SSC CGL 2025 Tier-II (Held On: 19 Jan, 2026) (Previous Year Paper)

Shortcut Trick

The standard deviation of any given data set is completely independent of the change of origin.

This means adding or subtracting a constant value from all observations does not change the standard deviation, as the overall spread of the data remains the same.

Since each original value is uniformly decreased by 7, the new standard deviation will be exactly equal to the original standard deviation.

∴ The correct answer is 5.

Alternate Method

Given:

Standard deviation of n values X₁, X₂, ..., X_n = 5

New values: X₁ − 7, X₂ − 7, ..., X_n − 7

Formula Used:

If a new dataset is formed by the linear transformation Y_i = aX_i + b, then SD(Y) = |a| × SD(X).

Calculations:

⇒ In the new series, each value is generated by shifting the original values by −7.

⇒ Comparing with the general transformation Y_i = aX_i + b, we get a = 1 and b = −7.

⇒ New Standard Deviation = |1| × Original Standard Deviation

⇒ New Standard Deviation = 1 × 5 = 5

⇒ The subtraction of a constant solely shifts the mean position of the data curve but does not affect the inner dispersion (spread) of the elements.

∴ The correct answer is 5.

Additional Information

Change of Origin (Addition/Subtraction)

Standard deviation and variance are completely independent of the change of origin. Adding or subtracting a constant value to all data points shifts the central mean but perfectly preserves the gaps between the data points.

Change of Scale (Multiplication/Division)

Standard deviation is heavily dependent on the change of scale. If every single observation is identically multiplied or divided by a constant 'k', the new standard deviation inherently becomes |k| times the original standard deviation.

Variance vs Standard Deviation

Variance mathematically behaves as the square of the standard deviation (σ²). Therefore, while the standard deviation formally scales by a factor of |k|, variance fundamentally scales by a factor of k² when values are multiplied by 'k'.

Shortcut Trick

Area of the square base = (Diagonal)² ÷ 2 = (√512)² ÷ 2 = 512 ÷ 2 = 256 m².

Volume of the pyramid = (1/3) × Base Area × Height

Volume = (1/3) × 256 × 8 = 2048 ÷ 3 ≈ 682.67 m³.

Rounding to the nearest integer gives 683 m³.

∴ The correct answer is 683 m³.

Alternate Method

Given:

Height of the pyramid (h) = 8 m

Diagonal of the square base (d) = √512 m

Formula Used:

Area of square = d² ÷ 2

Volume of pyramid = (1/3) × Area of base × Height

Calculations:

⇒ Area of base = (√512)² ÷ 2

⇒ Area of base = 512 ÷ 2 = 256 m²

⇒ Volume of pyramid = (1/3) × Area of base × Height

⇒ Volume of pyramid = (1/3) × 256 × 8

⇒ Volume of pyramid = 2048 ÷ 3

⇒ Volume of pyramid ≈ 682.66 m³

⇒ The approximate volume is 683 m³.

∴ The correct answer is 683 m³.

Additional Information

Volume of a Pyramid

The general formula V = (1/3) × Base Area × Height holds true for any pyramid, regardless of whether its base is triangular, square, or polygonal.

Area of a Square

Usually found using Side², but when only the diagonal is known, directly applying Area = (Diagonal)² ÷ 2 is much faster.

Lateral Surface Area (LSA)

For a regular pyramid, LSA = (1/2) × Perimeter of the Base × Slant Height.

Total Surface Area (TSA)

The Total Surface Area is the sum of the Lateral Surface Area and the Area of the Base.

Shortcut Trick

Put θ = 45°. We know tan 45° = 1, cot 45° = 1, sin 45° = 1/√2

Numerator = 1² − (1/√2)² = 1 − 1/2 = 1/2

Denominator = 2 + 1² + 1² = 4

Expression value = (1/2) ÷ 4 = 1/8

Check options: Only sin⁶ 45° = (1/√2)⁶ = 1/8 matches the expression value.

∴ The correct answer is sin⁶ θ.

Alternate Method

Given:

Expression = (tan² θ − sin² θ) / (2 + tan² θ + cot² θ)

Formula Used:

⇒ 1 + tan² θ = sec² θ and 1 + cot² θ = csc² θ

Calculations:

⇒ Numerator = (sin² θ / cos² θ) − sin² θ

⇒ Numerator = sin² θ × (1 / cos² θ − 1)

⇒ Numerator = sin² θ × (sec² θ − 1) = sin² θ × tan² θ

⇒ Denominator = (1 + tan² θ) + (1 + cot² θ)

⇒ Denominator = sec² θ + csc² θ

⇒ Denominator = (1 / cos² θ) + (1 / sin² θ)

⇒ Denominator = (sin² θ + cos² θ) / (sin² θ × cos² θ) = 1 / (sin² θ × cos² θ)

⇒ Expression = Numerator ÷ Denominator

⇒ Expression = (sin² θ × sin² θ / cos² θ) × (sin² θ × cos² θ)

⇒ Expression = sin⁶ θ

∴ The correct answer is sin⁶ θ.

Additional Information

Pythagorean Identities

The three fundamental identities are sin² θ + cos² θ = 1, sec² θ − tan² θ = 1, and csc² θ − cot² θ = 1.

Value Substitution Method

For expressions containing multiple trigonometric ratios, substituting standard angles like 30°, 45°, or 60° usually gives the fastest result while avoiding zero denominators.

Key Shortcut Combinations

Always remember: tan² θ − sin² θ = tan² θ × sin² θ and cot² θ − cos² θ = cot² θ × cos² θ.

Shortcut Trick

There is a direct property in geometry: when a quadrilateral is formed by joining the midpoints of the sides of a rectangle, it is always a rhombus.

The area of this inscribed rhombus is exactly half the area of the original rectangle.

Given the area of the rectangle is 48 cm², the perimeter data is not needed.

Area of rhombus = 48 ÷ 2 = 24 cm².

∴ The correct answer is 24 cm².

Alternate Method

Given:

Area of rectangle = L × B = 48 cm²

Perimeter of rectangle = 2(L + B) = 28 cm

Formula Used:

(L − B)² = (L + B)² − 4LB

Area of Rhombus = 1/2 × d₁ × d₂

Calculations:

⇒ 2(L + B) = 28 ⇒ L + B = 14 cm

⇒ (L − B)² = 14² − 4 × 48

⇒ (L − B)² = 196 − 192 = 4

⇒ L − B = √4 = 2 cm

⇒ Adding equations (L + B) and (L − B): 2L = 16 ⇒ L = 8 cm

⇒ B = 14 − 8 = 6 cm

When midpoints of a rectangle are joined to form a rhombus, the diagonals of the rhombus are equal to the sides of the rectangle.

⇒ Diagonal d₁ = L = 8 cm

⇒ Diagonal d₂ = B = 6 cm

⇒ Area of rhombus = 1/2 × 8 × 6 = 24 cm²

∴ The correct answer is 24 cm².

Additional Information

Varignon's Theorem

The quadrilateral formed by joining the midpoints of the sides of any quadrilateral is always a parallelogram, and its area is exactly half of the original quadrilateral.

Special Quadrilaterals from Midpoints

Midpoints of a Rectangle → Rhombus

Midpoints of a Rhombus → Rectangle

Midpoints of a Square → Square

Rhombus Properties

All four sides are equal. The diagonals bisect each other at right angles (90°). Area = 1/2 × d₁ × d₂.

Shortcut Trick

Let Assumed Mean (A) = 42 (middle class mark).

Deviations (d_i) of class marks from A: −15, −10, −5, 0, +5, +10, +15.

Total deviation = Σf_id_i = −150 − 120 − 40 + 0 + 75 + 100 + 225 = +90.

Total frequency (Σf_i) = 75.

Mean = A + (Σf_id_i ÷ Σf_i) = 42 + (90 ÷ 75) = 42 + 1.2 = 43.2.

∴ The correct answer is 43.2 years.

Alternate Method

Given:

Frequencies corresponding to age intervals are provided in the data.

Formula Used:

Mean = Σ(f_i × x_i) ÷ Σf_i

Class mark (x_i) = (Lower limit + Upper limit) ÷ 2

Calculation:

⇒ Class marks (x_i): 27, 32, 37, 42, 47, 52, 57.

⇒ Frequencies (f_i): 10, 12, 8, 5, 15, 10, 15.

⇒ Products (f_i × x_i): 270, 384, 296, 210, 705, 520, 855.

⇒ Total Frequency (Σf_i) = 10 + 12 + 8 + 5 + 15 + 10 + 15 = 75

⇒ Sum of Products (Σf_ix_i) = 270 + 384 + 296 + 210 + 705 + 520 + 855 = 3240

⇒ Mean = 3240 ÷ 75

⇒ Mean = 43.2

∴ The correct answer is 43.2 years.

