RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

Perimeter of the rectangle = 2 × (24 + 28) = 104 cm.

Since the rhombus has the same perimeter, the side length of the rhombus (a) = 104 / 4 = 26 cm.

Area of a rhombus with side length 'a' and interior angle 'θ' is given by the formula: Area = a² × sin(θ)

Substituting a = 26 cm and θ = 120°: Area = 26² × sin(120°) = 676 × (√3 / 2) = 338√3 cm²

Explanation

Average Marks = Sum of Marks Scored by All Students ÷ Total Number of Students

Sum of marks scored in Computer Science by students A, B, C, D, and E = 33 + 39 + 34 + 41 + 32 = 179

Average marks = 179 ÷ 5 = 35.8

Explanation

Total expenditure in 2018 = 783 + 203 + 479 + 81 + 69 = 1615 million

Admin expenditure in 2018 = 203 million

Percentage = (Admin expenditure / Total expenditure) × 100 = (203 / 1615) × 100 = 12.569% ≈ 12.57%

Explanation

For half-yearly compounding, the annual interest rate is halved and the number of conversion periods is doubled. Rate per half-year (r) = 4% ÷ 2 = 2% Number of half-year periods (n) = 1 × 2 = 2

Amount (A) = P × (1 + r/100)ⁿ A = 15,000 × (1 + 2/100)² A = 15,000 × (1.02)² A = 15,000 × 1.0404 = ₹15,606

Compound Interest = Amount - Principal CI = 15,606 - 15,000 = ₹606

Explanation

In Badminton, the highest number of students is 78 from School D, which exceeds School B's count of 67. For all other games listed in the table, School B has the highest number of participating students among all schools.

Explanation

From the given table, the sales of printer model L3 over the 5 years are: 2014: 340 2015: 235 2016: 423 2017: 548 2018: 674

Maximum sales for model L3 = 674 Minimum sales for model L3 = 235

Difference = Maximum sales - Minimum sales = 674 - 235 = 439

Explanation

If the ratio of three numbers is a:b:c where a, b, and c are co-prime to each other, the numbers are given by a × HCF, b × HCF, and c × HCF.

Since HCF = 7 and the ratio is 3:4:5, the required numbers are: 3 × 7 = 21 4 × 7 = 28 5 × 7 = 35

Explanation

A number x lies between 9.2 and 10.5 if it satisfies the inequality 9.2 < x < 10.5. Comparing 9.55 with the boundary values gives 9.20 < 9.55 < 10.50, which satisfies the given condition. Additionally, 9.55 is a terminating decimal that can be written in p/q form as 191/20, making it a rational number.

Explanation

Using complementary angle rules, cot 75° = tan 15° and tan 75° = cot 15°.

The expression simplifies as: tan 15° cot 75° + tan 75° cot 15° = tan² 15° + cot² 15°

Given tan 15° = 2 - √3, its reciprocal cot 15° = 1 / (2 - √3) = 2 + √3.

Adding the two terms gives: tan 15° + cot 15° = (2 - √3) + (2 + √3) = 4

Squaring both sides and using the identity (a + b)² = a² + b² + 2ab: (tan 15° + cot 15°)² = tan² 15° + cot² 15° + 2(tan 15° cot 15°) 4² = tan² 15° + cot² 15° + 2 16 = tan² 15° + cot² 15° + 2 tan² 15° + cot² 15° = 14

Explanation

From the given equation 0.02x = 0.5y, the ratio of x to y is: x / y = 0.5 / 0.02 = 50 / 2 = 25 / 1

Substituting x = 25 and y = 1 into the expression (x - y) / (x + y): (x - y) / (x + y) = (25 - 1) / (25 + 1) = 24 / 26 = 12 / 13

Explanation

Selling Price = Cost Price × (100 - Loss%) / 100

Substituting CP = ₹70 and Loss% = 10%: Selling Price = ₹70 × (100 - 10) / 100 = ₹70 × 90 / 100 = ₹63

Explanation

Let the daily work done by a boy be B and by a girl be G.

