RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

By the Remainder Theorem, when a polynomial f(a) is divided by a linear divisor (a - ½), the remainder is given by f(½).

Setting the divisor to zero: a - ½ = 0 a = ½

Substituting a = ½ into f(a) = 4a³ - 12a² + 14a - 3: f(½) = 4(½)³ - 12(½)² + 14(½) - 3 f(½) = 4(1/8) - 12(1/4) + 14(1/2) - 3 f(½) = ½ - 3 + 7 - 3 f(½) = ½ + 1 = 3/2

Explanation

To find the greatest number that divides given numbers leaving specific remainders, subtract the respective remainders from the numbers and calculate the Highest Common Factor (HCF) of the results:

Required Number = HCF(13 - 1, 26 - 2, 39 - 3) = HCF(12, 24, 36)

Since 12 completely divides both 24 and 36, HCF(12, 24, 36) = 12.

Explanation

According to the algebraic identity: a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)

Given that the sum of the three numbers is zero, i.e., a + b + c = 0: a³ + b³ + c³ - 3abc = 0 × (a² + b² + c² - ab - bc - ca) = 0 a³ + b³ + c³ = 3abc

Substituting a + b + c = 0 simplifies the sum of the cubes, a³ + b³ + c³, directly to 3abc, which is three times the product of the three numbers.

Explanation

The Organisation for Economic Co-operation and Development (OECD) is an intergovernmental organisation established in 1961 to promote economic growth, social well-being, and international trade. Its official headquarters are located at the Château de la Muette in Paris, France.

Explanation

Adenosine Triphosphate (ATP) is known as the energy currency of the cell because it captures chemical energy released from the breakdown of food molecules and releases it to fuel cellular processes. The energy is stored within the high-energy phosphate bonds connecting its three phosphate groups. When a cell requires energy for metabolic reactions, active transport, or mechanical work, the terminal phosphate bond is broken through hydrolysis, releasing usable energy.

Explanation

For any two positive integers x and y, the product of the numbers equals the product of their LCM and HCF:

x × y = LCM × HCF

Substituting the given values into the formula:

189 × y = 2079 × 27

y = (2079 × 27) / 189

Since 189 / 27 = 7:

y = 2079 / 7 = 297

Explanation

Amir Khusrau, the iconic Sufi musician, poet, and scholar, was born in 1253 CE at Patiyali (located in present-day Kasganj district, formerly part of Etah district) in Uttar Pradesh. Regarded as the "Parrot of India" (Tuti-e-Hind), he was a devoted disciple of Hazrat Nizamuddin Auliya and made pioneering contributions to Persian and Hindavi literature, Qawwali, and Indian classical music.

Explanation

Using place value powers of 10:

0.0003618 = 36.18 × 10⁻⁵

Therefore, 1 / 0.0003618 = 1 / (36.18 × 10⁻⁵) = (1 / 36.18) × 10⁵ = 0.0276 × 100000 = 2760

Explanation

The relationship between maturity amount (A), principal sum (P), rate of interest (R), and time period (T) in simple interest is given by:

A = P × [1 + (R × T) / 100]

Substituting A = 175, R = 5%, and T = 8 years into the formula:

175 = P × [1 + (5 × 8) / 100] 175 = P × (1 + 0.4) 175 = 1.4 × P P = 175 / 1.4 = 125

Explanation

Using the work equivalence formula M₁ × D₁ = M₂ × D₂, where M is the number of workers and D is the number of days taken to complete the work.

Since each worker has one broom in both scenarios, the individual work rate remains constant.

Substituting the given values: 12 × 4 = 8 × D₂ 48 = 8 × D₂ D₂ = 48 / 8 = 6 days.

Explanation

Following the Revolt of 1857, the British Parliament passed the Government of India Act 1858 to end the rule of the East India Company. On November 1, 1858, Lord Canning, the first Viceroy of India, read out Queen Victoria's Proclamation at a Grand Durbar in Allahabad. This proclamation formally announced that India would henceforth be governed directly by and in the name of the British Monarch, administered through the newly created post of the Secretary of State for India.

Explanation

Let the measure of the angle be x. Since two supplementary angles add up to 180°, the supplementary angle is 180° - x.

According to the question: x = ½ × (180° - x) 2x = 180° - x 3x = 180° x = 60°

Explanation

Using the standard algebraic identity: a³ + b³ = (a + b)³ - 3ab(a + b)

Let a = cos θ and b = sec θ. Since cos θ × sec θ = 1, substituting the given value cos θ + sec θ = √3 yields:

cos³θ + sec³θ = (cos θ + sec θ)³ - 3(cos θ)(sec θ)(cos θ + sec θ) = (√3)³ - 3(1)(√3) = 3√3 - 3√3 = 0

Explanation

Any factor of 2³ × 3³ × 5⁴ × 7² is of the form 2ᵃ × 3ᵇ × 5ᶜ × 7ᵈ.