Additional Information

Class Mark

The mid-point of a class interval is used as the representative value. Class mark = (Lower Limit + Upper Limit) ÷ 2.

Direct Method for Mean

Mean = Σ(f_ix_i) ÷ Σf_i, where x_i represents the class marks and f_i represents the corresponding frequencies.

Assumed Mean Method

Mean = A + (Σf_id_i ÷ Σf_i), where A is the assumed mean and d_i = x_i − A.

Shortcut Trick

A's profit parts = (2 × 4) + (3 × 8) = 8 + 24 = 32

B's profit parts = (3 × 4) + (2 × 8) = 12 + 16 = 28

C's profit parts = 5 × 12 = 60

Profit Ratio (A : B : C) = 32 : 28 : 60 = 8 : 7 : 15

Total profit = 8 + 7 + 15 = 30 units = Rs. 33,000 ⇒ 1 unit = Rs. 1,100

B's share = 7 × 1,100 = Rs. 7,700

∴ The correct answer is Rs. 7,700.

Alternate Method

Given:

Initial investment ratio = 2 : 3 : 5

Total profit at the end of the year = Rs. 33,000

Formula Used:

Profit Ratio = Ratio of Equivalent Capitals

Equivalent Capital = ∑ (Capital × Time in months)

Calculations:

⇒ Let the initial capitals of A, B, and C be 2x, 3x, and 5x respectively.

After 4 months, the new capital values are:

⇒ A's new capital = 2x + (50% of 2x) = 2x + x = 3x

⇒ B's new capital = 3x − (1/3 of 3x) = 3x − x = 2x

⇒ C's capital remains unchanged at 5x throughout the year.

Calculating the equivalent capital for 12 months:

⇒ A's equivalent capital = (2x × 4) + (3x × 8) = 8x + 24x = 32x

⇒ B's equivalent capital = (3x × 4) + (2x × 8) = 12x + 16x = 28x

⇒ C's equivalent capital = 5x × 12 = 60x

⇒ Ratio of profits (A : B : C) = 32x : 28x : 60x = 8 : 7 : 15

⇒ Sum of ratio parts = 8 + 7 + 15 = 30 parts

⇒ B's share = (7 / 30) × 33,000 = 7 × 1,100 = Rs. 7,700

∴ The correct answer is Rs. 7,700.

Additional Information

Simple vs Compound Partnership

In simple partnership, capitals are invested for the same time duration, and profit ratio directly equals capital ratio. In compound partnership, the time periods differ, so Profit Ratio = C₁T₁ : C₂T₂.

Active and Sleeping Partners

An active partner works directly in managing the business and receives a fixed salary or percentage of the total profit first. The remaining profit is then divided among all partners in the ratio of their respective investments.

Changes in Capital

When a partner modifies their capital mid-term by adding or withdrawing funds, their total equivalent capital is the sum of each distinct capital amount multiplied by the exact number of months it remained invested.

Shortcut Trick

Since T lies on the perpendicular bisector of PQ, TP = TQ, making ΔTPQ isosceles with ∠TQP = ∠TPQ.

The angle ∠PQR is the same as ∠TQP, so the total angle ∠P is equal to ∠PQR + 38°.

Using the angle sum property for the main ΔPQR: ∠P + ∠Q + ∠R = 180°.

(∠PQR + 38°) + ∠PQR + 62° = 180°.

2 × ∠PQR + 100° = 180° ⇒ 2 × ∠PQR = 80°.

∴ The correct answer is 40°.

Alternate Method

Given:

∠R = 62°

∠TPR = 38°

ST is the perpendicular bisector of the side PQ, where S lies on PQ and T lies on QR.

Formula Used:

Any point on the perpendicular bisector of a segment is equidistant from its endpoints.

Angles opposite to equal sides in a triangle are equal.

Sum of interior angles in a triangle is 180°.

Calculations:

⇒ Since T lies on the perpendicular bisector of PQ, it is equidistant from points P and Q, so TP = TQ.

⇒ In ΔTPQ, because the sides TP and TQ are equal, the triangle is isosceles and ∠TQP = ∠TPQ.

⇒ Let ∠PQR = x. Because Q, T, and R are collinear, ∠TQP is the same as ∠PQR, meaning ∠TQP = x.

⇒ Therefore, ∠TPQ = x.

⇒ In the full ΔPQR, the total angle at vertex P is ∠TPQ + ∠TPR = x + 38°.

⇒ Applying the angle sum property to ΔPQR:

⇒ ∠P + ∠Q + ∠R = 180°

⇒ (x + 38°) + x + 62° = 180°

⇒ 2x + 100° = 180°

⇒ 2x = 80°

⇒ x = 40°

∴ The correct answer is 40°.

Additional Information

Perpendicular Bisector Theorem

A point lies on the perpendicular bisector of a line segment if and only if it is completely equidistant from the two endpoints of the segment. This instantly defines isosceles triangles within larger shapes.

Isosceles Triangle Properties

In any triangle where two sides are equal in length, the angles directly opposite to those sides are also equal in measure. The altitude dropped to the unequal side acts as an angle bisector, a median, and a perpendicular bisector simultaneously.

Angle Sum Property

Regardless of the shape or size of a triangle, the sum of all its three interior angles will constantly equal exactly 180°.

Shortcut Trick

Since 39° and 51° are complementary angles (39° + 51° = 90°).

Apply direct properties: cos θ / sin(90° − θ) = 1 and sin² θ + sin²(90° − θ) = 1.

Substitute these directly into the expression: 3/2 × (1) − √1

We get: 3/2 − 1 = 1/2

∴ The correct answer is 1/2.

Alternate Method

Given:

Expression: 3/2 × (cos 39° / sin 51°) − √(sin² 39° + sin² 51°)

Formula Used:

sin(90° − θ) = cos θ

sin² θ + cos² θ = 1

Calculations:

⇒ We can express 51° as (90° − 39°)

⇒ sin 51° = sin(90° − 39°) = cos 39°

⇒ sin² 51° = sin²(90° − 39°) = cos² 39°

⇒ Substituting these converted values into the given expression:

⇒ 3/2 × (cos 39° / cos 39°) − √(sin² 39° + cos² 39°)

⇒ 3/2 × (1) − √(1)

⇒ 3/2 − 1

⇒ (3 − 2) / 2 = 1/2

∴ The correct answer is 1/2.

Additional Information

Trigonometric Identities

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

Complementary Angles

Two angles are complementary if their sum equals 90°. Key trigonometric relations are: sin(90° − θ) = cos θ, tan(90° − θ) = cot θ, and sec(90° − θ) = cosec θ.

Supplementary Angles

Two angles are supplementary if their sum equals 180°. Key trigonometric relations are: sin(180° − θ) = sin θ, and cos(180° − θ) = −cos θ.

Shortcut Trick

Time = 1.5 years ⇒ 3 half-yearly cycles.

Ratio of Amount to Principal = 19683 / 15625 = (27 / 25)³.

Multiplying factor per cycle = 27 / 25.

Half-yearly rate = (27 − 25) / 25 × 100 = 8%.

Annual rate = 8% × 2 = 16%.

∴ The correct answer is 16%.

Alternate Method

Given:

Principal (P) = ₹15,625

Amount (A) = ₹19,683

Time (T) = 1 year 6 months = 1.5 years

Formula Used:

When interest is compounded half-yearly: A = P(1 + R/200)^2T, where R is the annual interest rate.

Calculations:

⇒ Number of half-yearly cycles (n) = 1.5 × 2 = 3

⇒ 19683 = 15625(1 + R/200)³

⇒ 19683 / 15625 = (1 + R/200)³

⇒ (27 / 25)³ = (1 + R/200)³

By taking the cube root on both sides:

⇒ 27 / 25 = 1 + R/200

⇒ R / 200 = (27 / 25) − 1

⇒ R / 200 = 2 / 25

⇒ R = (2 × 200) / 25

⇒ R = 400 / 25 = 16

∴ The correct answer is 16%.

Additional Information

Compound Interest Formula

A = P(1 + R/100)ⁿ where A is amount, P is principal, R is rate per period, and n is number of periods.

Half-Yearly Compounding

When interest is compounded half-yearly, the rate is halved (R/2) and the time in years is doubled (2T) to find the number of periods.

Effective Annual Rate (EAR)

The true annual return on an investment when compounded more than once a year. Formula: EAR = (1 + R/n)ⁿ − 1. For 16% compounded half-yearly, EAR = (1.08)² − 1 = 16.64%.

Shortcut Trick

Volume of a pyramid = (1/3) × Base Area × Height

Base Area (Equilateral Triangle) = (√3 / 4) × (side)² = (√3 / 4) × 10² = 25√3 cm²

Volume = (1/3) × 25√3 × 24 = 200√3 cm³

∴ The correct answer is 200√3 cm³.

