HPAS Prelims 2024 CSAT (Paper II) with Answers & Explanations

Explanation

For any geometric figure standing on a base b and situated between two parallel lines, the perpendicular distance between the parallel lines represents its height h. The formula for the area of a parallelogram is base × height. Since the parallelogram with area A₁ and the rhombus with area A₂ stand on the same base b and lie between the same parallels, they both have the same base and height. Therefore, A₁ = b × h and A₂ = b × h, giving A₁ = A₂. The formula for the area of a triangle is ½ × base × height. Since the triangle with area A₃ stands on the same base b and lies between the same parallels, its area is A₃ = ½ × b × h. Multiplying both sides of this equation by 2 gives 2A₃ = b × h. Equating these expressions yields the relation A₁ = A₂ = 2A₃.

Explanation

The area of a triangle is given by the formula: Area = ½ × base × height

For the given triangle ABC, all sub-triangles formed by vertex A and any segment along the base BC share the same perpendicular height h from A to BC.

The n equidistant points A₁, A₂, ..., Aₙ on the base BC divide it into n + 1 segments of equal length: BA₁, A₁A₂, A₂A₃, ..., AₙC. Let the length of each of these equal segments be x. The total length of the base BC is therefore (n + 1)x.

Since the segment A₅A₆ is one of these equal segments, its length is x. The area of triangle AA₅A₆ is given by: k = ½ × A₅A₆ × h = ½ × x × h

Using this relation, the area of triangle ABC is: Area(ABC) = ½ × BC × h = ½ × (n + 1)x × h = (n + 1) × (½ × x × h) = (n + 1)k

Explanation

Pinky is 21st from the right end. Since Rinky is 10th to the left of Pinky, Rinky's position from the right end is 21 + 10 = 31st.

Minky is 17th from the left end and is fourth to the right of Rinky. This means Rinky is fourth to the left of Minky. Therefore, Rinky's position from the left end is 17 - 4 = 13th.

To find the total number of girls in the row, we use the standard position relation: Total = Position from left + Position from right - 1

Substituting the positions of Rinky into the formula: Total = 13 + 31 - 1 = 43

Explanation

Let the given function be f(x) = sinᵐ x sin mx + cosᵐ x cos mx.

To find the maximum value of f(x), we can apply the Cauchy-Schwarz inequality:

(sinᵐ x sin mx + cosᵐ x cos mx)² ≤ (sin²ᵐ x + cos²ᵐ x) (sin² mx + cos² mx)

Since sin² mx + cos² mx = 1, this simplifies to:

(sinᵐ x sin mx + cosᵐ x cos mx)² ≤ sin²ᵐ x + cos²ᵐ x

For any integer m ≥ 1, we have 2m ≥ 2. Since |sin x| ≤ 1 and |cos x| ≤ 1, it follows that sin²ᵐ x ≤ sin² x and cos²ᵐ x ≤ cos² x.

Thus, we get:

sin²ᵐ x + cos²ᵐ x ≤ sin² x + cos² x = 1

This implies:

(sinᵐ x sin mx + cosᵐ x cos mx)² ≤ 1

Consequently, the absolute maximum value that the function can attain is 1.

Let us evaluate the function at the values given in the options:

For x = π: f(π) = sinᵐ π sin mπ + cosᵐ π cos mπ f(π) = 0 + (-1)ᵐ cos mπ

If m is an even integer, (-1)ᵐ = 1 and cos mπ = 1, which gives f(π) = 1. If m is an odd integer, (-1)ᵐ = -1 and cos mπ = -1, which gives f(π) = (-1) × (-1) = 1.

Thus, for any integer m, the function always achieves its maximum value of 1 at x = π.

Evaluating at x = π/4 for other options gives: For m = 2: f(π/4) = (1/√2)² sin(π/2) + (1/√2)² cos(π/2) = ½ For m = 4: f(π/4) = (1/√2)⁴ sin(π) + (1/√2)⁴ cos(π) = -¼ For m = 6: f(π/4) = (1/8) sin(3π/2) + (1/8) cos(3π/2) = -⅛

Explanation

The pattern of the number series is division by 4. Starting from the first term:

24576 ÷ 4 = 6144 6144 ÷ 4 = 1536 1536 ÷ 4 = 384 384 ÷ 4 = 96 96 ÷ 4 = 24

According to this pattern, the fourth term must be 384 instead of 386, making 386 the wrong number in the series.

Explanation

The analogy represents a celestial body and its scientific classification. The Moon is a natural satellite of the Earth, making satellite its defining category. Following the same relationship, the Earth is classified as a planet.

Explanation

To determine the positions of the seven people sitting around the circular table facing the center, we can establish their relative positions step-by-step:

Let Hari be at position 1.

According to the conditions, Kirti is second to the left of Hari, which places Kirti at position 3.

Since Kirti is the neighbour of Tulsi and Ravi, they must occupy positions 2 and 4. We are given that Hari is not a neighbour of Tulsi, which means Tulsi cannot be at position 2. Therefore, Tulsi must be at position 4, and Ravi must be at position 2.

The remaining three positions, 5, 6, and 7, must be occupied by Ram, Shyam, and Mohan. Since Ram is the neighbour of both Shyam and Mohan, Ram must occupy the middle seat at position 6, while Shyam and Mohan occupy positions 5 and 7.

Since Hari at position 1 cannot be a neighbour of Mohan, Mohan cannot be at position 7. Thus, Mohan must be at position 5, and Shyam must be at position 7.

This gives the complete clockwise seating arrangement: Hari Ravi Kirti Tulsi Mohan Ram Shyam

In this circular arrangement, Shyam (at position 7) sits directly between Ram (at position 6) and Hari (at position 1).

Explanation

To solve the seating arrangement, we can place the seven people around a circular table in positions 1 to 7 in a clockwise direction. Assuming everyone is facing the center of the table, clockwise movement represents the left direction, while counter-clockwise movement represents the right direction.

First, let Hari sit at position 1. Since Kirti is second to the left of Hari, she sits at position 3. Since Kirti is a neighbor of Tulsi and Ravi, they must occupy positions 2 and 4 in some order.

Second, Ram is the neighbor of Shyam and Mohan. This requires Ram to sit between Shyam and Mohan. The only remaining three consecutive seats are 5, 6, and 7, meaning Ram must sit at position 6, while Shyam and Mohan occupy positions 5 and 7.

Third, Hari at position 1 cannot be adjacent to Mohan or Tulsi. Since position 7 is adjacent to position 1, it cannot be occupied by Mohan, meaning Shyam is at position 7 and Mohan is at position 5. Similarly, position 2 is adjacent to position 1 and cannot be occupied by Tulsi, meaning Ravi is at position 2 and Tulsi is at position 4.

This gives the clockwise seating sequence: Hari (1) - Ravi (2) - Kirti (3) - Tulsi (4) - Mohan (5) - Ram (6) - Shyam (7)

Since right-hand side movement corresponds to the counter-clockwise direction, the person on the immediate right of Kirti at position 3 is Ravi at position 2. Therefore, in the pair of Kirti and Ravi, the second person (Ravi) is sitting on the immediate right of the first person (Kirti).

Explanation

To compare the numbers 4¹/³, 5¹/⁴, 2¹/², and 3¹/⁶, find the least common multiple (LCM) of the denominators of the fractional powers. The denominators are 3, 4, 2, and 6, and their LCM is 12.

Raising each number to the power of 12 to eliminate the fractional exponents: (4¹/³)¹² = 4⁴ = 256 (5¹/⁴)¹² = 5³ = 125 (2¹/²)¹² = 2⁶ = 64 (3¹/⁶)¹² = 3² = 9

Comparing these integer values gives 256 > 125 > 64 > 9, which directly corresponds to the order 4¹/³ > 5¹/⁴ > 2¹/² > 3¹/⁶.

Explanation

Let the third number be 100.

The first number is 20% of the third number, which is 20. The second number is 50% of the third number, which is 50.

To find what percent the first number is of the second number, use the percentage formula:

Percentage = (First Number / Second Number) × 100%

Substituting the values:

Percentage = (20 / 50) × 100% = (2/5) × 100% = 40%

Explanation

The simple interest formula is: SI = (P × R × T) / 100

where SI is the simple interest, P is the principal, R is the rate of interest, and T is the time in years.

