RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

Using the combined time formula T = (a × b) / (a + b), where a = 10 days and b = 4 days:

T = (10 × 4) / (10 + 4) = 40 / 14 = 20 / 7 days.

Explanation

Roman numerals represent specific numerical values: M = 1000, D = 500, C = 100, L = 50, and X = 10. Arranging these values in descending order yields 1000 > 500 > 100 > 50 > 10, which corresponds to M > D > C > L > X.

Explanation

Each term in the denominator is equal to one-tenth of the corresponding term in the numerator: 0.001 = 0.01 ÷ 10 0.022 = 0.22 ÷ 10 0.0333 = 0.333 ÷ 10 0.04444 = 0.4444 ÷ 10

Squaring both sides for each term: (0.001)² = (0.01)² ÷ 100 (0.022)² = (0.22)² ÷ 100 (0.0333)² = (0.333)² ÷ 100 (0.04444)² = (0.4444)² ÷ 100

Factoring 1 ÷ 100 out of the denominator gives: Denominator = [(0.01)² + (0.22)² + (0.333)² + (0.4444)²] ÷ 100 = Numerator ÷ 100

Thus, Expression = Numerator ÷ (Numerator ÷ 100) = 100.

Explanation

Factorizing the expression gives x⁶⁵ - x = x(x⁶⁴ - 1).

Since x is a positive even number, x is divisible by 2, so 2 divides x(x⁶⁴ - 1).

For divisibility by 3, by Fermat's Little Theorem, x² ≡ 1 (mod 3) for any integer x not divisible by 3. Thus, x⁶⁴ = (x²)³² ≡ 1³² ≡ 1 (mod 3), meaning x⁶⁴ - 1 is divisible by 3. If x is divisible by 3, then x(x⁶⁴ - 1) is directly divisible by 3. Therefore, 3 always divides x⁶⁵ - x.

Since x⁶⁵ - x is divisible by both coprime numbers 2 and 3, it is always divisible by 2 × 3 = 6.

Explanation

To find the capacity of the largest measuring can that can measure the given quantities exactly without any remainder, we compute the Highest Common Factor (HCF) of 72, 90, and 144.

72 = 18 × 4 90 = 18 × 5 144 = 18 × 8

The highest common factor of 72, 90, and 144 is 18 litres.

Explanation

Time taken to complete a job is inversely proportional to efficiency. Since Y is 40% more efficient than X, the ratio of efficiency of X to Y is 100 : 140 = 5 : 7. Consequently, the ratio of time taken by X to Y is 7 : 5.

Using the inverse proportion formula: Days taken by Y = Days taken by X × (Efficiency of X / Efficiency of Y) Days taken by Y = 14 × (100 / 140) = 14 × (5 / 7) = 10 days.

Explanation

The standard formula relating distance D, original speed s, speed change Δs, and time change Δt is: D = s(s + Δs₁) / Δs₁ × Δt₁ = s(s - Δs₂) / Δs₂ × Δt₂

Given Δs₁ = 8 km/h, Δt₁ = 20/60 = 1/3 h, Δs₂ = 4 km/h, and Δt₂ = 40/60 = 2/3 h:

s(s + 8) / (8 × 3) = s(s - 4) / (4 × 3/2) (s + 8) / 24 = 2(s - 4) / 12 s + 8 = 4(s - 4) s + 8 = 4s - 16 3s = 24 s = 8 km/h

Substituting s = 8 km/h back into the distance formula: D = 8 × (8 + 8) / (8 × 3) = 16/3 km

Explanation

Let the length and breadth of the rectangle be 11x and 9x respectively.

Area = length × breadth 396 = 11x × 9x 99x² = 396 x² = 4 x = 2

Therefore, length = 22 cm and breadth = 18 cm.

Perimeter = 2 × (length + breadth) = 2 × (22 + 18) = 2 × 40 = 80 cm.

Explanation

The prime numbers strictly greater than 5 and less than 18 are 7, 11, 13, and 17.

Sum of these prime numbers = 7 + 11 + 13 + 17 = 48

One-third of the sum = 48 ÷ 3 = 16

Since 16 = 4², this value is the square of 4.