Alternate Method

Given:

Side of the equilateral triangle base (a) = 10 cm

Vertical height of the pyramid (h) = 24 cm

Formula Used:

Volume of a Pyramid = (1/3) × Base Area × Height

Area of an Equilateral Triangle = (√3 / 4) × a²

Calculations:

⇒ Area of the base = (√3 / 4) × 10²

⇒ Area of the base = (√3 / 4) × 100

⇒ Area of the base = 25√3 cm²

⇒ Volume of the pyramid = (1/3) × 25√3 × 24

⇒ Volume of the pyramid = 25√3 × 8

⇒ Volume of the pyramid = 200√3 cm³

∴ The correct answer is 200√3 cm³.

Additional Information

Lateral Surface Area of a Pyramid

The sum of the areas of all lateral triangular faces. LSA = (1/2) × Base Perimeter × Slant Height.

Total Surface Area of a Pyramid

The total area covered by the pyramid surfaces. TSA = Lateral Surface Area + Base Area.

Volume of a Cone

Similar to a pyramid, the volume of a cone is strictly one-third the volume of its corresponding cylinder. V = (1/3) × πr²h.

Shortcut Trick

Total numbers = 120.

Multiples of 6 = 20, Multiples of 8 = 15, Multiples of 24 (LCM) = 5.

Required numbers = Only 6's + Only 8's = (20 − 5) + (15 − 5) = 25.

Required Probability = 25/120.

∴ The correct answer is 25/120.

Alternate Method

Given:

Total numbers in sample space, n(S) = 120

Formula Used:

n(A Δ B) = n(A) + n(B) − 2 × n(A ∩ B)

P(E) = Favorable Outcomes ÷ Total Outcomes

Calculations:

Let A be the set of numbers divisible by 6.

⇒ n(A) = 120 ÷ 6 = 20

Let B be the set of numbers divisible by 8.

⇒ n(B) = 120 ÷ 8 = 15

Let A ∩ B be the set of numbers divisible by 24 (LCM of 6 and 8).

⇒ n(A ∩ B) = 120 ÷ 24 = 5

Favorable outcomes are numbers divisible by 6 or 8 but not both.

⇒ Favorable outcomes = n(A) + n(B) − 2 × n(A ∩ B)

⇒ Favorable outcomes = 20 + 15 − 2 × 5

⇒ Favorable outcomes = 35 − 10 = 25

⇒ Probability = 25/120

∴ The correct answer is 25/120.

Additional Information

Probability of an Event

The probability of an event E is the ratio of favorable outcomes to the total possible outcomes: P(E) = n(E) ÷ n(S).

Union of Two Sets

The number of elements in the union of sets A and B is given by: n(A ∪ B) = n(A) + n(B) − n(A ∩ B).

Symmetric Difference of Sets

The symmetric difference (A Δ B) consists of elements that belong to either A or B, but not both. n(A Δ B) = n(A) + n(B) − 2 × n(A ∩ B).

Shortcut Trick

Volume to be emptied = ½ × (⅔ × π × r³) = ⅓ × π × r³

Required Volume = ⅓ × (22/7) × 2.1³ = 9.702 m³ = 9702 Litres

Time required = Total Volume ÷ Rate = 9702 ÷ 7 = 1386 seconds

Time in minutes = 1386 ÷ 60 = 23.1 ≈ 23 minutes

∴ The correct answer is 23 minutes.

Alternate Method

Given:

Internal radius of the tank (r) = 2.1 m

Rate of emptying by the pipe = 7 Litres/second

1 m³ = 1000 Litres

Formula Used:

Volume of a hemisphere = ⅔ × π × r³

Time = Volume ÷ Rate

Calculation:

⇒ Total volume of the tank = ⅔ × (22/7) × 2.1 × 2.1 × 2.1

⇒ Total volume = ⅔ × (22/7) × 9.261 = 19.404 m³

⇒ Volume of water to be emptied (half of the tank) = 19.404 ÷ 2 = 9.702 m³

⇒ Convert volume from m³ to Litres = 9.702 × 1000 = 9702 Litres

⇒ Time required (in seconds) = Total Volume ÷ Emptying Rate = 9702 ÷ 7 = 1386 seconds

⇒ Convert time from seconds to minutes = 1386 ÷ 60 = 23.1 minutes

⇒ The approximate time required to empty half the tank is 23 minutes.

∴ The correct answer is 23 minutes.

Additional Information

Volume of a Sphere

A full sphere has a volume given by the formula V = &frac43; × π × r³.

Surface Area of a Hemisphere

The Curved Surface Area (CSA) is 2 × π × r², and the Total Surface Area (TSA) including the flat circular base is 3 × π × r².

Capacity Conversion Units

Always remember standard conversions: 1 m³ = 1000 Litres, and 1 Litre = 1000 cm³. These are frequently tested in 3D mensuration problems involving liquids.

Shortcut Trick

The mode is the value that appears most frequently in a data set.

By observing the given data set, the term (a + 6) appears 3 times, which is the maximum frequency.

∴ Mode = a + 6

Equating it to the given mode: a + 6 = 26

⇒ a = 20

∴ The correct answer is a = 20.

Alternate Method

Given:

Data set = {a, a + 2, a + 4, a + 6, a + 6, a + 6, a + 8}

Mode = 26

Concept Used:

Mode is the value that appears most frequently in a given set of data points.

Calculations:

⇒ Frequency of a = 1

⇒ Frequency of a + 2 = 1

⇒ Frequency of a + 4 = 1

⇒ Frequency of a + 6 = 3

⇒ Frequency of a + 8 = 1

⇒ Since the term (a + 6) has the highest frequency (3 times), it is the mode of the data set.

⇒ Mode = a + 6

⇒ It is given that the mode is 26.

⇒ a + 6 = 26

⇒ a = 26 − 6 = 20

∴ The correct answer is a = 20.

Additional Information

Mean

The mean is the arithmetic average of the numbers, calculated as the sum of all observations divided by the total number of observations.

Median

The median is the exact middle value when a data set is ordered from least to greatest. For an even number of observations, it is the average of the two middle values.

Empirical Relationship

For a moderately skewed distribution, the relationship is given by: Mode = 3 × Median − 2 × Mean.

Bimodal Data

A data set can have more than one mode. If two different values share the highest frequency, the data set is termed bimodal.

Shortcut Trick

Total work = LCM(20, 30) = 60 units. Efficiency of P = 3 units/day, Q = 2 units/day.

Work done by P and Q in 6 days = 6 × 5 = 30 units ⇒ Remaining work = 60 − 30 = 30 units.

Efficiency of (P + R) = 30 ÷ 8 = 3.75 units/day ⇒ R's efficiency = 3.75 − 3 = 0.75 units/day.

Time for R to do 60% of total work (36 units) = 36 ÷ 0.75 = 48 days.

∴ The correct answer is 48 days.

Alternate Method

Given:

Time taken by P to complete the work = 20 days

Time taken by Q to complete the work = 30 days

P and Q work together for 6 days.

Rest of the work is completed by P and R in 8 days.

Calculations:

⇒ Work done by P and Q in 1 day = 3 + 2 = 5 units

⇒ Work done by P and Q in 6 days = 6 × 5 = 30 units

⇒ Remaining work = Total Work − Completed Work = 60 − 30 = 30 units

⇒ This remaining work is completed by P and R together in 8 days.

⇒ Combined efficiency of P and R = Remaining work ÷ Time taken = 30 ÷ 8 = 3.75 units/day

⇒ We know P's efficiency is 3 units/day.

⇒ Efficiency of R = Combined efficiency − P's efficiency

⇒ Efficiency of R = 3.75 − 3 = 0.75 units/day (or 3/4 units/day)

⇒ Now, we need to find the time R alone takes to complete 60% of the work.

⇒ 60% of Total Work = 60 × (60 ÷ 100) = 36 units

⇒ Time taken by R alone = Required Work ÷ Efficiency of R = 36 ÷ 0.75 = 48 days

∴ The correct answer is 48 days.

Additional Information

Pipes & Cisterns

This concept follows the same rules as Time and Work. The only difference is that filling pipes are treated as positive efficiency, while emptying pipes or leaks have negative efficiency.

Work & Wages

Wages are always distributed in proportion to the work actually done. If individuals work for the same number of days, their wages are directly distributed in the ratio of their efficiencies.

Alternate Day Work

When two people work on alternate days, calculate the combined work done in a 2-day cycle, multiply by the maximum possible integer before the limit, and then assign the remaining work sequentially.

Shortcut Trick

In an equilateral triangle, due to perfect symmetry, the medians, altitudes, angle bisectors, and perpendicular bisectors are all identical line segments.

Consequently, all the primary centers of the triangle — Centroid, Circumcenter, Incenter, and Orthocenter — coincide at exactly the same point.

Thus, the circumcenter corresponds directly to the centroid.

