RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

To convert 5.37 hours into hours, minutes, and seconds, keep the whole number part as 5 hours. Convert the fractional part 0.37 hours into minutes by multiplying by 60, which yields 0.37 × 60 = 22.2 minutes. The whole number part gives 22 minutes. Finally, convert the remaining 0.2 minutes into seconds by multiplying by 60, giving 0.2 × 60 = 12 seconds. Combining these results gives 5 hours, 22 minutes, 12 seconds.

Explanation

Total age of the initial 11 players = 11 × 18 = 198 years.

After 4 new players join, the team size increases to 11 + 4 = 15 players, and the new average age becomes 18 + 2 = 20 years.

Total age of all 15 players = 15 × 20 = 300 years.

Sum of the ages of the 4 new members = 300 - 198 = 102 years.

Average age of the new members = 102 / 4 = 25.5 years.

Explanation

The greatest six-digit number is 999999 and the smallest six-digit number is 100000.

Difference = 999999 − 100000 = 899999.

Explanation

The volume of a cylinder is V = πr²h and its curved surface area is CSA = 2πrh.

Dividing the volume by the curved surface area gives: V / CSA = (πr²h) / (2πrh) = r / 2

Substituting the given values: 231 / 66 = r / 2 7 / 2 = r / 2 r = 7 m

Explanation

By Euclid's Lemma, if a prime number P divides Q², then P must divide Q.

Since P is a prime, P ≥ 2. Two consecutive integers Q and Q + 1 are always coprime, meaning gcd(Q, Q + 1) = 1. Since P divides Q, it cannot divide Q + 1, as no prime factor can be shared between consecutive integers. Therefore, P will never divide Q + 1.

Explanation

The sum of the interior angles of an n-sided polygon is given by the formula (n - 2) × 180°. For a hexagon, n = 6. Substituting this into the formula gives (6 - 2) × 180° = 4 × 180° = 720°.

Explanation

Using the sum and difference of cubes formulas, (1 + a)(1 - a + a²) = 1 + a³ and (1 - a)(1 + a + a²) = 1 - a³.

Substituting these into the expression: (a + 1)(1 - a)(1 - a + a²)(1 + a + a²)(1 + a⁶) = (1 + a³)(1 - a³)(1 + a⁶)

Using the difference of squares formula, (1 + a³)(1 - a³) = 1 - a⁶.

Multiplying the remaining terms: (1 - a⁶)(1 + a⁶) = 1 - a¹²

Explanation

By Pythagoras theorem, the height at which the tree breaks forms the perpendicular side of a right-angled triangle, where the broken upper part is the hypotenuse (10 m) and the distance from the foot to the top is the base (6 m).

Perpendicular height = √(Hypotenuse² - Base²) = √(10² - 6²) = √(100 - 36) = √64 = 8 m

Total height of the tree = Height of standing part + Length of broken upper part = 8 m + 10 m = 18 m

Explanation

Remaining provisions after 10 days are sufficient for 150 men for (45 - 10) = 35 days.

Total remaining man-days = 150 × 35 = 5250 man-days.

Number of remaining men = 150 - 25 = 125 men.

Number of days the remaining provisions will last = 5250 / 125 = 42 days.

Explanation

Squaring both sides of X = √(1 + √3/2) + √(1 - √3/2):

X² = (1 + √3/2) + (1 - √3/2) + 2 × √[(1 + √3/2)(1 - √3/2)]

Simplifying the terms: X² = 2 + 2 × √(1 - 3/4) X² = 2 + 2 × √(1/4) X² = 2 + 2 × ½ X² = 3

Since X is the sum of two positive real square roots, taking the positive square root gives X = √3.

Explanation

YOUTHSAT, launched on April 20, 2011 via PSLV-C16, is an experimental satellite developed jointly by ISRO and Roscosmos with participation from university students. Weighing 92 kg, it belongs to ISRO's Indian Mini Satellite (IMS) series and was designed to conduct aeronomy experiments studying solar variability and its effects on the Earth's upper atmosphere, thermosphere, and ionosphere.

Explanation

Prime factorization of 3675 gives 3¹ × 5² × 7².

