RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

To minimize the number of rows, each row must contain the maximum possible number of trees of a single variety. This capacity per row is given by the Highest Common Factor (HCF) of 24, 56, and 72.

HCF(24, 56, 72) = 8 trees per row.

Number of rows required: Mango tree rows = 24 ÷ 8 = 3 Apple tree rows = 56 ÷ 8 = 7 Orange tree rows = 72 ÷ 8 = 9

Total minimum number of rows = 3 + 7 + 9 = 19

Explanation

Let the two numbers be 16a and 16b, where a and b are co-prime positive integers with gcd(a, b) = 1.

16a + 16b = 288 16(a + b) = 288 a + b = 18

We select pairs of co-prime numbers (a, b) such that a < b and a + b = 18: Pair 1: (1, 17) with gcd(1, 17) = 1, giving numbers 16 and 272. Pair 2: (5, 13) with gcd(5, 13) = 1, giving numbers 80 and 208. Pair 3: (7, 11) with gcd(7, 11) = 1, giving numbers 112 and 176.

Other pairs summing to 18, such as (2, 16), (3, 15), (4, 14), (6, 12), (8, 10), and (9, 9), share common factors greater than 1, which would change the overall HCF.

Explanation

Distance = (s₁ × s₂) / (s₂ - s₁) × Δt

Given: Speed 1 (s₁) = 30 km/h Speed 2 (s₂) = 40 km/h Time difference (Δt) = 5 min late + 3 min early = 8 min = 8/60 h

Substituting the values into the formula: Distance = (30 × 40) / (40 - 30) × (8 / 60) Distance = (1200 / 10) × (8 / 60) Distance = 120 × (2 / 15) Distance = 16 km

Explanation

In a right-angled triangle, the hypotenuse c is calculated using the Pythagoras theorem: c = √(10² + 24²) = √(100 + 576) = √676 = 26 cm

The radius r of an inscribed circle in a right-angled triangle with perpendicular sides a, b and hypotenuse c is given by: r = (a + b - c) / 2

Substituting the values: r = (10 + 24 - 26) / 2 = 8 / 2 = 4 cm

Alternatively, using r = Area / s: Area = ½ × 10 × 24 = 120 cm² Semi-perimeter s = (10 + 24 + 26) / 2 = 30 cm r = 120 / 30 = 4 cm

Explanation

Let the longer diagonal of the rhombus be d. The shorter diagonal is 65% of d, which equals (65/100)d = (13/20)d.

Area of rhombus = ½ × diagonal₁ × diagonal₂ = ½ × d × (13/20)d = (13/40)d²

Since the side of the square is equal to the longer diagonal d: Area of square = d²

Ratio of the area of the rhombus to the area of the square = [(13/40)d²] / d² = 13 / 40 = 13:40

Explanation

In an isosceles triangle, the perpendicular altitude drawn to the base bisects the base into two equal halves.

For equal sides a = 26 cm and base b = 20 cm: Half of the base = 20 / 2 = 10 cm

Using the Pythagorean theorem: Altitude h = √(26² - 10²) = √(676 - 100) = √576 = 24 cm

Area of triangle ABC = ½ × base × altitude Area = ½ × 20 × 24 = 240 cm²

Explanation

Sum of weights of P, Q and R = 3 × 58 = 174 kg Sum of weights of P and Q = 2 × 54 = 108 kg Sum of weights of Q and R = 2 × 48 = 96 kg

Sum of (P + Q) and (Q + R) = P + 2Q + R = 108 + 96 = 204 kg Weight of Q = (P + 2Q + R) - (P + Q + R) = 204 - 174 = 30 kg

Explanation

To find the smallest number by which 35280 must be divided to make the quotient a perfect square, prime factorize 35280:

35280 = 2⁴ × 3² × 5¹ × 7²

For a number to be a perfect square, the powers of all its prime factors must be even numbers. In the prime factorization of 35280, the exponents of 2, 3, and 7 are even (4, 2, and 2), while the exponent of 5 is odd (1).

