RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

To find the total participation for each sport across all five schools, sum the number of students playing each game:

Badminton: 34 + 67 + 23 + 78 + 56 = 258 Table Tennis: 21 + 40 + 26 + 32 + 29 = 148 Football: 12 + 22 + 19 + 16 + 21 = 90 Basketball: 9 + 13 + 7 + 12 + 6 = 47 Cricket: 98 + 121 + 78 + 84 + 67 = 448

Basketball has the overall minimum participation with 47 students, and Cricket has the overall maximum participation with 448 students.

Explanation

The greatest length that can measure the given lengths completely is their Highest Common Factor (HCF).

Prime factorisation of the given numbers: 24 = 2³ × 3 32 = 2⁵ 42 = 2 × 3 × 7

The highest common factor is the lowest power of the common prime factor, which is 2¹ = 2 m.

Explanation

0.000327 can be written as 327 × 10⁻⁶. The largest perfect square less than 327 is 324, which is 18².

Therefore, the largest perfect square less than 0.000327 is: 324 × 10⁻⁶ = (18 × 10⁻³)² = (0.018)² = 0.000324

Subtracting this perfect square from the given number: 0.000327 − 0.000324 = 0.000003

Explanation

The percentage of people who actually travelled out of those who booked tickets for each year is calculated using the formula: Percentage = (Actually Travelled / Tickets Booked) × 100

2015: (2200 / 2400) × 100 = 91.67% 2016: (2000 / 2300) × 100 = 86.96% 2017: (2500 / 2600) × 100 = 96.15% 2018: (2450 / 2800) × 100 = 87.50%

Comparing the percentages, 86.96% in 2016 is the lowest value among all years.

Explanation

To find the average percentage, sum the actual marks obtained by each student across all six subjects (out of 500 total maximum marks per student):

• Pundir: 67 + 56 + 57 + 72 + 33 + 40 = 325
• Anil: 84 + 89 + 79 + 69 + 39 + 45 = 405
• Devesh: 65 + 72 + 62 + 80 + 34 + 47 = 360
• Ramit: 90 + 88 + 92 + 82 + 41 + 45 = 438

Total marks obtained by all 4 students = 1528 out of a total maximum of 2000. Average percentage = (1528 / 2000) * 100 = 76.4%, which is approximately 75%.

Explanation

Percentage share = (Sales of L-1102 in 2015 / Total sales in 2015) × 100

From the given table: Sales of printer model L-1102 in 2015 = 822 Total sales of printers in 2015 = 3086

Required percentage = (822 / 3086) × 100 = 26.63% ≈ 27%

Explanation

Sum of the three numbers = 3 × 29 = 87.

Sum of the other two numbers = 2 × 13 = 26.

Third number = 87 - 26 = 61.

Explanation

The volume of a cube with side length s is given by the formula V = s³.

When the length of each side is doubled, the new side length becomes 2s. Substituting this into the formula gives the new volume V':

V' = (2s)³ = 8s³ = 8V

Explanation

Total Cost Price = ₹3,00,000 + ₹1,00,000 = ₹4,00,000

Gain on car = 10% of ₹3,00,000 = ₹30,000 Loss on bike = 20% of ₹1,00,000 = ₹20,000

Net Gain = ₹30,000 - ₹20,000 = ₹10,000

Net Gain Percentage = (Net Gain / Total Cost Price) × 100 = (10,000 / 4,00,000) × 100 = 2.5% gain

Explanation

Relative speed of two objects moving towards each other is the sum of their individual speeds: Relative speed = 10 km/h + 14 km/h = 24 km/h

Time taken to meet = Total distance / Relative speed = 24 km / 24 km/h = 1 h

Explanation

The compound interest formula is A = P(1 + r/100)ᵗ, where A is the total amount, P is the principal, r is the rate of interest per annum, and t is the time period in years.

Substituting the given values: 7986 = 6000(1 + r/100)³ 7986 / 6000 = (1 + r/100)³

Simplifying the fraction by dividing both numerator and denominator by 6: 1331 / 1000 = (1 + r/100)³

Taking the cube root on both sides: (11 / 10)³ = (1 + r/100)³ 11 / 10 = 1 + r/100 1.1 = 1 + r/100 r / 100 = 0.1 r = 10%

Explanation

Rearranging the given equation: cos x = 3 sin x + √5 sin x cos x = (3 + √5) sin x

Dividing both sides by cos x: 1 = (3 + √5) tan x tan x = 1 / (3 + √5)

Rationalizing the denominator by multiplying the numerator and denominator by (3 - √5): tan x = (3 - √5) / ((3 + √5)(3 - √5)) tan x = (3 - √5) / (3² - (√5)²) tan x = (3 - √5) / (9 - 5) tan x = (3 - √5) / 4

Explanation

Time = File Size ÷ Speed

File size in MB = 1.75 × 1000 MB = 1750 MB. Transfer speed = 2 MB/s.