For a number to be a perfect square, every prime factor in its prime factorization must have an even exponent. Here, the exponent of 3 is 1 (odd), while the exponents of 5 and 7 are already even.

To get the least perfect square containing 3675 as a factor, 3675 must be multiplied by 3:

3675 × 3 = 3² × 5² × 7² = (3 × 5 × 7)² = 105² = 11025

Explanation

Let the incomes of A and B be 3x and 2x respectively. Since each saves ₹2,000, their expenditures are (3x - 2000) and (2x - 2000).

Given the expenditure ratio is 5 : 3: (3x - 2000) / (2x - 2000) = 5 / 3

Cross-multiplying: 3 × (3x - 2000) = 5 × (2x - 2000) 9x - 6000 = 10x - 10000 x = 4000

The difference between their incomes is 3x - 2x = x = 4,000.

Alternatively, multiplying the income ratio 3 : 2 by 2 gives 6 : 4, making the difference between income and expenditure equal for both (6 - 5 = 1 unit, and 4 - 3 = 1 unit). Since 1 unit = ₹2,000, the income difference of 2 units (6 - 4) is 2 × 2,000 = 4,000.

Explanation

Simplifying the expression (√2 + 1/√8)² gives:

√2 + 1/√8 = √2 + 1/(2√2) = (4 + 1)/(2√2) = 5/(2√2)

Squaring this value:

(5/(2√2))² = 25/(4 × 2) = 25/8

Since 25/8 is the ratio of two integers, it is a rational number.

Explanation

To find the HCF of decimal numbers, convert all numbers to have the same number of decimal places: 16.50, 0.45, and 15.00

Multiplying each by 100 gives the integers 1650, 45, and 1500.

Finding the prime factorizations: 1650 = 2 × 3 × 5² × 11 45 = 3² × 5 1500 = 2² × 3 × 5³

The highest common factor of 1650, 45, and 1500 is 3 × 5 = 15.

Dividing by 100 to adjust for the decimal shift gives HCF = 15 / 100 = 0.15.

Explanation

Simple Interest SI = (P × R × T) / 100

Given rate R = 4% per annum and time T = 8 years, the simple interest is: SI = (P × 4 × 8) / 100 = 0.32P

The simple interest is ₹340 less than the sum lent (P): P - SI = 340 P - 0.32P = 340 0.68P = 340 P = 340 / 0.68 = ₹500

Explanation

Since the required service time per interval is identical (5/6 h), the daily service time required is inversely proportional to the servicing interval.

Fraction of service time saved = [ (1/90) - (1/120) ] / (1/90) = 1 - (90 / 120) = 1 - 3/4 = 1/4.

Explanation

Given tan α = 1/x, consider a right-angled triangle with perpendicular = 1 and base = x.

By the Pythagorean theorem, hypotenuse = √(1² + x²) = √(1 + x²).

Using the basic trigonometric ratios: sin α = Perpendicular / Hypotenuse = 1 / √(1 + x²) cos α = Base / Hypotenuse = x / √(1 + x²)

Adding both terms yields: sin α + cos α = 1 / √(1 + x²) + x / √(1 + x²) = (1 + x) / √(1 + x²)

Explanation

Ratio of work efficiency of A and B is given by 2A = 3B, which means A : B = 3 : 2 = 15 : 10. Ratio of efficiency of B and C is given by 4B = 5C, which means B : C = 5 : 4 = 10 : 8. Combining these gives the efficiency ratio A : B : C = 15 : 10 : 8.

Let daily efficiencies of A, B, and C be 15, 10, and 8 units respectively. Total work = A's efficiency × Time taken by A = 15 × 22 = 330 units.

Combined daily efficiency of A, B, and C = 15 + 10 + 8 = 33 units/day.

Time taken working together = Total work / Combined efficiency = 330 / 33 = 10 days.

Explanation

To find the median, arrange the 10 observations in ascending order: 8, 10, 13, 15, 16, 18, 20, 20, 25, 30.

Since the number of observations n = 10 is even, the median is the average of the (n/2)th term and the (n/2 + 1)th term, which correspond to the 5th and 6th terms in the ordered list.