Total work W = (3B + 5G) × 6 = 18B + 30G Also, total work W = (2B + 7G) × 8 = 16B + 56G

Equating the two expressions for work: 18B + 30G = 16B + 56G 2B = 26G B = 13G

Substituting B = 13G into the total work equation: W = 18(13G) + 30G = 234G + 30G = 264G

Daily work rate of 8 girls = 8G.

Number of days required by 8 girls = Total work / Daily work rate = 264G / 8G = 33 days

Explanation

Let the number be x.

According to the given condition: (1/4) × (1/8) × x = 300 x / 32 = 300 x = 300 × 32 = 9600

Now, one-fifth of the same number: (1/5) × 9600 = 1920

Explanation

To solve the given expression 684 ÷ 9 ÷ 0.5, apply the BODMAS rule of performing sequential division operations from left to right.

Step 1: Divide 684 by 9 684 ÷ 9 = 76

Step 2: Divide 76 by 0.5 Since dividing a number by 0.5 is equivalent to multiplying it by 2: 76 ÷ 0.5 = 76 × 2 = 152

Explanation

Let the speed of the train be v m/s and its length be L meters.

When the train and a person move in opposite directions, their relative speed is the sum of their speeds.

For the first person moving at 4 m/s: L = (v + 4) × 10 = 10v + 40

For the second person moving at 10 m/s: L = (v + 10) × 8 = 8v + 80

Equating the two length equations: 10v + 40 = 8v + 80 2v = 40 v = 20 m/s

Explanation

Let the breadth of the rectangle be x units, so the length is 2x units.

Perimeter = 2 × (Length + Breadth) = 2 × (2x + x) = 6x Area = Length × Breadth = 2x × x = 2x²

Given the ratio of the perimeter to the area is 3:4: Perimeter / Area = 3 / 4 6x / 2x² = 3 / 4 3 / x = 3 / 4 x = 4

Length of the rectangle = 2x = 2 × 4 = 8 units.

Explanation

Convert each mixed fraction into an improper fraction: 4 ⅓ = 13/3 2 ⅘ = 14/5 2 ⅓ = 7/3

Substitute these values into the expression and change division to multiplication by taking the reciprocal: (13/3) × (14/5) ÷ (7/3) = (13/3) × (14/5) × (3/7)

Canceling out common factors in the numerator and denominator: = (13 × 14 × 3) / (3 × 5 × 7) = (13 × 2) / 5 = 26/5

Converting 26/5 back to a mixed fraction yields 5 ⅕.

Explanation

Let the daily work efficiency of B be 1 unit per day. Since worker A is thrice as efficient as B, A's daily efficiency is 3 units per day.

Their combined efficiency is 3 + 1 = 4 units per day.

Total work = Combined efficiency × Days taken together = 4 × 18 = 72 units.

Time taken by A alone = Total work / Efficiency of A = 72 / 3 = 24 days.

Explanation

With total students equal to 60 and the ratio of girls to boys as 5:7, the sum of the ratio terms is 5 + 7 = 12.

Number of girls = (5 ÷ 12) × 60 = 25 Number of boys = (7 ÷ 12) × 60 = 35

When 5 boys leave, the number of boys remaining is 35 - 5 = 30.

The new ratio of boys to girls is 30 : 25 = 6 : 5.

Explanation

Let the breadth of the rectangle be b cm, so its length is (b + 2) cm.

Perimeter of a rectangle = 2 × (length + breadth) 20 = 2 × (b + 2 + b) 20 = 2 × (2b + 2) 10 = 2b + 2 2b = 8 b = 4 cm

Length = 4 + 2 = 6 cm

Area of the rectangle = length × breadth = 6 × 4 = 24 cm².

Explanation

The area of a triangle is related to its inradius and semi-perimeter by the formula Area = r × s, where r is the inradius and s is the semi-perimeter.

