RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

Let the required number be x.

According to the problem statement: [5 × (x + 7)] / 9 - 3 = 12

Solving the equation step by step: 5 × (x + 7) / 9 = 12 + 3 = 15 5 × (x + 7) = 15 × 9 = 135 x + 7 = 135 / 5 = 27 x = 27 - 7 = 20

Explanation

Let total work be completed by (Man + Boy) together in 24 days. Since the man works alone for the last 6 days of the total 26 days, both worked together for the first 26 - 6 = 20 days.

Fraction of work completed together in 20 days = 20/24 = 5/6. Remaining work = 1 - 5/6 = 1/6.

The man completes this remaining 1/6 work alone in 6 days. Man's 1-day work = (1/6) ÷ 6 = 1/36.

Boy's 1-day work = Combined 1-day work - Man's 1-day work = 1/24 - 1/36 = (3 - 2) / 72 = 1/72.

Time taken by the boy working alone = 1 ÷ (1/72) = 72 days.

Explanation

Substitute a = 0.9 and b = 0.1 into the given expression step by step:

a - b = 0.9 - 0.1 = 0.8 = 8/10

1/√(ab) = 1/√(0.9 × 0.1) = 1/√0.09 = 1/0.3 = 10/3

√(a/b) = √(0.9/0.1) = √9 = 3

Adding these terms together: 0.9 ⊕ 0.1 = 8/10 + 10/3 + 3 = 4/5 + 10/3 + 3

Taking 30 as the common denominator: = (24 + 100 + 90) / 30 = 214/30

Explanation

Since the angles A, B, and C of triangle ABC are in arithmetic progression and their sum is 180°, we have A + B + C = 3B = 180°, which gives B = 60°.

In the right-angled triangle AMB where AM is perpendicular to BC (∠AMB = 90°):

cos(B) = BM / AB

Substituting B = 60°:

BM / AB = cos(60°) = ½

Explanation

Rearranging the given equation: a = b² / (b - a) a(b - a) = b² ab - a² = b² a² - ab + b² = 0

Using the standard algebraic identity: a³ + b³ = (a + b)(a² - ab + b²)

Substituting a² - ab + b² = 0 gives: a³ + b³ = (a + b) × 0 = 0

Explanation

By trigonometric identity, sin A sin B + cos A cos B = cos(A - B).

Since angle A = angle B, the difference (A - B) = 0°.

Therefore, cos(A - B) = cos 0° = 1.

Alternatively, since triangle ABC is right-angled at C and angle A = angle B, both A and B equal 45°. Substituting these values gives sin 45° sin 45° + cos 45° cos 45° = (1/√2 × 1/√2) + (1/√2 × 1/√2) = ½ + ½ = 1.

Explanation

Given that 40% × (a - b) = 20% × (a + b).

Dividing both sides by 20%: 2(a - b) = 1(a + b) 2a - 2b = a + b a = 3b

To find what percentage b is of a: (b / a) × 100 = (1 / 3) × 100 = 100/3 %

Explanation

Let Vikas' initial salary be ₹100. Since he spends 80% of his salary: Initial Expenditure = ₹80 Initial Savings = 100 - 80 = ₹20

After the increases: New Salary = ₹100 × 1.25 = ₹125 New Expenditure = ₹80 × 1.15 = ₹92 New Savings = 125 - 92 = ₹33

Increase in savings = 33 - 20 = ₹13 Percentage increase in savings = (13 / 20) × 100 = 65%

Explanation

Let the initial ratio of the two numbers be 9 : 11 and the new ratio after subtracting 4 from each be 7 : 9.

The difference in ratio units for both numbers is: (9 - 7) = (11 - 9) = 2 units

Since subtracting 4 reduces each number by 2 ratio units: 2 units = 4 1 unit = 2

The sum of the two numbers in ratio units is 9 + 11 = 20 units.

Sum = 20 × 2 = 40

Alternatively, since the ratio is 9 : 11, the sum of the numbers must be a multiple of (9 + 11) = 20, which directly gives 40.

