RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

Convert all terms to base 5 using the exponent rules (aᵐ)ⁿ = aᵐⁿ and aᵐ × aⁿ ÷ aᵖ = aᵐ⁺ⁿ⁻ᵖ:

Left side: 25³.⁸ × 5⁴.² ÷ 5⁶.⁴ = (5²)³.⁸ × 5⁴.² ÷ 5⁶.⁴ = 5⁷.⁶ × 5⁴.² ÷ 5⁶.⁴ = 5⁷.⁶ ⁺ ⁴.² ⁻ ⁶.⁴ = 5⁵.⁴

Right side: 25ˣ = (5²)ˣ = 5²ˣ

Equating the powers of 5: 2x = 5.4 x = 2.7

Explanation

GEO stands for Geostationary Earth Orbit, which is a circular orbit positioned approximately 35,786 kilometers above Earth's equator. Satellites placed in this orbit revolve around the Earth with an orbital period equal to Earth's rotational period of 23 hours, 56 minutes, and 4 seconds. Consequently, the satellite moves in the same direction and at the same rate as Earth rotates, making it appear stationary relative to a fixed position on the Earth's surface. This makes it ideal for telecommunications, weather monitoring, and satellite broadcasting.

Explanation

The maximum length of a stick that can fit inside a rectangular box (cuboid) is given by the length of its space diagonal:

Diagonal = √(l² + b² + h²)

Given dimensions l = 15 cm, b = 8 cm, and h = 6 cm:

Diagonal = √(15² + 8² + 6²) Diagonal = √(225 + 64 + 36) Diagonal = √325 Diagonal = √(25 × 13) = 5√13 cm

Explanation

To convert the given fractional ratio 2/5 : 1/3 : 3/5 into integers, multiply each term by the LCM of the denominators, which is 15: (2/5 × 15) : (1/3 × 15) : (3/5 × 15) = 6 : 5 : 9

The total number of ratio parts = 6 + 5 + 9 = 20 parts.

Since 5 is the smallest ratio term, the length of the shortest side = (5 / 20) × 110 = 27.5 cm.

Explanation

For two numbers in the ratio a : b where a and b are co-prime, the relationship between HCF and LCM is given by the formula:

LCM = HCF × a × b

Given the ratio a : b = 5 : 6 and HCF = 6:

LCM = 6 × 5 × 6 = 180

Explanation

To find the least perfect square completely divisible by 4, 5, 9, and 12, first find the LCM of the numbers by prime factorization:

4 = 2² 5 = 5¹ 9 = 3² 12 = 2² × 3¹

LCM(4, 5, 9, 12) = 2² × 3² × 5¹ = 180

For a number to be a perfect square, all prime factors must have even exponents. In 180, the prime factor 5 appears with an odd power (5¹). Multiplying 180 by 5 yields:

2² × 3² × 5² = 900

Explanation

Initial number of apples = 100

Apples given to Amar = 10% of 100 = 10 Apples given to Kenhav = 15% of 100 = 15 Apples given to Rinku = (2/5) × 100 = 40

Total apples distributed = 10 + 15 + 40 = 65

Apples left with Atulit = 100 - 65 = 35

Explanation

For a rectangle with length l and width w: Perimeter = 2(l + w) = 62 m, which gives l + w = 31 m. Area = l × w = 168 m².

The diagonal d is given by the formula d = √(l² + w²).

Using the algebraic identity (l + w)² = l² + w² + 2lw: l² + w² = (l + w)² - 2lw l² + w² = 31² - 2 × 168 = 961 - 336 = 625

Substituting this into the diagonal formula: d = √625 = 25 m.

Explanation

Using the algebraic identity (x - 1/x)³ = x³ - 1/x³ - 3(x - 1/x), we can express x³ - 1/x³ as:

x³ - 1/x³ = (x - 1/x)³ + 3(x - 1/x)

Substituting the given value x - 1/x = 3 into the formula:

x³ - 1/x³ = 3³ + 3(3) = 27 + 9 = 36

Explanation

Convert all time intervals into minutes: 30 min, 45 min, 1 h = 60 min, and 1½ h = 90 min.