Median = (5th term + 6th term) / 2 = (16 + 18) / 2 = 34 / 2 = 17

Explanation

Total simple interest received in 1 year = 8,280 ÷ 3 = ₹2,760.

If the entire sum of ₹30,000 was lent at 8% per annum, the interest for 1 year would be: 8% of 30,000 = ₹2,400.

The excess interest obtained is 2,760 - 2,400 = ₹360.

This difference of ₹360 is due to the additional 2% (10% - 8%) interest rate applied on the sum lent at 10%.

2% of Amount at 10% = ₹360 Amount at 10% = 360 × 100 ÷ 2 = ₹18,000

Explanation

Total distance covered = Speed × Time = 45 km/h × 4 h = 180 km = 180,000 m.

Number of intervals between adjacent poles = Total distance / Distance between two poles = 180,000 m / 50 m = 3600.

Including both the starting pole and the final pole, the total number of poles passed is Number of intervals + 1 = 3600 + 1 = 3601.

Explanation

In the right-angled triangle formed by the house wall, ground, and ladder, the height h of the house is related to the ladder length L and angle of inclination θ by the formula:

h = L × sin θ

Substituting L = 5.2 m and θ = 45°: h = 5.2 × sin 45° h = (26/5) × (1 / √2) h = 26 / (5√2)

Rationalizing the denominator by multiplying the numerator and denominator by √2: h = (26√2) / (5 × 2) h = (13√2)/5 m

Explanation

Using the relation between distance D, original speed v, increase in speed Δv, and time difference Δt:

D = [v(v + Δv) × Δt] / Δv

Substituting D = 300 km, Δv = 5 km/h, and Δt = 2 h:

300 = [v(v + 5) × 2] / 5 750 = v² + 5v v² + 5v - 750 = 0

Factoring the quadratic equation: (v + 30)(v - 25) = 0

Since speed must be a positive quantity, v = 25 km/h.

Explanation

Given x = a(b - c), y = b(c - a), and z = c(a - b), dividing by a, b, and c respectively gives: x/a = b - c y/b = c - a z/c = a - b

Adding these three terms together: x/a + y/b + z/c = (b - c) + (c - a) + (a - b) = 0

By the algebraic identity, if u + v + w = 0, then u³ + v³ + w³ = 3uvw.

Applying this identity with u = x/a, v = y/b, and w = z/c: (x/a)³ + (y/b)³ + (z/c)³ = 3(x/a)(y/b)(z/c) = 3xyz / abc

Explanation

Perimeter of the circle = 2πr = 2 × (22/7) × 7 = 44 cm.

Since the same wire is bent into a square, the perimeter of the square equals the circumference of the circle: 4 × side (a) = 44 cm a = 11 cm

Diagonal of the square = a√2 = 11√2 cm.

Explanation

Using the algebraic identity a³ + b³ = (a + b)(a² - ab + b²), the given expression can be written in terms of powers of 0.4 and 0.3:

Numerator: 0.064 + 0.027 = (0.4)³ + (0.3)³ Denominator: 0.16 - 0.12 + 0.09 = (0.4)² - (0.4 × 0.3) + (0.3)²

Let a = 0.4 and b = 0.3. The expression simplifies to: (a³ + b³) / (a² - ab + b²) = [(a + b)(a² - ab + b²)] / (a² - ab + b²) = a + b

Substituting the values gives: 0.4 + 0.3 = 0.7

Explanation

Volume of a cylinder is given by V = π r² h.

Let the initial volume be V₁ = π r² h. When the radius is doubled (r' = 2r) and the height is halved (h' = h/2), the new volume becomes: V₂ = π (2r)² (h/2) = π (4r²) (h/2) = 2 π r² h = 2 V₁

The ratio of the new volume to the previous volume is V₂ / V₁ = (2 V₁) / V₁ = 2 / 1, which gives 2:1.