∴ The correct answer is the centroid of the triangle.

Alternate Method

Given:

An equilateral triangle with all sides equal and all angles equal to 60°.

Concept Used:

⇒ The circumcenter is the point of intersection of the perpendicular bisectors of the sides.

⇒ The centroid is the point of intersection of the medians.

Calculations:

⇒ By definition, a median connects a vertex to the midpoint of its opposite side.

⇒ A perpendicular bisector intersects the midpoint of a side at exactly 90°.

⇒ In an equilateral triangle, because of symmetry, the line connecting any vertex to the midpoint of the opposite side is inherently perpendicular to that side.

⇒ Therefore, the medians and perpendicular bisectors act as the exact same line segments.

⇒ As a result, the centroid (intersection of medians) and the circumcenter (intersection of perpendicular bisectors) are mathematically the exact same point.

∴ The correct answer is the centroid of the triangle.

Additional Information

Centroid

The centroid is the point where all three medians of a triangle intersect. It always divides each median internally in the ratio of 2:1 and acts as the center of gravity of the triangle.

Circumcenter

The circumcenter is the intersection of the perpendicular bisectors of the sides. It is equidistant from all three vertices and serves as the center of the circumcircle that passes through all the vertices.

Orthocenter & Incenter

The orthocenter is the intersection of altitudes, while the incenter is the intersection of internal angle bisectors. In an equilateral triangle, these two points also perfectly coincide with the centroid and circumcenter.

Euler Line

In any general non-equilateral triangle, the orthocenter, centroid, and circumcenter are collinear and form what is known as the Euler line. However, in an equilateral triangle, this line shrinks down to a single distinct point.

Shortcut Trick

Height of trapezium = √(13² − 5²) = 12 cm

Base area = 1/2 × (10 + 20) × 12 = 180 cm²

Prism height = Volume ÷ Base area = 3240 ÷ 180 = 18 cm

TSA = 2 × Base area + Perimeter × Prism height

TSA = 2 × 180 + (10 + 20 + 13 + 13) × 18 = 360 + 1008 = 1368 cm²

∴ The correct answer is 1368 cm².

Alternate Method

Given:

Parallel sides of the isosceles trapezium base, a = 10 cm, b = 20 cm

Non-parallel sides, c = 13 cm

Volume of the right prism, V = 3240 cm³

Formula Used:

Area of trapezium = 1/2 × (Sum of parallel sides) × Height

Volume of prism = Area of base × Height of prism

Total Surface Area of prism = 2 × (Area of base) + (Perimeter of base) × Height of prism

Calculations:

⇒ The base of the right-angled triangle formed by dropping a perpendicular = (20 − 10) ÷ 2 = 5 cm

⇒ By Pythagoras theorem, height of trapezium, h_t = √(13² − 5²) = √(169 − 25) = √144 = 12 cm

⇒ Area of trapezium base = 1/2 × (10 + 20) × 12 = 1/2 × 30 × 12 = 180 cm²

⇒ Perimeter of trapezium base = 10 + 20 + 13 + 13 = 56 cm

⇒ Volume of prism = Area of base × Height of prism

⇒ 3240 = 180 × Height of prism

⇒ Height of prism = 3240 ÷ 180 = 18 cm

⇒ Lateral Surface Area = Perimeter of base × Height of prism = 56 × 18 = 1008 cm²

⇒ Total Surface Area = 2 × Area of base + Lateral Surface Area

⇒ Total Surface Area = 2 × 180 + 1008 = 360 + 1008 = 1368 cm²

∴ The correct answer is 1368 cm².

Additional Information

Prism Properties

A right prism is a 3D solid with two identical and parallel polygon bases. The lateral faces are rectangles perfectly perpendicular to the bases.

Volume of Prism

Volume is universally calculated as V = Base Area × Height. This holds true for any uniform cross-section shape.

Isosceles Trapezium

A quadrilateral with exactly one pair of parallel sides and non-parallel sides of equal length. The base angles are equal, and the diagonals are also of equal length.

Total Surface Area (TSA)

TSA of any prism = Lateral Surface Area + 2 × Base Area. LSA is calculated by multiplying the perimeter of the base by the height of the prism.

Shortcut Trick

Total frequency N = 30, so N/2 = 15.

Cumulative frequency crosses 15 in the 20-30 interval (5 + 8 = 13, then 13 + 6 = 19).

Directly apply: Median = Lower limit + [ (N/2 − cf) ÷ f ] × h

Median = 20 + [ (15 − 13) ÷ 6 ] × 10

Median = 20 + 3.33 = 23.33

∴ The correct answer is 23.3 marks.

Alternate Method

Given:

Total frequency (N) = 5 + 8 + 6 + 6 + 5 = 30

Value of N/2 = 30 ÷ 2 = 15

Finding Cumulative Frequencies (cf):

⇒ 0-10: 5

⇒ 10-20: 5 + 8 = 13

⇒ 20-30: 13 + 6 = 19 (This is the median class since cf ≥ 15)

⇒ 30-40: 19 + 6 = 25

⇒ 40-50: 25 + 5 = 30

Formula Used:

Median = L + [ (N/2 − cf) ÷ f ] × h

Where L = Lower limit of median class = 20

cf = Cumulative frequency of preceding class = 13

f = Frequency of median class = 6

h = Class width = 10

Calculation:

⇒ Median = 20 + [ (15 − 13) ÷ 6 ] × 10

⇒ Median = 20 + (2 ÷ 6) × 10

⇒ Median = 20 + 10 ÷ 3

⇒ Median = 20 + 3.33

⇒ Median = 23.33

∴ The correct answer is 23.3 marks.

Additional Information

Mode of Grouped Data

Mode = L + [ (f₁ − f₀) ÷ (2 × f₁ − f₀ − f₂) ] × h, where f₁ is the frequency of the modal class, f₀ is the frequency of the preceding class, and f₂ is the frequency of the succeeding class.

Mean of Grouped Data

Mean = (Σf_ix_i) ÷ Σf_i, where x_i represents the class marks (midpoints) of the respective intervals.

Empirical Relationship

In a moderately skewed distribution, the relationship between measures of central tendency is: Mode = 3 × Median − 2 × Mean.

Shortcut Trick

Using the Extended Sine Rule for a circle: Chord Length = 2 × Radius × sin(θ)

Where θ is the angle subtended by the chord at the circumference.

Length of chord = 2 × r × sin(60°) = 2 × r × (√3 ÷ 2) = r√3

Required Ratio = Radius : Chord = r : r√3 = 1 : √3

∴ The correct answer is 1 : √3.

Alternate Method

Given:

⇒ Angle subtended by chord at circumference = 60°

⇒ Radius of the circle = r

Formula Used:

⇒ Angle at center = 2 × Angle at circumference.

⇒ sin(θ) = Perpendicular ÷ Hypotenuse

Calculations:

⇒ Angle subtended by chord at the center ∠AOB = 2 × 60° = 120°.

⇒ In ΔOAB, OA = OB = r (radii).

⇒ Draw perpendicular OD from center O to chord AB.

⇒ In right-angled ΔOAD, ∠AOD = 120° ÷ 2 = 60°.

⇒ sin(60°) = AD ÷ OA

⇒ √3 ÷ 2 = AD ÷ r

⇒ AD = r√3 ÷ 2

⇒ Total length of chord AB = 2 × AD = 2 × (r√3 ÷ 2) = r√3.

⇒ Ratio of radius to length of chord = r : r√3 = 1 : √3.

∴ The correct answer is 1 : √3.

Additional Information

Extended Sine Rule

In any triangle inscribed in a circle, the ratio of a side to the sine of its opposite angle equals the diameter of the circle (a ÷ sinA = 2R).

Angle at Center Theorem

The angle subtended by an arc at the center is double the angle subtended by it at any remaining part of the circle.

Chord Bisection Theorem

A perpendicular drawn from the center of a circle to a chord bisects the chord into two equal halves, and also bisects the central angle.

Shortcut Trick

When a sphere of radius r is cut into 8 identical pieces by 3 mutually perpendicular cuts through its center, each piece forms a spherical octant.

Total Surface Area of 1 piece = Curved Surface Area + 3 × Flat Quadrant Areas

Total Area = (1/8 × 4πr²) + 3 × (1/4 × πr²) = 5/4 × πr²

Given r = 21 cm, Area = 5/4 × π × (21)² = 5/4 × 441π

Area = 2205π ÷ 4 = 551.25π cm²

∴ The correct answer is 551.25π cm².

Alternate Method

Given:

Radius of the sphere, r = 21 cm.

The sphere is cut into 8 identical pieces (octants) by three mutually perpendicular cuts.

Formula Used:

Surface area of a sphere = 4πr²

Area of a circle = πr²

Calculation:

⇒ Total parts created = 8 identical spherical octants.