Explanation

Using the complementary angle identity cos²(90° - θ) = sin² θ, we can pair terms as cos² θ + cos²(90° - θ) = cos² θ + sin² θ = 1.

The sequence has terms from 3° to 87°, giving total number of terms N = 87 - 3 + 1 = 85.

The terms from 3° to 44° pair with terms from 87° down to 46°, forming 42 pairs that each sum to 1. The middle term cos² 45° remains unpaired.

Sum = 42 × 1 + cos² 45° = 42 + (1/√2)² = 42 + ½ = 85/2.

Explanation

Let Marked Price (MP) = 100.

When sold at 90% of the marked price, Selling Price (SP) = 90.

Since there is a loss of 7%, the Cost Price (CP) is given by: CP = SP / (1 - Loss%) = 90 / (1 - 0.07) = 90 / 0.93 = 3000 / 31

When the article is sold at the marked price, SP = 100.

Profit percentage = [(SP - CP) / CP] × 100 = [(100 - 3000/31) / (3000/31)] × 100 = (100/31) / (3000/31) × 100 = (100 / 3000) × 100 = 10 / 3 % = 3.33%

Explanation

Let the two numbers be 3x and 2x. Since 3 and 2 are coprime, the LCM of 3x and 2x is 6x.

6x = 96 x = 16

The numbers are: 3 × 16 = 48 2 × 16 = 32

Explanation

The coefficient of variation (CV) is calculated using the formula: Coefficient of Variation = (Standard Deviation / Arithmetic Mean) × 100

Substituting the given values: CV = (3 / 6) × 100 = 50

Explanation

Production of red lentil in state B in 2017 = 210 tons Production of red lentil in state D in 2018 = 240 tons

Required ratio = 210 : 240 = 7 : 8

Explanation

To convert 9.11 hours into hours, minutes, and seconds:

The whole number part gives 9 hours.

Multiply the decimal part by 60 to convert it into minutes: 0.11 × 60 = 6.6 minutes

The whole number part gives 6 minutes.

Multiply the remaining decimal part by 60 to convert it into seconds: 0.6 × 60 = 36 seconds

Combining these gives 9 hours, 6 minutes, and 36 seconds.

Explanation

For the given arithmetic progression, the first term a = 15, common difference d = 13 - 15 = -2, and total sum Sₙ = -105.

Using the sum formula for an AP: Sₙ = (n/2)[2a + (n - 1)d]

-105 = (n/2)[2(15) + (n - 1)(-2)] -105 = (n/2)[30 - 2n + 2] -105 = n(16 - n) n² - 16n - 105 = 0 (n - 21)(n + 5) = 0

Since the number of terms n must be positive, n = 21.

The last term k is: k = a + (n - 1)d k = 15 + (21 - 1)(-2) = 15 - 40 = -25

Explanation

Using the BODMAS rule, solve the brackets from innermost to outermost:

First, evaluate the innermost parenthesis: 17 - 2 = 15

Next, evaluate the expression inside the curly braces: 45 ÷ 15 = 3

Then, perform the multiplication: 5 × 3 = 15

Finally, perform addition and subtraction from left to right: 24 + 13 - 15 = 37 - 15 = 22

Explanation

For a right-angled triangle with legs a and b, and hypotenuse c, the inradius r and circumradius R are given by the standard formulas:

r = (a + b - c) / 2 R = c / 2

Since 3² + 4² = 5², the given triangle is a right-angled triangle with legs 3 cm and 4 cm, and hypotenuse 5 cm.

r = (3 + 4 - 5) / 2 = 1 cm R = 5 / 2 = 2.5 cm

Ratio of inradius to circumradius = r : R = 1 : 2.5 = 2 : 5

Explanation

Let h be the height of the tower and d be the horizontal distance between the building and the tower.

