RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

Each term in the numerator is 10 times the corresponding term in the denominator: 0.03 = 10 × 0.003 0.51 = 10 × 0.051 0.083 = 10 × 0.0083

Rewriting the numerator: (0.03)² + (0.51)² + (0.083)² = (10 × 0.003)² + (10 × 0.051)² + (10 × 0.0083)² = 10² × [(0.003)² + (0.051)² + (0.0083)²] = 100 × [(0.003)² + (0.051)² + (0.0083)²]

Dividing the numerator by the denominator: [100 × ((0.003)² + (0.051)² + (0.0083)²)] / [(0.003)² + (0.051)² + (0.0083)²] = 100

Explanation

To find a number that is 40% less than 80, calculate 60% of 80:

Value = 80 × (1 - 0.40) = 80 × 0.60 = 48.

Alternatively, calculate 40% of 80 which is (40 / 100) × 80 = 32, and subtract it from 80: 80 - 32 = 48.

Explanation

The mid-value of a class interval is calculated using the formula:

Mid-value = (Lower boundary + Upper boundary) / 2

Substituting the given values into the formula: 35 = (30 + Upper boundary) / 2 70 = 30 + Upper boundary Upper boundary = 70 - 30 = 40

Explanation

Average = Total number of candidates / Total number of years

Sum of candidates from Kerala across all five years (2012 to 2016) = 440 + 400 + 350 + 380 + 420 = 1990

Average = 1990 / 5 = 398

Explanation

The total number of smaller cubes formed by dividing a larger cube is given by n³, where n is the ratio of the side length of the larger cube to that of a smaller cube.

n = Side of larger cube / Side of smaller cube = 8 / 2 = 4

Total number of smaller cubes = n³ = 4³ = 64

Explanation

Amount A under compound interest is calculated using the formula A = P(1 + r/100)ⁿ, where P is the principal amount, r is the annual interest rate, and n is the time period in years.

Given: P = ₹8000 r = 5% n = 2 years

Substituting the values into the formula: A = 8000 × (1 + 5/100)² A = 8000 × (21/20)² A = 8000 × (441 / 400) A = 20 × 441 A = ₹8820

Explanation

Selling Price = Labeled Price × (100 - d₁)/100 × (100 - d₂)/100

Substituting the given values: Selling Price = 12,000 × (100 - 25)/100 × (100 - 15)/100 = 12,000 × (3/4) × (17/20) = 150 × 51 = ₹7,650

Explanation

For two numbers with a known sum S and difference D, the larger number is given by the formula (S + D) / 2.

Substituting S = 33 and D = 13: Larger number = (33 + 13) / 2 = 46 / 2 = 23.

Explanation

Let x = √(6 + √(6 + √(6 + ...)))

Squaring both sides yields the quadratic equation: x² = 6 + x x² - x - 6 = 0

Factoring the quadratic equation gives: (x - 3)(x + 2) = 0

Since the expression represents a positive real value (x > 0), taking the positive root gives x = 3.

Alternatively, using the standard formula for an infinite radical series of the form √(n + √(n + ...)): x = (1 + √(1 + 4n)) / 2

Substituting n = 6: x = (1 + √(1 + 4 × 6)) / 2 = (1 + √25) / 2 = (1 + 5) / 2 = 3

Explanation

The diameter of the largest sphere that can be carved out from a solid cube is equal to the side length of the cube.

Given side length a = 6 cm, the radius of the sphere r = a / 2 = 3 cm.

Volume of a sphere V = (4/3)πr³

Substituting r = 3 cm: V = (4/3) × π × 3³ = (4/3) × π × 27 = 36π cm³

Explanation

Let Rajesh's marks be x. Then Ravi's marks are x + 8, and the total sum of their marks is 2x + 8.

Given that Ravi's marks are 52% of the total sum: x + 8 = 0.52 × (2x + 8) x + 8 = 1.04x + 4.16 0.04x = 3.84 x = 96

Thus, Rajesh obtained 96 marks and Ravi obtained 96 + 8 = 104 marks.