Given: Perimeter = 64 cm Semi-perimeter (s) = 64 / 2 = 32 cm Inradius (r) = 8 cm

Area = 8 × 32 = 256 cm²

Explanation

Since 60² = 3600 and 70² = 4900, the square root of 4096 lies between 60 and 70.

Using the expansion (a + b)² = a² + 2ab + b²: 64² = (60 + 4)² = 3600 + 2 × 60 × 4 + 16 = 3600 + 480 + 16 = 4096

Thus, √4096 = 64.

Explanation

The expression 4x² + 4xy + y² represents a perfect square trinomial following the algebraic identity (a + b)² = a² + 2ab + b², where a = 2x and b = y.

4x² + 4xy + y² = (2x + y)²

Substituting the given values x = 3 and y = 5:

2x + y = 2(3) + 5 = 11

(2x + y)² = 11² = 121

Explanation

Time taken by Person A = 250 / 6 = 41⅔ hours = 41 hours 40 minutes.

For Person B: Pages read at normal speed = 250 - 72 = 178 pages. Time for normal pages = 178 / 8 = 22.25 hours = 22 hours 15 minutes. Since taking double time per page implies half the reading speed, speed for difficult pages = 8 / 2 = 4 pages per hour. Time for difficult pages = 72 / 4 = 18 hours. Total time taken by Person B = 22 hours 15 minutes + 18 hours = 40 hours 15 minutes.

Difference in time = 41 hours 40 minutes - 40 hours 15 minutes = 1 hour 25 minutes.

Explanation

Using the trigonometric identity: (sin θ + cos θ)² + (sin θ - cos θ)² = 2(sin² θ + cos² θ) = 2

Substituting sin θ + cos θ = √7/2: (√7/2)² + (sin θ - cos θ)² = 2 7/4 + (sin θ - cos θ)² = 2 (sin θ - cos θ)² = 2 - 7/4 = 1/4 sin θ - cos θ = ½

Explanation

Total revenue from selling mangoes = Cost Price + Profit = ₹600 + ₹210 = ₹810.

Number of mangoes sold = Total revenue ÷ Selling price per mango = 810 ÷ 3 = 270.

Number of rotten mangoes thrown away = Total purchased - Mangoes sold = 300 - 270 = 30.

Percentage of mangoes thrown away = (30 ÷ 300) × 100 = 10%.

Explanation

For any real angle θ and constants a and b, the fundamental trigonometric identity gives: (a sin θ + b cos θ)² + (a cos θ - b sin θ)² = a² + b²

Substituting a = 5, b = 12, and a sin θ + b cos θ = 13 into the identity: 13² + (5 cos θ - 12 sin θ)² = 5² + 12² 169 + (5 cos θ - 12 sin θ)² = 25 + 144 169 + (5 cos θ - 12 sin θ)² = 169 (5 cos θ - 12 sin θ)² = 0 5 cos θ - 12 sin θ = 0

Explanation

Let the cost price of 1 g of flowers be ₹1.

Cost price of 20 g = ₹20

Selling price of 25 g at 4% loss = 25 × (100 - 4) / 100 = ₹24

Since he actually gives only 20 g, his selling price for that amount is ₹24.

Profit = ₹24 - ₹20 = ₹4

Percentage Profit = (4 / 20) × 100 = 20%

Alternatively, using the faulty weight formula: Percentage Profit = [(100 - Loss%) × (True Weight / False Weight)] - 100 = [(100 - 4) × (25 / 20)] - 100 = (96 × 1.25) - 100 = 20%

Explanation

Using the algebraic identity (x + y)² = x² + y² + 2xy:

(x + y)² = 41 + 2(20) = 41 + 40 = 81 x + y = √81 = 9

The required expression is: 1/x + 1/y = (x + y) / xy = 9/20

Explanation

Let the two numbers be 8a and 8b, where a and b are co-prime positive integers.

Using the standard relation: Product of two numbers = HCF × LCM 8a × 8b = 8 × 160 64ab = 1280 ab = 20

The co-prime factor pairs of 20 are (1, 20) and (4, 5).