From the top of the 200 m building, the height above the building to the top of the tower is (h - 200). tan 30° = (h - 200) / d ⇒ d = (h - 200)√3

From the bottom of the building: tan 45° = h / d ⇒ d = h

Equating the two expressions for d: h = (h - 200)√3 h = h√3 - 200√3 h(√3 - 1) = 200√3 h = 200√3 / (√3 - 1)

Rationalizing the denominator: h = [200√3 × (√3 + 1)] / [(√3 - 1)(√3 + 1)] h = 200√3(√3 + 1) / 2 = 100√3(√3 + 1) m

Explanation

Perform the addition and subtraction step by step:

1.45 + 0.312 = 1.762

1.762 - 1.112 = 0.65

Converting the decimal 0.65 into a simplified fraction:

0.65 = 65/100 = 13/20

Explanation

Convert the mixed fractions into improper fractions: 3(21/23) = (3 × 23 + 21) / 23 = 90/23 3(15/31) = (3 × 31 + 15) / 31 = 108/31

Dividing the two fractions: (90/23) ÷ (108/31) = (90/23) × (31/108)

Simplifying 90/108 by dividing the numerator and denominator by 18: = (5/23) × (31/6) = (5 × 31) / (23 × 6) = 155/138

Explanation

Using the percentage formula: (x / 100) × 10.8 = 32.4

Solving for x: x = (32.4 × 100) / 10.8 x = 3 × 100 = 300

Explanation

First, rationalizing the expression inside the parentheses by multiplying the numerator and denominator by (2 + √3):

(2 + √3) / (2 - √3) = (2 + √3)² / (2² - (√3)²) = (4 + 3 + 4√3) / (4 - 3) = 7 + 4√3

Squaring this result:

(7 + 4√3)² = 7² + (4√3)² + 2 × 7 × 4√3 = 49 + 48 + 56√3 = 97 + 56√3

Explanation

Let y = 3ˣ. Rewriting the terms using exponent laws gives 3²ˣ⁺¹ = 3y², 3ˣ⁺³ = 27y, and 3² = 9. Substituting these into the given equation yields:

3y² - y = 27y - 9

3y² - 28y + 9 = 0

Factoring the quadratic expression gives:

(3y - 1)(y - 9) = 0

Solving for y yields y = 9 or y = 1/3. Substituting back y = 3ˣ:

3ˣ = 9 = 3² ⇒ x = 2

3ˣ = 1/3 = 3⁻¹ ⇒ x = -1

Thus, the values of x are 2 and -1.

Explanation

To find the time when all three traffic signals change simultaneously again, calculate the Least Common Multiple (LCM) of 72, 108, and 48 seconds.

Prime factorization of each interval: 72 = 2³ × 3² 108 = 2² × 3³ 48 = 2⁴ × 3¹

LCM(72, 108, 48) = 2⁴ × 3³ = 16 × 27 = 432 seconds

Converting 432 seconds into minutes and seconds: 432 seconds = 7 minutes and 12 seconds

Adding 7 minutes and 12 seconds to the initial time of 9:30:00 am gives 9:37:12 am.

Explanation

The difference (D) between compound interest and simple interest for 3 years is given by the formula: D = P × (r/100)² × (300 + r)/100

Substituting D = 244 and r = 5: 244 = P × (5/100)² × (300 + 5)/100 244 = P × (1/400) × (305/100) 244 = P × (61/8000) P = (244 × 8000) ÷ 61 P = 4 × 8000 = 32,000

Explanation

The mean of grouped data is calculated using the formula Mean = Σ(f_i × x_i) / Σf_i, where x_i represents the class midpoint and f_i represents the corresponding frequency.

The midpoints (x_i) for the class intervals 0-10, 10-20, 20-30, 30-40, and 40-50 are 5, 15, 25, 35, and 45 respectively.

Total frequency Σf_i = 8 + 10 + k + 6 + 12 = 36 + k

Sum of products Σ(f_i × x_i) = (8 × 5) + (10 × 15) + (k × 25) + (6 × 35) + (12 × 45) = 40 + 150 + 25k + 210 + 540 = 940 + 25k

Given Mean = 26: 26 = (940 + 25k) / (36 + k) 26 × (36 + k) = 940 + 25k 936 + 26k = 940 + 25k 26k - 25k = 940 - 936 k = 4

Explanation

The volume of a sphere is proportional to the cube of its radius (V ∝ r³), while its surface area is proportional to the square of its radius (A ∝ r²).

