RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

Total work = LCM(56, 70) = 280 units.

Efficiency of A = 280 / 56 = 5 units/day Efficiency of B = 280 / 70 = 4 units/day

Work completed by B alone in the last 34 days = 34 × 4 = 136 units. Remaining work completed by A and B together = 280 - 136 = 144 units.

Combined efficiency of A and B = 5 + 4 = 9 units/day. Number of days A worked with B = 144 / 9 = 16 days.

Explanation

Given p : q = 9 : 2, substitute p = 9 and q = 2 directly into the given expression:

4p + 7q = 4(9) + 7(2) = 36 + 14 = 50 4p - 7q = 4(9) - 7(2) = 36 - 14 = 22

Therefore, the ratio (4p + 7q) : (4p - 7q) = 50 : 22 = 25 : 11.

Explanation

Let CP be the cost price and SP be the selling price of one item.

Given, 150 × SP = 125 × CP SP / CP = 125 / 150 = 5 / 6

Loss = CP - SP = 6 - 5 = 1

Loss Percentage = (Loss / CP) × 100 = (1 / 6) × 100 = 16⅔%

Explanation

Convert all mixed fractions to improper fractions: 1 1/5 = 6/5 3 2/4 = 7/2 1 3/4 = 7/4 3 1/8 = 25/8 1 1/7 = 8/7

Substitute these into the given equation: 6/5 - [7/2 ÷ 7/4 ÷ (x + 25/8) ÷ 8/7] = 1

First, simplify (7/2 ÷ 7/4): (7/2) × (4/7) = 2

The equation simplifies to: 6/5 - [2 ÷ (x + 25/8) ÷ 8/7] = 1 6/5 - 1 = 2 ÷ (x + 25/8) ÷ (8/7) 1/5 = 2 × (1 / (x + 25/8)) × (7/8) 1/5 = (14/8) / (x + 25/8) 1/5 = 14 / (8x + 25)

Cross-multiply to solve for x: 8x + 25 = 70 8x = 45 x = 45/8 = 5 5/8

Explanation

When the difference between each divisor and its corresponding remainder is constant (2 - 1 = 1, 3 - 2 = 1, 8 - 7 = 1), the numbers in the sequence follow the formula:

N = LCM(2, 3, 8) × k - 1

Calculating LCM(2, 3, 8) = 24 gives the general term N = 24k - 1.

For the second term (k = 2): N = 24 × 2 - 1 = 47

Explanation

Using the angle difference formula sin(A - B) = sin A cos B - cos A sin B:

sin 15° = sin(45° - 30°) = sin 45° cos 30° - cos 45° sin 30°

Substituting the standard trigonometric values: sin 45° = 1/√2, cos 45° = 1/√2 sin 30° = ½, cos 30° = √3/2

sin 15° = (1/√2) × (√3/2) - (1/√2) × (½) = (√3 - 1) / (2√2)

Multiplying the numerator and denominator by √2 to rationalize: sin 15° = (√6 - √2) / 4

Explanation

Using the algebraic identity a² - b² = (a + b)(a - b): 75 = 5 × (a - b) a - b = 15

Using the identity 4ab = (a + b)² - (a - b)²: 4ab = 5² - 15² 4ab = 25 - 225 4ab = -200 ab = -50

Explanation

Using the identity (a+b)/(a-b) + (a-b)/(a+b) = 2(a² + b²)/(a² - b²), let a = √3 and b = 1.

x + y = [(√3 + 1)² + (√3 - 1)²] / [(√3)² - 1²] x + y = 2((√3)² + 1²) / (3 - 1) = 2(3 + 1) / 2 = 4

Therefore, 3(x + y) = 3 × 4 = 12.

Explanation

The National Highways Authority of India (NHAI) was set up under the National Highways Authority of India Act, 1988. It functions as an autonomous agency under the Ministry of Road Transport and Highways, responsible for the development, maintenance, and management of the national highway network across the country.

Explanation

Let x be the side length of the inner square park. Since the path has a uniform width of 1.5 m all around, the side length of the outer square including the path is x + 2(1.5) = x + 3 m.