Explanation

Total number of ways to select a 3-member delegation from 5 salesmen is ⁵C₃ = 10.

Number of favorable delegations that include A requires selecting 2 more members from the remaining 4 salesmen, which is ⁴C₂ = 6.

Probability that A gets selected = ⁴C₂ / ⁵C₃ = 6 / 10 = 3/5.

Explanation

Let the initial selling price of the TV be ₹100.

Anisha's cost price (CP) = ⅘ × 100 = ₹80 Anisha's selling price (SP) = 100 + 4% of 100 = ₹104

Gain = SP - CP = 104 - 80 = ₹24 Gain percentage = (Gain / CP) × 100 = (24 / 80) × 100 = 30%

Explanation

Evaluating the numerator: 19 × 9 + 31 × 10 + 32 × 11 = 171 + 310 + 352 = 833

Evaluating the denominator: 233.25 × 2 - 50 = 466.5 - 50 = 416.5

Value of the expression = 833 ÷ 416.5 = 2

Explanation

Using complementary angle relations, cot 71° = tan(90° - 71°) = tan 19°, 1/tan 71° = tan 19°, and 1/sec² 69° = cos² 69° = sin²(90° - 69°) = sin² 21°.

Substituting these into the bracketed expression: cot 71° cos² 21° + 1 / (tan 71° sec² 69°) = tan 19° cos² 21° + tan 19° sin² 21° = tan 19° (cos² 21° + sin² 21°)

Using the identity cos² θ + sin² θ = 1, the inner expression simplifies to tan 19° × 1 = tan 19°.

Multiplying by the outer term: cot 19° × tan 19° = 1

Explanation

LCM(5, 6, 8, 9, 12) = 360.

Any number that leaves a remainder of 1 when divided by 5, 6, 8, 9, and 12 is of the form N = 360k + 1, where k is a positive integer.

Rewriting 360k + 1 as (13 × 27k + 9k + 1), the number N is divisible by 13 when (9k + 1) is divisible by 13.

Testing values for k: For k = 10, 9(10) + 1 = 91, which is divisible by 13 (91 ÷ 13 = 7).

Thus, the required least positive number is N = 360 × 10 + 1 = 3601.

Explanation

To find (x, y), rationalize the denominator by multiplying both numerator and denominator by (2 + √3):

(5 + 2√3) / (2 - √3) = [(5 + 2√3)(2 + √3)] / [(2 - √3)(2 + √3)]

The denominator simplifies using the identity (a - b)(a + b) = a² - b²: 2² - (√3)² = 4 - 3 = 1

The numerator expands as: (5 × 2) + (5 × √3) + (2√3 × 2) + (2√3 × √3) = 10 + 5√3 + 4√3 + 6 = 16 + 9√3

Comparing 16 + 9√3 with x + y√3 gives x = 16 and y = 9, so (x, y) = (16, 9).

Explanation

Area of a parallelogram = base × altitude.

Let the base be b and the altitude be h. Given that h = 2b, which means b = h / 2.

Area = (h / 2) × h = h² / 2 288 = h² / 2 h² = 576 h = √576 = 24 m

Explanation

For a right-angled isosceles triangle with hypotenuse h, the area is given by the formula Area = h² / 4.

Substituting h = 8 cm into the formula: Area = 8² / 4 = 64 / 4 = 16 cm²

Alternatively, if the two equal sides are of length a, then by the Pythagorean theorem, a² + a² = 8², which yields 2a² = 64, or a² = 32. The area of the triangle is ½ × a² = ½ × 32 = 16 cm².

Explanation

To find the largest number that divides given numbers leaving the same remainder in each case, calculate the Highest Common Factor (HCF) of the absolute differences between the numbers.

Differences between the numbers: 3739 - 2270 = 1469 6677 - 3739 = 2938 6677 - 2270 = 4407

Finding HCF(1469, 2938, 4407): 2938 = 2 × 1469 4407 = 3 × 1469

Since 1469 divides both 2938 and 4407 completely, the highest common factor is 1469.