Given the volume ratio V₁/V₂ = 8/27: r₁/r₂ = ∛(8/27) = 2/3

The ratio of their surface areas is: A₁/A₂ = (r₁/r₂)² = (2/3)² = 4/9

Explanation

Let B = 100. Since A is 70% more than B, A = 170.

The percentage by which B is less than A is given by the formula: Percentage Less = [(A - B) / A] × 100

Substituting the values: Percentage Less = [(170 - 100) / 170] × 100 = (70 / 170) × 100 = 700 / 17 % = 41 3/17 %

Explanation

Using the successive percentage change formula for profit:

Final Price = Original Cost Price × (1 + p₁/100) × (1 + p₂/100)

Substituting the given values into the formula:

225 = Cost Price × (1 + 20/100) × (1 + 25/100) 225 = Cost Price × (6/5) × (5/4) 225 = Cost Price × 1.5 Cost Price = 225 / 1.5 = 150

Explanation

The first car gets a 15-minute (0.25 hour) head start, covering an initial distance of 45 × 0.25 = 11.25 km.

The relative speed of the second car with respect to the first car is 54 - 45 = 9 km/h.

Time required for the second car to catch up = Initial Distance / Relative Speed = 11.25 / 9 = 1.25 hours.

Distance from city A where both cars meet = Speed of second car × Catch-up time = 54 × 1.25 = 67.5 km.

Explanation

When individuals work together for the entire duration of a task, total wages are divided in the ratio of their daily work efficiency (rate of work).

Ratio of efficiency (Ram : Shyam) = (1/12) : (1/18) = 18 : 12 = 3 : 2

Total ratio parts = 3 + 2 = 5

Ram's share = (3/5) × ₹15,000 = ₹9,000 Shyam's share = (2/5) × ₹15,000 = ₹6,000

Explanation

For a square field of side length s and area A, A = s². Given A = 2 km², the side length s = √2 km.

The length of the fence constructed along the diagonal d of the square is given by d = s√2.

Substituting s = √2 km: d = √2 × √2 = 2 km.

Explanation

From the given equation: 1.5X = 0.05Y

Solving for Y in terms of X: Y = (1.5 / 0.05)X Y = 30X

Substitute Y = 30X into the expression: (Y - X) / (Y + X) = (30X - X) / (30X + X) = 29X / 31X = 29 / 31

Calculating the decimal value: 29 / 31 ≈ 0.93548...

Rounding to three decimal places gives 0.935.

Explanation

To minimize the total number of groups, the number of members in each group must be as large as possible, which equals the Highest Common Factor (HCF) of the number of boys and girls.

HCF(391, 323) = 17

Number of boys' groups = 391 ÷ 17 = 23 Number of girls' groups = 323 ÷ 17 = 19

Minimum total number of groups = 23 + 19 = 42

Explanation

When a value is increased by x% and subsequently decreased by x%, the net overall change is always a reduction given by the formula:

Overall percentage reduction = x²/100 %

Substituting x = 15:

Overall percentage reduction = 15²/100 = 225/100 = 2.25%

Explanation

First, calculate m from the first set of six numbers: m = (2 + 4 + 5 + 8 + 2 + 3) / 6 = 24 / 6 = 4

Substituting m = 4 into the second set gives seven numbers: 4, 3, 3, 5, 4, 3, and p. The mean of this set is m + 1 = 5. Using the mean formula: (4 + 3 + 3 + 5 + 4 + 3 + p) / 7 = 5 22 + p = 35 p = 13

Arranging the second set in ascending order: 3, 3, 3, 4, 4, 5, 13. The median q is the middle (4th) term, so q = 4. The mode r is the most frequently occurring value, so r = 3.

Evaluating the required expression: p + q - r = 13 + 4 - 3 = 14

Explanation

Let marked price MP = 5x and cost price CP = 3x.