Area of the path = (Outer side)² − (Inner side)² 225 = (x + 3)² − x² 225 = x² + 6x + 9 − x² 225 = 6x + 9 6x = 216 x = 36 m

Perimeter of the park = 4 × x = 4 × 36 = 144 m.

Explanation

If 82% of the number is 738, then 1% of the number is 738 ÷ 82 = 9.

Therefore, 90% of the number is 90 × 9 = 810.

Explanation

Under simple interest, the interest earned is directly proportional to time. When a principal P triples in 8 years, the simple interest earned in 8 years is 3P - P = 2P.

Interest earned per year = 2P / 8 = P / 4. Interest earned in 12 years = 12 × (P / 4) = 3P.

Total amount after 12 years = Principal + Interest = P + 3P = 4P.

Explanation

Using the algebraic identity (a - 1/a)² = a² + 1/a² - 2:

Substituting a² + 1/a² = 18 gives: (a - 1/a)² = 18 - 2 = 16

Taking the square root on both sides yields: a - 1/a = √16 = 4

Explanation

The cosecant function is defined as cosec x = 1 / sin x. For cosec x to be well-defined, the denominator sin x must not be equal to zero. Since sin x = 0 whenever x is an integral multiple of π (i.e., x = nπ where n ∈ Z), these values must be excluded from the set of all real numbers R. Thus, the domain of cosec x is R - {nπ | n ∈ Z}.

Explanation

To determine if a number is prime, check for divisibility by prime numbers up to its square root. For 173, √173 ≈ 13.15, so the prime numbers to test are 2, 3, 5, 7, 11, and 13. Since 173 is not divisible by any of these prime numbers (173 ÷ 7 = 24 remainder 5, 173 ÷ 11 = 15 remainder 8, 173 ÷ 13 = 13 remainder 4), it has no positive factors other than 1 and 173, making it a prime number.

Explanation

For a moderately asymmetrical distribution, Karl Pearson's empirical formula relating mean, median, and mode is:

Mode = 3 × Median - 2 × Mean

Substituting the given values into the formula: 25.2 = 3 × Median - 2 × 27.5 25.2 = 3 × Median - 55 3 × Median = 80.2 Median = 80.2 / 3 = 26.733 ≈ 26.73

Explanation

Mean = (Sum of all values) / (Total number of values)

Sum of the given values = 2.2 + 4.2 + 6.4 + 8.3 + 10.5 = 31.6 Total number of values = 5

Mean = 31.6 / 5 = 6.32

Explanation

Sikkim is located in the northeastern part of India, nestled in the Himalayas. It shares international borders with three countries: Nepal to its west, China (Tibet Autonomous Region) to its north and northeast, and Bhutan to its east. Within India, it shares its southern border with West Bengal.

Explanation

The given expression is a telescoping series of the form 1/(a × b) where the difference between factors in each denominator is d = 3.

Sum = (1/d) × (1/first factor - 1/last factor)

Here, d = 3, the first factor is 5, and the last factor is 17.

Sum = (1/3) × (1/5 - 1/17) Sum = (1/3) × ((17 - 5) / (5 × 17)) Sum = (1/3) × (12 / 85) Sum = 4 / 85

Alternatively, for a telescoping series with n terms: Sum = n / (first factor × last factor) = 4 / (5 × 17) = 4 / 85

Explanation

Solving the system of simultaneous linear equations:

From 2x + y = 15 and x + 2y = 9, multiply the first equation by 2 and subtract the second equation: 2(2x + y) - (x + 2y) = 2(15) - 9 3x = 21 x = 7

Substitute x = 7 into 2x + y = 15: 2(7) + y = 15 y = 1

Substitute y = 1 into y + 2z = 17: 1 + 2z = 17 2z = 16 z = 8

Evaluating the required expression: 4x + 3y + z = 4(7) + 3(1) + 8 = 28 + 3 + 8 = 39

Explanation

The sum of an interior angle and an exterior angle of a regular polygon is 180°. Therefore, each exterior angle = 180° - 165° = 15°.