Explanation

Simplify the expression by converting the numbers into whole numbers and combining their powers of 10:

Numerator = 3.6 × 0.48 × 2.5 = (36 × 48 × 25) × 10⁻⁴ Denominator = 0.12 × 0.09 × 0.5 = (12 × 9 × 5) × 10⁻⁵

Dividing the numerator by the denominator: Value = [(36 ÷ 9) × (48 ÷ 12) × (25 ÷ 5)] × 10⁻⁴⁺⁵ Value = (4 × 4 × 5) × 10¹ Value = 80 × 10 = 800

Explanation

Passing marks = Marks obtained + Marks failed by = 60 + 60 = 120

Since the required passing criteria is 60% of the maximum marks: 60% of Maximum Marks = 120 (60 / 100) × Maximum Marks = 120 Maximum Marks = (120 × 100) / 60 = 200

Explanation

Using the compound angle formula tan(A + B) = (tan A + tan B) / (1 - tan A tan B), let A = 100° and B = 125°.

Since A + B = 225°, tan(225°) = tan(180° + 45°) = tan 45° = 1.

Substituting these values into the formula: 1 = (tan 100° + tan 125°) / (1 - tan 100° tan 125°)

Cross-multiplying: tan 100° + tan 125° = 1 - tan 100° tan 125°

Rearranging the terms: tan 100° + tan 125° + tan 100° tan 125° = 1

Explanation

Using the fundamental trigonometric identity cosec²θ - cot²θ = 1, we can factorize it as: (cosecθ - cotθ)(cosecθ + cotθ) = 1

Given cosecθ - cotθ = a, substituting this yields: cosecθ + cotθ = 1/a

Adding the two equations: (cosecθ - cotθ) + (cosecθ + cotθ) = a + 1/a 2 cosecθ = a + 1/a cosecθ = ½(a + 1/a)

Explanation

To find the Lowest Common Multiple (LCM) of 22, 34, and 40, determine the prime factorization of each number:

22 = 2 × 11 34 = 2 × 17 40 = 2³ × 5

Taking the highest power of each prime factor present: LCM = 2³ × 5 × 11 × 17 = 8 × 5 × 11 × 17 = 40 × 187 = 7480

Explanation

Total Simple Interest percentage = (R₁ × T₁) + (R₂ × T₂) + (R₃ × T₃)

For 11 years: First 3 years at 6% = 3 × 6% = 18% Next 5 years at 9% = 5 × 9% = 45% Remaining 3 years (11 - 8) at 13% = 3 × 13% = 39%

Total Interest Rate = 18% + 45% + 39% = 102%

Principal = (Total Interest × 100) / Total Interest Rate Principal = (8160 × 100) / 102 = 8,000

Explanation

Passing Marks = Obtained Marks + Shortfall = 120 + 30 = 150

Total Marks = (Passing Marks / Passing Percentage) × 100 Total Marks = (150 / 40) × 100 = 375

Explanation

Evaluating the continued fraction components step-by-step:

First term: 3 + 1/3 = 10/3, so 1/(3 + 1/3) = 3/10 = 0.3. This gives: 3 + 1/(3 + 1/3) = 3 + 0.3 = 3.3.

Second term: 2 + 1/2 = 5/2, so 1/(2 + 1/2) = 2/5 = 0.4. This gives: 2 + 1/(2 + 1/2) = 2 + 0.4 = 2.4.

Substituting these into the original expression: 35.7 - 3.3 - 2.4 = 35.7 - 5.7 = 30.

Explanation

The empirical relationship between mode, median, and mean is given by the formula:

Mode = 3 × Median - 2 × Mean

Given that Mode - Mean = 9, we have Mean = Mode - 9. Substituting this into the empirical formula:

Mode = 3 × Median - 2 × (Mode - 9) Mode = 3 × Median - 2 × Mode + 18 3 × Mode - 3 × Median = 18 3 × (Mode - Median) = 18 Mode - Median = 6

Explanation

Using the product-to-sum identity sin A cos B = ½ [sin(A + B) + sin(A - B)]: sin 21° cos 9° = ½ [sin 30° + sin 12°] = ½ [½ + sin 12°] = ¼ + ½ sin 12°

Since cos 84° = sin(90° - 84°) = sin 6°: cos 84° cos 6° = sin 6° cos 6° = ½ (2 sin 6° cos 6°) = ½ sin 12°

Subtracting the two terms: (¼ + ½ sin 12°) - ½ sin 12° = ¼

Explanation

Using the algebraic identity (a + b)(a - b) = a² - b²:

(7 + 3√5)(7 - 3√5) = 7² - (3√5)² = 49 - 45 = 4

Taking the square root of this product yields √4 = 2.