⇒ Total Surface Area (TSA) of each piece = Curved Surface Area + Flat Surface Area

⇒ Curved surface area = Total sphere curved area ÷ 8 = 4πr² ÷ 8 = πr²/2

⇒ Three mutually perpendicular cuts create 3 flat planar surfaces for each octant.

⇒ Each flat surface is exactly one quadrant (1/4^th) of a circle = πr²/4

⇒ Total flat surface area = 3 × (πr²/4) = 3πr²/4

⇒ TSA of one piece = πr²/2 + 3πr²/4 = 5πr²/4

⇒ Substitute r = 21 cm:

⇒ TSA = 5/4 × π × (21)²

⇒ TSA = (5 × 441π) ÷ 4

⇒ TSA = 2205π ÷ 4 = 551.25π cm²

∴ The correct answer is 551.25π cm².

Additional Information

Hemisphere (1 Cut)

When a sphere is cut into two equal halves, the Total Surface Area of each hemisphere is 3πr² (Curved area 2πr² + Base area πr²).

Quarter Sphere (2 Cuts)

When cut by two mutually perpendicular planes, it forms 4 identical pieces. The Total Surface Area of each quarter is 2πr² (Curved area πr² + 2 flat semi-circles πr²).

Spherical Octant (3 Cuts)

Formed by three mutually perpendicular cuts through the center, producing 8 identical pieces. Total Surface Area of each octant is 5πr²/4.

Shortcut Trick

Ratio of A : B : C : D = 3 : 5 : 7 : 11

Ratio part of E = (A + D) ÷ 2 = (3 + 11) ÷ 2 = 7

Total expenditure = 3 + 5 + 7 + 11 + 7 = 33 parts

Difference (C − B) = 7 − 5 = 2 parts = ₹18,000 ⇒ 1 part = ₹9,000

Total expenditure = 33 × 9,000 = ₹2,97,000

∴ The correct answer is ₹2,97,000.

Alternate Method

Given:

Ratio of expenditures on A, B, C, and D = 3 : 5 : 7 : 11

Expenditure on E is the arithmetic mean of A and D.

Difference in expenditure of C and B = ₹18,000

Calculations:

⇒ Let the respective expenditures on A, B, C, D, and E be 3x, 5x, 7x, 11x, and Ex.

⇒ Expenditure on E = (3x + 11x) ÷ 2 = 14x ÷ 2 = 7x

⇒ Expenditure of C − Expenditure of B = 7x − 5x = 2x

⇒ 2x = 18,000

⇒ x = 9,000

⇒ Total expenditure = 3x + 5x + 7x + 11x + 7x = 33x

⇒ Total expenditure = 33 × 9,000 = 2,97,000

∴ The correct answer is ₹2,97,000.

Additional Information

Arithmetic Mean

The arithmetic mean of two numbers a and b is their average, calculated as (a + b) ÷ 2.

Proportions and Common Multipliers

When values are given in a proportional format like a : b : c, their exact values can be written as ax, bx, and cx where x acts as a common variable multiplier.

Compound Ratio

If independent ratios a : b and c : d are given, their combined compound ratio is obtained by multiplying the corresponding parts, leading to (a × c) : (b × d).

Shortcut Trick

Direct formula: If x + 1/x = k, then x³ + 1/x³ = k³ − 3k.

Given, k = 2√3

Result = (2√3)³ − 3(2√3)

Result = 24√3 − 6√3 = 18√3

∴ The correct answer is 18√3.

Alternate Method

Given:

x + 1/x = 2√3

Formula Used:

(a + b)³ = a³ + b³ + 3ab(a + b)

Calculations:

⇒ Cubing both sides of the given equation:

⇒ (x + 1/x)³ = (2√3)³

⇒ x³ + 1/x³ + 3 × x × (1/x) × (x + 1/x) = 8 × 3√3

⇒ x³ + 1/x³ + 3(2√3) = 24√3

⇒ x³ + 1/x³ + 6√3 = 24√3

⇒ x³ + 1/x³ = 24√3 − 6√3

⇒ x³ + 1/x³ = 18√3

∴ The correct answer is 18√3.

Additional Information

Square Identity

If x + 1/x = k, then x² + 1/x² = k² − 2.

Minus Power Identity

If x − 1/x = k, then x³ − 1/x³ = k³ + 3k.

Fifth Power Expansion

To calculate x⁵ + 1/x⁵, multiply the results of (x² + 1/x²) and (x³ + 1/x³), then subtract (x + 1/x).

Shortcut Trick

For a system of linear equations to have no solution, the ratio of x-coefficients must equal the ratio of y-coefficients.

Ratio of x-coefficients = 3 ÷ 9 = 1/3

Ratio of y-coefficients = (−k) ÷ (−18) = k/18

Equating the ratios: k/18 = 1/3 ⇒ k = 6

∴ The correct answer is 6.

Alternate Method

Given:

Equation 1: 3x − ky + 8 = 0

Equation 2: 9x − 18y + 20 = 0

Formula Used:

For two linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 to have no solution:

a₁ / a₂ = b₁ / b₂ ≠ c₁ / c₂

Calculation:

⇒ Here, a₁ = 3, b₁ = −k, c₁ = 8

⇒ And, a₂ = 9, b₂ = −18, c₂ = 20

⇒ 3 / 9 = (−k) / (−18) ≠ 8 / 20

⇒ 1 / 3 = k / 18 ≠ 2 / 5

⇒ Since 1 / 3 ≠ 2 / 5 is true, we equate the first two terms:

⇒ k = 18 ÷ 3

⇒ k = 6

∴ The correct answer is 6.

Additional Information

Unique Solution (Intersecting Lines)

The system of equations has exactly one solution when the lines intersect at a single point. Condition: a₁ / a₂ ≠ b₁ / b₂.

Infinite Solutions (Coincident Lines)

The system of equations has infinitely many solutions when the lines overlap completely. Condition: a₁ / a₂ = b₁ / b₂ = c₁ / c₂.

No Solution (Parallel Lines)

The system of equations has no solution when the lines are parallel and never intersect. Condition: a₁ / a₂ = b₁ / b₂ ≠ c₁ / c₂.

Shortcut Trick

A number has exactly 3 divisors if and only if it is the square of a prime number.

We need to find prime numbers whose squares are ≤ 100.

The prime numbers up to 10 are 2, 3, 5, and 7.

Their respective squares are 4, 9, 25, and 49.

There are exactly 4 such integers in the given set.

∴ The correct answer is 4.

Alternate Method

Given:

Set of integers from 1 to 100.

Formula Used:

If N = p^a × q^b × r^c ..., where p, q, and r are prime factors, then:

Total number of divisors = (a + 1) × (b + 1) × (c + 1) ...

Calculations:

⇒ For exactly 3 divisors, the product of powers (a + 1)(b + 1)... must equal 3.

⇒ Since 3 is a prime number, it can only be written as (2 + 1).

⇒ Thus, the number must be of the form N = p², where p is a prime number.

⇒ We need to find all such numbers ≤ 100.

⇒ p² ≤ 100

⇒ p ≤ 10

⇒ The prime numbers ≤ 10 are 2, 3, 5, and 7.

⇒ Their squares are 2² = 4, 3² = 9, 5² = 25, and 7² = 49.

⇒ Total count of such integers is 4.

∴ The correct answer is 4.

Additional Information

Total Number of Divisors

For any integer N = p^a × q^b, the total number of divisors is given by the formula (a + 1)(b + 1). This includes 1 and the number itself.

Divisors of Perfect Squares

A perfect square always has an odd number of total divisors because all the powers of its prime factors are even numbers, making each (a + 1) an odd number.

Prime Numbers

A prime number has exactly 2 distinct divisors: 1 and the number itself. The only even prime number is 2, and all prime numbers greater than 2 are odd.

Shortcut Trick

Total ratio parts = 3 + 4 + 5 = 12 parts

12 parts = Rs. 4800 ⇒ 1 part = 4800 ÷ 12 = Rs. 400

Difference between B and A = 4 parts − 3 parts = 1 part

Required difference = 1 × 400 = Rs. 400

∴ The correct answer is Rs. 400.

Alternate Method

Given:

Total sum = Rs. 4800

Ratio of shares of A, B, and C = 3 : 4 : 5

Calculations:

⇒ Let the shares of A, B, and C be 3x, 4x, and 5x respectively.

⇒ Total sum = 3x + 4x + 5x

⇒ 12x = 4800

⇒ x = 4800 ÷ 12

⇒ x = 400

⇒ Difference between the shares of B and A = 4x − 3x

⇒ Difference = x

⇒ Difference = Rs. 400

∴ The correct answer is Rs. 400.

Additional Information

Proportion Basics

If four numbers are in proportion (a : b = c : d), then the product of extremes equals the product of means (a × d = b × c).