Explanation

Total maximum marks for both students across all four subjects = 4 × 50 = 200.

Raj's aggregate marks = 37 + 45 + 42 + 32 = 156 Raj's aggregate percentage = (156 / 200) × 100 = 78%

Rohit's aggregate marks = 48 + 30 + 45 + 35 = 158 Rohit's aggregate percentage = (158 / 200) × 100 = 79%

Difference in aggregate percentage = 79% − 78% = 1%

Alternatively, difference in total marks = 158 − 156 = 2 Percentage difference = (2 / 200) × 100 = 1%

Explanation

Total production of wheat in 2015 = 7.2 + 5.6 + 6.2 = 19 tonnes

Total production of wheat in 2016 = 7.8 + 6.2 + 6 = 20 tonnes

Difference = 20 - 19 = 1 tonne

Explanation

Argument P is weak because simply stating that a policy is followed in other countries is an example-based argument that fails to provide a logical reason or justification for why it should be implemented.

Argument Q is strong because government interference directly compromises media independence and objectivity, causing news to lose its credibility and public trust, which directly addresses a fundamental consequence of the issue.

Explanation

The English alphabet has 26 letters. Deleting every third letter (3rd, 6th, 9th, ..., 24th) removes 8 letters, leaving a total of 18 letters in the new series.

The 9th letter from the right in an 18-letter series is equivalent to the (18 - 9 + 1) = 10th letter from the left.

The remaining letters in order from the left are A, B, D, E, G, H, J, K, M, N, P, Q, S, T, V, W, Y, Z. Counting to the 10th position from the left gives N.

Explanation

At a rate of 5% per annum, the fractional equivalent is 1/20.

For a period of 3 years, let the Principal P = 20³ = 8000 units. The Amount A after 3 years = 21³ = 9261 units.

Compound Interest (CI) = A - P = 9261 - 8000 = 1261 units.

Given CI = ₹6,305: 1261 units = ₹6,305 1 unit = 6305 / 1261 = ₹5

Simple Interest (SI) = (P × R × T) / 100 SI = (8000 units × 5 × 3) / 100 = 1200 units

Value of Simple Interest = 1200 × 5 = ₹6,000

Explanation

Converting the fractions into their approximate decimal equivalents:

11/13 ≈ 0.846 13/14 ≈ 0.928 15/17 ≈ 0.882 17/19 ≈ 0.894

Comparing the decimal values shows that 0.846 is the least value. Alternatively, cross-multiplying 11/13 with any of the other fractions confirms it is smaller in every comparison (for example, 11 × 14 = 154 < 13 × 13 = 169, and 11 × 17 = 187 < 13 × 15 = 195).

Explanation

Substitute the standard trigonometric values into the given expression: sin 45° = 1/√2 sec 45° = √2 tan 45° = 1 cos 60° = ½ sin 30° = ½ sin 90° = 1

Evaluate each component: First term = sin 45° × sec 45° × tan 45° = (1/√2) × √2 × 1 = 1 Second term = cos 60° × sin 30° × sin 90° = ½ × ½ × 1 = ¼ Third term = -4 cos 60° = -4 × ½ = -2

Combine the terms: 1 + ¼ - 2 = 5/4 - 2 = -3/4

Explanation

For a sum of money to triple (n = 3) under simple interest, the interest earned equals twice the principal.

The rate of simple interest is given by the formula: R = [(n - 1) / T] × 100%

Substituting n = 3 and T = 12.5 years: R = [(3 - 1) / 12.5] × 100 R = (2 / 12.5) × 100 = 16%

Explanation

Dividing the polynomial 6x⁴ + 8x³ + 17x² + 21x + 7 by 3x² + 4x + 1 using polynomial division:

First step: 6x⁴ / 3x² = 2x² 2x² × (3x² + 4x + 1) = 6x⁴ + 8x³ + 2x² Subtracting this from the dividend gives: (6x⁴ + 8x³ + 17x² + 21x + 7) - (6x⁴ + 8x³ + 2x²) = 15x² + 21x + 7

Second step: 15x² / 3x² = 5 5 × (3x² + 4x + 1) = 15x² + 20x + 5 Subtracting this gives the remainder: (15x² + 21x + 7) - (15x² + 20x + 5) = x + 2

Comparing the remainder x + 2 with ax + b gives a = 1 and b = 2.