Total time = 1750 MB ÷ 2 MB/s = 875 seconds.

Converting seconds to minutes and seconds: 875 ÷ 60 = 14 minutes with a remainder of 35 seconds, since 14 × 60 = 840 seconds and 875 − 840 = 35 seconds.

Explanation

In the number 275413, the digit 2 is in the hundred-thousands place, so its place value is 2 × 100000 = 200000. The digit 4 is in the hundreds place, so its place value is 4 × 100 = 400.

Difference = 200000 − 400 = 199600.

Explanation

Using the algebraic identity a² - b² = (a - b)(a + b):

(544)² - (256)² = (544 - 256)(544 + 256) = 288 × 800 = 230400

Taking the square root:

√(230400) = √(2304 × 100) = 48 × 10 = 480

Explanation

Since x:y = 2:3, substitute x = 2 and y = 3 directly into the given expression:

4x + 3y = 4(2) + 3(3) = 8 + 9 = 17 2x + 5y = 2(2) + 5(3) = 4 + 15 = 19

Evaluating the fraction (4x + 3y) / (2x + 5y) gives 17/19, which equals 17:19.

Explanation

Using the algebraic identity (a - b)² = a² - 2ab + b² with a = 380 and b = 2:

378² = (380 - 2)² = 380² - 2 × 380 × 2 + 2² = 144400 - 1520 + 4 = 142884

Taking the square root on both sides gives √142884 = 378.

Explanation

Let x = a sin θ + b cos θ.

Squaring and adding the two equations: (a cos θ - b sin θ)² + (a sin θ + b cos θ)² = a² cos² θ + b² sin² θ - 2ab sin θ cos θ + a² sin² θ + b² cos² θ + 2ab sin θ cos θ = a²(cos² θ + sin² θ) + b²(sin² θ + cos² θ) = a² + b²

Substituting the given equation c = a cos θ - b sin θ: c² + x² = a² + b² x² = a² + b² - c² x = ± √(a² + b² - c²)

Explanation

0.008 / x = 0.01

x = 0.008 / 0.01

Multiplying both numerator and denominator by 100:

x = 0.8 / 1 = 0.8

Explanation

Let the total cost price of the goods be C.

Net profit percentage = (60% × 4%) - (40% × 2%) = 2.4% - 0.8% = 1.6%

Given that the total profit is 250: 1.6% of C = 250 (1.6 / 100) × C = 250 C = (250 × 100) / 1.6 = 25000 / 1.6 = 15,625

Explanation

Given 13 sin A - 12 = 0, we have sin A = 12/13. Since A lies in the second quadrant, cos A is negative: cos A = -√(1 - sin² A) = -√(1 - (12/13)²) = -5/13

Using the trigonometric identity sec² A - 1 = tan² A, the expression simplifies directly: tan² A - (sec² A - 1) = tan² A - tan² A = 0

Substituting the values into the given expression: 5 sin A + 12 cos A + tan² A - (sec² A - 1) = 5(12/13) + 12(-5/13) + 0 = 60/13 - 60/13 = 0

Explanation

Ratio of efficiency of Raju to Ravi = 3 : 1. Since efficiency is inversely proportional to time, the ratio of time taken by Raju and Ravi is 1 : 3.

Let the time taken by Raju be x days and Ravi be 3x days. Difference in time = 3x - x = 2x days

Given, 2x = 20 days, which gives x = 10 days. Time taken by Ravi = 3x = 3 × 10 = 30 days.

Explanation

Correct Mean = Incorrect Mean + [(Sum of correct observations - Sum of incorrect observations) ÷ Number of observations]

Substitute the given values into the formula: Correct Mean = 50 + [(13 + 24 - 31 - 42) ÷ 20] Correct Mean = 50 + [(-36) ÷ 20] Correct Mean = 50 - 1.80 = 48.20

Explanation

Let the two numbers be 4a and 4b, where a and b are co-prime integers.

Product of the numbers = 4a × 4b = 16ab Given that the product is 48: 16ab = 48 ab = 3

Since 3 is a prime number, the only co-prime pair for (a, b) is (1, 3). Substituting these values gives the numbers as 4 × 1 = 4 and 4 × 3 = 12.

