RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

Using the algebraic identity (a - b)(a + b) = a² - b²: (1 - cos P)(1 + cos P) = 1 - cos² P

Given 1 - cos² P = 3/4: cos² P = 1 - 3/4 = 1/4

Since P is an acute angle, cos P is positive: cos P = 1/2

Therefore, sec P = 1 / cos P = 1 / (1/2) = 2.

Explanation

Let the efficiency of Bahi be 1 unit per day, which makes Anil's efficiency 2 units per day. Their combined efficiency is 1 + 2 = 3 units per day. Since together they finish the work in 12 days, the total work is 3 × 12 = 36 units. Therefore, the time taken by Anil alone to complete the work is 36 / 2 = 18 days.

Explanation

Using the standard profit and loss formula: Selling Price = Cost Price × (100 ± Percentage) / 100

First, determine the Cost Price (CP) from the 15% loss transaction: CP × (100 - 15) / 100 = 170 CP × 85 / 100 = 170 CP = (170 × 100) / 85 = ₹200

Next, calculate the new selling price required to earn a 10% profit: New Selling Price = 200 × (100 + 10) / 100 = 200 × 1.10 = ₹220

Alternatively, using the direct shortcut formula: New Selling Price = Given Selling Price × (100 + Profit%) / (100 - Loss%) New Selling Price = 170 × 110 / 85 = 2 × 110 = ₹220

Explanation

Using the algebraic expansion formula (a + b)³ = a³ + 3a²b + 3ab² + b³:

101³ = (100 + 1)³ = 100³ + 3 × 100² × 1 + 3 × 100 × 1² + 1³ = 1000000 + 30000 + 300 + 1 = 1030301

Explanation

The prime factorizations of the given numbers are: 81 = 3⁴ 91 = 7 × 13 101 = 101 (prime) 111 = 3 × 37

The numbers 81 and 91 share no common prime factors, as 81 has only 3 as its prime factor while 91 is composed of 7 and 13. Furthermore, 101 is a prime number. Since there is no prime factor common to all four numbers, their Highest Common Factor (HCF) is 1.

Explanation

The smallest fraction that each of the given fractions will divide exactly is their Least Common Multiple (LCM).

LCM of fractions = LCM of Numerators / HCF of Denominators

For the given fractions 5/14, 6/7, and 10/21: LCM of Numerators = LCM(5, 6, 10) = 30 HCF of Denominators = HCF(14, 7, 21) = 7

Required fraction = 30/7

Explanation

Let C be the total number of chairs and P be the total number of people present in the hall.

Since 30 chairs are empty, 5/14 of the available chairs are vacant: (5/14) × C = 30 C = 84 chairs

Number of occupied chairs = 84 - 30 = 54 chairs

Since 6/11 of the total people present are sitting, each occupying one chair: (6/11) × P = 54 P = 99 people

The fraction of people standing is 1 - 6/11 = 5/11 of the total people: Number of standing people = (5/11) × 99 = 45

Explanation

Radha receives an active management allowance of ₹637.50 per month for 3 years (36 months). Total management allowance = 637.50 × 36 = ₹22,950

Subtracting this allowance from Radha's total earnings gives her share from the actual investment profit: Radha's profit share = 31,500 - 22,950 = ₹8,550

The investments of Radha and Vatsal are ₹36,000 and ₹42,000 respectively, which simplifies to an investment ratio of: Radha : Vatsal = 36,000 : 42,000 = 6 : 7

Since profit is shared in proportion to investment: Vatsal's profit = (7 ÷ 6) × 8,550 = 7 × 1,425 = ₹9,975

Explanation

Sum of the 6 positive integers = 6 × 55 = 330.

Let the 6 integers in ascending order be x₁ ≤ x₂ ≤ x₃ ≤ x₄ ≤ x₅ ≤ x₆. Given that x₆ - x₁ = 24, we get x₁ = x₆ - 24.

To maximize the largest integer x₆, the middle four integers x₂, x₃, x₄, and x₅ must be minimized. Since the integers are in ascending order, the minimum possible value for each of x₂, x₃, x₄, and x₅ is x₁.

Substituting x₂ = x₃ = x₄ = x₅ = x₁ into the sum equation: 5x₁ + x₆ = 330

Substituting x₁ = x₆ - 24: 5(x₆ - 24) + x₆ = 330 6x₆ - 120 = 330 6x₆ = 450 x₆ = 75

Explanation

Total work = 14 men × 9 days = 126 man-days = 16 women × 7 days = 112 woman-days.

Equating total work gives 126 M = 112 W, which simplifies to 9 M = 8 W.

Converting 6 men and 8 women into men: 6 men + 8 women = 6 men + 9 men = 15 men.

Using total work = Men × Days: 14 × 9 = 15 × Days Days = (14 × 9) ÷ 15 = 126 ÷ 15 = 8.4 days.