Explanation

Rewrite the given expression: Numerator = (0.2)³ + (0.04)³ = (2 × 0.1)³ + (2 × 0.02)³ = 2³ × [(0.1)³ + (0.02)³] = 8 × [(0.1)³ + (0.02)³]

Denominator = (0.1)² × 0.1 + (0.02) × (0.02)² = (0.1)³ + (0.02)³

Dividing the numerator by the denominator: Value = 8 × [(0.1)³ + (0.02)³] / [(0.1)³ + (0.02)³] = 8

Explanation

The HCF of a given set of fractions is calculated using the formula:

HCF of fractions = (HCF of numerators) / (LCM of denominators)

For the given fractions 2/3, 8/9, 16/81, and 10/27: HCF of numerators (2, 8, 16, 10) = 2 LCM of denominators (3, 9, 81, 27) = 81

Substituting these values into the formula gives HCF = 2/81.

Explanation

Let total work = LCM(12, 16) = 48 units.

Daily work rate: A = 48 ÷ 12 = 4 units/day B = 48 ÷ 16 = 3 units/day

In a 2-day alternate cycle (A on day 1, B on day 2), work completed = 4 + 3 = 7 units.

In 6 complete cycles (12 days), work completed = 6 × 7 = 42 units. Remaining work = 48 - 42 = 6 units.

On the 13th day, A works and completes 4 units, leaving 6 - 4 = 2 units of work. On the 14th day, B works. Since B's rate is 3 units/day, B takes ⅔ day to finish the remaining 2 units.

Total time taken = 12 + 1 + ⅔ = 13⅔ days.

Explanation

Let the required number be x. According to the given condition, the difference between the number and its two-fifths is 60:

x - (2/5)x = 60 (3/5)x = 60 x = (60 × 5) / 3 x = 100

Explanation

The slant height l of a frustum of a cone is given by the formula l = √[h² + (R - r)²], where h is the height, R is the larger radius, and r is the smaller radius.

Given h = 8 cm, R = 9 cm, and r = 3 cm:

R - r = 9 - 3 = 6 cm

Substituting the values into the formula: l = √(8² + 6²) = √(64 + 36) = √100 = 10 cm

Explanation

To find the required number, first determine the Least Common Multiple (LCM) of 3, 4, 5, and 6.

LCM(3, 4, 5, 6) = 60

Divide 2495 by 60 to find the remainder: 2495 ÷ 60 = 41 with a remainder of 35

The least number required to be added to 2495 to make the sum divisible by 60 is: 60 − 35 = 25

Explanation

If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram. In any parallelogram, adjacent angles are supplementary, meaning their sum is 180°. Therefore, ∠A + ∠B = 180°. Substituting ∠A = 45° yields ∠B = 180° − 45° = 135°.

Explanation

Difference in speeds = 60 km/h - 50 km/h = 10 km/h

Since the bus travels 10 km more than the car in every hour, the time taken for the bus to cover 50 km more than the car is: Time = 50 / 10 = 5 hours

Total distance between x and y = Relative speed × Time Total distance = (60 + 50) × 5 = 110 × 5 = 550 km

Explanation

Let each equal segment of distance be d km, making the total distance 3d km.

Total time = d/10 + d/15 + d/20 65/60 hours = d × (6 + 4 + 3)/60 65/60 = 13d/60 13d = 65 d = 5 km

Total distance = 3 × 5 = 15 km.