Dividing 35280 by 5 removes the unpaired prime factor, leaving 2⁴ × 3² × 7² = 7056, which is a perfect square (84²).

Explanation

Using the effective percentage change formula for successive changes: Effective change = -20 + 15 + (-20 × 15)/100 = -8%

This represents a net decrease of 8% from the original number.

Given that an 8% decrease corresponds to a reduction of 64: Original number = (64 / 8) × 100 = 800

Explanation

Using the trigonometric identity tan θ + cot θ = 2 / sin 2θ, substitute θ = 15°:

tan 15° + cot 15° = 2 / sin(2 × 15°) = 2 / sin 30°

Since sin 30° = ½:

tan 15° + cot 15° = 2 / (½) = 4

Alternatively, using standard trigonometric values tan 15° = 2 - √3 and cot 15° = 2 + √3:

tan 15° + cot 15° = (2 - √3) + (2 + √3) = 4

Explanation

Let the daily work rate of a man be M and that of a boy be B.

Since total work equals daily efficiency multiplied by the number of days: 10 × (20M + 15B) = 9 × (25M + 10B) 200M + 150B = 225M + 90B 25M = 60B M / B = 60 / 25 = 12 / 5

So, M : B = 12 : 5.

Explanation

Pass percentage of boys in school B = 32 × (3 / (3 + 5)) = 32 × (3 / 8) = 12%

Pass percentage of boys in school C = 24 × (1 / (1 + 2)) = 24 × (1 / 3) = 8%

Ratio of pass percentage of boys in school B to school C = 12 : 8 = 3 : 2

Explanation

Selling Price is directly proportional to the percentage of Cost Price (CP).

120% of CP = 1,20,000 1% of CP = 1,000 130% of CP = 130 × 1,000 = 1,30,000

Thus, to earn a profit of 30%, the car must be sold for 1,30,000.

Explanation

By Archimedes' principle, the total volume of the N submerged lead spheres equals the volume of water displaced in the cylindrical beaker.

Volume of one sphere = (4/3)πr³ Given diameter of sphere = 6 cm, so radius r = 3 cm. Volume of one sphere = (4/3) × π × 3³ = 36π cm³

Volume of displaced water = πR²h Given beaker diameter = 9 cm, so radius R = 9/2 cm, and rise in water level h = 32 cm. Volume of displaced water = π × (9/2)² × 32 = π × (81/4) × 32 = 648π cm³

Number of spheres N = Volume of displaced water ÷ Volume of one sphere N = 648π ÷ 36π = 18

Explanation

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

Perform division: 6.3 ÷ 7 = 0.9

Perform multiplication: 3 × 1.3 = 3.9

Substitute these values back into the expression: 15 - 0.9 + 3.9 - 2

Perform addition and subtraction from left to right: 15 - 0.9 + 3.9 - 2 = 14.1 + 3.9 - 2 = 18 - 2 = 16

Explanation

From √x = 8, squaring both sides gives x = 64.

Squaring x gives x² = 64² = 4096.

Substituting x² into the equation x² + y = 4112: 4096 + y = 4112 y = 4112 - 4096 y = 16

Therefore, √y = √16 = 4.

Explanation

Following the BODMAS rule:

First, evaluate the expression in brackets: 5.4 ÷ 9 = 0.6

Next, perform multiplication operations: 4 × 0.6 = 2.4 6 × 1.9 = 11.4

Finally, perform subtraction and addition from left to right: 17 - 2.4 + 11.4 = 14.6 + 11.4 = 26

Explanation

Following the BODMAS rule, simplify each term of the expression separately:

First term: 25 ÷ 10 = 5/2 = 2.5

Second term: (7/4 × 1/3) × 6/5 = 7/12 × 6/5 = 7/10 = 0.7

Third term: 14/3 × 9/10 = 42/10 = 4.2

Fourth term: 1/5 ÷ 1/25 = 1/5 × 25 = 5

Substituting these evaluated values back into the expression: 2.5 - 0.7 + 4.2 - 5 = 1.8 + 4.2 - 5 = 6 - 5 = 1

Explanation

By modular arithmetic properties, if the remainder when n is divided by 9 is 4, then the remainder when 12n is divided by 9 equals the remainder when 12 × 4 is divided by 9.