Mean Proportion

The mean proportional between two numbers a and b is given by the square root of their product: √(ab).

Third Proportion

For two numbers a and b, the third proportion c is calculated using the relation a : b = b : c, which results in c = b² ÷ a.

Compound Ratio

The compound ratio of two individual ratios a : b and c : d is found by multiplying the respective terms: (a × c) : (b × d).

Shortcut Trick

Given sin θ / x = cos θ / y ⇒ sin θ / cos θ = x / y ⇒ tan θ = x / y

For a right-angled triangle, if tan θ = Perpendicular / Base = x / y

Then Hypotenuse = √(x² + y²)

sin θ = P / H = x / √(x² + y²) and cos θ = B / H = y / √(x² + y²)

⇒ sin θ − cos θ = (x − y) / √(x² + y²)

∴ The correct answer is (x−y)/√(x²+y²).

Alternate Method

Given:

sin θ / x = cos θ / y

Formula Used:

tan θ = sin θ / cos θ = Perpendicular / Base

sin θ = Perpendicular / Hypotenuse

cos θ = Base / Hypotenuse

Hypotenuse = √(Perpendicular² + Base²)

Calculations:

⇒ sin θ / x = cos θ / y

⇒ sin θ / cos θ = x / y

⇒ tan θ = x / y

⇒ Let the corresponding sides of the right-angled triangle be Perpendicular = x and Base = y.

⇒ Hypotenuse = √(x² + y²)

⇒ Substituting values for sine and cosine:

⇒ sin θ = x / √(x² + y²)

⇒ cos θ = y / √(x² + y²)

⇒ Therefore, sin θ − cos θ = [x / √(x² + y²)] − [y / √(x² + y²)]

⇒ sin θ − cos θ = (x − y) / √(x² + y²)

∴ The correct answer is (x−y)/√(x²+y²).

Additional Information

Fundamental Trigonometric Identities

sin² θ + cos² θ = 1 | sec² θ − tan² θ = 1 | cosec² θ − cot² θ = 1

Pythagoras Theorem

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the perpendicular and the base: H² = P² + B².

Basic Trigonometric Ratios

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

Shortcut Trick

The sum of the squares of three adjacent face diagonals is twice the sum of the squares of the sides.

⇒ l² + w² + h² = (5 + 10 + 13) ÷ 2 = 14

⇒ Squares of sides: l² = 14 − 10 = 4, w² = 14 − 13 = 1, h² = 14 − 5 = 9

⇒ Sides are l = 2, w = 1, h = 3

⇒ Volume = l × w × h = 2 × 1 × 3 = 6

∴ The correct answer is 6 cm³.

Alternate Method

Given:

Face diagonals of a rectangular parallelepiped (cuboid) are √5 cm, √10 cm, and √13 cm.

Formula Used:

Length of face diagonals = √(l² + w²), √(w² + h²), √(l² + h²)

Volume of cuboid = l × w × h

Calculations:

Squaring the given face diagonals, we get the following system of equations:

⇒ l² + w² = 5 ... (i)

⇒ w² + h² = 10 ... (ii)

⇒ l² + h² = 13 ... (iii)

Adding equations (i), (ii), and (iii):

⇒ 2(l² + w² + h²) = 5 + 10 + 13 = 28

⇒ l² + w² + h² = 14 ... (iv)

Subtracting equation (ii) from (iv) to find l:

⇒ l² + 10 = 14 ⇒ l² = 4 ⇒ l = 2 cm

Subtracting equation (iii) from (iv) to find w:

⇒ w² + 13 = 14 ⇒ w² = 1 ⇒ w = 1 cm

Subtracting equation (i) from (iv) to find h:

⇒ h² + 5 = 14 ⇒ h² = 9 ⇒ h = 3 cm

Now, substitute the sides to calculate the volume:

⇒ Volume = l × w × h

⇒ Volume = 2 × 1 × 3 = 6 cm³

∴ The correct answer is 6 cm³.

Additional Information

Space Diagonal of a Cuboid

The longest distance inside a rectangular parallelepiped is its main space diagonal, given by the formula D = √(l² + w² + h²).

Total Surface Area (TSA)

The total surface area of a cuboid represents the sum of areas of all its 6 faces, which is calculated as TSA = 2(lw + wh + hl).

Lateral Surface Area (LSA)

The area of the 4 vertical walls excluding the top and bottom rectangular faces is calculated as LSA = 2h(l + w).

Shortcut Trick

List marks in ascending order: 5, 6, 7, 7, 7, 8, 8, 9, 10.

Mode is the most frequent value ⇒ 7 occurs three times, so Mode = 7.

Average = Sum ÷ Count = 67 ÷ 9 ≈ 7.44.

Marks above 7.44 are 8, 8, 9, 10 ⇒ 4 students.

∴ The correct answer is Mode = 7, 4 students.

Alternate Method

Given:

Marks of 9 students: 5, 8, 7, 9, 6, 7, 8, 7, 10

Formula Used:

Average = Sum of observations ÷ Total number of observations

Calculations:

⇒ Sum of marks = 5 + 8 + 7 + 9 + 6 + 7 + 8 + 7 + 10 = 67

⇒ Total number of students = 9

⇒ Average marks = 67 ÷ 9 ≈ 7.44

⇒ To determine the students above average, observe the marks greater than 7.44: 8, 9, 8, 10.

⇒ Number of students scoring above average = 4

⇒ To find the mode, determine the frequency of each distinct mark:

⇒ Mark 5 occurs 1 time

⇒ Mark 6 occurs 1 time

⇒ Mark 7 occurs 3 times

⇒ Mark 8 occurs 2 times

⇒ Mark 9 occurs 1 time

⇒ Mark 10 occurs 1 time

⇒ The mark with the highest frequency is 7.

⇒ Therefore, Mode = 7

∴ The correct answer is Mode = 7, 4 students.

Additional Information

Mean (Average)

The mean is calculated by adding all the values in a dataset and dividing the sum by the total number of observations. It represents the central point of the data.

Median

The median is the exact middle value of a sorted dataset. For an odd number of observations, it is the center value. For an even number, it is the average of the two central values.

Mode

The mode is the value that appears most frequently in a dataset. A set of data may have one mode (unimodal), multiple modes (multimodal), or no mode if all values occur with the same frequency.

Shortcut Trick

Using the deviation method, assume the average is 350.

Deviations: A(−100), B(+50), C(0), D(+150), E(−50) → Sum of deviations = +50.

Actual Average = 350 + (50 ÷ 5) = 350 + 10 = $360.

Families spending more than the average ($360) are B ($400) and D ($500).

Difference = Highest (500) − Lowest (250) = $250.

∴ The correct answer is Average = $360, Families above average = B, D, Difference =$250.

Alternate Method

Given:

Weekly expenditure: Family A = $250, Family B =$400, Family C = $350, Family D =$500, Family E = $300.

Formula Used:

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

Difference = Maximum value − Minimum value

Calculations:

⇒ Total expenditure = 250 + 400 + 350 + 500 + 300 = $1800

⇒ Average expenditure = 1800 ÷ 5 = $360

⇒ Comparing each family's expenditure with the calculated average ($360):

⇒ Family B ($400) and Family D ($500) spend more than the average.

⇒ Highest expenditure = $500 (Family D)

⇒ Lowest expenditure = $250 (Family A)

⇒ Difference = 500 − 250 = $250

∴ The correct answer is Average = $360, Families above average = B, D, Difference =$250.

Additional Information

Average (Arithmetic Mean)

It represents the central value of a finite set of numbers, computed as the sum of the values divided by the number of values.

Deviation Method for Averages

Instead of adding large numbers, choose an assumed mean. Calculate the individual deviations, find their average, and add it back to the assumed mean to find the exact average.

Range

The range of a set of data is the difference between the highest and lowest values, indicating the overall spread of the data set.

Shortcut Trick

In a right-angled triangle, the orthocenter (H) is at the vertex containing the 90° angle.

The circumcenter (O) is at the midpoint of the hypotenuse, making the median HO = Hypotenuse ÷ 2.

Hypotenuse = √(6² + 8²) = 10 cm ⇒ Median HO = 5 cm.

The centroid (G) lies on this median and divides it in a 2:1 ratio from the vertex.

Distance HG = ²/₃ × HO = ²/₃ × 5 = 10/3 cm.

∴ The correct answer is 10/3 cm.

Alternate Method

Given:

Right-angled triangle ABC, ∠B = 90°, AB = 6 cm, BC = 8 cm.

Formula Used:

Distance formula = √((x₂ − x₁)² + (y₂ − y₁)²)

Centroid G = ((x₁ + x₂ + x₃) ÷ 3, (y₁ + y₂ + y₃) ÷ 3)

Calculations:

⇒ Let B be the origin (0, 0). Since ∠B = 90°, the orthocenter H lies exactly at B(0, 0).