According to the question, the simple interest is 2/5 of the principal: SI = 2/5 × P

The rate of interest is R = 8% per annum. Substituting these values into the formula yields: 2/5 × P = (P × 8 × T) / 100

Dividing both sides by P: 2/5 = 8T / 100

Multiplying both sides by 100: (2 × 100) / 5 = 8T 40 = 8T T = 5

Explanation

The mathematical operation represented by the multiplication symbol in this puzzle is determined by summing the digits of each individual number and then multiplying those two sums together.

For the first term: Sum of digits of 324 = 3 + 2 + 4 = 9 Sum of digits of 150 = 1 + 5 + 0 = 6 Product of the digit sums = 9 × 6 = 54

For the second term: Sum of digits of 251 = 2 + 5 + 1 = 8 Sum of digits of 402 = 4 + 0 + 2 = 6 Product of the digit sums = 8 × 6 = 48

For the third term: Sum of digits of 523 = 5 + 2 + 3 = 10 Sum of digits of 246 = 2 + 4 + 6 = 12 Product of the digit sums = 10 × 12 = 120

Applying this rule to 651 × 345: Sum of digits of 651 = 6 + 5 + 1 = 12 Sum of digits of 345 = 3 + 4 + 5 = 12 Product of the digit sums = 12 × 12 = 144

Explanation

The sequence MPON deviates from the pattern shared by the other options. In a standard sequence of this group, the first and third letters are consecutive in alphabetical order (+1), and the second and fourth letters are also consecutive in alphabetical order (+1). Additionally, the first and fourth letters, as well as the second and third letters, are pairs of opposite letters in the alphabet whose position values sum to 27.

In the sequence MPON, where M = 13, P = 16, O = 15, and N = 14, the first and third letters (M and O) increase by 2, and the second and fourth letters (P and N) decrease by 2. Furthermore, the second and third letters (P and O) are not alphabetical opposites because the opposite of P is K, and the opposite of O is L. Therefore, this sequence is the odd one out.

Explanation

During the night, photosynthesis stops in trees due to the absence of sunlight, but respiration continues. This respiration process consumes oxygen and releases carbon dioxide (CO₂). Sleeping under a tree at night exposes a person to a higher concentration of CO₂ in the immediate surrounding air, which can cause breathing difficulties and is injurious to health. Therefore, statement (II) acts as the logical and scientifically valid reason behind the given advice, making it implicit.

Explanation

Let the reading positions of the five men from first to last be numbered 1, 2, 3, 4, and 5.

The first reader gives the newspaper to C, which means C is the second reader at position 2.

The last reader at position 5 takes the newspaper from A, meaning A is the fourth reader at position 4.

At this stage, the sequence is _ , C, _ , A, _ .

Since E is neither the first nor the last reader, the only available position for E is position 3.

The sequence becomes _ , C, E, A, _ .

There are two readers between B and A. Since A is at position 4, B must be at position 1 so that C and E are the two readers between them.

The sequence is now B, C, E, A, _ .

The only remaining person, D, must be the fifth and final reader at position 5. This completes the sequence as B, C, E, A, D.

Explanation

The net percentage change in daily earnings, which is the product of working hours and wages per hour, is calculated using the successive percentage change formula:

Net percentage change = a + b + (a × b) / 100

Here, the increase in working hours (a) is 20% and the increase in wages per hour (b) is 15%.

Substituting these values into the formula:

Net percentage change = 20 + 15 + (20 × 15) / 100 Net percentage change = 35 + 300 / 100 Net percentage change = 35 + 3 = 38%

Explanation

The coding pattern splits the ten-letter word into two equal halves of five letters each.

For the first half of the word, STABI, the letters are reversed to become IBATS, and then each letter is shifted forward by one position (+1): I → J B → C A → B T → U S → T This yields JCBUT.

For the second half of the word, LISED, the letters are reversed to become DESIL, and then each letter is shifted backward by one position (-1): D → C E → D S → R I → H L → K This yields CDRHK.

Combining both halves results in JCBUTCDRHK.

Explanation

January 1, 1991 fell on a Tuesday and 1991 is a non-leap year. Tracking day-of-week advances (+1 for a normal year, +2 for a leap year), the next non-leap year that also starts on a Tuesday is 2002 — but 2002 is absent from the given options. This question is defective due to a likely print typo (2002 printed as 2003). Option B (2003) is exactly one year off from the mathematically correct answer and is the best available choice.

Explanation

The angle θ between the hour hand and the minute hand of a clock is given by the formula:

θ = |30H - (11/2)M|

where H is the hour and M is the minutes.

Substituting H = 9 and M = 45:

θ = |30 × 9 - (11/2) × 45| θ = |270 - 247.5| θ = 22.5°

This simplifies to 22½°.

Explanation

Average Speed = Total Distance ÷ Total Time

First, the total distance covered by the cyclist is calculated: Total Distance = 24 km + 36 km = 60 km

Next, the time taken for each part of the journey is determined: Time for the first part = 24 km ÷ 16 km/h = 1.5 hours Time for the second part = 36 km ÷ 15 km/h = 2.4 hours

The total time for the entire journey is the sum of these times: Total Time = 1.5 hours + 2.4 hours = 3.9 hours

Using these values, the average speed is: Average Speed = 60 km ÷ 3.9 hours ≈ 15.38 km/h

Explanation

The population after n years with a compound growth rate of r% per year is given by:

A = P × (1 + r/100)ⁿ

Here, A is the present population, P is the population n years ago, r is the annual growth rate, and n is the time in years. Substituting the given values:

10648 = P × (1 + 10/100)³ 10648 = P × (11/10)³ 10648 = P × (1331/1000) P = (10648 × 1000) / 1331 P = 8 × 1000 P = 8000

Explanation

Let the total number of voters be 100x.

Initially, the voters promised to vote as follows: Promised to Bansi = 40% of 100x = 40x Promised to Pappu = 60% of 100x = 60x

On election day, some voters changed their votes: Voters who switched from Bansi to Pappu = 15% of 40x = 6x Voters who switched from Pappu to Bansi = 25% of 60x = 15x

Now, calculate the final votes obtained by each candidate: Final votes for Bansi = 40x - 6x + 15x = 49x Final votes for Pappu = 60x - 15x + 6x = 51x

The difference in votes between Pappu and Bansi is: 51x - 49x = 2x

Since Bansi lost by 2 votes, we set up the equation: 2x = 2 x = 1

The total number of voters is 100x = 100 × 1 = 100.

Explanation

Let the selling price of one table be S.

The selling price of 27 tables is 27S.

The gain obtained by selling 27 tables is equal to the selling price of 5 tables, which is 5S.

Using the relation: Cost Price = Selling Price - Gain

The cost price of 27 tables is: Cost Price = 27S - 5S = 22S

The gain percentage is calculated as: Gain Percentage = (Gain / Cost Price) × 100 Gain Percentage = (5S / 22S) × 100 Gain Percentage = 500/22 % = 250/11 % = 22 8/11 %

Explanation

Let S, SS, and US denote the daily work output of one skilled, one semiskilled, and one unskilled worker, respectively.

Since 5 skilled workers build the home in 20 days, the daily work of 1 skilled worker is: S = 1 / (5 × 20) = 1/100

Since 8 semiskilled workers build the home in 25 days, the daily work of 1 semiskilled worker is: SS = 1 / (8 × 25) = 1/200

Since 10 unskilled workers build the home in 30 days, the daily work of 1 unskilled worker is: US = 1 / (10 × 30) = 1/300

The combined daily work rate of a team of 2 skilled, 6 semiskilled, and 5 unskilled workers is: Combined Rate = 2S + 6SS + 5US Combined Rate = 2/100 + 6/200 + 5/300 Combined Rate = 1/50 + 3/100 + 1/60

Using 300 as the common denominator: Combined Rate = (6 + 9 + 5) / 300 Combined Rate = 20/300 Combined Rate = 1/15

Since the team completes 1/15 of the work in one day, the number of days required to build the entire home is 15.

Explanation

Let the initial price of the article be 100.

After a 30% increase, the price becomes: 100 × (1 + 0.30) = 130

Applying the first discount of 10% gives: 130 × (1 - 0.10) = 130 × 0.90 = 117

Applying the second discount of 10% gives: 117 × (1 - 0.10) = 117 × 0.90 = 105.3

The net change from the original price of 100 to the final price of 105.3 represents a percentage increase of: 105.3 - 100 = 5.3%

Explanation

Let the two numbers be 29a and 29b, where a and b are co-prime integers. Since both numbers are greater than 29, both a and b must be greater than 1.