12 × 4 = 48

Dividing 48 by 9 gives 48 = 9 × 5 + 3, so the remainder is 3.

Explanation

Let the monthly incomes of Amar and Komal be 5x and 4x respectively.

Since Income − Expenditure = Savings, their expenditures are (5x − 6000) and (4x − 6000).

Given the ratio of their expenditures is 2 : 1: (5x − 6000) / (4x − 6000) = 2 / 1

Cross-multiplying: 5x − 6000 = 2(4x − 6000) 5x − 6000 = 8x − 12000 3x = 6000 x = 2000

Amar's income = 5x = 5 × 2000 = 10,000.

Explanation

Using the trigonometric identity sec² θ - tan² θ = 1:

Given: sec θ = 5x => x² = sec² θ / 25 tan θ = 5/x => 1/x² = tan² θ / 25

Subtracting the two equations: x² - 1/x² = (sec² θ - tan² θ) / 25 = 1 / 25

Multiplying both sides by 10: 10(x² - 1/x²) = 10 × (1 / 25) = 10/25 = 2/5

Explanation

The numbers that leave a remainder of 5 when divided by 8, 12, and 16 are of the form LCM(8, 12, 16) × k + 5.

LCM(8, 12, 16) = 48

The general expression for these numbers is 48k + 5.

Finding the values of 48k + 5 between 400 and 500: For k = 9: 48 × 9 + 5 = 432 + 5 = 437 For k = 10: 48 × 10 + 5 = 480 + 5 = 485

Sum of these numbers = 437 + 485 = 922.

Explanation

Out of every 18 eggs bought, 3 are rotten, which leaves 18 - 3 = 15 good eggs.

The fraction of good eggs is 15/18 = 5/6.

For 690 eggs bought, the total number of good eggs = 690 × (5/6) = 115 × 5 = 575.

Explanation

Using the identity (x + x⁻¹)³ = x³ + x⁻³ + 3(x + x⁻¹), we can express x³ + x⁻³ as:

x³ + x⁻³ = (x + x⁻¹)³ - 3(x + x⁻¹)

Given x + x⁻¹ = 7:

x³ + x⁻³ = 7³ - 3(7) = 343 - 21 = 322

Explanation

The compound interest formula relating amount A, principal P, interest rate r, and time n is A/P = (1 + r/100)ⁿ.

Substituting A/P = 14641/10000 and n = 4:

(1 + r/100)⁴ = 14641 / 10000

Taking the fourth root on both sides:

1 + r/100 = 11 / 10

r/100 = 11/10 - 1 = 1/10

r = 10%

Explanation

Let the cost prices of the two TV sets be C₁ and C₂. Since their selling prices are equal:

120% of C₁ = 85% of C₂ C₁ / C₂ = 85 / 120 = 17 / 24

Total cost price = 17x + 24x = 41x = ₹41,000 x = 1,000

Thus, C₁ = ₹17,000 and C₂ = ₹24,000.

Profit on the first TV = 20% of 17,000 = ₹3,400 Loss on the second TV = 15% of 24,000 = ₹3,600

Overall loss = ₹3,600 - ₹3,400 = ₹200

Explanation

Using the standard algebraic identity: a³ + b³ + c³ - 3abc = (a + b + c)[(a + b + c)² - 3(ab + bc + ca)]

Substituting the given values into the formula: a³ + b³ + c³ - 3(15) = 14 × [14² - 3(47)] a³ + b³ + c³ - 45 = 14 × (196 - 141) a³ + b³ + c³ - 45 = 14 × 55 a³ + b³ + c³ - 45 = 770 a³ + b³ + c³ = 770 + 45 = 815

Explanation

Simple interest is calculated using the formula SI = (P × R × T) / 100, where P is principal, R is rate of interest per annum, and T is time in years.