⇒ The coordinates of the vertices are A(0, 6), B(0, 0), and C(8, 0).

⇒ The coordinates of centroid G = ((0 + 0 + 8) ÷ 3, (6 + 0 + 0) ÷ 3) = (8/3, 2).

⇒ The distance between orthocenter H(0, 0) and centroid G(8/3, 2) is the length HG.

⇒ HG = √((8/3 − 0)² + (2 − 0)²)

⇒ HG = √(64/9 + 4)

⇒ HG = √((64 + 36) ÷ 9) = √(100/9)

⇒ HG = 10/3 cm.

∴ The correct answer is 10/3 cm.

Additional Information

Orthocenter

The orthocenter is the intersection of altitudes. In a right-angled triangle, it always lies at the vertex containing the right angle.

Circumcenter

The circumcenter is the intersection of perpendicular bisectors. In a right-angled triangle, it lies exactly at the midpoint of the hypotenuse.

Euler Line

In any triangle, the Orthocenter (H), Centroid (G), and Circumcenter (O) are collinear. The centroid divides the Euler line such that HG = 2 × GO.

Length of Median to Hypotenuse

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

Shortcut Trick

Convert the decimals to their nearest standard fractions for fast calculation:

Numerator = 1/6 × 5/6 × 1/3 = 5/108

Denominator = 2/9 × 2/3 × 1/8 = 4/216 = 1/54

Expression = (5/108) ÷ (1/54) = 5/108 × 54/1 = 5/2 = 2.5

∴ The correct answer is 2.5.

Alternate Method

Given:

Expression: (0.1667 × 0.8333 × 0.3333) ÷ (0.2222 × 0.6667 × 0.1250)

Method Used:

Convert the recurring and standard decimal values to exact fractions to simplify the multiplication and division.

Calculations:

⇒ 0.1667 ≈ 1/6

⇒ 0.8333 ≈ 5/6

⇒ 0.3333 ≈ 1/3

⇒ 0.2222 ≈ 2/9

⇒ 0.6667 ≈ 2/3

⇒ 0.1250 = 1/8

⇒ Substitute these fractional values into the original expression:

⇒ Numerator = (1/6) × (5/6) × (1/3)

⇒ Numerator = (1 × 5 × 1) / (6 × 6 × 3) = 5/108

⇒ Denominator = (2/9) × (2/3) × (1/8)

⇒ Denominator = (2 × 2 × 1) / (9 × 3 × 8) = 4/216

⇒ Simplify the denominator: 4/216 = 1/54

⇒ Now, divide the Numerator by the Denominator:

⇒ Result = (5/108) ÷ (1/54)

⇒ Result = (5/108) × 54

⇒ Result = 5/2

⇒ Result = 2.5

∴ The correct answer is 2.5.

Additional Information

Common Fraction to Decimal Conversions

Fractions involving 3: 1/3 ≈ 0.3333, 2/3 ≈ 0.6667

Fractions of Base 6 & 9

Base 6: 1/6 ≈ 0.1667, 5/6 ≈ 0.8333 | Base 9: 1/9 ≈ 0.1111, 2/9 ≈ 0.2222

Fractions of Base 8

1/8 = 0.125, 3/8 = 0.375, 5/8 = 0.625, 7/8 = 0.875

BODMAS Rule in Simplification

When solving such simplification problems, replacing recurring decimals with fractions not only makes calculation significantly faster but also prevents accuracy loss from rounding off intermediate values.

The position and positional values or English Alphabets are:

Given: ALMIRAH is written as BNPMWGO

The logic followed here is:

Similarly,

Thus, CLOSGOT would be written as DNRWLUA.

Hence, the correct answer is "Option 2".

The logic followed here is:

Given:

COMPUTER is coded as OCPMTURE

Similarly,

Thus, KEYBOARD is coded as EKBYAODR in that code.

Hence, the correct answer is "Option 2".

BODMAS Rule:

Decoding the given information:

Given equation: 36 − 2 + 4 × 10 ÷ 5

Now replacing the sign from left to right using BODMAS Rule, we get:

⇒ 36 × 2 ÷ 4 + 10 - 5 = ?

⇒ 36 × 0.5 + 10 - 5 = ?

⇒ 18 + 10 - 5 = ?

⇒ 28 - 5 = ?

⇒ 23 = ?

Hence, the correct answer is "Option 1".

The paper when folded will appear as shown below:

Hence, the correct answer is "Option 2".

The position and positional values or English Alphabets are:

Given: EV, GT, JQ, OL,?

The logic followed here is:

Thus, VE will replace the question mark.

Hence, the correct answer is "Option 3".

The logic is:

 → First two elements 'TC' rotates 180 degree clockwise. Thirst element '()' shifts to 4^th position in figure B and is replaced by a solid black color filled star. 4^th element 'S' shifts to 3^rd position in figure B and is rotated 90 degree clockwise.

Similarly,

 → First two elements rotates 180 degree clockwise. Thirst element rectangle shifts to 4^th position in figure B and is replaced by a solid black color filled circle. 4^th element '+' shifts to 3^rd position in figure B and is rotated 90 degree clockwise.

Hence, the correct option is “Option 2”.

Family Chart:

Given: P is married to M. W is the wife of S. G is the daughter of A. M is the mother of S. A is the sister of P.

Thus, P is W's Father-in-law.

Hence, the correct answer is "Option 4".

The mirror image of the given question figure, when mirror is placed at the right side of the question figure is shown below:

Hence, the correct answer is "Option 2".

Family Chart:

Given: Pointing towards Sakshi, I said, "She is the daughter of my wife's son's brother."

Thus, I am grandfather of Sakshi.

Hence, the correct answer is "Option 3".

BODMAS Rule:

Given equation: 20 ÷ 5 × 3 + 10 − 4 = 20

1. 20 and 10

Given equation: 20 ÷ 5 × 3 + 10 − 4 = 20

Swapping the numbers in the equation, we get:

⇒ 10 ÷ 5 × 3 + 20 − 4 = 20

⇒ 2 × 3 + 20 − 4 = 20

⇒ 6 + 20 − 4 = 20

⇒ 26 − 4 = 20

⇒ 22 ≠ 20

⇒ LHS ≠ RHS

2. 3 and 4

Given equation: 20 ÷ 5 × 3 + 10 − 4 = 20

Swapping the numbers in the equation, we get:

⇒ 20 ÷ 5 × 4 + 10 − 3 = 20

⇒ 4 × 4 + 10 − 3 = 20

⇒ 16 + 10 − 3 = 20

⇒ 26 − 3 = 20

⇒ 23 ≠ 20

⇒ LHS ≠ RHS

3. 4 and 5

Given equation: 20 ÷ 5 × 3 + 10 − 4 = 20

Swapping the numbers in the equation, we get:

⇒ 20 ÷ 4 × 3 + 10 − 5 = 20

⇒ 5 × 3 + 10 − 5 = 20

⇒ 15 + 10 − 5 = 20

⇒ 25 − 5 = 20

⇒ 20 = 20

⇒ LHS = RHS

4. 5 and 10

Given equation: 20 ÷ 5 × 3 + 10 − 4 = 20

Swapping the numbers in the equation, we get:

⇒ 20 ÷ 10 × 3 + 5 − 4 = 20

⇒ 2 × 3 + 5 − 4 = 20

⇒ 6 + 5 − 4 = 20

⇒ 11 − 4 = 20

⇒ 7 ≠ 20

⇒ LHS ≠ RHS

Hence, the correct answer is "Option 3".

Given: cd_fc_efcdefcde_cdefcde_cdef

Let's check each option one by one:

1. fcde → cdff/ccef/cdef/cde**d/cdef/cdee/**cdef → No pattern is repeated.
2. cddf → cdcf/cdef/cdef/cde**d/cdef/cdef/**cdef → No pattern is repeated.
3. efdf → cdef/cfef/cdef/cde**d/cdef/cdef/**cdef → No pattern is repeated.
4. edff → cdef/cdef/cdef/cde**f/cdef/cdef/**cdef → 'cdef' is repeated.

Hence, the correct answer is "Option 4".

Statements:

All televisions are radios.

Some radios are transistors.

The least possible Venn diagram for the given statement is:

Conclusions:

I. All radios are televisions. - False (Because All televisions are radios, which means some radios are televisions.)

II. Some televisions are transistors. - False (There is no definite relation between televisions and transistors, it can be possible but it is not definite)

Thus, Neither conclusion I nor II follows.

Hence, the correct answer is "Option 1".

The logic followed here is:

• All Fathers are Males.
• Some Males can be Teachers.
• Some Teachers can be Males.
• Some Fathers may also be Teachers (but not necessarily all).

So the correct Venn relation is:

Hence, the correct answer is "Option 1".