The relationship between the least common multiple (LCM), highest common factor (HCF), and the co-prime factors is given by: LCM = HCF × a × b

Substituting the given values into the formula: 4147 = 29 × a × b a × b = 4147 / 29 a × b = 143

The factors of 143 are 1 and 143, or 11 and 13. Since both a and b are greater than 1, we choose: a = 11 b = 13

The two numbers are: 29 × 11 = 319 29 × 13 = 377

The sum of these two numbers is: 319 + 377 = 696

Explanation

Let the four numbers be x₁, x₂, x₃, and x₄.

The average of the first three numbers is thrice the fourth number: (x₁ + x₂ + x₃) / 3 = 3x₄ x₁ + x₂ + x₃ = 9x₄

The average of all four numbers is 5: (x₁ + x₂ + x₃ + x₄) / 4 = 5 x₁ + x₂ + x₃ + x₄ = 20

Substituting the first equation into the second: 9x₄ + x₄ = 20 10x₄ = 20 x₄ = 2

Explanation

To calculate the total number of male members in the family, we sum the males across all generations. First, the patriarch of the family represents 1 male. Second, the patriarch has 4 sons, which adds 4 males to the count. Third, each of these 4 sons has 3 sons of their own, contributing another 4 × 3 = 12 males. Summing these values gives the total number of male members as 1 + 4 + 12 = 17.

Explanation

The expression P ÷ R + S + Q is decoded from left to right using the given definitions:

P ÷ R means P is the daughter of R. This implies P is female.

R + S means R is the father of S. Since both P and S are children of R, they are siblings. Since P is female, P is the sister of S.

S + Q means S is the father of Q.

Since S is the father of Q and P is the sister of S, P is the father's sister of Q. Therefore, P is the aunt of Q.

Explanation

The expression P × R ÷ Q can be decoded step-by-step using the given definitions. First, P × R indicates that P is the brother of R, which means P is male. Second, R ÷ Q indicates that R is the daughter of Q. Since P is the brother of R and R is the daughter of Q, P and R are siblings who share the same parent, Q. Because P is male, P must be the son of Q.

Explanation

The relation is represented by the expression P - R + Q.

From the given definitions, P - R means P is the wife of R. This establishes that P is female and R is her husband.

Further, R + Q means R is the father of Q.

Since R is the father of Q and P is married to R, P is the mother of Q.

Explanation

The given expression is P × R - Q.

Using the given codes: P × R means P is the brother of R, which implies P is a male. R - Q means R is the wife of Q, which implies R is a female and Q is her husband (male).

Since P is the brother of Q's wife R, P is the brother-in-law of Q.

Explanation

To divide the cuboid into 72 cubes (64 smaller cubes of side 1 and 8 bigger cubes of side 2), the dimensions of the solid cuboid must be 4 × 4 × 8, giving a total volume of 128 units: 64 × 1³ + 8 × 2³ = 64 + 64 = 128

The 8 bigger cubes of size 2 × 2 × 2 are placed in the middle section of the cuboid, spanning from z = 2 to z = 6. The 64 smaller cubes of size 1 × 1 × 1 are placed at the two ends, from z = 0 to z = 2 (32 cubes) and from z = 6 to z = 8 (32 cubes).

To satisfy the condition that none of the bigger cubes have any Black colored surfaces, the two opposite faces of size 4 × 4 at the ends (z = 0 and z = 8) must be colored Black. Since the bigger cubes are positioned in the middle region (z = 2 to z = 6), they never touch the Black faces.

The remaining four faces of size 4 × 8 are colored Red and Green in opposite pairs. Let the two opposite faces at x = 0 and x = 4 be colored Red.

To find the number of cubes with at least one Red surface, we count the cubes touching either x = 0 or x = 4.

For the 8 bigger cubes in the middle section, the width along the x-axis is 4. Since each bigger cube has a side length of 2, there are only 2 cubes along the x-direction. Thus, all 8 bigger cubes touch either the face x = 0 or x = 4. This contributes 8 cubes.

For the smaller cubes at each end, the total number of cubes in each 4 × 4 × 2 section is 32. The cubes that do not touch the Red faces lie in the inner columns along the x-axis (x ∈ [1, 3]). In each section, there are 2 (along x) × 4 (along y) × 2 (along z) = 16 such inner cubes. Consequently, the number of smaller cubes touching the Red faces in each end section is 32 − 16 = 16. This contributes 16 + 16 = 32 cubes.

Adding these together, the total number of cubes with at least one Red surface is 8 + 32 = 40.

Explanation

To find the number of cubes with only one face colored, we first determine the dimensions of the cuboid.

Let the side of a smaller cube be s and that of a bigger cube be 2s. The volume of a smaller cube is s³ and that of a bigger cube is 8s³.

The cuboid is cut into 64 smaller cubes and 8 bigger cubes. The total volume in terms of s³ is: Total Volume = 64 × 1s³ + 8 × 8s³ = 128s³

Since the cuboid is cut into these cubes, its dimensions must multiply to 128. Since we must fit bigger cubes of side 2s, the dimensions must be even. The only integer dimensions satisfying this are 4 × 4 × 8.

The cuboid is colored as follows: Two opposite faces of size 4 × 4 are colored Black. Two opposite faces of size 4 × 8 are colored Red. Two opposite faces of size 4 × 8 are colored Green.

Since the 8 bigger cubes cannot have any Black-colored surfaces, they cannot touch the Black faces of size 4 × 4. Thus, the Black faces must be at the ends of the length of 8 (at x = 0 and x = 8).

The 8 bigger cubes (each of size 2 × 2 × 2) form a 2 × 2 × 2 block of total size 4 × 4 × 4. This block fits perfectly in the middle of the cuboid (from x = 2 to x = 6), which prevents them from touching the Black faces at x = 0 and x = 8. This leaves two layers of smaller cubes on each end (from x = 0 to 2, and from x = 6 to 8).

Now, we count the cubes with only one face colored:

Black-only cubes: The Black faces are at x = 0 and x = 8, each of size 4 × 4. The cubes on these faces that do not touch any other faces are the 4 central cubes in the 2 × 2 middle. Since there are 2 Black faces, we have: 2 × 4 = 8 Black-only cubes.

Red-only cubes: The Red faces are at y = 0 and y = 4, each of size 4 × 8 (along x and z). For the Red face at y = 0: The columns at x ∈ [0, 1] and x ∈ [7, 8] touch the Black faces. The rows at z ∈ [0, 1] and z ∈ [3, 4] touch the Green faces. The middle region x ∈ [2, 6] contains the bigger cubes, which touch the Green faces at z = 0 or z = 4, so none of the bigger cubes have only one face colored. The remaining smaller cubes with only one face colored are in the columns x ∈ [1, 2] and x ∈ [6, 7], within the rows z ∈ [1, 3]. This yields 2 columns of 2 cubes each, which is 4 cubes. With 2 Red faces, we have: 2 × 4 = 8 Red-only cubes.

Green-only cubes: By symmetry with the Red faces, each of the 2 Green faces of size 4 × 8 contains 4 smaller cubes with only one face colored. This gives: 2 × 4 = 8 Green-only cubes.

Summing these counts gives the total number of cubes with only one face colored: 8 + 8 + 8 = 24

Explanation

The solid cuboid (with dimensions 4 × 4 × 8) is divided into two halves of size 4 × 4 × 4: One half is cut into 8 bigger cubes of size 2 × 2 × 2. The other half is cut into 64 smaller cubes of size 1 × 1 × 1.

Cubes with three of their faces colored can only be located at the corners of the original cuboid, as only corner positions have three faces exposed to the exterior.

Since a cuboid has exactly 8 corners: 4 of these corners are occupied by the bigger cubes at one end of the cuboid. 4 of these corners are occupied by the smaller cubes at the opposite end of the cuboid.

Thus, there are exactly 4 + 4 = 8 cubes that have three of their faces colored.

Explanation

To determine the number of unpainted cubes, we first find the dimensions of the solid cuboid. Let the side length of each of the 64 smaller cubes be 1 unit (volume = 1 cubic unit), and the side length of each of the 8 bigger cubes be 2 units (volume = 2³ = 8 cubic units).

The total volume of the cuboid is the sum of the volumes of all the cut cubes: Total Volume = (64 × 1) + (8 × 8) = 64 + 64 = 128 cubic units.