Interest from Dubey = (7200 × R × 3) / 100 = 216R Interest from Raghav = (8400 × R × 4) / 100 = 336R

Total interest received = 216R + 336R = 552R

Given total interest = 4968 552R = 4968 R = 4968 / 552 = 9%

Explanation

The greatest five-digit number is 99999. To find the largest five-digit number divisible by 468, divide 99999 by 468 to determine the remainder:

99999 ÷ 468 = 213 with a remainder of 315 99999 = (468 × 213) + 315

Subtracting the remainder 315 from 99999 yields:

99999 - 315 = 99684

Explanation

When two objects move in opposite directions, the relative speed at which the distance between them increases is equal to the sum of their individual speeds.

Sum of speeds = Total Distance / Time = 4500 km / 9 h = 500 km/h

Explanation

Let A be the average age of the 11-member team. The total age of the team is 11A.

The age of the captain is 35 years and the age of the wicket keeper is 40 years (35 + 5), making their combined age 75 years.

Excluding these 2 players leaves 9 players with an average age of A - 3 years. Setting up the equation for the total age of the team:

11A = 9(A - 3) + 75 11A = 9A - 27 + 75 2A = 48 A = 24 years

Explanation

A cuboid is a three-dimensional solid shape bounded by six rectangular faces. It has 8 vertices and 12 straight line segments known as edges. These 12 straight lines comprise 4 edges forming the top rectangular face, 4 edges forming the bottom rectangular face, and 4 vertical edges connecting the top and bottom faces.

Explanation

Using the identity cot A cot B = 1, we have cos(A + B) = 0.

Here, A = 3θ and B = 6θ, so cos(3θ + 6θ) = 0, which gives cos 9θ = 0.

The general solution is 9θ = 90° + k × 180°.

Taking k = 2 gives 9θ = 450°, so θ = 50°.

Verifying: cot(3 × 50°) × cot(6 × 50°) = cot 150° × cot 300° = (-√3) × (-1/√3) = 1.

Substituting θ = 50° into tan 15θ: tan(15 × 50°) = tan 750° = tan(2 × 360° + 30°) = tan 30° = 1/√3.

Explanation

Let Cost Price (CP) = 100.

Since Profit = 150% of CP, Profit = 150

Selling Price (SP) = CP + Profit = 100 + 150 = 250

Ratio of Cost Price to Selling Price (CP : SP) = 100 : 250 = 2 : 5

Explanation

The sum of an interior angle and an exterior angle of a regular polygon is always 180°. Given the ratio of interior angle to exterior angle is 4:1, the measure of the exterior angle is 180° × 1 / (4 + 1) = 36°.

The number of sides n of a regular polygon is given by the formula n = 360° / exterior angle = 360° / 36° = 10.

Explanation

Alok's present age = 62 - 20 = 42 years.

Given the ratio of present ages of Alok and Anil is 3 : 4, Anil's present age = (4 / 3) × 42 = 56 years.

Explanation

Ratio of efficiency of A to B = 5 : 1. Since efficiency is inversely proportional to time taken, the ratio of time taken by A to B = 1 : 5.

Difference in time units = 5 - 1 = 4 units. Given that 4 units = 60 days, 1 unit = 15 days. Therefore, A takes 15 days and B takes 75 days to complete the job individually.

When working together, total time taken = (A × B) / (A + B) = (15 × 75) / (15 + 75) = 1125 / 90 = 12.5 = 12½ days.

Explanation

Total work = 10 × 18 = 180 man-days.

Work completed in the first 6 days by 10 men = 10 × 6 = 60 man-days.

Remaining work = 180 − 60 = 120 man-days.

After 6 days, 5 more men join, making a total of 10 + 5 = 15 men.

Days required to complete remaining work = Remaining man-days / Total men = 120 / 15 = 8 days.

Explanation

The side length of the equilateral triangle is equal to its perimeter divided by 3, which gives 36 / 3 = 12 m.