To find when all four devices will beep together again, calculate the Least Common Multiple (LCM) of the four time intervals: LCM(30, 45, 60, 90) = 180 minutes

Converting 180 minutes into hours: 180 / 60 = 3 hours

Adding 3 hours to the initial time of 10.45 a.m.: 10.45 a.m. + 3 hours = 1.45 p.m.

Explanation

Using the depreciation formula: Present Value = P × (1 - r/100)ⁿ

where P is the value of the car 2 years ago, r = 12%, and n = 2 years.

7,74,400 = P × (1 - 12/100)² 7,74,400 = P × (88/100)² 7,74,400 = P × (22/25)² 7,74,400 = P × (484/625)

P = (7,74,400 × 625) / 484 P = 1600 × 625 = 10,00,000

Explanation

To find the value of √(8/5), multiply the numerator and denominator under the square root by 5 to introduce 40 into the expression:

√(8/5) = √((8 × 5) / (5 × 5)) = √(40/25) = √(40) / 5

Substituting the given value of √(40) = 6.3245:

√(8/5) = 6.3245 / 5 = 1.2649

Explanation

The decimal 1.25 can be written as the fraction 125/100. Dividing both the numerator and denominator by their greatest common divisor, 25, gives the simplest fraction form:

1.25 = 125/100 = 5/4

Here, the numerator is 5 and the denominator is 4. The difference between the numerator and denominator is:

5 - 4 = 1

Explanation

Using the BODMAS rule (Brackets, Division/Multiplication from left to right, Addition):

First, evaluate the expression inside the brackets: (3 × 2) = 6

This simplifies the expression to: 5 + 2 × 3 ÷ 6 × 3

Next, execute the multiplication and division sequentially from left to right: 2 × 3 = 6 6 ÷ 6 = 1 1 × 3 = 3

Finally, perform the addition: 5 + 3 = 8

Explanation

Given tan θ = 1 for an acute angle θ, θ = 45°.

Substituting θ = 45° into the expression: 2 sin 45° cos 45° - cosec² 45° = sin 90° - cosec² 45° = 1 - (√2)² = 1 - 2 = -1

Explanation

When an amount doubles itself under simple interest, the interest earned (SI) is equal to the principal (P).

Using the simple interest formula SI = (P × R × T) / 100: P = (P × R × 4) / 100 1 = (R × 4) / 100 R = 100 / 4 = 25% per annum

Explanation

When a fixed distance is covered at two different speeds S₁ and S₂, the distance D is given by the formula:

D = (S₁ × S₂ / |S₁ - S₂|) × Δt

Given S₁ = 45 km/h, S₂ = 50 km/h, and time difference Δt = 32 minutes = 32/60 hours = 8/15 hours.

Substituting these values into the formula:

D = (45 × 50 / 5) × (8 / 15) D = 450 × (8 / 15) D = 240 km

Explanation

A's 1-day work = 1/24.

Work done by A in 21 days = 21/24 = 7/8. Remaining work = 1 - 7/8 = 1/8.

B finishes the remaining 1/8 work in 5 days, so B's 1-day work = (1/8) ÷ 5 = 1/40.

Combined 1-day work of A and B = 1/24 + 1/40 = (5 + 3)/120 = 8/120 = 1/15.

Time taken by A and B together to complete the work = 1 ÷ (1/15) = 15 days.

Explanation

Multiplying the given factors: (4y - 1)(y + 4) = 4y² + 16y - y - 4 = 4y² + 15y - 4

Comparing the coefficients of 4y² + 15y - 4 with py² + 15y - q: p = 4 -q = -4 ⇒ q = 4

Since p = 4 and q = 4, it follows that p = q.

Explanation

√(0.000081) = 0.009, since 0.009 × 0.009 = 0.000081.

Multiplying this value by 4 gives: 4 × 0.009 = 0.036

Explanation

Circumference of a circle is 2πr and area is πr², where r is the radius. Equating the numerical values of both gives:

2πr = πr²

Dividing both sides by πr yields r = 2 units.