To accommodate both types of cubes under the given conditions, the dimensions of the cuboid must be 4 × 4 × 8. Let the length of the cuboid along the X-axis be 8 units, the width along the Y-axis be 4 units, and the height along the Z-axis be 4 units.

The opposite faces at the two ends of the X-axis (at x = 0 and x = 8, each of size 4 × 4) are colored Black. The other four faces along the Y and Z axes (at y = 0, y = 4, z = 0, and z = 4, each of size 4 × 8) are colored Red and Green.

Since the 8 bigger cubes cannot have any Black-colored surfaces, they must not touch the two outer Black faces at x = 0 and x = 8. This means they are packed inside the middle region of the cuboid from x = 2 to x = 6 (forming a 4 × 4 × 4 space containing 2 × 2 × 2 = 8 bigger cubes).

Because each of these 8 bigger cubes has a side length of 2 units and the total width and height of the cuboid are both 4 units, every bigger cube must touch either the outer boundary at y = 0 or y = 4 (colored Red) and z = 0 or z = 4 (colored Green). Consequently, all 8 bigger cubes have at least some of their surfaces colored, meaning 0 bigger cubes are completely unpainted.

The remaining 64 smaller cubes are located in the two outer end regions of size 4 × 4 × 2 (one at x ∈ [0, 2] and the other at x ∈ [6, 8]).

For a smaller cube of size 1 × 1 × 1 to have no colored surfaces, it must not touch any of the outer faces of the cuboid: In the first end region (x ∈ [0, 2]), the unpainted smaller cubes must have: x-coordinate in the range [1, 2] (1 layer, to avoid the Black face at x = 0) y-coordinate in the range [1, 3] (2 layers, to avoid the Red faces at y = 0 and y = 4) z-coordinate in the range [1, 3] (2 layers, to avoid the Green faces at z = 0 and z = 4)

This gives 1 × 2 × 2 = 4 unpainted smaller cubes in the first end region.

By symmetry, in the second end region (x ∈ [6, 8]), the unpainted smaller cubes must have: x-coordinate in the range [6, 7] (1 layer, to avoid the Black face at x = 8) y-coordinate in the range [1, 3] (2 layers) z-coordinate in the range [1, 3] (2 layers)

This gives 1 × 2 × 2 = 4 unpainted smaller cubes in the second end region.

Adding these together, the total number of cubes with no color on any of their surfaces is 4 + 4 = 8.

Explanation

The solid cuboid has dimensions 4 × 4 × 8, which can be represented as a grid of unit spaces of size 1 × 1 × 1.

The colors are assigned to the faces as follows: The two opposite end faces of dimensions 4 × 4 (at z = 0 and z = 8) are colored Black. Two opposite side faces of dimensions 4 × 8 (at x = 0 and x = 4) are colored Red. The remaining two opposite side faces of dimensions 4 × 8 (at y = 0 and y = 4) are colored Green.

To obtain 72 cubes in total consisting of 64 smaller cubes of size 1 × 1 × 1 and 8 bigger cubes of size 2 × 2 × 2: The middle portion of the cuboid (from z = 2 to z = 6, of dimensions 4 × 4 × 4) is cut into 8 bigger cubes of size 2 × 2 × 2. This satisfies the condition that no bigger cube touches the Black faces at z = 0 and z = 8. The two end portions (from z = 0 to 2, and from z = 6 to 8, each of dimensions 4 × 4 × 2) are cut into smaller unit cubes, yielding 32 smaller cubes at each end, totaling 64 smaller cubes.

To find the number of cubes with at least one surface colored Green, we count those touching the Green faces at y = 0 and y = 4:

For the smaller cubes: In each of the two end portions of dimensions 4 × 4 × 2: The layer of smaller cubes touching the Green face y = 0 consists of 4 (along x) × 2 (along z) = 8 cubes. The layer of smaller cubes touching the Green face y = 4 also consists of 4 (along x) × 2 (along z) = 8 cubes. Since the y-dimension is 4 units and the smaller cubes have a side length of 1, these two layers do not overlap. Thus, each end portion has 8 + 8 = 16 smaller cubes touching a Green face. For both end portions combined, there are 16 + 16 = 32 smaller cubes.

For the bigger cubes: In the middle portion of dimensions 4 × 4 × 4: The 8 bigger cubes are arranged in a 2 × 2 × 2 grid. Along the y-axis, there are exactly 2 bigger cubes of side 2, occupying y from 0 to 2 and y from 2 to 4. This means every bigger cube lies either in the layer touching y = 0 or the layer touching y = 4. Consequently, all 8 bigger cubes touch a Green face.

Adding these together, the total number of cubes with at least one surface colored Green is 32 smaller cubes + 8 bigger cubes = 40.

Explanation

In the original cuboid, the Red faces are opposite to each other, and the Green faces are opposite to each other. This means that the Red and Green faces of the cuboid are perpendicular to each other.

Any cube obtained by cutting the cuboid will have its faces parallel to the outer faces of the cuboid. Consequently, if an individual cube has both a Red-painted surface and a Green-painted surface, these two surfaces must be perpendicular (adjacent) to each other, sharing an edge.

Since the Red and Green surfaces on any cut cube can only be adjacent, it is geometrically impossible for any cube to have a Red surface and a Green surface opposite to each other. Therefore, the number of such cubes is 0.

Explanation

To find the number of cubes, we first determine the dimensions of the cuboid and the arrangement of the cuts.

The cuboid is cut into 72 cubes in total: 64 smaller cubes of side 1 unit (total volume = 64 × 1³ = 64 cubic units) 8 bigger cubes of side 2 units (total volume = 8 × 2³ = 64 cubic units)

The total volume of the cuboid is 64 + 64 = 128 cubic units. Since the smaller cubes form a 4 × 4 × 4 block and the bigger cubes form another 4 × 4 × 4 block, the dimensions of the solid cuboid are 4 × 4 × 8.

We are given that the bigger cubes must not have any of their surfaces colored Black. Since opposite faces are colored with the same color, the two opposite 4 × 4 end faces (at z = 0 and z = 8) must be colored Black.

To ensure the bigger cubes do not touch these Black faces: The 8 bigger cubes (each of side 2) are placed in the middle section of the cuboid (z-coordinates from 2 to 6). The 64 smaller cubes are placed at the two ends (32 cubes from z = 0 to 2, and 32 cubes from z = 6 to 8). The remaining four opposite 4 × 8 side faces are colored Red and Green respectively.

Cubes with adjacent Red and Green surfaces (and no other colored faces) must lie along the 4 long edges (each of length 8) of the cuboid, where the Red and Green faces meet.

Along each of these 4 long edges: Smaller cubes at z ∈ [0, 1] and z ∈ [7, 8] touch the Black end faces and thus have 3 faces painted (Red, Green, Black). Smaller cubes at z ∈ [1, 2] and z ∈ [6, 7] touch only the Red and Green faces, so they have exactly 2 faces painted. This gives 2 smaller cubes per edge. Both bigger cubes in the middle section (z ∈ [2, 4] and z ∈ [4, 6]) touch only the Red and Green side faces, meaning they have exactly 2 faces painted. This gives 2 bigger cubes per edge.

Thus, along each of the 4 long edges, there are 2 smaller cubes and 2 bigger cubes with exactly Red and Green colored surfaces.

Total number of such cubes = 4 edges × (2 smaller cubes + 2 bigger cubes) = 16 cubes.

Explanation

The volume of a cone is given by the formula V = ⅓ π r² h, where r is the radius of the base and h is the height.

Let the radius and height of the original cone be r and h. Its volume is V₁ = ⅓ π r² h.

For the new cone, the radius is doubled to 2r and the height remains h. Its volume is V₂ = ⅓ π (2r)² h = 4/3 π r² h.

The ratio of the volume of the new cone to the original cone is V₂ : V₁ = (4/3 π r² h) / (⅓ π r² h) = 4 : 1.

Explanation

Using the principle of set theory, the percentage of students who passed in at least one of the two subjects is given by the formula:

P(M ∪ P) = P(M) + P(P) - P(M ∩ P)

Where P(M) represents the percentage of students who passed in Mathematics, P(P) is the percentage of students who passed in Physics, and P(M ∩ P) is the percentage of students who passed in both.