Explanation

The Third Battle of Panipat took place on 14 January 1761 at Panipat between the Maratha Empire, led by Sadashivrao Bhau, and the invading Afghan army under Ahmad Shah Durrani (Abdali), supported by regional allies including Najib-ud-Daula and Shuja-ud-Daula. The conflict resulted in a decisive defeat for the Marathas and significantly checked their expansion in northern India.

Explanation

Rate of work done by (P + Q) in 1 day = 1/9 Rate of work done by (Q + R) in 1 day = 1/12 Rate of work done by (P + R) in 1 day = 1/18

Adding the daily work rates: 2 × (P + Q + R) = 1/9 + 1/12 + 1/18 2 × (P + Q + R) = (4 + 3 + 2) / 36 = 9/36 = 1/4 P + Q + R = 1/8

Since P, Q, and R together complete 1/8 of the field in 1 day, they will take 8 days to complete the entire field working together.

Explanation

e-Sampark is an initiative launched by the Government of India under the Digital India campaign to establish direct communication with citizens. Developed by the National Informatics Centre (NIC), it enables central and state government departments to send personalized and bulk information regarding policies, public services, advisories, and welfare schemes directly to citizens using electronic channels like SMS, outbound dialing (OBD), and emails.

Explanation

Following the Gaya session of the Indian National Congress in December 1922, a resolution to enter the legislative councils was defeated. Chittaranjan Das, who presided over the session, resigned along with Motilal Nehru and established the Swaraj Party on January 1, 1923, to contest council elections and obstruct British policy from within the legislature.

Explanation

Compound interest is calculated using the formula: CI = P × [(1 + r/100)ⁿ - 1]

Given: Principal (P) = ₹ 10,000 Rate of interest (r) = 10% per annum Time period (n) = 3 years

Substituting the given values into the formula: CI = 10,000 × [(1 + 10/100)³ - 1] CI = 10,000 × [(1.1)³ - 1] CI = 10,000 × [1.331 - 1] CI = 10,000 × 0.331 = ₹ 3,310

Alternatively, using the 3:3:1 ratio method for 3 years: First year interest = 10% of 10,000 = 1,000 Second year interest on interest = 10% of 1,000 = 100 Third year interest on interest = 10% of 100 = 10

CI = (3 × 1,000) + (3 × 100) + (1 × 10) = 3,000 + 300 + 10 = ₹ 3,310

Explanation

The sum of percentages of football, kabaddi, and chess players is 82% + 7% + 4% = 93%.

The percentage of cricket players is 100% - 93% = 7%.

Number of cricket players = 7% of 3600 = (7 / 100) × 3600 = 7 × 36 = 252.

Explanation

To increase a principal sum P by 40% through simple interest, the simple interest earned SI must equal 0.40 × P.

Using the simple interest formula SI = (P × R × T) / 100:

0.40 × P = (P × 10 × T) / 100 40 = 10 × T T = 4 years

Explanation

Using the algebraic expansion formula: (a - b)² = a² + b² - 2ab

Substituting the given values a² + b² = 29 and ab = 10: (a - b)² = 29 - 2(10) = 29 - 20 = 9

Explanation

Average Speed = Total Distance / Total Time

Distance covered by P = 1 h × 40 km/h = 40 km Distance covered by Q = 2 h × 50 km/h = 100 km Distance covered by R = 3 h × 60 km/h = 180 km

Total Distance = 40 + 100 + 180 = 320 km Total Time = 1 + 2 + 3 = 6 hours

Mean Speed = 320 / 6 = 53.33 km/h

Explanation

In the number 128935, the digit 2 is in the ten-thousands place, giving it a place value of 20000. The digit 3 is in the tens place, giving it a place value of 30.