Alternatively, average speed for 3 equal distances: Average Speed = 3 / (1/10 + 1/15 + 1/20) = 180/13 km/h

Total Distance = Average Speed × Total Time Total Distance = (180/13) × (65/60) = 15 km

Explanation

Simplify the terms in the denominator: 2√12 = 2 × 2√3 = 4√3 √32 = 4√2 √50 = 5√2

Substituting these into the denominator gives: 5√3 - 4√3 - 4√2 + 5√2 = √3 + √2

Factor out √3 from the numerator: 3 + √6 = √3(√3 + √2)

Divide the numerator by the denominator: √3(√3 + √2) / (√3 + √2) = √3

Explanation

Cost Price per kg = Total Cost / Quantity = 320 / 80 = ₹4. Selling Price per kg = ₹4.50. Gain per kg = Selling Price - Cost Price = 4.50 - 4 = ₹0.50.

Gain % = (Gain / Cost Price) × 100 = (0.50 / 4) × 100 = 12.5% or 12½%

Explanation

Let x be the number of ₹2 coins and y be the number of ₹5 coins.

x + y = 60 2x + 5y = 240

Multiplying the first equation by 2 gives 2x + 2y = 120.

Subtracting this from the second equation: 3y = 120 y = 40

Substituting y = 40 into the first equation: x = 60 - 40 = 20

Thus, Rakesh has 20 coins of ₹2 and 40 coins of ₹5.

Explanation

Let total work be 1 unit.

Work done by Manvi and Darshan together in 25 days = 25/40 = ⅝ unit.

Remaining work = 1 - ⅝ = ⅜ unit.

Manvi completes ⅜ unit of work in 24 days.

Time taken by Manvi alone to complete 1 unit of work = 24 × (8/3) = 64 days.

Explanation

Let the present ages of A and B be 3x and 1x years respectively.

After 15 years, their ages will be (3x + 15) and (x + 15) years.

According to the given condition: (3x + 15) / (x + 15) = 2 / 1

Cross-multiplying: 3x + 15 = 2(x + 15) 3x + 15 = 2x + 30 x = 15

Present age of A = 3x = 3 × 15 = 45 years Present age of B = 1x = 1 × 15 = 15 years

Alternatively, balancing the difference of ratio terms: Present ratio = 3 : 1 (difference = 2) Future ratio = 2 : 1 = 4 : 2 (difference = 2)

The change of 1 ratio unit (4 - 3) corresponds to 15 years. A's present age = 3 units × 15 = 45 years B's present age = 1 unit × 15 = 15 years

Explanation

Number of candidates from MP in 2015 = 400

Total number of candidates from MP across all years (2012 to 2016) = 250 + 280 + 240 + 400 + 540 = 1710

Required percentage = (400 / 1710) × 100 = 23.39% ≈ 23%

Explanation

Production of rice in Bihar in 2016 = 250 hundred tonnes Production of rice in UP in 2017 = 225 hundred tonnes

Required ratio = 250 : 225 = 10 : 9

Explanation

The cost price (CP) is calculated using the formula: CP = [SP1 / (100 - Loss%)] × 100

Substituting SP1 = ₹31 and Loss = 7%: CP = [31 / (100 - 7)] × 100 = (31 / 93) × 100 = 100 / 3 = ₹33.33

When the article is sold at SP2 = ₹35, the profit percentage is: Profit% = [(SP2 - CP) / CP] × 100 Profit% = [(35 - 100/3) / (100/3)] × 100 = (5/3 ÷ 100/3) × 100 = 5%

Explanation

Let B and G represent the initial number of boys and girls, respectively.