Explanation

The difference between the two maturity amounts represents the additional simple interest earned due to the 2% higher rate of interest over 7.2 years.

Additional Interest = ₹6,272 - ₹5,768 = ₹504

Using the simple interest formula for the additional interest: Additional Interest = (Principal × Extra Rate × Time) / 100

Substituting the given values: 504 = (Principal × 2 × 7.2) / 100 504 = Principal × 14.4 / 100 Principal = (504 × 100) / 14.4 = ₹3,500

Explanation

The Least Common Multiple (LCM) of fractions is given by the formula: LCM of fractions = (LCM of numerators) / (HCF of denominators)

For the fractions 6/25, 4/45, and 3/35: Numerators = 6, 4, 3 LCM(6, 4, 3) = 12

Denominators = 25, 45, 35 HCF(25, 45, 35) = 5

Substituting these values gives: LCM = 12/5

Explanation

Simple Interest = (P × R × T) / 100

Given: Principal (P) = ₹10,000, Rate (R) = 15% per annum, Time (T) = 2 years.

Simple Interest = (10,000 × 15 × 2) / 100 = ₹3,000

Explanation

In a right isosceles triangle with equal sides of length a, the hypotenuse is given by a√2.

Given hypotenuse = 34√2 cm: a√2 = 34√2 a = 34 cm

The area of the triangle is given by the formula: Area = ½ × base × height = ½ × a² Area = ½ × 34 × 34 = 17 × 34 = 578 cm²

Explanation

Selling Price = Cost Price × (100 + Gain%) / 100

Substituting the given values: Selling Price = 2,450 × (110 / 100) = 2,450 × 1.1 = ₹2,695

Explanation

Volume of a sphere of radius r is (4/3)πr³ and volume of a right circular cylinder of radius r and height h is πr²h.

Total volume = (4/3)πr³ + πr²h = (7/3)πr³

Subtracting (4/3)πr³ from both sides: πr²h = (7/3 - 4/3)πr³ πr²h = πr³

Dividing both sides by πr² gives h = r, which yields h/r = 1.

Explanation

Curved surface area of a cylindrical pillar = 2πrh Given radius r = 28 cm = 0.28 m, height h = 4 m, and π = 22/7.

Curved surface area of 1 pillar = 2 × (22/7) × 0.28 × 4 = 7.04 m²

Total curved surface area of 24 pillars = 24 × 7.04 = 168.96 m²

Total cost of painting at the rate of 8 per m² = 168.96 × 8 = 1,351.68

Explanation

The total number of factors of a composite number N = pᵃ × qᵇ is given by the formula (a + 1)(b + 1), where p and q are prime factors.

Expressing 12288 in its prime factorized form: 12288 = 4096 × 3 = 2¹² × 3¹

Applying the formula using exponents a = 12 and b = 1: Total factors = (12 + 1) × (1 + 1) = 13 × 2 = 26

Explanation

Using the right-angled triangle definitions for trigonometric ratios: cot θ = Adjacent / Opposite = 12 / 35

By Pythagoras theorem, Hypotenuse = √(Adjacent² + Opposite²) = √(12² + 35²) = √(144 + 1225) = √1369 = 37

sec θ = Hypotenuse / Adjacent = 37 / 12

Explanation

Speed of the train in m/s = 69 × (5/18) = 115/6 m/s.

The time taken by the train to cross the platform alone is the difference between the time taken to cross the platform and the time taken to cross the pole: 32 - 8 = 24 seconds.

Length of the platform = Speed × Time = (115/6) × 24 = 115 × 4 = 460 m.

Explanation

Compound Interest (CI) = P[(1 + R/100)ⁿ - 1]

Given P = ₹5,000, R = 6%, and n = 3 years:

CI = 5000 × [(1 + 6/100)³ - 1] CI = 5000 × [(1.06)³ - 1] CI = 5000 × [1.191016 - 1] CI = 5000 × 0.191016 = ₹955.08

Rounding off ₹955.08 to the nearest integer gives ₹955.

Explanation

Percentage increase = [(Final Value - Initial Value) / Initial Value] × 100

At the secondary level: Employment rate in 1995 = 12% Employment rate in 2000 = 18%

Percentage increase between 1995 and 2000 = [(18 - 12) / 12] × 100 = (6 / 12) × 100 = 50%

Explanation

Ratio = Expense on food / Expense on house rent = 1600 / 3600 = 4 / 9 = 4:9.

Explanation

The relationship between two numbers, their HCF, and their LCM is given by the formula:

Product of two numbers = HCF × LCM

Let the other number be x. Substituting the given values into the formula:

125 × x = 5 × 750

x = (5 × 750) ÷ 125 x = 3750 ÷ 125 x = 30

Explanation

Total score is calculated as 2W + 1D + 0L, where W, D, and L are the number of wins, draws, and losses respectively.