Difference = 20000 - 30 = 19970

Explanation

The lateral surface area of a cylinder is given by the formula LSA = 2πrh.

Given, radius (r) = 25 cm and height (h) = 42 cm.

Substituting the values with π = 22/7: LSA = 2 × (22/7) × 25 × 42 LSA = 2 × 22 × 25 × 6 LSA = 6600 cm²

Explanation

The initial ratio of students in the three classes is 2 : 3 : 5, and after adding 30 students to each class, the new ratio becomes 4 : 5 : 7.

The increase in ratio units for each class is 4 - 2 = 5 - 3 = 7 - 5 = 2 units.

Since an increase of 2 units corresponds to 30 students: 1 unit = 15 students

The total number of ratio units after the increase is 4 + 5 + 7 = 16 units.

Total students after the increase = 16 × 15 = 240.

Explanation

Product of two numbers = Product of their LCM and HCF

Let the two numbers be 5x and 4x.

(5x) × (4x) = 18000 20x² = 18000 x² = 900 x = 30

Sum of the two numbers = 5x + 4x = 9x = 9 × 30 = 270.

Explanation

The total surface area of a cube with side length a is given by the formula: Total Surface Area = 6a²

Given side length a = 2 m: Total Surface Area = 6 × 2² = 6 × 4 = 24 m²

Explanation

The product of two numbers is equal to the product of their LCM and HCF.

Let the two positive numbers be 7x and 9x.

(7x) × (9x) = 25200 63x² = 25200 x² = 400 x = 20

The sum of the numbers is 7x + 9x = 16x. Substituting x = 20: Sum = 16 × 20 = 320

Explanation

To find the least number of trees that can be planted in 105, 210, or 420 rows without any leftover, the total number of trees must be the Least Common Multiple (LCM) of 105, 210, and 420.

Since 420 is a multiple of both 105 and 210: 105 × 4 = 420 210 × 2 = 420 420 × 1 = 420

LCM(105, 210, 420) = 420

Explanation

Ramu's selling price to Rahul at 5% profit = 2,00,000 × (105 / 100) = ₹2,10,000

Rahul's selling price back to Ramu at 2% loss = 2,10,000 × (98 / 100) = ₹2,05,800

Since Ramu gets the car back, his net financial gain is the difference between the cash received and the cash spent: Net profit = 2,10,000 - 2,05,800 = ₹4,200

Explanation

When x + 1/x = k, the algebraic relation for the sum of cubes is: x³ + 1/x³ = k³ - 3k

Substituting k = 2: x³ + 1/x³ = 2³ - 3(2) = 8 - 6 = 2

Alternatively, solving x + 1/x = 2 gives x = 1, which directly yields 1³ + 1/1³ = 2.

Explanation

To completely pass the canal, the train must cover a total distance equal to the sum of its length and the length of the canal.

Total distance = 150 m + 102 m = 252 m

Speed = Total distance ÷ Time = 252 ÷ 7 = 36 m/s

Explanation

For three consecutive natural numbers, the middle number equals the average of the set:

Middle number = 120 / 3 = 40

Since the numbers are consecutive, the three numbers are 39, 40, and 41. The largest number in this set is 41.

Explanation

Expanding (√5 + 1)² using the formula (x + y)² = x² + 2xy + y²:

(√5 + 1)² = (√5)² + 2(√5)(1) + 1² = 5 + 2√5 + 1 = 6 + 2√5

Equating this to a + b√5 gives: a = 6 and b = 2

Since 6 > 2, the condition a > b is satisfied.

Explanation

Total distance to be covered = Sum of train lengths = 131 + 89 = 220 m

Since the trains are moving in opposite directions, relative speed = 42 + 30 = 72 km/h

Converting speed to m/s = 72 × (5 / 18) = 20 m/s

Time taken to completely clear each other = Total distance / Relative speed = 220 / 20 = 11 s

Explanation

Calculate each part: (4/5) × 3500 = 2800 (1/3) × 6000 = 2000

Subtract the second value from the first to find the difference: 2800 - 2000 = 800