Taking (a, b) = (4, 5): First number = 8 × 4 = 32 Second number = 8 × 5 = 40

Sum of the numbers = 32 + 40 = 72

Explanation

Given that 20% of A = 25% of B, express A in terms of B: 0.20 × A = 0.25 × B A = (25 / 20) × B = 1.25 × B

Substitute A into 28% of A: 28% of A = 0.28 × A = 0.28 × (1.25 × B) = 0.35 × B = 35% of B

Explanation

Let the speed of the rider be v km/h and the length of the train be L meters.

The distance covered while passing the rider in both cases equals the length of the train L.

In opposite directions: L = (70 + v) × (5/18) × 2

In the same direction: L = (70 - v) × (5/18) × 5

Equating the two expressions for L: (70 + v) × 2 = (70 - v) × 5 140 + 2v = 350 - 5v 7v = 210 v = 30 km/h

Explanation

Let the first number be x and the second number be y.

According to the given condition: 3% of x + 2% of y = ¾ × (3% of x + 5% of y)

Multiplying through by 100 to clear the percentages: 3x + 2y = ¾ × (3x + 5y)

Multiplying both sides by 4: 4 × (3x + 2y) = 3 × (3x + 5y) 12x + 8y = 9x + 15y 12x - 9x = 15y - 8y 3x = 7y x : y = 7 : 3

Explanation

Since 6.5 = 65 / 10, the given expression can be rewritten as:

6.5 × 6.5 = (65 / 10) × (65 / 10) = (65 × 65) / 100

Substituting the given value 65 × 65 = 4225:

4225 / 100 = 42.25

Alternatively, since each factor has one decimal place, the product must have a total of two decimal places, placing the decimal point two digits from the right in 4225 to give 42.25.

Explanation

The expression can be split into integer and fractional parts:

(777 + 1/5) + (777 + 2/5) + (777 + 3/5) + (777 + 4/5)

= (777 × 4) + (1/5 + 2/5 + 3/5 + 4/5) = 3108 + (10/5) = 3108 + 2 = 3110

Explanation

Given: A = (2/3)B ⇒ A : B = 2 : 3 B = (1/3)C ⇒ B : C = 1 : 3

To combine the ratios, make the value corresponding to B common in both ratios by multiplying B : C by 3: B : C = (1 × 3) : (3 × 3) = 3 : 9

Combining A : B and B : C gives: A : B : C = 2 : 3 : 9

Explanation

Total students from U.P. in 2013 and 2014 = 400 + 450 = 850.

Total students from M.P. in 2013 and 2014 = 280 + 240 = 520.

Required ratio = 850 : 520 = 85 : 52.

Explanation

The overall percentage increase in the village population is ((4500 - 4000) / 4000) × 100 = 12.5%.

Using the rule of alligation: Male increase rate = 10% Female increase rate = 20% Overall increase rate = 12.5%

Ratio of Males to Females = (20 - 12.5) : (12.5 - 10) = 7.5 : 2.5 = 3 : 1

Initial number of males = (3 / 4) × 4000 = 3000.

Explanation

Using the standard order of operations (BODMAS/PEMDAS):

Evaluate the term inside the parenthesis / division: 45 ÷ 9 = 5

Perform multiplication: 8 × (-4) = -32

Combine addition and subtraction from left to right: 9 + 5 - (-32) = 14 + 32 = 46

Explanation

Passing marks = Marks obtained + Deficit = 70 + 20 = 90 marks.

Since 40% of the maximum marks equals the passing marks (90), the maximum marks = (90 / 40) × 100 = 225.

Explanation

Let the man buy 30 apples of each type (LCM of 5 and 6), making a total of 60 apples.