Explanation

Using the BODMAS rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction), evaluate the expression step-by-step from the innermost brackets outward:

Innermost parentheses: (2 + 2) = 4 Curly braces: 2 - 2(4) = 2 - 8 = -6 Square brackets: 2 - (-6) = 2 + 6 = 8 Outer expression: 2 - 8 = -6

Explanation

To arrange the fractions in descending order, convert each fraction to its decimal form:

16/17 ≈ 0.9412 6/7 ≈ 0.8571 11/13 ≈ 0.8462 5/6 ≈ 0.8333

Comparing the values: 0.9412 > 0.8571 > 0.8462 > 0.8333, giving the order 16/17, 6/7, 11/13, 5/6.

Explanation

Using the algebraic expansion (x + y)³ = x³ + 3x²y + 3xy² + y³:

(√2 + 1)³ = (√2)³ + 3(√2)²(1) + 3(√2)(1)² + 1³ = 2√2 + 6 + 3√2 + 1 = 7 + 5√2

Comparing this with a + b√2 gives a = 7 and b = 5.

Substituting these values yields a - b = 7 - 5 = 2.

Explanation

From 5A - 2B = 0, express B in terms of A: B = 5A / 2

Substitute B into the second equation, 4B - 5C = 0: 4(5A / 2) - 5C = 0 10A - 5C = 0 5C = 10A C / A = 10 / 5 = 2 / 1

Thus, C:A = 2:1.

Explanation

The required number must be a multiple of the least common multiple (LCM) of 3, 4, 5, and 8.

LCM(3, 4, 5, 8) = 120

The greatest three-digit number is 999. Dividing 999 by the LCM: 999 = 120 × 8 + 39

Subtracting the remainder 39 from 999 gives the greatest three-digit number divisible by 3, 4, 5, and 8: 999 - 39 = 960

Explanation

The exterior angle of a regular polygon is 180° - 108° = 72°.

The number of sides n is given by: n = 360° / exterior angle = 360° / 72° = 5.

Explanation

The volume of a hemisphere is given by the formula V = (2/3) × π × r³, where r is the radius.

Substituting the given values: 19404 = (2/3) × (22/7) × r³ 19404 = (44/21) × r³ r³ = (19404 × 21) ÷ 44 r³ = 441 × 21 r³ = 21³ r = 21 cm

Explanation

Daily earning of (X + Y + Z) = 2,400 / 15 = ₹160 Daily earning of (X + Y) = 1,840 / 16 = ₹115 Daily earning of (Y + Z) = 1,530 / 18 = ₹85

Daily earning of Y = (Daily earning of X + Y) + (Daily earning of Y + Z) - (Daily earning of X + Y + Z) = 115 + 85 - 160 = ₹40

Explanation

Let the two numbers be x and y, where x > y.

Given x + y = 20 and xy = 64.

Using the algebraic identity (x - y)² = (x + y)² - 4xy: (x - y)² = 20² - 4(64) = 400 - 256 = 144 x - y = 12

Solving x + y = 20 and x - y = 12 gives: x = 16 and y = 4

Since the first number is greater than the second, the required ratio is 16 : 4 = 4 : 1.

Explanation

Percentage of failed students = 100% - 90% = 10%

Let the total number of students be N. 10% of N = 260 (10 / 100) × N = 260 N = (260 × 100) / 10 = 2600

Explanation

Let the principal amount invested in each bank be P.

Total Simple Interest = P × (R₁ + R₂) × T / 100

Substituting the given values: 1,650 = P × (10 + 12) × 1 / 100 1,650 = P × 22 / 100 P = (1,650 × 100) / 22 = 7,500

Explanation

The least common multiple (LCM) of prime numbers 2, 3, 5, and 7 is 2 × 3 × 5 × 7 = 210.

Dividing 1549 by 210 gives a quotient of 7 and a remainder of 79: 1549 = 210 × 7 + 79

The least number that must be added to 1549 to make it a multiple of 210 is: 210 - 79 = 131

Explanation

Arranging the given 8 numbers in ascending order: 16, 27, 32, 35, 45, 58, 64, 76

Since the number of terms n = 8 (even), the median is the average of the middle two terms (the 4th and 5th terms):

Median = (35 + 45) / 2 = 80 / 2 = 40

Explanation

When a solid object is completely immersed in liquid, the volume of displaced liquid equals the volume of the immersed object.

Volume of the metal cube = edge³ = 8³ = 512 cm³

The radius of the glass cylinder r = 20 / 2 = 10 cm.

Let h be the height by which the water level rises in the cylinder.