Explanation

Let the two numbers be 3x and 2x.

According to the given condition: (3x - 8) / (2x - 6) = 8 / 5

Cross-multiplying gives: 5(3x - 8) = 8(2x - 6) 15x - 40 = 16x - 48 x = 8

Therefore, the numbers are: First number = 3 × 8 = 24 Second number = 2 × 8 = 16

Explanation

20% of 75 = (20 / 100) × 75 = 15

Substituting this into the given equation: 15 = 225 - (x / 100) × 420

Rearranging the terms: (x / 100) × 420 = 225 - 15 4.2x = 210 x = 210 / 4.2 = 50

Explanation

Let the present ages of Kunal and Kirti be x and y years, respectively.

According to the given conditions: y + 6 = 2(x + 6) ⇒ y = 2x + 6 y - 4 = 4(x - 4) ⇒ y = 4x - 12

Equating the two expressions for y: 2x + 6 = 4x - 12 2x = 18 x = 9

Explanation

First value = (3/5) × 200 = 120 Second value = (1/2) × 300 = 150

Difference = 150 - 120 = 30

Explanation

Convert the given mixed fractions into improper fractions: 2(1/5) = 11/5 5(2/5) = 27/5

Let x be the required multiplier. (11/5) × x = 27/5 x = (27/5) ÷ (11/5) x = 27/11 = 2(5/11)

Explanation

Sum of the interior angles of a triangle is always 180°. Given the ratio of the angles as 2 : 3 : 4, the total number of ratio parts is 2 + 3 + 4 = 9.

Value of 1 ratio part = 180° / 9 = 20°.

The largest angle corresponds to the largest ratio part, 4, so its measure is 4 × 20° = 80°.

Explanation

The least common multiple (LCM) of 6, 8, and 12 is 24.

Any number that leaves a remainder of 1 when divided by 6, 8, and 12 is of the form 24k + 1, where k is a positive integer.

For k = 1: 24(1) + 1 = 25, which is not a multiple of 7. For k = 2: 24(2) + 1 = 49, which is divisible by 7 (49 = 7 × 7).

Explanation

Cost Price for A = Final Price / ((1 + Profit₁%)(1 + Profit₂%))

Cost Price for A = 360 / (1.20 × 1.20) = 360 / 1.44 = ₹250.

Explanation

Total outcomes = (92 - 33) + 1 = 60 cards.

The perfect squares in the range 33 to 92 are 36 (6²), 49 (7²), 64 (8²), and 81 (9²). Number of favorable outcomes = 4.

Probability = Favorable outcomes / Total outcomes = 4 / 60 = 1 / 15.

Explanation

Prime factorization of the given numbers: 30 = 2 × 3 × 5 40 = 2³ × 5 45 = 3² × 5

The LCM is the product of the highest power of each prime factor involved: LCM(30, 40, 45) = 2³ × 3² × 5 = 8 × 9 × 5 = 360

Explanation

The formula for compound amount with quarterly compounding is A = P × (1 + R/400)⁽4t), where P is the principal, A is the final amount, R is the annual interest rate, and t is the time in years.

Substituting P = 3200, A = 3362, and R = 10 into the formula: 3362 = 3200 × (1 + 10/400)⁽4t) 3362 / 3200 = (41/40)⁽4t) 1681 / 1600 = (41/40)⁽4t) (41/40)² = (41/40)⁽4t)

Equating exponents: 4t = 2 t = ½ year

Explanation

Work done by A, B, and C together in 1 day = 1/14 Work done by C alone in 1 day = 1/42

Work done by A and B together in 1 day = 1/14 - 1/42 = (3 - 1)/42 = 2/42 = 1/21

Alternatively, using the combined work formula: Time taken = (14 × 42) / (42 - 14) = 588 / 28 = 21 days

Explanation

Using direct substitution into the expression:

a² - b² = (-5)² - (5)²

Since (-5)² = 25 and 5² = 25, substituting these values gives:

a² - b² = 25 - 25 = 0

Alternatively, using the algebraic identity a² - b² = (a + b)(a - b):

a² - b² = (-5 + 5)(-5 - 5) = 0 × (-10) = 0

Explanation

Using the algebraic identity, if a + b + c = 0, then a³ + b³ + c³ - 3abc = 0.