With D = 3 draws out of 20 matches, the remaining matches are W + L = 17, with constraints W ≥ 1 and L ≥ 1.

For the maximum score, set L to its minimum value of 1, giving W = 16: Maximum Score = 2(16) + 1(3) + 0(1) = 35

For the minimum score, set W to its minimum value of 1, giving L = 16: Minimum Score = 2(1) + 1(3) + 0(16) = 5

Explanation

When a person travels equal distances at two different speeds, a and b, the average speed for the entire journey is given by the formula:

Average Speed = 2ab / (a + b)

Substituting the given values, a = 15 km/hr and b = 10 km/hr:

Average Speed = (2 × 15 × 10) / (15 + 10) = 300 / 25 = 12 km/hr

Explanation

Average Speed = Total Distance / Total Time

Time taken for the first part = Distance / Speed = 30 / 6 = 5 hours Time taken for the second part = 5 hours Total time taken = 5 + 5 = 10 hours

Total distance covered = 30 + 40 = 70 km

Average speed = 70 / 10 = 7 km/hr

Explanation

Divide the numerator and denominator of the given expression by cos θ:

(p sin θ - q cos θ) / (p sin θ + q cos θ) = (p tan θ - q) / (p tan θ + q)

Substituting tan θ = p / q gives:

[p(p / q) - q] / [p(p / q) + q] = (p² / q - q) / (p² / q + q) = (p² - q²) / (p² + q²)

Explanation

For any two positive integers, the product of the numbers is equal to the product of their HCF and LCM.

First number × Second number = HCF × LCM

Substituting the given values: 55 × Second number = 11 × 330 Second number = (11 × 330) / 55 = 330 / 5 = 66

Explanation

HUDCO pays simple interest on the borrowed amount and receives compound interest on the lent amount over three years.

Simple Interest paid = (P × R × T) / 100 = (1,000,000 × 10 × 3) / 100 = ₹300,000

Compound Interest received = P[(1 + R/100)ᵀ - 1] = 1,000,000 × [(1 + 10/100)³ - 1] = 1,000,000 × (1.331 - 1) = ₹331,000

Net earning of HUDCO = Compound Interest received - Simple Interest paid = ₹331,000 - ₹300,000 = ₹31,000

Explanation

Let the initial quantities of sand and cement be 9x kg and 2x kg respectively.

According to the given condition: (9x + 20) / (2x + 10) = 4 / 1 9x + 20 = 8x + 40 x = 20

Initial quantity of cement = 2x = 2 × 20 = 40 kg.

Explanation

Volume of a cuboid = length × breadth × height = 54 × 18 × 6 cm³.

Factoring the dimensions: Volume = (3 × 18) × 18 × (18 / 3) = 18 × 18 × 18 = 18³ cm³.

Since the volume of the cube equals the volume of the cuboid: a³ = 18³ a = 18 cm.

Explanation

Converting the boundary fractions into decimal form: 5/7 ≈ 0.714 3/4 = 0.750

The required range for the fractions is between 0.714 and 0.750.

Converting 8/11 and 11/15 into decimal form: 8/11 ≈ 0.727 11/15 ≈ 0.733

Since 0.714 < 0.727 < 0.733 < 0.750, both 8/11 and 11/15 lie strictly between 5/7 and 3/4.

Explanation

Profit is shared in the ratio of the product of capital and investment period.

Total annual profit = ₹1200 Raja's profit share = ₹800 Ranga's profit share = 1200 − 800 = ₹400

Ratio of Raja's profit to Ranga's profit = 800 : 400 = 2 : 1

Let Ranga invest for t months. Raja invested for the full year (12 months):

(4400 × 12) / (2400 × t) = 2 / 1 52800 / 2400t = 2 4800t = 52800 t = 11 months

Since Ranga invested his capital for 11 months, he joined 12 − 11 = 1 month after Raja started the business.

Explanation

Using the standard work-time formula, if A and B together take x days and A alone takes y days, the time taken by B alone is given by:

Time taken by B = (x × y) / (y - x)

Substituting x = 6 and y = 9:

Time taken by B = (6 × 9) / (9 - 6) = 54 / 3 = 18 days.

Alternatively, using unit work rate: Rate of B = 1/6 - 1/9 = (3 - 2) / 18 = 1/18 work per day.

Explanation

Profit is distributed in the ratio of the product of capital invested and time period:

Profit Ratio = (Capital of A × Time of A) : (Capital of B × Time of B)

= (7500 × 8) : (8000 × 10) = 60000 : 80000 = 3 : 4

Explanation

Converting the fractions to have a common denominator of 4:

-3/2 = -6/4 3/2 = 6/4 5/2 = 10/4 11/4 = 11/4

Comparing the numerators (-6, 6, 10, 11), the largest numerator is 11, making 11/4 (or 2.75 in decimal form) the greatest number.