For a factor to be divisible by 50 (2¹ × 5²), we require a ≥ 1 and c ≥ 2. For it to not be divisible by 100 (2² × 5²), the exponent of 2 cannot be 2 or greater. Thus, a = 1 (1 choice).

The exponent of 3 (b) can be 0, 1, 2, or 3 (4 choices). The exponent of 5 (c) can be 2, 3, or 4 (3 choices). The exponent of 7 (d) can be 0, 1, or 2 (3 choices).

By the fundamental counting principle, the total number of such factors is 1 × 4 × 3 × 3 = 36.

Explanation

Selling price of each article = ₹4,000 Total selling price = ₹8,000

Cost price of the first article = 4000 / 1.15 = ₹3,478.26 Cost price of the second article = 4000 / 0.85 = ₹4,705.88

Total cost price = 3478.26 + 4705.88 = ₹8,184.14

Total loss = Total cost price - Total selling price Total loss = 8184.14 - 8000 = ₹184.14

Rounding to the nearest integer gives a loss of ₹184.

Explanation

To find the total number of divisors of a composite number, express it as the product of its prime factors:

120 = 2³ × 3¹ × 5¹

Using the formula for the number of divisors d(N) = (a + 1)(b + 1)(c + 1)... where a, b, c are the exponents of the prime factors:

d(120) = (3 + 1) × (1 + 1) × (1 + 1) = 4 × 2 × 2 = 16

Explanation

Value per kg is calculated using the formula: Value per kg = Total Value (in ₹) ÷ Total Quantity (in kg)

Calculating the value per kg for each year: For 1993: 230 ÷ 110 = 2.09 ₹/kg For 1994: 210 ÷ 95 = 2.21 ₹/kg For 1995: 340 ÷ 125 = 2.72 ₹/kg For 1996: 430 ÷ 130 = 3.31 ₹/kg

Comparing these results, the maximum value per kg is achieved in 1996, which is approximately 3.3 ₹/kg.

Explanation

When a cube is remolded into a cuboid, its volume remains conserved at 324 cm³.

Let the length, width, and height of the cuboid be 3x, 2x, and 2x respectively.

Volume of cuboid = length × width × height (3x) × (2x) × (2x) = 324 12x³ = 324 x³ = 27 x = 3

Length of the cuboid = 3x = 3 × 3 = 9 cm.

Explanation

Find the least common multiple (LCM) of the denominators 9, 12, 16, and 8, which is 144.

Convert each fraction to an equivalent fraction with denominator 144: 7/9 = 112/144 11/12 = 132/144 13/16 = 117/144 1/8 = 18/144

Substitute these into the expression: (112 - 132 + 117 - 18) / 144 = 79/144

Explanation

Amount of spirit in the original liquid = 20% of 50 L = (20 / 100) × 50 = 10 L.

When 10 L of water is added, the total volume of the resulting mixture becomes 50 L + 10 L = 60 L.

Since no additional spirit was added, the quantity of spirit remains 10 L.

Percentage of spirit in the new mixture = (Amount of spirit / Total volume) × 100% = (10 / 60) × 100% = 100 / 6 % = 16⅔%.

Explanation

Total number of balls in the bowl = 9 + 7 + 5 = 21.

The total number of ways to draw 2 balls out of 21 is ²¹C₂ = (21 × 20) / 2 = 210.

For exactly one ball to be white, one ball must be drawn from the 7 white balls and the other ball from the 14 non-white balls (9 blue + 5 black).

Number of favorable ways = ⁷C₁ × ¹⁴C₁ = 7 × 14 = 98.

Probability = Favorable outcomes / Total outcomes = 98 / 210 = 7/15.

Explanation

Coast Redwood (Sequoia sempervirens), native to coastal California and southwestern Oregon, is the tallest living tree species on Earth. Individual coast redwoods can reach heights of over 115 meters (380 feet), with the tallest known living tree named Hyperion measuring 115.92 meters. These conifers thrive in cool, humid coastal climates where thick oceanic fog provides essential moisture for their immense growth.

Explanation

The 18 km additional distance covered at the original speed instead of the reduced speed saves 45 min - 36 min = 9 min = 3/20 h.

Let original speed be v km/h. The reduced speed is 4/5 v km/h.

Time taken to cover 18 km at reduced speed = 18 / (4/5 v) = 22.5 / v hours. Time taken to cover 18 km at original speed = 18 / v hours.

