 units.

Remaining work = 42 − 24 = 18 units.

Time for Q and R to finish = 18 ÷ (2 + 1) = 6 days.

Total time = 4 + 6 = 10 days.

∴ The correct answer is 10 days.

Alternate Method

Given: Time taken by P = 14 days, Q = 21 days, and R = 42 days.

⇒ All three started together, so combined efficiency = 3 + 2 + 1 = 6 units/day.

⇒ Work completed by P, Q, and R in 4 days = 6 × 4 = 24 units.

⇒ Remaining work to be done = 42 − 24 = 18 units.

⇒ Since P left, Q and R complete the remaining work.

⇒ Combined efficiency of Q and R = 2 + 1 = 3 units/day.

⇒ Time taken by Q and R = 18 ÷ 3 = 6 days.

⇒ Total days taken for the wall to be painted = 4 + 6 = 10 days.

∴ The correct answer is 10 days.

Additional Information

Efficiency and Time

Efficiency is inversely proportional to the time taken to complete a task. If the ratio of time is a:b, then the ratio of efficiency is b:a.

Work and Wages

Wages are generally distributed in the ratio of the total work done by each individual. If they work for the same time, it is the ratio of their efficiencies.

Pipes and Cisterns

This concept follows the same logic as Time & Work, where an inlet pipe does positive work and an outlet pipe (leakage) does negative work.

Shortcut Trick

Assume the mean 13 as the balance point for all deviations.

Net deviation from 13: (6.5 − 13) + (10.8 − 13) + (20 − 13) + (a − 13) = 0

⇒ −6.5 − 2.2 + 7.0 + (a − 13) = 0

⇒ −1.7 + (a − 13) = 0

⇒ a − 13 = 1.7 ⇒ a = 13 + 1.7 = 14.7

∴ The correct answer is 14.7.

Alternate Method

Given: Numbers = 6.5, 10.8, 20, a | Total Count (n) = 4 | Average = 13

Formula Used: Average = (Sum of all observations) ÷ (Total number of observations)

⇒ Sum of observations = Average × Count

⇒ 6.5 + 10.8 + 20 + a = 13 × 4

⇒ 37.3 + a = 52

⇒ a = 52 − 37.3

⇒ a = 14.7

∴ The correct answer is 14.7.

Additional Information

Arithmetic Mean Properties

If each observation is increased or decreased by a constant k, the new average also increases or decreases by k.

Sum of Deviations

The sum of deviations of all observations from their mean is always zero (Σ(x_i − x̄) = 0).

Weighted Average

When two groups with averages A₁ and A₂ and sizes n₁ and n₂ are combined, the combined average = (n₁A₁ + n₂A₂) ÷ (n₁ + n₂).

Shortcut Trick

Length of 4 identical cubes placed in a row = 256 cm.

Side of a single cube (a) = 256 ÷ 4 = 64 cm.

Volume of the cube = a³ = 64 × 64 × 64.

Volume = 262144 cm³.

∴ The correct answer is 262144 cm³.

Alternate Method

Given:

Total length of the cuboid formed by 4 cubes = 256 cm.

Formula Used:

Volume of a cube = (side)³

Calculations:

⇒ Let the side of each cube be a.

⇒ When 4 cubes are placed side-by-side, length (L) = 4a.

⇒ 4a = 256

⇒ a = 256 ÷ 4

⇒ a = 64 cm

⇒ Volume of a single cube = a × a × a

⇒ Volume = 64 × 64 × 64

⇒ Volume = 262144 cm³

∴ The correct answer is 262144 cm³.

Additional Information

Total Surface Area of Cube

The total surface area of a cube with side a is given by 6a².

Diagonal of a Cube

The length of the longest diagonal in a cube is a√3.

Volume of a Cuboid

For a cuboid with dimensions L, B, and H, the volume is L × B × H.

Shortcut Trick

When two trains move in opposite directions, their relative speed is the sum of their individual speeds.

Relative Speed = 8 + 10 = 18 m/s.

Combined Length = Relative Speed × Time = 18 × 8 = 144 m.

∴ The correct answer is 144 m.