After two successive discounts of 25% and 10%, the selling price is: SP = MP × (1 - 0.25) × (1 - 0.10) SP = 5x × 0.75 × 0.90 = 3.375x

Profit percentage = [(SP - CP) / CP] × 100 Profit percentage = [(3.375x - 3x) / 3x] × 100 Profit percentage = (0.375 / 3) × 100 = 12.5%

Explanation

To convert 7.16 hours into hours, minutes, and seconds, separate the whole hours from the decimal portion: Whole hours = 7 hours

Convert the decimal portion 0.16 hours into minutes by multiplying by 60: 0.16 × 60 = 9.6 minutes

Keep the whole number 9 as minutes and convert the remaining decimal portion 0.6 minutes into seconds by multiplying by 60: 0.6 × 60 = 36 seconds

Combining these components yields 7 hours, 9 minutes, and 36 seconds.

Explanation

Combined rate of work for A, B, and C = (1/10) + (1/14) = 12/70 = 6/35 work per day.

Time required to complete 3/5 of the work = (3/5) ÷ (6/35) = (3/5) × (35/6) = 7/2 days.

Explanation

Let the two numbers be 5x and 3x.

Given that the difference between the two numbers is 50: 5x - 3x = 50 2x = 50 x = 25

The two numbers are: First number = 5 × 25 = 125 Second number = 3 × 25 = 75

Product of the numbers = 125 × 75 = 9375

Explanation

Four years ago, the ratio of Ram's age to Shyam's age was 13 : 9, where the ratio difference is 4 units.

Eight years from now, their age ratio becomes 4 : 3. Multiplying this ratio by 4 gives 16 : 12 to maintain a uniform difference of 4 units between both age ratios.

The total time gap between 4 years ago and 8 years from now is 4 + 8 = 12 years.

The change in Ram's ratio units is 16 - 13 = 3 units, which corresponds to 12 years. Thus, 1 unit = 4 years.

Since the difference between their ages remains constant over time, the difference between their present ages is 4 units × 4 = 16 years.

Explanation

Total maximum marks for both students = 50 + 50 + 50 + 50 = 200

Total marks obtained by A = 42 + 45 + 40 + 33 = 160 Aggregate percentage of A = (160 / 200) × 100 = 80%

Total marks obtained by B = 35 + 33 + 40 + 42 = 150 Aggregate percentage of B = (150 / 200) × 100 = 75%

Difference between the aggregate percentage marks of A and B = 80% - 75% = 5%

Explanation

The total number of decimal places in the product equals the sum of the decimal places in the factors: 2 decimal places in 3.31 plus 3 decimal places in 0.226, giving a total of 5 decimal places.

Shifting the decimal point 5 places to the left from 74806 gives: 3.31 × 0.226 = 74806 / 100000 = 0.74806

Explanation

Let the number to be subtracted be x.

8/5 - x = 5/8

x = 8/5 - 5/8

Taking the LCM of 5 and 8 as 40:

x = (64 - 25) / 40 = 39/40

Explanation

Let the total capacity of the bottle be 1 unit. Filled part = 3/5 Empty part = 1 - 3/5 = 2/5

Ratio of filled part to empty part = (3/5) : (2/5) = 3 : 2.

Explanation

For two numbers in the simplest ratio a : b with Highest Common Factor H, their Least Common Multiple is given by the formula:

LCM = a × b × H

Given a = 3, b = 7, and H = 3:

LCM = 3 × 7 × 3 = 63

Alternatively, the two numbers are 3 × 3 = 9 and 7 × 3 = 21, and the LCM of 9 and 21 is 63.

Explanation

Substitute x = 6 into the linear equation 4x - 3y = 9:

4(6) - 3y = 9 24 - 3y = 9 3y = 24 - 9 3y = 15 y = 5

Explanation

Present Value = Purchase Price × (1 - r/100)ⁿ

Here, depreciation rate r = 20%, time n = 3 years, and Present Value = ₹12,288.

12,288 = Purchase Price × (1 - 20/100)³ 12,288 = Purchase Price × (4/5)³ 12,288 = Purchase Price × 64/125

Purchase Price = (12,288 × 125) / 64 = 192 × 125 = ₹24,000

Explanation

The sum of the cubes of the first n natural numbers is given by the formula [n(n + 1) / 2]². Substituting n = 4 into the formula gives [4 × (4 + 1) / 2]² = [4 × 5 / 2]² = 10² = 100. Alternatively, adding the individual cubes yields 1³ + 2³ + 3³ + 4³ = 1 + 8 + 27 + 64 = 100.