Let a = 2x, b = y, and c = -3.

Given that a + b + c = 2x + y - 3 = 0, we substitute these into the identity:

a³ + b³ + c³ - 3abc = (2x)³ + y³ + (-3)³ - 3(2x)(y)(-3) = 8x³ + y³ - 27 + 18xy = 8x³ + 18xy + y³ - 27 = 0

Explanation

New amount in the first friend's account = ₹47,000 + ₹2,000 = ₹49,000 New amount in the second friend's account = ₹54,000 + ₹2,000 = ₹56,000

Ratio = 49,000 : 56,000 = 49 : 56

Dividing both terms by their greatest common divisor (7): 49 ÷ 7 = 7 and 56 ÷ 7 = 8, giving the simplified ratio 7 : 8.

Explanation

Total cards in a standard deck = 52. Number of black Kings and Queens removed = 2 Kings (Spades, Clubs) + 2 Queens (Spades, Clubs) = 4 cards. Remaining total cards = 52 - 4 = 48.

Original number of spade cards = 13. Since the King of Spades and Queen of Spades were removed, remaining spade cards = 13 - 2 = 11.

Probability of drawing a spade card = Favorable outcomes / Total outcomes = 11/48.

Explanation

Ascending order means arranging numbers from the smallest value to the largest value. Comparing the whole number parts and decimal place values: 0.0882 is the smallest since its whole number part is 0 and tenths digit is 0, followed by 0.25 which has a tenths digit of 2, then 1.24 which has a whole number part of 1, and finally 2.67 which has the largest whole number part of 2. The sequence from smallest to largest is 0.0882 < 0.25 < 1.24 < 2.67.

Explanation

Total maximum marks for 5 subjects = 5 × 50 = 250. Since the maximum possible marks are equal for all students, the student with the lowest total marks will have the lowest overall percentage.

Total marks obtained by B = 29 + 33 + 22 + 15 + 30 = 129 Overall percentage for B = (129 / 250) × 100 = 51.6%

With a total of 129 marks out of 250, student B secures the lowest total marks and consequently the lowest overall percentage of 51.6%.

Explanation

Total maximum marks for the 5 subjects = 5 × 50 = 250.

Calculating the total marks for student A gives 48 + 47 + 49 + 46 + 44 = 234, which is the highest total score among all students.

Maximum overall percentage = (234 / 250) × 100 = 93.60%.

Explanation

To find the two students with the least difference in marks, calculate the total marks obtained by each student across all subjects:

Total marks of A = 48 + 47 + 49 + 46 = 190 Total marks of B = 29 + 33 + 22 + 15 = 99 Total marks of C = 35 + 36 + 42 + 40 = 153 Total marks of D = 46 + 49 + 48 + 48 = 191 Total marks of E = 32 + 30 + 25 + 21 = 108

The difference in total marks between A and D is 191 - 190 = 1, which is the smallest possible difference between any pair of students.

Explanation

Calculating total marks in all listed subjects for each student: D: 46 + 49 + 48 + 48 = 191 A: 48 + 47 + 49 + 46 = 190 C: 35 + 36 + 42 + 40 = 153 E: 32 + 30 + 25 + 21 = 108 B: 29 + 33 + 22 + 15 = 99

Rank order in all subjects: 1st D, 2nd A, 3rd C, 4th E, 5th B.

Calculating marks in Physics, Chemistry, and Maths only: A: 48 + 47 + 49 = 144 D: 46 + 49 + 48 = 143 C: 35 + 36 + 42 = 113 E: 32 + 30 + 25 = 87 B: 29 + 33 + 22 = 84

Rank order in PCM: 1st A, 2nd D, 3rd C, 4th E, 5th B.