Cost Price (CP): Cost of 30 apples of first type = 30 × (50 / 5) = ₹300 Cost of 30 apples of second type = 30 × (50 / 6) = ₹250 Total CP = ₹300 + ₹250 = ₹550

Selling Price (SP): Selling price of 60 apples at 11 for ₹100 = 60 × (100 / 11) = ₹6000 / 11

Loss = CP - SP = 550 - (6000 / 11) = ₹50 / 11

Loss percentage = (Loss / CP) × 100 = (50 / 11) / 550 × 100 = (100 / 121)%

Explanation

The LCM of 8, 12, and 16 is 48. Any number that leaves a remainder of 3 when divided by 8, 12, and 16 is of the form N = 48k + 3, where k is a positive integer.

For k = 1: N = 48(1) + 3 = 51 For k = 2: N = 48(2) + 3 = 99 For k = 3: N = 48(3) + 3 = 147

Since 147 ÷ 7 = 21, 147 is a multiple of 7 and is the smallest such number.

Explanation

Let the selling price (SP) of 1 chair be ₹x.

Total SP of 25 chairs = 25x Profit = SP of 5 chairs = 5x Total Cost Price (CP) = ₹37,500

Using the profit formula: Profit = Total SP - Total CP 5x = 25x - 37,500 20x = 37,500 x = 37,500 / 20 = ₹1,875

Therefore, the selling price of one chair is ₹1,875.

Explanation

Using the algebraic identity (a - b)² = a² + b² - 2ab: 4² = 10 - 2ab 16 = 10 - 2ab 2ab = -6 ab = -3

Next, using the identity for difference of cubes: a³ - b³ = (a - b)(a² + b² + ab)

Substituting the known values: a³ - b³ = 4 × (10 - 3) a³ - b³ = 4 × 7 = 28

Explanation

Rearrange the given equation to isolate the question mark (?):

8432 + 937 + 260 - ? = 5605 + 713

First, sum the numbers on the left-hand side: 8432 + 937 + 260 = 9629

Next, sum the numbers on the right-hand side: 5605 + 713 = 6318

Now, substitute these sums back into the equation: 9629 - ? = 6318 ? = 9629 - 6318 ? = 3311

Explanation

Volume of a sphere V = (4/3)πr³

Substituting V = 38808 cm³ and π = 22/7: 38808 = (4/3) × (22/7) × r³ 38808 = (88/21) × r³ r³ = (38808 × 21) / 88 r³ = 441 × 21 r³ = 21³ r = 21 cm

Explanation

By the Factor Theorem, if (x + 1) and (x + 2) are factors of the polynomial P(x) = ax³ + 3x² + bx, then P(-1) = 0 and P(-2) = 0.

Evaluating P(-1) = 0: a(-1)³ + 3(-1)² + b(-1) = 0 -a + 3 - b = 0 a + b = 3

Evaluating P(-2) = 0: a(-2)³ + 3(-2)² + b(-2) = 0 -8a + 12 - 2b = 0 4a + b = 6

Subtracting the first equation from the second yields 3a = 3, which gives a = 1. Substituting a = 1 back into the first equation yields 1 + b = 3, so b = 2.

Explanation

Prime factorization of the given numbers: 12 = 2² × 3 39 = 3 × 13 54 = 2 × 3³

The Highest Common Factor (HCF) is determined by taking the product of the lowest powers of the common prime factors. The only prime factor common to all three numbers is 3, with its lowest exponent being 1.

HCF(12, 39, 54) = 3¹ = 3.

Explanation

Let S and K be the present ages of Sunil and Kamal, respectively.

From the first condition (six years hence): S + 6 = 2(K + 6) S - 2K = 6

From the second condition (two years ago): S - 2 = 4(K - 2) S - 4K = -6

Subtracting the second equation from the first equation: (S - 2K) - (S - 4K) = 6 - (-6) 2K = 12 K = 6

Explanation

Between any two distinct real numbers, there exists an infinite density of decimal numbers. Between 0.225 and 0.227, one can form infinitely many numbers such as 0.2251, 0.22501, 0.225001, and so on, simply by extending the decimal places. This property of real numbers ensures that an unlimited number of values lie between any given pair of distinct decimals.