Explanation

Multiply numerator and denominator of 1/(x + 1) by x: 1/(x + 1) = x / (x² + x)

Substitute x² = y + z into the denominator: x / (x² + x) = x / (x + y + z)

By symmetry, the other two terms become: 1/(y + 1) = y / (x + y + z) 1/(z + 1) = z / (x + y + z)

Adding all three expressions: 1/(x + 1) + 1/(y + 1) + 1/(z + 1) = (x + y + z) / (x + y + z) = 1

Explanation

According to the BODMAS rule, operations inside brackets are evaluated first, followed by division and multiplication from left to right, and finally addition.

Evaluating the bracket first: (5 × 5) = 25

Substituting this back into the expression: 5 + 5 × 5 ÷ 25 × 5

Executing multiplication and division from left to right: 5 + 25 ÷ 25 × 5 = 5 + 1 × 5 = 5 + 5

Performing final addition: 5 + 5 = 10

Explanation

Let x = 0.666... Multiplying both sides by 10 gives 10x = 6.666...

Subtracting x from 10x: 10x - x = 6.666... - 0.666... 9x = 6 x = 6/9 = 2/3

Explanation

Distance = Speed × Time

Time taken by the train = 10:30 p.m. - 1:30 p.m. = 9 hours Average speed = 90 km/h

Distance = 90 km/h × 9 h = 810 km

Explanation

Amount saved due to the 10% reduction = 10% of ₹2250 = ₹225.

This saved amount allows the person to buy 10 additional mangoes at the reduced price.

Reduced price per mango = ₹225 ÷ 10 = ₹22.50.

Reduced price per dozen (12 mangoes) = 12 × ₹22.50 = ₹270.

Explanation

To find the greatest number that leaves remainders 9 and 7 when dividing 285 and 1249 respectively, subtract the remainders from the given numbers and calculate their Highest Common Factor (HCF).

First number = 285 - 9 = 276 Second number = 1249 - 7 = 1242

Required number = HCF(276, 1242)

Using Euclid's division algorithm: 1242 = 276 × 4 + 138 276 = 138 × 2 + 0

The last non-zero divisor is 138, which is the HCF.

Explanation

Since total money remains ₹1, the quantity of rubber bands sold is inversely proportional to the selling price percentage.

Initial SP percentage = 100% − 40% = 60% Desired SP percentage = 100% + 20% = 120%

Ratio of Selling Prices = 60 : 120 = 1 : 2

Ratio of Quantities = 2 : 1

Given that 2 parts correspond to 32 rubber bands: 1 part = 32 × ½ = 16 rubber bands

Explanation

Let the total cost price of the commodity be CP.

Profit from 1/3 part = (1/3) × 15% = 5% of CP Profit from 1/4 part = (1/4) × 20% = 5% of CP

Remaining part = 1 - (1/3 + 1/4) = 1 - (7/12) = 5/12 Profit from 5/12 part = (5/12) × 24% = 10% of CP

Total profit percentage = 5% + 5% + 10% = 20% of CP

Given total profit = ₹62: 20% of CP = ₹62 CP = (62 × 100) / 20 = ₹310

Explanation

The maximum score in the examination is 200 marks. 20% of 200 = 40 marks.

The candidates scoring less than 20% marks fall in the 0–40 score interval, which contains 2219 candidates in GK out of a total of 10000 candidates.

Required percentage = (2219 / 10000) × 100 = 22.19%.

Explanation

The maximum possible score in English is 200. Cut-off marks for English = 40% of 200 = 80 marks.

To be evaluated for LR, a candidate must score more than 80 marks in English, which corresponds to the candidates in the score intervals 81-120, 121-160, and 161-200.

Total candidates = 3562 + 2412 + 1217 = 7191.

Explanation

Maximum score in each subject is 200.

80% of 200 = 160 marks.

Scores above 80% correspond to marks strictly greater than 160, which falls into the 161-200 score interval.

Sum of candidates scoring in the 161-200 interval across all subjects: GK = 856 LR = 218 English = 1217 Computer = 191

Total = 856 + 218 + 1217 + 191 = 2482

Explanation

The maximum score in each subject is 200, making the 40% cut-off equal to 80 marks. The provided table shows subject-wise score distributions independently, but does not provide candidate-level cross-tabulation to track individual performance across multiple subjects. Without knowing the overlap of candidates scoring above 80 marks across all four subjects, the exact number of candidates who cleared the cut-off in all subjects simultaneously cannot be calculated.