Alternate Method

Given:

Speed of the first train (v₁) = 8 m/s

Speed of the second train (v₂) = 10 m/s

Time taken to pass each other (t) = 8 sec

Formula Used:

Total Distance (Sum of lengths) = Relative Speed × Time

Calculations:

⇒ When two objects move in opposite directions, Relative Speed (S_R) = v₁ + v₂

⇒ S_R = 8 + 10 = 18 m/s

⇒ Combined Length (L₁ + L₂) = S_R × t

⇒ Combined Length = 18 × 8 = 144 m

∴ The correct answer is 144 m.

Additional Information

Relative Speed in Same Direction

When two objects move in the same direction with speeds u and v (where v), the relative speed is u − v.

Crossing a Stationary Object

When a train crosses a stationary object of negligible length (like a pole), the distance covered is the length of the train.

Unit Conversion Shortcut

To convert speed from km/hr to m/s, multiply by 5/18. To convert m/s to km/hr, multiply by 18/5.

Shortcut Trick

The greatest number that divides the given numbers exactly is their Highest Common Factor (HCF).

Find the differences between the consecutive numbers:

105 − 75 = 30

150 − 105 = 45

The HCF must divide these differences. The greatest common divisor of 30 and 45 is 15.

Since 15 also divides 75, 105, and 150 exactly, the greatest divisor is 15.

∴ The correct answer is 15.

Alternate Method

Given:

The numbers are 75, 105, and 150.

Formula/Concept Used:

To find the greatest number that divides a set of numbers exactly, we calculate their Highest Common Factor (HCF) using the Prime Factorization Method.

Shortcut Trick

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

⇒ SP = 3000 × (1 − 15/100) × (1 − 25/100)

⇒ SP = 3000 × 0.85 × 0.75

⇒ SP = 3000 × 0.6375 = ₹1,912.50

∴ The correct answer is ₹1,912.50.

Alternate Method

Given:

Marked Price (MP) = ₹3,000

First Discount (d₁) = 15%

Second Discount (d₂) = 25%

Calculations:

⇒ Price after first discount = 3000 − (15% of 3000)

⇒ Price after first discount = 3000 − (15/100 × 3000) = 3000 − 450 = ₹2,550

⇒ Price after second discount = 2550 − (25% of 2550)

⇒ Price after second discount = 2550 − (25/100 × 2550) = 2550 − 637.50

⇒ Final Selling Price = ₹1,912.50

∴ The correct answer is ₹1,912.50.

Additional Information

Single Equivalent Discount Formula

For two successive discounts of x% and y%, the single equivalent discount is (x + y − xy/100)%.

Order of Successive Discounts

The final selling price remains the same regardless of the order in which the successive discounts are applied (e.g., 15% then 25% is the same as 25% then 15%).

Marked Price vs Selling Price

Selling Price = Marked Price − Discount. Discounts are always calculated on the Marked Price (or the remaining balance in successive cases).

Shortcut Trick

To find the larger value (Elder Sibling) when the Sum and Difference are given:

Elder Sibling Age = (Sum + Difference) ÷ 2

⇒ Age = (28 + 8) ÷ 2

⇒ Age = 36 ÷ 2 = 18 years

∴ The correct answer is 18 years.

Alternate Method

Given:

Sum of ages = 28 years

Difference of ages = 8 years

Method: Linear Equations

⇒ Let the age of the elder sibling be x years and the younger sibling be y years.

⇒ x + y = 28 --- (Equation 1)

⇒ x − y = 8 --- (Equation 2)

⇒ Adding Equation 1 and Equation 2:

⇒ (x + y) + (x − y) = 28 + 8

⇒ 2x = 36

⇒ x = 36 ÷ 2

⇒ x = 18 years

∴ The correct answer is 18 years.

Additional Information

Sum and Difference Formula

For any two numbers x and y, if Sum = S and Difference = D, then the larger number x = (S + D) ÷ 2 and the smaller number y = (S − D) ÷ 2.

Age Ratio Problems

If the ratio of ages of two people is a : b, their ages can be taken as ax and bx. The difference between their ages remains constant throughout their lives.

Average Age

The average age of n persons is the sum of their ages divided by n. If each person's age increases by 'k' years, the average also increases by 'k' years.

Shortcut Trick

Ratio of Amount to Principal = 1728 : 1000.

Since interest is compounded semi-annually for 1.5 years, there are 3 half-yearly periods.

Take the cube root for one period: √³1728 : √³1000 = 12 : 10.

Rate per half-year = (12 − 10)/10 × 100 = 20%.

Annual Interest Rate = 20% × 2 = 40%.