The logic followed here is:

Logic: a³ + a = b

3 : 30 → 3³ + 3 = 27 + 3 = 30

4 : 68 → 43 + 4 = 64 + 4 = 68

Similarly,

5 : ? → 53 + 5 = ?

⇒ 125 + 5 = ?

⇒ 130 = ?

Hence, the correct answer is "Option 2".

The logic followed here is:

Logic: (A + B + C + D) ) × 2 = Result

Row 1: (5 + 2 + 3 + 10) × 2 = 20 × 2 = 40

Row 2: (4 + 3 + 5 + 8) × 2 = 20 × 2 = 40

Similarly,

Row 3: (6 + 4 + 2 + 12) × 2 = 24 × 2 = 48

Thus, '48' will come in the place of ?.

Hence, the correct answer is "Option 3".

Given: S_ KB_FH_BSF_K_SF _B

Let's check each option:

1. FHSKBHHK → SFHKB / SFHKB / SFBKH / SFHKB → No pattern is repeated. 2. FHSKHKHK → SFHKB / SFHKB / SFHKK / SFHKB → No pattern is repeated. 3. FHSKHBHK → SFHKB / SFHKB / SFHKB / SFHKB → SFHKB is repeated. 4. FHSHKBHK → SFHKB / SFHHB / SFKKB / SFHKB → No pattern is repeated.

Thus, 'FHSKHBHK' completes the series logically.

Hence, the correct answer is "Option 3".

The image embedded in given figure is as shown below:

Hence, the correct answer is "Option 2".

The position and positional values or English Alphabets are:

Given: ZYA, WVD, TSG, QPJ,?

The logic followed here is:

Thus, NMM should come next in the sequence.

Hence, the correct answer is "Option 3".

The logic followed here is:

Logic: a : (a + 3)²

4 : 49 → 4 : (4 + 3)² = 4 : 7² = 4 : 49

6 : 81 → 6 : (6 + 3)² = 6 : 92 = 6 : 81

Similarly,

8 : ?

⇒ ? = (8 + 3)2

⇒ ? = 112

⇒ ? = 121

Hence, the correct answer is "Option 4".

Shortcut Trick

In mathematical operations, always check the division symbol (Q) first to eliminate options with fractional results.

In Option 3, 5 Q 10 simplifies exactly to 1/2.

Applying 16 S (multiply by 16) gives 16 × 1/2 = 8, a whole number.

Solving the rest: 8 + 4 − 3 = 9, which matches the RHS perfectly.

∴ The correct answer is 16 S 5 Q 10 P 4 R 3 = 9.

Alternate Method

Given:

P = +

Q = ÷

R = −

S = ×

Method Used:

Apply the BODMAS rule to check each option after substituting the operators.

Calculations:

Checking Option 1: 16 P 5 R 10 S 4 Q 3 = 19

⇒ 16 + 5 − 10 × 4 ÷ 3

⇒ 21 − 40/3 = 23/3 ≈ 7.66 ≠ 19

Checking Option 2: 16 S 5 R 10 P 4 Q 3 = 9

⇒ 16 × 5 − 10 + 4 ÷ 3

⇒ 80 − 10 + 1.33 = 71.33 ≠ 9

Checking Option 3: 16 S 5 Q 10 P 4 R 3 = 9

⇒ 16 × 5 ÷ 10 + 4 − 3

⇒ 16 × (1/2) + 4 − 3

⇒ 8 + 4 − 3

⇒ 12 − 3 = 9 (Correct)

Checking Option 4: 16 Q 5 P 10 S 4 R 3 = 12

⇒ 16 ÷ 5 + 10 × 4 − 3

⇒ 3.2 + 40 − 3 = 40.2 ≠ 12

∴ The correct answer is 16 S 5 Q 10 P 4 R 3 = 9.

Additional Information

BODMAS Rule

It dictates the correct order of mathematical operations: Brackets, Orders, Division, Multiplication, Addition, and Subtraction.

Elimination Technique

Always identify the division operations first. If a division results in a non-terminating fraction, and the final answer is an integer, that option can usually be eliminated quickly.

Sign Substitution

Ensure accurate replacement of symbols before applying BODMAS. Substituting one wrong sign completely changes the equation value.

The position and positional values or English Alphabets are:

The logic followed here is:

WOODLAND : ZIRXOUQX

PERIODIC : SYUCRXLW

Similarly,

Thus, MAGAZINE is related to PUJUCCQY.

Hence, the correct answer is "Option 4".

Given word: PROCRASTINATE

Let's check each option:

1. PATTERN → PROCRASTINATE → Can be formed. 2. PRINTED → PROCRASTINATE → Cannot be formed as D is missing. 3. TRACTOR → PROCRASTINATE → Can be formed. 4. PAINTER → PROCRASTINATE → Can be formed.

Hence, the correct answer is "Option 2".

The figure which would most closely resemble the unfolded form of figure Z is shown below:

Thus, figure B resembles the unfolded form of figure Z.

Hence, the correct answer is "Option 2".

The number of squares present in the figure is shown below:

The figure has 19 squares.

Hence, the correct answer is "Option 4".

Given: 

Now, checking each option:

1. 2, 5, 1, 8, 7, 3, 4, 6, 9

The above option does not represent a meaningful word. 

2. 5, 1, 2, 3, 7, 8, 6, 4, 9

The above option represents a meaningful word. Indicator means a thing that indicates the state or level of something.

3. 4, 9, 1, 3, 2, 8, 6, 7, 5

The above option does not represent a meaningful word.

4. 5, 2, 1, 9, 8, 4, 3, 7, 6

The above option does not represent a meaningful word.

Hence, the correct answer is "Option 2".

The logic is:

1. 31, 76, 24, 96 → A, V, F, X 2. 20, 59, 41, 88 → D, S, L, T 3. 33, 68, 31, 75 → J, O, A, N 4. 42, 57, 75, 11 → D, U, N, G

So, DUNG is represented by '42, 57, 75, 11'.

Hence, the correct answer is "Option 4".

Statement: Humans cannot match the data processing speed of supercomputers, which perform billions of calculations per second.

Inferences:

I. Humans are less efficient than supercomputers in data processing. - Follows (Because, humans cannot match the data processing speed of supercomputers, which perform billions of calculations per second. So, we can conclude that humans are less efficient than supercomputers in data processing.)

II. Humans and supercomputers differ in their capabilities. - Follows (Because, humans cannot match the processing speed of supercomputers, there is clearly a difference in capability.)

Thus, both I and II follows.

Hence, the correct answer is "Option 2".

The number of squares present in the figure is shown below:

The figure has 16 squares.

Hence, the correct answer is "Option 1".

For find the opposite surface of '6', we need to compare the common surface in the given dices, Now, numbers '3' and '5' are common in Dice 1 and Dice 2,

So, From Dice 1 and Dice 2,

→ 3 and 5 is the common surface. So, the remaining '1' will be the opposite face of '6'.

Hence, the correct answer is "Option 4".

The logic followed here is:

2 × 2² = 2 × 4 = 8

2 × 42 = 2 × 16 = 32

2 × 62 = 2 × 36 = 72

2 × 82 = 2 × 64 = 128

2 × 102 = 2 × 100 = 200

Thus, 200 is the missing number in the series.

 

Hence, the correct answer is "Option 2".

The correct answer is Option 2.

 

Key Points

• The primary issue in the sentence is the omission of the word "than" after the term "more."
• When comparing amounts or indicating a quantity exceeding a certain limit, the comparative "more" must be followed by "than."
• The correct phrase should be "invest more than," ensuring grammatical correctness and clarity.
• Thus, the error is located in part A of the sentence.

Therefore, the correct answer is Option 2.

 

Correct Sentence: "The company will invest more than six hundred crores in the next five years to expand its operations in Britain."

 

Additional Information

• B: "six hundred crores" – This part of the sentence is grammatically correct and does not require any changes.
• C: "in the next five years to expand" – This part of the sentence is grammatically correct and does not require any changes.
• D: "its operations in Britain" – This part of the sentence is grammatically correct and does not require any changes.

The correct answer is Option 1.

 

Key Points

• The sentence logically flows from introducing a main subject (extreme weather events) to describing where it happened, and finally explaining its cause.
• Part B is the first part as it introduces the main subject of the sentence:
• "the unprecedented frequency of extreme weather events"
• Part D follows as it acts as a modifier describing where and when these events were observed:
• "observed across various continents over the last decade"
• Part A is the third part because it introduces the verb phrase and the agents performing the action:
• "has been attributed by many climate scientists"
• Part C is the fourth as it completes the phrase "attributed to" by providing the object or cause:
• "to the steady rise in global atmospheric temperatures"
• The correct sequence is BDAC.

 

Coherent sentence after rearrangement:

"The unprecedented frequency of extreme weather events observed across various continents over the last decade has been attributed by many climate scientists to the steady rise in global atmospheric temperatures."