Substituting the given values:

P(M ∪ P) = 65 + 48 - 30 P(M ∪ P) = 83%

This means 83% of the students passed in at least one of the subjects. The percentage of students who failed in both subjects is the complement:

100% - 83% = 17%

Explanation

The total expense on sugar is the product of its price and consumption: Expense = Price × Consumption.

When the price of a commodity increases by r%, the percentage decrease in consumption required to keep the total expenditure unchanged is given by the direct formula:

Decrease in consumption % = [r / (100 + r)] × 100 %

Substituting r = 25 into the formula:

Decrease in consumption % = [25 / (100 + 25)] × 100 % Decrease in consumption % = [25 / 125] × 100 % Decrease in consumption % = 1/5 × 100 % = 20 %

Explanation

The series S₁ is the alternating harmonic series, which can be evaluated using the Maclaurin series expansion for log(1 + x):

log(1 + x) = x - x²/2 + x³/3 - x⁴/4 + ...

Substituting x = 1 into this formula gives:

log(2) = 1 - 1/2 + 1/3 - 1/4 + ...

Since this series is conditionally convergent by the alternating series test, the sum S₁ converges to log 2.

The series S₂ is the standard harmonic series:

S₂ = 1 + 1/2 + 1/3 + 1/4 + ...

Using the integral test, we can compare this series to the integral of 1/x from 1 to infinity. Since this integral diverges to infinity, the harmonic series also diverges. Consequently, the sum S₂ does not exist as a finite real number.

Explanation

To determine the scheduling of the lectures across the seven days of the week (Monday to Sunday), we can analyze the given conditions step-by-step:

We have 6 lectures (Organizational Behaviour, Psychology, Statistics, Computer Science, Research Methods, Economics) and 1 Off day. The Off day is scheduled immediately before Economics, which forms a consecutive block of (Off day, Economics). Since Monday cannot be the Off day, this block must occur later in the week. Psychology is scheduled immediately after Organizational Behaviour, forming another consecutive block of (Organizational Behaviour, Psychology). There is a one-day gap between Statistics and Research Methods, meaning they are separated by exactly one day. Statistics cannot be scheduled on Friday, and Computer Science cannot be scheduled on Tuesday or Saturday.

By aligning these constraints, we get the following unique weekly schedule:

Monday: Organizational Behaviour Tuesday: Psychology Wednesday: Statistics Thursday: Computer Science Friday: Research Methods Saturday: Off day Sunday: Economics

This arrangement satisfies all given criteria: Computer Science is on Thursday, avoiding Tuesday and Saturday. Statistics is on Wednesday, avoiding Friday. The gap between Statistics (Wednesday) and Research Methods (Friday) is exactly one day (Thursday). The Off day is Saturday, which is immediately prior to Economics on Sunday, and Monday is a lecture day.

Consequently, the last lecture scheduled in the week is Economics on Sunday.

Explanation

To find the schedule, we analyze the constraints step-by-step for the 7 days from Monday to Sunday:

Psychology is immediately after Organizational Behaviour, forming a consecutive block (Organizational Behaviour, Psychology). One day prior to Economics is the Off day, forming another consecutive block (Off day, Economics). Since Monday is not the Off day, the Off day can only be from Tuesday to Saturday. Statistics and Research Methods have a one-day gap between them (days d and d+2). Since Statistics cannot be on Friday, let us place Statistics on Wednesday and Research Methods on Friday.

Now we place the remaining events: With Wednesday and Friday occupied, the remaining open days are Monday, Tuesday, Thursday, Saturday, and Sunday. Since Computer Science cannot be scheduled on Tuesday or Saturday, and we need to fit the consecutive block (Organizational Behaviour, Psychology), they must occupy Monday and Tuesday respectively. This leaves Thursday, Saturday, and Sunday. Computer Science is scheduled on Thursday. The consecutive block (Off day, Economics) must occupy Saturday and Sunday, making Saturday the Off day and Sunday the day for Economics.

This gives the unique, consistent weekly schedule: Monday: Organizational Behaviour Tuesday: Psychology Wednesday: Statistics Thursday: Computer Science Friday: Research Methods Saturday: Off Day Sunday: Economics

Therefore, Research Methods is scheduled on Friday.

Explanation

To find the number of lectures scheduled between Economics and Psychology, we can construct the daily schedule from Monday to Sunday using the given conditions:

Psychology is scheduled immediately after Organisational Behaviour. One day prior to Economics is the Off day (no lecture), and Monday is not the Off day. Statistics is not on Friday, and there is a one-day gap between Statistics and Research Methods. Computer Science is not scheduled on Tuesday or Saturday.

By organizing the conditions sequentially, the unique daily schedule for the week is:

Monday: Organisational Behaviour (Lecture 1) Tuesday: Psychology (Lecture 2) Wednesday: Statistics (Lecture 3) Thursday: Computer Science (Lecture 4) Friday: Research Methods (Lecture 5) Saturday: Off Day (No Lecture) Sunday: Economics (Lecture 6)

Evaluating this schedule against the conditions confirms all rules are satisfied: Psychology immediately follows Organisational Behaviour; Computer Science is on Thursday (not Tuesday or Saturday); Statistics is on Wednesday (not Friday); there is a one-day gap (Thursday) between Statistics and Research Methods; and the Off day (Saturday) is immediately prior to Economics (Sunday).

The days between Tuesday (Psychology) and Sunday (Economics) are Wednesday, Thursday, Friday, and Saturday. Among these days, the scheduled lectures are Statistics (Wednesday), Computer Science (Thursday), and Research Methods (Friday), while Saturday is an Off day with no lecture.

Therefore, there are exactly 3 lectures scheduled between Economics and Psychology.

Explanation

To find the schedule, let us analyze the constraints step-by-step:

We have seven days from Monday to Sunday, with six lectures and one Off day.

Organizational Behaviour must be immediately followed by Psychology. This forms a block of two consecutive days: Organizational Behaviour followed by Psychology.

The day before Economics is the Off day. This forms another block: Off Day followed by Economics. Since Monday is not the Off day, this block cannot start on Monday.

Statistics and Research Methods have a one-day gap between them, and Statistics cannot be on Friday.

Computer Science cannot be on Tuesday or Saturday.

If we place the Off Day on Saturday and Economics on Sunday, the remaining days are Monday, Tuesday, Wednesday, Thursday, and Friday.

We can place the Organizational Behaviour and Psychology block on Monday and Tuesday respectively.

This leaves Wednesday, Thursday, and Friday. Since Statistics and Research Methods must have a one-day gap, they must occupy Wednesday and Friday. Since Statistics cannot be scheduled on Friday, Statistics must be on Wednesday and Research Methods must be on Friday.

The only remaining day for Computer Science is Thursday. This perfectly satisfies all conditions, including that Computer Science is not on Tuesday or Saturday. Thus, Computer Science is scheduled on Thursday.

Explanation

According to the provided passage, international competition forces each company or firm in industrially rich countries to produce only one, or at the most a few, varieties and styles of the same product rather than numerous different varieties and styles. This limitation in the produced varieties is a direct impact of international competition, which is essential for keeping unit costs low and achieving economies of scale.

Explanation

Lower product prices directly translate to an increase in the real income or purchasing power of consumers. When prices of differentiated goods decrease due to economies of scale, consumers are able to buy a larger quantity and variety of goods with the same amount of money, thereby raising their overall purchasing power and welfare.

Explanation

According to the passage, the development and use of more specific and effective machinery (specialised machinery) is achieved by focusing on fewer varieties to keep unit costs low. This specialization enables international trade where nations exchange different varieties of the same product. Consequently, this process leads to a wider range of choices and a significantly increased variety of goods available for consumers to buy.

Explanation

According to the passage, when tariffs and other obstructions to the flow of trade among members of the European Union were removed in 1958, the immediate result was that the volume of trade increased. This directly shows that the removal of trade barriers led to a surge in the volume of trade among the member states.

Explanation

The word noticeable refers to something that is easily seen, observed, or detected, making it clear or apparent. The word perceptible shares this exact meaning, as it describes something that is capable of being perceived, noticed, or recognized by the senses or the mind. Therefore, perceptible is a synonym of noticeable.

Explanation

Implicit communication refers to the transmission of messages, feelings, and intentions without the use of explicit or spoken words. Body language, encompassing gestures, posture, facial expressions, and eye movement, is a primary form of nonverbal communication. Because these physical cues convey underlying meanings, attitudes, and emotional states indirectly, they require interpretation based on context rather than direct verbal expression, making body language fundamentally implicit.