Using the area formula for a triangle, Area = ½ × base × height: Area = ½ × 12 m × 6 m = 36 m²

Explanation

Prime factorization of each given number: 22 = 2 × 11 24 = 2³ × 3 48 = 2⁴ × 3 16 = 2⁴

To find the Least Common Multiple (LCM), multiply the highest powers of all prime factors involved: LCM = 2⁴ × 3¹ × 11¹ = 16 × 3 × 11 = 528

Explanation

Using the work equivalence formula (M₁ × D₁) / W₁ = (M₂ × D₂) / W₂:

Given: M₁ = 15 persons, D₁ = 7 days, W₁ = 35 boxes M₂ = required persons, D₂ = 5 days, W₂ = 65 boxes

Substituting the values into the formula: (15 × 7) / 35 = (M₂ × 5) / 65 105 / 35 = 5M₂ / 65 3 = 5M₂ / 65 3 = M₂ / 13 M₂ = 39

Explanation

The formula for the difference in Simple Interest is ΔSI = (P × ΔR × T) / 100, where P is the principal, T is the time in years, and ΔR is the difference between the annual interest rates.

Substituting the given values: 10.80 = (1200 × ΔR × 3) / 100 10.80 = 36 × ΔR ΔR = 10.80 / 36 = 0.3%

Explanation

When one number in a given set of numbers is a multiple of all the other numbers, that largest number is itself the least common multiple (LCM).

Here, 144 = 36 × 4 and 144 = 72 × 2. Since 144 is divisible by both 36 and 72, LCM(36, 72, 144) = 144.

Explanation

The area of a square with diagonal d is given by Area = d² / 2. Since the area is directly proportional to the square of the diagonal, the ratio of their areas is equal to the square of the ratio of their diagonals.

Ratio of areas = (3 / 5)² = 9 / 25 = 9:25

Explanation

The fundamental division formula is: Dividend = (Divisor × Quotient) + Remainder.

Given that Remainder = 8: Divisor = 6 × Remainder = 6 × 8 = 48 Quotient = Divisor / 2 = 48 / 2 = 24

Substituting these values into the division formula: Dividend = (48 × 24) + 8 = 1152 + 8 = 1160

Explanation

A number is divisible by 9 if the sum of its digits is a multiple of 9.

Sum of digits of 2346x45 = 2 + 3 + 4 + 6 + x + 4 + 5 = 24 + x

The smallest non-negative integer value for x that makes (24 + x) a multiple of 9 is 3, since 24 + 3 = 27, which is divisible by 9.

Explanation

Let the two natural numbers be 6x and 8x.

Given that their product is 768: (6x) × (8x) = 768 48x² = 768 x² = 768 / 48 = 16 x = 4

The smaller number is 6x = 6 × 4 = 24.

Explanation

Using the standard algebraic identity (x + 1/x)² = x² + 1/x² + 2:

x² + 1/x² = (x + 1/x)² - 2

Substituting x + 1/x = 4:

x² + 1/x² = 4² - 2 = 16 - 2 = 14

Explanation

Expressing both equations with the common base 2:

2⁽x + y) = 2⁴ ⇒ x + y = 4 (2⁶)⁽x - y) = 2¹ ⇒ 6(x - y) = 1 ⇒ x - y = 1/6

Adding the two equations yields: 2x = 4 + 1/6 = 25/6 x = 25/12

Subtracting the second equation from the first yields: 2y = 4 - 1/6 = 23/6 y = 23/12

This yields a unique solution where x = 25/12 and y = 23/12.

Explanation

To minimize the number of rows, the maximum possible number of trees must be planted in each row. Since each row must contain trees of only one variety, the number of trees per row is the Highest Common Factor (HCF) of 7, 28, and 42.

HCF(7, 28, 42) = 7 trees per row

Number of rows required for each variety: Orange trees = 7 ÷ 7 = 1 row Apple trees = 28 ÷ 7 = 4 rows Mango trees = 42 ÷ 7 = 6 rows

Total minimum number of rows = 1 + 4 + 6 = 11 rows