Comparing both rankings, the 1st and 2nd positions swap between A and D, while the 3rd, 4th, and 5th positions (C, E, B) remain unchanged. Thus, only the top two ranks are different.

Explanation

To convert 4.14 hours into hours, minutes, and seconds:

Whole hours = 4 hours.

Convert the decimal part to minutes: 0.14 × 60 = 8.4 minutes, which gives 8 whole minutes.

Convert the remaining decimal part of the minute to seconds: 0.4 × 60 = 24 seconds.

Combining these gives 4 hours, 8 minutes, 24 seconds.

Explanation

Let the cost price of the first article be CP1 = 100. Since it is sold at a 20% profit, its selling price is SP1 = 100 × 1.20 = 120.

The cost price of the second article equals SP1, so CP2 = 120. The second article is sold at a 20% loss, so its selling price is SP2 = 120 × 0.80 = 96.

Total Cost Price = CP1 + CP2 = 100 + 120 = 220. Total Selling Price = SP1 + SP2 = 120 + 96 = 216.

Total Loss = 220 - 216 = 4. Loss percentage = (4 ÷ 220) × 100 = 20/11 % ≈ 1.818% ≈ 1.8% loss.

Explanation

Percentage change in area when two successive dimensions increase by x% and y% is given by the formula: Net percentage change = x + y + (x × y) / 100

Since area of a square is side × side, both dimensions increase by 20%, so x = 20 and y = 20.

Substituting these values into the formula: Net percentage change = 20 + 20 + (20 × 20) / 100 = 40 + 400 / 100 = 40 + 4 = 44%

Alternatively, assuming initial side length as 10 units, initial area is 10 × 10 = 100 square units. With a 20% increase, the new side length becomes 12 units, making the new area 12 × 12 = 144 square units. The percentage increase is ((144 - 100) / 100) × 100 = 44%.

Explanation

The LCM of 6, 12, and 18 is 36.

The largest four-digit number is 9999. Dividing 9999 by 36 yields a quotient of 277 and a remainder of 27 (9999 = 36 × 277 + 27).

Subtracting the remainder gives the largest four-digit number divisible by 36: 9999 - 27 = 9972

To find the number that leaves a remainder of 5 in each case, add 5 to this multiple: 9972 + 5 = 9977

Explanation

For two numbers in the ratio a : b (where a and b are co-prime) and having a Highest Common Factor (HCF) of h, the Least Common Multiple (LCM) is given by the formula:

LCM = HCF × a × b

Substituting the given values into the formula:

LCM = 3 × 7 × 5 = 105

Explanation

To rationalize the denominator of (7 + 2√5) / (11 + 3√5), multiply the numerator and denominator by the conjugate of the denominator, (11 - 3√5):

Numerator = (7 + 2√5)(11 - 3√5) = 77 - 21√5 + 22√5 - 30 = 47 + √5

Denominator = (11 + 3√5)(11 - 3√5) = 11² - (3√5)² = 121 - 45 = 76

Thus, the fraction simplifies to: (47 + √5) / 76 = (47 / 76) + (1 / 76)√5

Comparing this with a + b√5 yields: a = 47/76 and b = 1/76

Explanation

The volume of a hemisphere is (2/3)πR³ and the volume of a cone is (1/3)πr²h.

Number of bottles required = Volume of hemisphere / Volume of one conical bottle = [(2/3)πR³] / [(1/3)πr²h] = (2R³) / (r²h)

Given: Radius of hemisphere (R) = 12 cm Radius of cone (r) = 3/2 = 1.5 cm Height of cone (h) = 4 cm

Number of bottles = (2 × 12 × 12 × 12) / (1.5 × 1.5 × 4) = 3456 / 9 = 384

Explanation

Amount received as a percentage of the original car price = 80% × 90% = 72%

Percentage difference between car price and amount received = 100% - 72% = 28%

Difference in amount = 28% × ₹6,25,000 = 0.28 × 6,25,000 = ₹1,75,000