Explanation

Non-verbal communication consists of conveying information through body language, facial expressions, gestures, posture, eye contact, and tone of voice. It is mostly unstructured because it lacks a formal system of rules, syntax, and organized grammar, which are standard in verbal language. Instead, these cues are spontaneous, highly contextual, and deeply tied to individual emotional states and cultural backgrounds, making them fluid and informal.

Explanation

Parallel communication refers to a method where multiple data signals or packets are transmitted simultaneously over different channels or paths. In the context of Internet chatting, users communicate in real time, and the network system facilitates the simultaneous, concurrent transmission of multiple data streams (such as text, media, and network protocol data) in parallel pathways. This enables immediate, multifaceted exchanges and parallel active sessions between multiple participants without sequential delays.

Explanation

Communication barriers are obstacles that distort or prevent the effective transmission and reception of messages. Physical noise, such as loud external environmental sounds or poor acoustics, directly interferes with the physical transmission of a message. Psychological noise, which includes internal cognitive distractions such as emotional stress, personal biases, anxiety, or preoccupation, hinders the objective understanding of the message by either the sender or the receiver. Together, these forms of noise represent significant external and internal barriers that disrupt the smooth flow of communication.

Explanation

In communication, the context refers to the physical, social, cultural, psychological, and temporal environment in which a message is exchanged. It establishes the frame of reference, background, and relationship between the sender and receiver, which directly guides how a message is decoded and understood. Without context, the meaning of a message can be easily misunderstood, as the same verbal or non-verbal cues can carry entirely different meanings under different circumstances.

Explanation

The word is misspelled because the correct spelling is piety, where the letter i precedes the letter e. Piety refers to the quality of being religious, reverent, or devout. The spelling in the option incorrectly transposes these two letters as peity.

Explanation

The word gregarious describes an individual who is fond of company, highly sociable, and enjoys being surrounded by others. The word reclusive describes a person who avoids the company of other people, preferring to live in solitude or isolation. Since these two terms represent directly opposing behavioral preferences regarding social interaction, reclusive is the exact antonym of gregarious.

Explanation

The word tremulous is an adjective used to describe someone or something that is shaking, quivering, or unsteady, often due to nervousness, fear, or physical weakness. The antonym of this term is steady, which means firm, stable, balanced, and free from shaking or movement.

Explanation

A patent is an official legal right granted by a government agency to an inventor, giving them the sole authority to manufacture, use, and sell their unique invention for a set number of years. This exclusive right prevents competitors from copying, making, or importing the patented invention without the consent of the patent holder, thereby protecting and rewarding the inventor's intellectual property and innovation.

Explanation

A hedonist is a person whose life is dedicated to the pursuit of pleasure and self-gratification. The term originates from the Greek word hedone, which translates directly to pleasure. In both philosophical and common usage, a hedonist believes that pleasure is the supreme good and the primary goal of human life, leading them to constantly seek enjoyable experiences and avoid pain.

Explanation

The idiom "to eat someone's salt" originates from the historical value of salt as a precious commodity, symbolizing hospitality, friendship, and loyalty. Partaking of someone's salt signifies accepting their hospitality and being their guest. Therefore, the phrase best expresses the meaning of being a guest.

Explanation

The idiom 'bury the hatchet' means to end a quarrel or conflict and make peace with someone. It originates from a Native American custom in which chiefs of warring tribes would literally bury their weapons — particularly hatchets — as a formal gesture symbolising the cessation of hostilities and the establishment of peaceful relations between them. Among the options given, only option D ('to make peace') correctly captures this meaning. Options A ('to maintain a secret'), B ('to bury the treasure'), and C ('to bury the bucket') are all unrelated to the idiomatic sense of the phrase.

Explanation

The idiom "kick up a row" means to create a noisy disturbance, raise a loud complaint, or make a major scene about something. In this context, it signifies that the police created a significant stir or made a great fuss over the situation.

Explanation

The idiom "tip of the iceberg" refers to a small, perceptible part of a much larger, hidden situation or problem. The metaphor comes from the physical nature of icebergs, where only a tiny fraction of the total mass is visible above the water line, while the massive bulk remains submerged and unseen. In this context, the technical glitch is merely a minor, visible symptom of a much larger and more complex set of challenges confronting the new project.

Explanation

The idiom "by fits and starts" refers to an activity that is carried out irregularly, characterized by sudden, brief periods of activity followed by intervals of inactivity or stopping. Because the work lacks a steady, continuous, or uniform momentum, it is performed inconsistently.

Explanation

Profit in a partnership is shared in proportion to each partner's capital multiplied by the duration for which it was invested (i.e. the 'capital-months' or equivalent investment).

All three partners invest Rs. 16,000 each for the first 3 months. After 3 months:

• Ram withdraws Rs. 2,000, so his capital becomes Rs. 14,000 for the remaining 9 months.
• Mohan withdraws Rs. 4,000, so his capital becomes Rs. 12,000 for the remaining 9 months.
• Sohan invests Rs. 8,000 more, so his capital becomes Rs. 24,000 for the remaining 9 months.

Equivalent investment (capital × months): Ram = (16,000 × 3) + (14,000 × 9) = 48,000 + 126,000 = 1,74,000 Mohan = (16,000 × 3) + (12,000 × 9) = 48,000 + 108,000 = 1,56,000 Sohan = (16,000 × 3) + (24,000 × 9) = 48,000 + 2,16,000 = 2,64,000

Profit-sharing ratio: Ram : Mohan : Sohan = 1,74,000 : 1,56,000 : 2,64,000 = 174 : 156 : 264 = 29 : 26 : 44 (dividing each by 6)

Sum of ratio parts = 29 + 26 + 44 = 99

Ram's share of profit = (29 / 99) × 41,580 = 29 × 420 = Rs. 12,180

The correct answer is option B.

Explanation

Let Ramchandani's salary be S.

The initially decided donation is 16% of the salary, which is: Decided Donation = 0.16S

The actual donation of Rs. 4,320 is three-fifth of the decided donation: 4,320 = (3/5) × 0.16S

Expressing this as a fraction to simplify: 4,320 = (3/5) × (16/100) × S 4,320 = (48/500) × S

Solving for S: S = (4,320 × 500) / 48 S = 90 × 500 S = 45,000

Explanation

Selling price of the mixture is Rs. 18 per liter, and the profit is 25 percent.

Cost price of the mixture = Selling price ÷ (1 + Profit percent) Cost price of the mixture = 18 ÷ 1.25 = 18 × 4/5 = Rs. 14.4 per liter

Using the rule of alligation, where the cost price of pure milk is Rs. 20 per liter and the cost price of water is Rs. 0 per liter:

Ratio of Milk to Water = (Mean Cost Price - Cost of Water) : (Cost of Milk - Mean Cost Price) Ratio of Milk to Water = (14.4 - 0) : (20 - 14.4) Ratio of Milk to Water = 14.4 : 5.6

Multiplying both sides by 10 gives 144 : 56. Dividing both terms by 8 simplifies the ratio to 18 : 7.

Explanation

The difference (D) between compound interest and simple interest on a principal sum (P) for two years at an annual interest rate (R%) is given by the formula:

D = P × (R/100)²

Substituting the given values, where D = 10 and R = 5:

10 = P × (5/100)²

10 = P × (1/20)²

10 = P × (1/400)

P = 10 × 400

P = 4000

Explanation

Let the starting point be the origin (0, 0), where the positive x-axis represents East and the positive y-axis represents North.

First, turning to his right (East), he walks 25 meters, reaching (25, 0). Second, turning to his left (North), he walks 30 meters, reaching (25, 30). Third, turning to his right (East), he walks 25 meters, reaching (50, 30). Fourth, turning to his right (South), he walks 55 meters, reaching (50, -25). Fifth, turning to his right (West), he walks 40 meters, reaching (10, -25).

His final position has a positive East coordinate (10 meters East) and a negative North coordinate (25 meters South), meaning he is in the South-East direction relative to his starting point.