Explanation

Mean Deviation = Σ|x - x̅| / n

First, find the mean (x̅) of the eight observations: x̅ = (6 + 7 + 10 + 12 + 13 + 4 + 8 + 12) / 8 = 72 / 8 = 9

Next, calculate the absolute deviations from the mean |x - x̅|: |6 - 9| = 3 |7 - 9| = 2 |10 - 9| = 1 |12 - 9| = 3 |13 - 9| = 4 |4 - 9| = 5 |8 - 9| = 1 |12 - 9| = 3

Sum of absolute deviations = 3 + 2 + 1 + 3 + 4 + 5 + 1 + 3 = 22

Mean Deviation = 22 / 8 = 2.75

Explanation

Pitch is the perceptual attribute of sound that allows the human brain to categorize sound as high or low based on the frequency of the sound wave. High-frequency sound waves produce a high-pitched sound, whereas low-frequency sound waves result in a low-pitched sound.

Explanation

In February 2021, Defence Minister Rajnath Singh launched the e-Chhawani portal to provide online municipal services to over 20 lakh residents residing across 62 Cantonment Boards in India. Developed jointly by the Directorate General Defence Estates, Bharat Electronics Limited, eGov Foundation, and National Informatics Centre, the portal enables citizens to easily access services such as property tax payment, trade license issuance, water and sewerage connections, and grievance redressal without visiting board offices.

Explanation

Using the BODMAS rule, evaluate the expression step-by-step starting from the innermost parentheses:

Inner bracket: (6 - 24 ÷ 6 × 8) = 6 - (4 × 8) = 6 - 32 = -26

Curly bracket: {25 - (-26)} = 25 + 26 = 51

Square bracket: [29 - 51] = -22

Final value: 55 + (-22) = 55 - 22 = 33

Explanation

Area = ½ × a × b × sin C 16√6 = ½ × 10 × 8 × sin C 16√6 = 40 sin C sin C = (2√6)/5

Using cos C = ±√(1 - sin² C): cos C = ±√(1 - 24/25) = ±1/5

By the Law of Cosines, c² = a² + b² - 2ab cos C: c² = 10² + 8² - 2(10)(8) cos C = 164 - 160 cos C

For cos C = 1/5: c² = 164 - 32 = 132 ⇒ c = 2√33 (irrational)

For cos C = -1/5: c² = 164 + 32 = 196 ⇒ c = √196 = 14 cm (rational)

Explanation

Sum of ages = Mean × Number of friends

Sum of the ages of 3 friends = 3 × 22 = 66 Sum of the ages of 4 friends = 4 × 24 = 96

Age of the fourth friend = (Sum of ages of 4 friends) - (Sum of ages of 3 friends) = 96 - 66 = 30 years.

Explanation

When a journey is divided into two equal distances, the average speed is given by the formula:

Average Speed = (2 × s₁ × s₂) / (s₁ + s₂)

Substituting the given speeds s₁ = 18 km/h and s₂ = 30 km/h:

Average Speed = (2 × 18 × 30) / (18 + 30) = 1080 / 48 = 22.5 km/h

Total Distance = Average Speed × Total Time Total Distance = 22.5 km/h × 12 h = 270 km

Explanation

Simple Interest (SI) = Amount - Principal = 14,632 - 12,400 = 2,232.

Using the formula SI = (P × R × T) / 100: 2,232 = (12,400 × R × 3) / 100 2,232 = 372 × R R = 2,232 / 372 = 6%

Explanation

Let the number be x. Subtracting 42 reduces the number to 70% of its original value, which means 42 corresponds to the remaining 30% (100% - 70%) of the number.

30% of x = 42 (30/100) × x = 42 x = (42 × 100) / 30 = 140

Two-fifths of the number is: (2/5) × 140 = 56

Explanation

Using the given percentage equality: 12% × a = 25% × b

Canceling the percentage sign from both sides: 12 × a = 25 × b a / b = 25 / 12

Thus, a : b = 25 : 12.