Explanation

Two numbers are co-prime if their greatest common divisor (GCD) is 1. Prime factorization gives 28 = 2² × 7 and 81 = 3⁴. Since 28 and 81 share no common prime factors, GCD(28, 81) = 1, making them a pair of co-prime numbers.

Explanation

In a right-angled isosceles triangle, the two perpendicular sides (legs) are equal in length.

By the Pythagorean theorem: Hypotenuse = √(a² + a²) = a√2

Given the side length a = 2 m: Hypotenuse = 2 × √2 = 2√2 m

Explanation

The smallest three-digit number is 100, which is composite as it is divisible by 2 and 5. Testing the next consecutive integer, 101, for primality involves checking divisibility by primes up to √101 ≈ 10, namely 2, 3, 5, and 7. Since 101 is not divisible by any of these primes, it is a prime number and the smallest three-digit prime number.

Explanation

The commutative property of multiplication states that for any two numbers a and b, a × b = b × a. Changing the order of the numbers being multiplied does not change the final product. In this statement, -3 and -7 are multiplied in different orders to yield the same result.

Explanation

Using the complementary angle identity sin(90° - θ) = cos θ, we rewrite the terms as sin 85° = cos 5° and sin 80° = cos 10°.

Substituting these into the expression gives: sin² 5° + sin² 10° + sin² 80° + sin² 85° = (sin² 5° + cos² 5°) + (sin² 10° + cos² 10°)

Applying the trigonometric identity sin² θ + cos² θ = 1: = 1 + 1 = 2

Explanation

To convert a percentage into a fraction, divide the percentage value by 100 and simplify:

48% = 48/100

Dividing both the numerator and denominator by their greatest common divisor, 4:

48/100 = (48 ÷ 4) / (100 ÷ 4) = 12/25

Explanation

By the division algorithm, dividend = (divisor × quotient) + remainder.

Here, 43 = qx + (x - 5), where q is a non-negative integer quotient. 48 = x(q + 1)

Since x is a divisor of 48, and the remainder (x - 5) must be non-negative (x - 5 ≥ 0), x must be a natural number greater than or equal to 5.

The positive divisors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.

Filtering for x ≥ 5 gives 6 values: 6, 8, 12, 16, 24, and 48. Thus, x has 6 possible solutions.

Explanation

The length of the tangent from an external point (x₁, y₁) to a circle with equation x² + y² = r² is given by the formula L = √(x₁² + y₁² - r²).

Substituting (x₁, y₁) = (2, 3) and r² = 4 into the formula: L = √(2² + 3² - 4) L = √(4 + 9 - 4) L = √9 = 3

Explanation

In simple interest, the relation between the multiple of sum (n) and time (T) is given by (n₁ - 1) / T₁ = (n₂ - 1) / T₂.

Given n₁ = 3, T₁ = 15 years, and n₂ = 5:

(3 - 1) / 15 = (5 - 1) / T₂

2 / 15 = 4 / T₂

T₂ = (4 × 15) / 2 = 30 years

Explanation

Let the two numbers be 18a and 18b, where a and b are co-prime integers.

LCM = HCF × a × b 378 = 18 × a × b a × b = 21

The pairs of co-prime factors of 21 are (1, 21) and (3, 7).

For the factor pair (3, 7): First number = 18 × 3 = 54 Second number = 18 × 7 = 126

Explanation

Using the BODMAS rule to solve the expression step-by-step:

Innermost parenthesis: 1 - 36 ÷ 6 × 8 = 1 - 6 × 8 = 1 - 48 = -47

Curly brackets: 45 - (-47) = 45 + 47 = 92

Square brackets: 37 - 92 = -55

Final expression: 79 + (-55) = 79 - 55 = 24

Explanation

Let the cost price of 1 orange be CP and selling price be SP.

12 × CP = 9 × SP SP / CP = 4 / 3

Taking CP = 3 and SP = 4: Profit per orange = SP - CP = 4 - 3 = 1

Discount on 10 oranges = Profit on 5 oranges: 10 × Discount = 5 × 1 Discount = 0.5

Marked Price (MP) = SP + Discount = 4 + 0.5 = 4.5

Discount Percentage = (Discount / MP) × 100 = (0.5 / 4.5) × 100 = (1 / 9) × 100 = 11.11%

Explanation

In a parallelogram PQRS, the diagonals PR and QS bisect each other at the same midpoint. Therefore, the sum of opposite vertices is equal: P + R = Q + S, which gives S = P + R - Q.

Substituting the coordinates of P(-1, -1), R(7, 5), and Q(8, 0): x = -1 + 7 - 8 = -2 y = -1 + 5 - 0 = 4

Thus, the coordinates of vertex S are (-2, 4).