Explanation

The relationship between the number of bulbs (B), days (D), hours (H), and cost (C) is given by the chain rule formula:

(B₁ × D₁ × H₁) / C₁ = (B₂ × D₂ × H₂) / C₂

Substitute the given values into the formula:

B₁ = 200 D₁ = 6 H₁ = 4 C₁ = 40

B₂ = x D₂ = 15 H₂ = 3 C₂ = 48

(200 × 6 × 4) / 40 = (x × 15 × 3) / 48

Simplify the left side of the equation:

(200 × 24) / 40 = 5 × 24 = 120

Now equate it to the right side:

120 = (x × 45) / 48

Solve for x:

120 = 15x / 16

15x = 120 × 16

x = (120 × 16) / 15

x = 8 × 16

x = 128

Explanation

The number of triangles that can be formed using n points in a plane where m points are collinear is given by the formula nC₃ - mC₃. This represents the total combinations of choosing any three points minus the invalid combinations of choosing three points that all lie on the same straight line.

Substituting the given values n = 11 and m = 5:

11C₃ - 5C₃

Calculating the combinations: 11C₃ = (11 × 10 × 9) / (3 × 2 × 1) = 165 5C₃ = (5 × 4 × 3) / (3 × 2 × 1) = 10

Subtracting the collinear combinations: 165 - 10 = 155

This yields 155 triangles.

Explanation

Let the original three-digit number be represented as 100a + 10b + c, where a is the hundreds digit, b is the tens digit, and c is the units digit.

When the last two digits are interchanged, the new number is represented as 100a + 10c + b.

According to the given condition, the new number exceeds the original number by 54: (100a + 10c + b) - (100a + 10b + c) = 54

Subtracting the terms yields: 9c - 9b = 54

Factoring out 9 gives: 9(c - b) = 54

Dividing both sides by 9 gives the difference between the last two digits: c - b = 6

Explanation

Given: a² + 1/a² = 98, with a > 0.

Step 1 – Find (a + 1/a). Using the algebraic identity (a + 1/a)² = a² + 1/a² + 2: (a + 1/a)² = 98 + 2 = 100 Since a > 0, taking the positive square root: a + 1/a = 10

Step 2 – Find a³ + 1/a³. Using the identity a³ + 1/a³ = (a + 1/a)³ − 3(a + 1/a): a³ + 1/a³ = 10³ − 3 × 10 = 1000 − 30 = 970

The correct answer is option D (970).

Explanation

To find the value of the expression, we can multiply the numerator and the denominator of each term by x, y, and z respectively:

1/(x+1) = x / (x² + x)

Substitute the given relation x² = y + z into the denominator: 1/(x+1) = x / (x + y + z)

Similarly, for the other two terms: 1/(y+1) = y / (y² + y) = y / (x + y + z) 1/(z+1) = z / (z² + z) = z / (x + y + z)

Adding these three expressions together: 1/(x+1) + 1/(y+1) + 1/(z+1) = x / (x + y + z) + y / (x + y + z) + z / (x + y + z) = (x + y + z) / (x + y + z) = 1

Alternatively, by symmetry, we can assume x = y = z. Substituting this into the first equation gives x² = 2x, which yields x = 2. Substituting x = y = z = 2 into the required expression gives: 1/3 + 1/3 + 1/3 = 1

Explanation

The given expression is cos 0° + cos 1° + cos 2° + ... + cos 179°.

Using the trigonometric identity cos(180° - x) = -cos x, we can pair the terms from cos 1° to cos 179° as follows:

cos 179° = -cos 1° cos 178° = -cos 2° and so on, up to cos 91° = -cos 89°

This pairing yields: cos 1° + cos 179° = 0 cos 2° + cos 178° = 0 and so on, up to cos 89° + cos 91° = 0

The terms cos 0° and cos 90° remain unpaired. Since cos 0° = 1 and cos 90° = 0, the total sum is:

cos 0° + (cos 1° + cos 179°) + (cos 2° + cos 178°) + ... + (cos 89° + cos 91°) + cos 90° = 1 + 0 + 0 + ... + 0 + 0 = 1

Explanation

Let the height of the tree be h. When the angle of elevation of the sun is 60°, the length of the shadow is x. When the angle of elevation is 30°, the length of the shadow is x + 5.

Using the trigonometric tangent function for both elevations: tan 60° = h / x x = h / √3

tan 30° = h / (x + 5) x + 5 = h√3

Substituting the value of x from the first equation into the second: h / √3 + 5 = h√3 5 = h√3 - h / √3 5 = h(√3 - 1 / √3) 5 = h(3 - 1) / √3 5 = 2h / √3 h = 5√3/2

Explanation

In right-angled triangle AEB, applying Pythagoras theorem: AB² = AE² + BE²

Since D is the midpoint of BC, we have: BD = BC/2

Since AC > AB, the altitude AE lies closer to B than to C, which means E lies between B and D. Therefore, the length BE can be written as: BE = BD - ED = BC/2 - ED

In right-angled triangle AED, applying Pythagoras theorem gives: AE² = AD² - ED²

Substituting the expressions for AE² and BE into the equation for AB²: AB² = (AD² - ED²) + (BC/2 - ED)² AB² = AD² - ED² + BC²/4 + ED² - 2 × (BC/2) × ED AB² = AD² - BC.ED + ¼ BC²

Explanation

Let the center of the circle be at the origin O(0,0). Since the two chords are perpendicular to each other, we can align them parallel to the coordinate axes.

Let the chord of length 2a be parallel to the x-axis and the chord of length 2b be parallel to the y-axis. Their point of intersection P has coordinates (x, y).

The distance from the center O to the point of intersection P is given as c. Therefore: x² + y² = c²

The perpendicular distance from the center O to the first chord (of length 2a) is the absolute value of y. Using Pythagoras' theorem on the right-angled triangle formed by the radius R, half of the chord length a, and the perpendicular distance from the center: R² = a² + y²

Similarly, the perpendicular distance from the center O to the second chord (of length 2b) is the absolute value of x. This gives the relation: R² = b² + x²

Adding these two equations: 2R² = a² + b² + x² + y²

Substituting x² + y² = c² into the equation: 2R² = a² + b² + c²

Dividing both sides by 2 and taking the square root: R = √((a² + b² + c²)/2)

Explanation

The boundaries of the region are defined by y = |x| - 1 and y = 1 - |x|. Finding their points of intersection gives |x| - 1 = 1 - |x|, which simplifies to 2|x| = 2, or |x| = 1. This yields the intersection points (1, 0) and (-1, 0) on the x-axis.

Evaluating the equations at x = 0 gives the vertices along the y-axis at (0, 1) and (0, -1).

Plotting these points reveals that the bounded region is a square with vertices at (1, 0), (0, 1), (-1, 0), and (0, -1). The diagonals of this square lie along the coordinate axes, with the horizontal diagonal d₁ extending from -1 to 1 (length 2) and the vertical diagonal d₂ extending from -1 to 1 (length 2).

Using the formula for the area of a quadrilateral with perpendicular diagonals:

Area = ½ × d₁ × d₂

Substituting the diagonal lengths:

Area = ½ × 2 × 2 = 2

Explanation

To find the value of √n + 1/√n, first express n as a perfect square: n = 7 + 4√3 = 4 + 3 + 2 × 2 × √3 = 2² + (√3)² + 2 × 2 × √3 = (2 + √3)²

Taking the square root on both sides: √n = 2 + √3

Now, calculate the reciprocal 1/√n by rationalizing the denominator: 1/√n = 1 / (2 + √3) = (2 - √3) / ((2 + √3)(2 - √3)) = (2 - √3) / (4 - 3) = 2 - √3

Substitute these values into the required expression: √n + 1/√n = (2 + √3) + (2 - √3) = 4

Explanation

Let the cylinder of radius r and height h be inscribed in a sphere of radius a. The relationship between the dimensions of the cylinder and the sphere is obtained from the right-angled triangle formed by the radius of the cylinder, half of its height, and the radius of the sphere:

r² + (h/2)² = a² r² = a² - h²/4

The volume V of the cylinder is given by:

V = π r² h

Substituting the expression for r² into the volume formula gives:

V = π (a² - h²/4) h V = π (a² h - h³/4)

To find the height that maximizes the volume, differentiate V with respect to h and set the derivative to zero:

dV/dh = π (a² - 3h²/4) = 0 a² - 3h²/4 = 0 3h²/4 = a² h² = 4a²/3 h = 2a/√3

Taking the second derivative:

d²V/dh² = -3π h / 2

Since h is positive, the second derivative is negative, confirming that the volume is maximized when the height is 2a/√3.

Explanation

A non-leap year consists of 365 days. Dividing 365 by 7 gives 52 weeks and a remainder of 1 day (365 = 52 × 7 + 1).

The 52 complete weeks guarantee that there will be exactly 52 Sundays in the year. The presence of a 53rd Sunday depends entirely on the single remaining day.