∴ The correct answer is 40%.

Alternate Method

Given: Principal (P) = ₹1,000, Amount (A) = ₹1,728, Time = 1 year 6 months.

Formula Used: A = P(1 + r/200)²ⁿ (for semi-annual compounding, where r is annual rate and n is years).

⇒ Total periods (n) = 1.5 years × 2 = 3 half-years.

⇒ 1728 = 1000 × (1 + r/200)³

⇒ 1728/1000 = (1 + r/200)³

⇒ 1.728 = (1 + r/200)³

⇒ √³(1.728) = 1 + r/200

⇒ 1.2 = 1 + r/200

⇒ r/200 = 1.2 − 1 = 0.2

⇒ r = 0.2 × 200 = 40%.

∴ The correct answer is 40%.

Additional Information

Compound Interest Compounding Frequency

For semi-annual compounding, the rate is halved (r/2) and the time periods are doubled (2n). For quarterly, the rate is divided by 4 and periods multiplied by 4.

Effective Interest Rate

The actual interest earned over a year when compounding occurs more than once. Formula: E = (1 + r/k)^k − 1, where k is compounding frequency per year.

Rule of 72

A quick way to estimate the time required to double an investment: Years to double ≈ 72 ÷ Annual Interest Rate.

Shortcut Trick

Let Cost Price (CP) = 100.

Marked Price (MP) is 150% above CP ⇒ 100 + 150 = 250.

To gain 110%, Selling Price (SP) must be 100 + 110 = 210.

Discount = MP − SP = 250 − 210 = 40.

Discount % = (40 ÷ 250) × 100 = 16%.

∴ The correct answer is 16%.

Alternate Method

Given:

Markup = 150% above Cost Price

Required Profit = 110%

Calculations:

⇒ Assume CP = 100 units

⇒ Marked Price (MP) = 100 + (150% of 100) = 100 + 150 = 250 units

⇒ Selling Price (SP) = 100 + (110% of 100) = 100 + 110 = 210 units

⇒ Discount Amount = MP − SP = 250 − 210 = 40 units

⇒ Discount % = (Discount Amount ÷ MP) × 100

⇒ Discount % = (40 ÷ 250) × 100

⇒ Discount % = 4 ÷ 25 × 100 = 16%

∴ The correct answer is 16%.

Additional Information

Relationship between CP, MP, Profit, and Discount

The ratio of MP to CP can be expressed as: MP/CP = (100 + Profit%) ÷ (100 − Discount%).

Markup Percentage

Markup % is the percentage by which the cost price is increased to set the marked price: ((MP − CP) ÷ CP) × 100.

Effective Discount

For two successive discounts d₁ and d₂, the single equivalent discount is (d₁ + d₂ − (d₁ × d₂) ÷ 100)%.

Shortcut Trick

Identify factors of the product 175 that add up to the sum 40.

Factors of 175: 1 × 175, 5 × 35, 7 × 25.

Check sum: 5 + 35 = 40 (Correct).

Calculate positive difference: 35 − 5 = 30.

∴ The correct answer is 30.

Alternate Method

Given: Sum (x + y) = 40 and Product (xy) = 175.

Formula: (x − y)² = (x + y)² − 4xy

⇒ (x − y)² = 40² − 4 × 175

⇒ (x − y)² = 1600 − 700

⇒ (x − y)² = 900

⇒ x − y = √900 = 30

∴ The correct answer is 30.

Additional Information

Algebraic Identity

The relationship between sum, product, and difference is given by the identity: (a+b)² − (a−b)² = 4ab.

Quadratic Equations

If the sum (S) and product (P) are known, the numbers are roots of the equation: t² − St + P = 0.

Difference of Roots

For a quadratic equation ax² + bx + c = 0, the positive difference of roots is given by √D / |a|, where D = b² − 4ac.

Given:

3^x − 3^x−1 = 18

Formula used:

a^m−n = a^m / aⁿ

Calculation:

3^x − 3^x−1 = 18

⇒ 3^x − (3^x / 3) = 18

⇒ 3^x (1 − 1/3) = 18

⇒ 3^x × (2/3) = 18

⇒ 3^x = (18 × 3) / 2

⇒ 3^x = 9 × 3

⇒ 3^x = 27

⇒ 3^x = 3³

⇒ x = 3

Substituting x in the expression (2x − 1) / (2x + 3):

⇒ (2 × 3 − 1) / (2 × 3 + 3)

⇒ (6 − 1) / (6 + 3)

⇒ 5/9

∴ The correct answer is option (2).