Volume of displaced water in cylinder = π × r² × h 3.142 × 10² × h = 512 314.2 × h = 512 h = 512 / 314.2 ≈ 1.629 cm

Correct to one decimal place, the height by which water rises is 1.6 cm.

Explanation

Initial rural voters = 80% of 50,000 = 40,000 Initial urban voters = 20% of 50,000 = 10,000

After shifting 25% of rural voters (25% of 40,000 = 10,000) to the urban area: Final rural voters = 40,000 - 10,000 = 30,000 Final urban voters = 10,000 + 10,000 = 20,000

Voters who are both honest and hardworking constitute 35% of the voters in each area and vote for NOTA: NOTA votes from rural area = 35% of 30,000 = 10,500 NOTA votes from urban area = 35% of 20,000 = 7,000 Total NOTA votes = 10,500 + 7,000 = 17,500

The remaining 65% of voters in each area vote for their respective candidates: Candidate A (rural voters) = 65% of 30,000 = 19,500 Candidate B (urban voters) = 65% of 20,000 = 13,000

Explanation

Percentage of remaining oranges = 100% - 60% = 40%.

Since 40% of the total number of oranges is equal to 380: Total oranges = (380 × 100) / 40 = 950.

Explanation

The smallest number that leaves a remainder r when divided by a set of numbers is given by:

Required Number = LCM(given numbers) + r

First, find the prime factorizations: 12 = 2² × 3 16 = 2⁴ 20 = 2² × 5 25 = 5² 30 = 2 × 3 × 5

LCM(12, 16, 20, 25, 30) = 2⁴ × 3¹ × 5² = 16 × 3 × 25 = 1200

Required Number = 1200 + 3 = 1203

Explanation

The percentage difference between a 16% profit (+16%) and a 12% loss (-12%) is 16% - (-12%) = 28%.

Given that 28% of the cost price equals 70:

Cost Price = (70 × 100) / 28 = 250

Explanation

Mean = (Sum of observations) ÷ (Total number of observations)

Sum = 8 + (-2) + 9 + 6 + 13 + 17 + 12 = 63 Total number of observations = 7

Mean = 63 ÷ 7 = 9

Explanation

Perimeter of a rectangle = 2 × (length + width)

100 = 2 × (29 + width) 50 = 29 + width width = 50 - 29 = 21 m

Explanation

Since time is constant at 30 hours, distance covered is directly proportional to speed. A reduction in speed by 1/15 results in a 1/15 reduction in the total distance covered.

Given that the reduction in distance is 10 km: 1/15 × Total Distance = 10 km Total Distance = 10 × 15 = 150 km

Speed = Total Distance / Time = 150 / 30 = 5 km/h.

Explanation

Following the BODMAS rule, evaluate the expression step-by-step from the innermost brackets:

Inner parenthesis: 12 + 4 + 3 = 19 Curly braces: 3 + 19 = 22 Square brackets: 2 + 22 = 24

Multiply by 2/5: (2/5) × 24 = 48/5

Converting 48/5 into a mixed fraction gives 9 3/5.

Explanation

Let the present ages of Naresh and Suresh be 6x and 5x years respectively.

Given that the sum of their ages is 44 years: 6x + 5x = 44 11x = 44 x = 4

Naresh's present age = 6 × 4 = 24 years Suresh's present age = 5 × 4 = 20 years

After 15 years: Naresh's age = 24 + 15 = 39 years Suresh's age = 20 + 15 = 35 years

Therefore, the ratio of their ages after 15 years is 39:35.

Explanation

Divide the given equation sin x - 5 cos x = 2√6 cos x by cos x: tan x - 5 = 2√6 tan x = 5 + 2√6

Since cot x = 1 / tan x: cot x = 1 / (5 + 2√6)

Rationalizing the denominator by multiplying both the numerator and denominator by (5 - 2√6): cot x = (5 - 2√6) / ((5 + 2√6)(5 - 2√6)) cot x = (5 - 2√6) / (25 - 24) cot x = 5 - 2√6

Explanation

Loss = Total Cost Price - Total Selling Price

Given that the loss on 18 toys equals the cost price of 4 toys: Loss = 18 CP - Total SP 4 CP = 18 CP - 980 18 CP - 4 CP = 980 14 CP = 980 CP = 980 / 14 = ₹70

Explanation

Let a/3 = b/4 = c/7 = k.

Then, a = 3k, b = 4k, and c = 7k.

Substituting these values into the given expression: (a + b + c) / c = (3k + 4k + 7k) / 7k = 14k / 7k = 2