From the first condition, when 12 girls leave, the ratio of boys to girls becomes 2:1: B / (G - 12) = 2 / 1 B = 2G - 24

From the second condition, when 30 boys leave after this, the ratio of girls to boys becomes 3:1: (G - 12) / (B - 30) = 3 / 1 G - 12 = 3B - 90

Substituting B = 2G - 24 into the second equation: G - 12 = 3(2G - 24) - 90 G - 12 = 6G - 162 5G = 150 G = 30

Substituting G = 30 into the expression for B gives: B = 2(30) - 24 = 36

Explanation

Using the algebraic identity a³ - b³ = (a - b)(a² + ab + b²), the expression simplifies to:

(a³ - b³) / (a² + ab + b²) = a - b

Taking a = 564 and b = 246:

a - b = 564 - 246 = 318

Explanation

By the principle of inclusion-exclusion, the percentage of students who failed in at least one subject is given by:

Fail(Hindi ∪ English) = Fail(Hindi) + Fail(English) - Fail(Both) = 28% + 32% - 20% = 40%

The percentage of students who passed in both subjects is the complement of those who failed in at least one subject:

Pass(Both) = 100% - 40% = 60%

Explanation

The surface area A of a sphere is directly proportional to the square of its radius r (A ∝ r²).

When the radius is increased by 50%, the new radius becomes 1.5 times the original radius.

New surface area = (1.5)² × original surface area = 2.25 × original surface area.

Expressed as a percentage, 2.25 × 100% = 225%.

Explanation

The coordinates of the centroid of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) are given by the formula:

((x₁ + x₂ + x₃) / 3, (y₁ + y₂ + y₃) / 3)

Substituting the given vertices (2, 5), (3, 9), and (4, 1):

x-coordinate = (2 + 3 + 4) / 3 = 9 / 3 = 3 y-coordinate = (5 + 9 + 1) / 3 = 15 / 3 = 5

This gives the centroid coordinates as (3, 5).

Explanation

When two objects move in opposite directions from the same point, their relative speed is the sum of their individual speeds.

Relative speed = 14 km/h + 21 km/h = 35 km/h

Time taken = Distance / Relative Speed Time = 43.75 km / 35 km/h = 1.25 hours

To convert the time into minutes: Time = 1.25 × 60 minutes = 75 minutes

Explanation

15% × (x - y) = 9% × (x + y)

Dividing both sides by 3%: 5(x - y) = 3(x + y) 5x - 5y = 3x + 3y 2x = 8y x = 4y

To find y as a percentage of x: (y / x) × 100 = (y / 4y) × 100 = 25%

Explanation

Volume of the entire object = Volume of cube + Volume of pyramid

Volume of cube = s³ = 10³ = 1000 cm³

The altitude of the pyramid is h = 3 × 10 = 30 cm, and its base is a square of side 10 cm, giving a base area of 10² = 100 cm².

Volume of pyramid = ⅓ × Base Area × Height = ⅓ × 100 × 30 = 1000 cm³

Total Volume = 1000 + 1000 = 2000 cm³

Explanation

Let the original price of the microwave oven be 100.

Sumanth buys it at a 15% discount, so his cost price = 100 - 15 = 85.

He sells it at a profit of 35% on his cost price: Selling price = 85 × (1 + 35/100) = 85 × 1.35 = 114.75.

Percentage by which the selling price is more than the original price = ((114.75 - 100) / 100) × 100 = 14.75%.

Explanation

Volume of a hemisphere = (2/3) × π × R³ Volume of a cylinder = π × r² × h

Given: Radius of hemispherical bowl (R) = 12 cm Radius of cylindrical bottle (r) = 3 cm Height of cylindrical bottle (h) = 4 cm

Volume of liquid in the bowl = (2/3) × π × 12³ = (2/3) × π × 1728 = 1152π cm³ Volume of one cylindrical bottle = π × 3² × 4 = 36π cm³

Number of bottles required = Volume of bowl / Volume of one bottle Number of bottles = 1152π / 36π = 32

Explanation

Let total work = LCM(20, 15) = 60 units.

Efficiency of A = 60 / 20 = 3 units/day Efficiency of B = 60 / 15 = 4 units/day

Work completed by A and B together in 5 days = 5 × (3 + 4) = 35 units Remaining work completed by C in 2 days = 60 - 35 = 25 units Efficiency of C = 25 / 2 = 12.5 units/day

Payment per unit of work = 2400 / 60 = ₹40

Daily wage of A = 3 × 40 = ₹120 Daily wage of C = 12.5 × 40 = ₹500

Difference between daily wages of C and A = 500 - 120 = ₹380