Shortcut Trick

To identify co-prime numbers, check if the two numbers have any common factor other than 1.

(15, 25) → Both are divisible by 5.

(14, 35) → Both are divisible by 7.

(21, 49) → Both are divisible by 7.

(9, 28) → 9 = 3×3 and 28 = 2×2×7. No common prime factor exists.

∴ The correct answer is (9, 28).

Alternate Method

Given: Pairs of numbers (15, 25), (14, 35), (9, 28), and (21, 49).

Formula Used: Two numbers are co-prime if their Highest Common Factor (HCF) is 1.

⇒ Factors of 15: 1, 3, 5, 15 | Factors of 25: 1, 5, 25. HCF = 5.

⇒ Factors of 14: 1, 2, 7, 14 | Factors of 35: 1, 5, 7, 35. HCF = 7.

⇒ Factors of 21: 1, 3, 7, 21 | Factors of 49: 1, 7, 49. HCF = 7.

⇒ Factors of 9: 1, 3, 9 | Factors of 28: 1, 2, 4, 7, 14, 28. HCF = 1.

∴ The correct answer is (9, 28).

Additional Information

Definition of Co-prime Numbers

A set of numbers is co-prime if they have no common positive factor other than 1. Their HCF is always 1.

Consecutive Numbers

Any two consecutive natural numbers (e.g., 14 and 15, or 20 and 21) are always co-prime to each other.

LCM of Co-prime Numbers

The Least Common Multiple (LCM) of two co-prime numbers is always equal to their product (a × b).

Prime Numbers Property

Any two prime numbers are always co-prime to each other since their only common factor is 1.

Shortcut Trick

Find the LCM of the given divisors: LCM(12, 8, 11) = 264.

Divide the largest 4-digit number (9999) by 264 to find the remainder: 9999 ÷ 264 gives a remainder of 231.

Subtract this remainder from 9999 to get the largest multiple: 9999 − 231 = 9768.

Add the required common remainder of 3 to the multiple: 9768 + 3 = 9771.

∴ The correct answer is 9771.

Alternate Method

Given:

Divisors = 12, 8, 11

Common remainder in each case = 3

Formula Used:

Required Number = (LCM of divisors × k) + Remainder, where k is an integer.

4-Digit Numbers 5-Digit

Shortcut Trick

Apply the Direct Distance Formula for Total Travel Time: Distance = (S₁ × S₂) ÷ (S₁ + S₂) × Total Time.

Here, S₁ = 60 km/hr, S₂ = 80 km/hr, and Total Time = 3.5 hr.

⇒ Distance = (60 × 80) ÷ (60 + 80) × 3.5

⇒ Distance = (4800 ÷ 140) × 3.5 = (480 ÷ 14) × 3.5

⇒ Distance = 480 ÷ 4 = 120 km.

∴ The correct answer is 120 km.

Alternate Method

Given: Speed from Gwalior to Jhansi = 60 km/hr, speed from Jhansi to Gwalior = 80 km/hr, and Total travel time = 3.5 hr.

Formula Used: Time = Distance ÷ Speed

Let the one-way distance be x km.

⇒ Time taken to travel from Gwalior to Jhansi = x ÷ 60

⇒ Time taken to travel from Jhansi to Gwalior = x ÷ 80

⇒ Total Time = (x ÷ 60) + (x ÷ 80) = 3.5

⇒ (4x + 3x) ÷ 240 = 3.5

⇒ 7x ÷ 240 = 3.5

⇒ 7x = 3.5 × 240

⇒ 7x = 840

⇒ x = 840 ÷ 7 = 120 km.

∴ The correct answer is 120 km.

Additional Information

Average Speed Formula

When a person travels the same distance at two different speeds S₁ and S₂, the Average Speed is given by (2 × S₁ × S₂) ÷ (S₁ + S₂).

Speed and Time Relation

For a constant distance, Speed is inversely proportional to Time (S₁ ÷ S₂ = T₂ ÷ T₁).

Distance when Time Difference is Given

If the difference in time (ΔT) is given instead of total time, the formula becomes: Distance = (S₁ × S₂) ÷ |S₁ − S₂| × ΔT.