Time saved = (22.5 / v) - (18 / v) = 4.5 / v = 3/20

Solving for v: v = (4.5 × 20) / 3 = 30 km/h.

Explanation

The place value of a digit in a positional base-10 number system is given by the formula:

Place value = Digit × 10ⁿ

where n is the number of digits to the right of the given digit. In the number 367892141, the digit 3 is followed by 8 digits (6, 7, 8, 9, 2, 1, 4, 1), so n = 8.

Thus, the place value of 3 is 3 × 10⁸.

Explanation

Using the BODMAS rule (Brackets, Orders, Division/Multiplication, Addition/Subtraction):

First, evaluate the expression inside the brackets: (6 + 4) = 10 4 / 2 = 2

Next, multiply the results: 10 × 2 = 20

Finally, perform addition and subtraction from left to right: 20 + 5 - 3 = 22

Explanation

The time required to finish a task is inversely proportional to the efficiency or work rate.

Given that X and Y together are 3 times as fast as Z: Efficiency of (X + Y) = 3 × Efficiency of Z

Adding Z's efficiency to both sides gives the combined efficiency of all three: Efficiency of (X + Y + Z) = 3 × Efficiency of Z + 1 × Efficiency of Z = 4 × Efficiency of Z

Since all three working together complete the task in 15 days, Z working alone with ¼ of the combined efficiency will take: Time taken by Z = 4 × 15 = 60 days

Explanation

Let height of the watch tower h = 200 m.

The angles of depression to the two cliffs are 45° and 30°, which correspond to their angles of elevation from the ground.

Distance of the first cliff from the base of the tower = h / tan 45° = 200 / 1 = 200 m.

Distance of the second cliff from the base of the tower = h / tan 30° = 200 / (1 / √3) = 200√3 m.

Since both cliffs lie on the same side of the tower, the distance between them is the difference of their distances from the base: 200√3 - 200 = 200(√3 - 1) m.

Explanation

The length of a chord subtending an angle θ at the center of a circle of radius R is given by L = 2R sin(θ/2).

For chord x subtending 60° at the center: x = 2R sin(30°) = 2R × ½ = R

For chord y subtending 90° at the center: y = 2R sin(45°) = 2R × (1/√2) = √2 R

Substituting R = x yields y = √2 x.

Explanation

Evaluating the expression using the standard order of operations:

Numerator: 6 × 2 = 12 Denominator: 8 - 1 + 5 = 7 + 5 = 12

Dividing the numerator by the denominator: 12 ÷ 12 = 1

Explanation

Let the total population be 100.

Number of people affected with asthma = 20% of 100 = 20. Number of asthma patients working in flour mills = 90% of 20 = 18. Total number of people working in flour mills = 30% of 100 = 30.

Percentage of flour mill workers suffering from asthma = (18 / 30) × 100 = 60%.

Explanation

Simplify each square root by factoring out perfect squares:

√75 = √(25 × 3) = 5√3 √147 = √(49 × 3) = 7√3

Adding the two terms together: √75 + √147 = 5√3 + 7√3 = 12√3

Explanation

Total work = 18 × 12 skilled worker-days = 24 × 15 semi-skilled worker-days.

216 Skilled worker-days = 360 Semi-skilled worker-days 3 Skilled workers = 5 Semi-skilled workers

To complete the remaining task in the stipulated time, the work rate of the 3 skilled workers who left must be fully compensated by the replacement workers. Since 3 skilled workers are equivalent in work rate to 5 semi-skilled workers, exactly 5 semi-skilled workers are needed to replace them.

Explanation

Let C₁ and C₂ be the cost prices of the first and second bikes respectively, where C₁ + C₂ = ₹1,00,000.

Since the selling prices of both bikes are equal: 1.20 × C₁ = 0.80 × C₂ C₁ / C₂ = 0.80 / 1.20 = 2 / 3

Thus, the ratio of C₁ to C₂ is 2 : 3.

Cost price of the first bike (C₁) = (2 / 5) × 1,00,000 = ₹40,000

Explanation

Total students who appeared from AP in 2016 and 2017 = 260 + 400 = 660

Total students who appeared from UP in 2016 and 2017 = 450 + 500 = 950

Required ratio = 660 : 950 = 66 : 95

Explanation

Percentage of students passing at least one subject = 100% - 15% = 85%.

Using the set formula for inclusion-exclusion: Percentage passing both subjects = Percentage passing Physics + Percentage passing Chemistry - Percentage passing at least one subject Percentage passing both subjects = 80% + 70% - 85% = 65%

Since 65% of the total students corresponds to 325: Total students = (325 × 100) / 65 = 500