This remaining day can be any one of the seven days of the week: Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, or Sunday. Each of these seven outcomes is equally likely.

For the year to have 53 Sundays, this remaining day must be a Sunday. Therefore, there is only 1 favorable outcome out of 7 possible outcomes, which gives a probability of 1/7.

Explanation

When two objects move in the same direction, the relative speed of the faster object with respect to the slower is the difference of their individual speeds.

Relative speed of train with respect to man = 55 km/h − 10 km/h = 45 km/h

Converting to metres per second (multiply by 5/18): Relative speed = 45 × (5/18) = 12.5 m/s

For the train to completely pass the man, the train must cover a distance equal to its own length (150 m) relative to the man.

Time = Distance / Relative speed Time = 150 / 12.5 = 12 seconds

The correct answer is option B (12 seconds).

Explanation

To form a committee of 5 members from 8 men and 5 women containing at least 3 women, we consider the following mutually exclusive cases:

Case 1: The committee has exactly 3 women and 2 men. The number of ways to choose 3 women from 5 is ⁵C₃, and the number of ways to choose 2 men from 8 is ⁸C₂. ⁵C₃ × ⁸C₂ = (5 × 4 / 2) × (8 × 7 / 2) = 10 × 28 = 280 ways.

Case 2: The committee has exactly 4 women and 1 man. The number of ways to choose 4 women from 5 is ⁵C₄, and the number of ways to choose 1 man from 8 is ⁸C₁. ⁵C₄ × ⁸C₁ = 5 × 8 = 40 ways.

Case 3: The committee has exactly 5 women and 0 men. The number of ways to choose 5 women from 5 is ⁵C₅, and the number of ways to choose 0 men from 8 is ⁸C₀. ⁵C₅ × ⁸C₀ = 1 × 1 = 1 way.

Adding the ways from all three cases gives the total number of ways: Total ways = 280 + 40 + 1 = 321 ways.

Explanation

We can find the probability that at least one of them passes the exam by subtracting the probability that both of them fail from 1.

The probability that the boy fails the exam is: P(B') = 1 - 3/5 = 2/5

The probability that the girl fails the exam is: P(G') = 1 - 2/5 = 3/5

Since these are independent events, the probability that both of them fail is: P(both fail) = P(B') × P(G') = 2/5 × 3/5 = 6/25

Thus, the probability that at least one of them passes the exam is: P(at least one passes) = 1 - P(both fail) = 1 - 6/25 = 19/25

Explanation

Time taken by A to complete one full round is distance divided by speed, which is 9/5 hours.

Time taken by B to complete one full round is 9/2 hours.

To find the time when both A and B will meet for the first time at the starting point, we calculate the Least Common Multiple (LCM) of the times they take to complete one round.

LCM of fractions is calculated as the LCM of numerators divided by the HCF of denominators.

LCM of 9/5 and 9/2 = LCM(9, 9) / HCF(5, 2)

LCM(9, 9) = 9 HCF(5, 2) = 1

LCM(9/5, 9/2) = 9/1 = 9 hours.

Explanation

This question is defective as stated. Here is the rigorous calculation:

The two trains move in the same direction, so their relative speed is the difference of their speeds: Relative speed = 45 − 35 = 10 km/h = 10 × (5/18) = 25/9 m/s

For the faster train to completely pass the slower train of equal length L, the distance covered relative to the slower train = L + L = 2L.

Using Time = Distance / Speed: 27 = 2L / (25/9) 2L = 27 × 25/9 = 75 m L = 37.5 m

A length of 37.5 m does not appear among any of the four options (350 m, 300 m, 250 m, 200 m). The question is therefore defective — either the time (27 seconds) or the speeds are misprinted.

Among the given options, option B (300 m) is the closest result obtainable by a coherent (though situationally inapplicable) formula: if the trains were travelling in opposite directions, the relative speed would be 45 + 35 = 80 km/h = 200/9 m/s, and 2L = (200/9) × 27 = 600 m, giving L = 300 m. This appears to be the calculation the question setters intended, suggesting the directional condition ('same direction') may have been misprinted. Option B (300 m) is therefore selected as the best available answer.

Explanation

The total work done is the product of the number of men and the number of days taken. This is expressed by the standard relation: M₁ × D₁ = M₂ × D₂

Let the original number of men be M. Initial days D₁ = 90 days New number of men M₂ = M - 8 New days D₂ = 90 + 10 = 100 days

Substituting these values into the formula: M × 90 = (M - 8) × 100 90M = 100M - 800 10M = 800 M = 80

Explanation

In an equilateral triangle, the medians, internal angle bisectors, perpendicular bisectors, and altitudes are all identical line segments. Because these line segments are the same, their points of concurrency—the centroid, incentre, circumcentre, and orthocentre—must coincide at the exact same point. This makes statement I correct.

In an obtuse-angled triangle, the circumcentre and the orthocentre always lie strictly outside the triangle. The circumcentre lies in the exterior region opposite the obtuse angle, while the orthocentre lies in the exterior region behind the vertex of the obtuse angle. This makes statement II incorrect.

Explanation

By the area theorem of similar triangles, the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

If the ratio of the corresponding sides of the two similar triangles is 9 : 4, then the ratio of their areas is:

(9 / 4)² = 9² / 4² = 81 / 16

This gives the ratio of their areas as 81 : 16.

Explanation

Using the trigonometric identity cot θ - tan θ = cos θ / sin θ - sin θ / cos θ = (cos² θ - sin² θ) / (sin θ cos θ) = cos 2θ / (½ sin 2θ) = 2 cot 2θ.

This can be rewritten as: tan θ = cot θ - 2 cot 2θ

Applying this identity to each term of the given expression: tan x = cot x - 2 cot 2x 2 tan 2x = 2(cot 2x - 2 cot 4x) = 2 cot 2x - 4 cot 4x 4 tan 4x = 4(cot 4x - 2 cot 8x) = 4 cot 4x - 8 cot 8x

Substituting these into the original expression: tan x + 2 tan 2x + 4 tan 4x + 8 cot 8x = (cot x - 2 cot 2x) + (2 cot 2x - 4 cot 4x) + (4 cot 4x - 8 cot 8x) + 8 cot 8x

The terms cancel out in a telescoping manner: -2 cot 2x cancels with 2 cot 2x, -4 cot 4x cancels with 4 cot 4x, and -8 cot 8x cancels with 8 cot 8x.

This leaves only cot x.

Explanation

To find the value of cosec x from the equation 5 cos x + 12 sin x = 13, we can divide the entire equation by 13:

(5/13) cos x + (12/13) sin x = 1

Since (5/13)² + (12/13)² = 25/169 + 144/169 = 169/169 = 1, we can define an angle θ such that cos θ = 5/13 and sin θ = 12/13.

Substituting these values into the equation:

cos θ cos x + sin θ sin x = 1

Using the trigonometric identity for the cosine of a difference, this simplifies to:

cos(x - θ) = 1

This equation is satisfied when x - θ = 2kπ for any integer k, which means x = θ + 2kπ. Therefore, the sine value remains the same:

sin x = sin θ = 12/13

Since cosec x is the reciprocal of sin x, we get:

cosec x = 1 / sin x = 13/12

Explanation

The arc length formula relates arc length (s), radius (r), and central angle (θ in radians): s = r × θ ⟹ r = s / θ

Given: Arc length s = 40 m Change in direction θ = 25°

Convert θ to radians: θ = 25 × (π / 180) = 25π / 180 = 5π / 36 radians

Substitute into the formula: r = 40 / (5π / 36) r = 40 × 36 / (5π) r = 1440 / (5π) r = 288 / π

Using the standard approximation π = 22/7: r = 288 × 7 / 22 r = 2016 / 22 r ≈ 91.636 m ≈ 91.64 m

The correct answer is option A (91.64 meters).

Explanation

The length of the direct common tangent L of two circles with radii r₁ and r₂ is given by the formula L = √(d² - (r₁ - r₂)²), where d is the distance between the centers of the two circles.

When the two circles touch each other externally, the distance d between their centers is equal to the sum of their radii, d = r₁ + r₂.

Substituting this into the formula: L = √((r₁ + r₂)² - (r₁ - r₂)²)

Using the algebraic identity (a + b)² - (a - b)² = 4ab, the equation simplifies to: L = √(4r₁r₂) L = 2√(r₁r₂)