Since the sum of all exterior angles of a convex polygon is 360°, the number of sides n = 360° / 15° = 24.

Explanation

Let the original number be x.

Subtracting 106 reduces the number to 47% of its value, which means the difference corresponds to 53% of the number: (100% - 47%) of x = 106 53% of x = 106 1% of x = 106 / 53 = 2 x = 200

Now, calculate 11.5% of 200: 11.5% of 200 = 11.5 × 2 = 23

Explanation

Using Heron's formula, the area of a triangle with side lengths a = 5 cm, b = 7 cm, and c = 11 cm is given by Area = √(s(s - a)(s - b)(s - c)), where s is the semi-perimeter.

s = (a + b + c) / 2 = (5 + 7 + 11) / 2 = 11.5 cm

Substitute the values into the formula: Area = √(11.5 × (11.5 - 5) × (11.5 - 7) × (11.5 - 11)) Area = √(11.5 × 6.5 × 4.5 × 0.5) Area = √168.1875 ≈ 12.97 cm²

Explanation

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

Given: Principal (P) = ₹10,000 Rate of interest (R) = 12% Time period (T) = 5 years

Substituting the given values into the formula: SI = (10,000 × 12 × 5) / 100 = 100 × 60 = ₹6,000

Explanation

Using the trigonometric identity cot(θ) - tan(θ) = 2 cot(2θ):

For θ = 15°: cot(15°) - tan(15°) = 2 cot(30°)

Since cot(30°) = √3: cot(15°) - tan(15°) = 2 × √3 = 2√3

Alternatively, substituting the standard values tan(15°) = 2 - √3 and cot(15°) = 2 + √3: (2 + √3) - (2 - √3) = 2√3

Explanation

(1/6) × 432 = 72

(3/4) × 216 = 3 × 54 = 162

Difference = 162 - 72 = 90

Explanation

To find the ratio A:D, multiply the given consecutive ratios:

A/D = (A/B) × (B/C) × (C/D)

Substituting the given values into the formula: A/D = (2/3) × (2/3) × (3/4) = 12/36 = 1/3

Explanation

Simple interest paid by Sunil for 1.5 years at 11% per annum: SI = (P × R × T) / 100 = (40,000 × 11 × 1.5) / 100 = ₹6,600

Compound interest paid by Kamal with semi-annual compounding at 5% per half-year for 3 half-years: Amount = P × (1 + r/100)ⁿ = 40,000 × (1 + 5/100)³ = 40,000 × (21/20)³ = ₹46,305 CI = Amount - Principal = 46,305 - 40,000 = ₹6,305

Difference = 6,600 - 6,305 = ₹295 extra paid by Sunil.

Explanation

Using the direct formula: New Selling Price = Old Selling Price × (100 + Gain%) / (100 - Loss%)

Substituting the given values: New Selling Price = 1980 × (100 + 15) / (100 - 10) New Selling Price = 1980 × 115 / 90 New Selling Price = 22 × 115 = ₹2,530

Explanation

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

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

Rationalizing the denominator by multiplying the numerator and denominator by (3 - √3): tan x = (3 - √3) / ((3 + √3)(3 - √3)) tan x = (3 - √3) / (3² - (√3)²) tan x = (3 - √3) / (9 - 3) tan x = (3 - √3) / 6

Explanation

Total Cost Price (CP) = 54 × 25 = ₹1350. Usable quantity of oranges = 54 - 4 = 50 kg. Total Selling Price (SP) = 50 × 36 = ₹1800. Profit = SP - CP = 1800 - 1350 = ₹450. Profit percentage = (Profit / CP) × 100 = (450 / 1350) × 100 = 33 1/3%.

Explanation

Let the length and breadth of the rectangular field be 7x and 6x respectively.

Given that the breadth is 35 m less than the length: 7x - 6x = 35 x = 35

The formula for the perimeter of a rectangle is: Perimeter = 2 × (Length + Breadth) Perimeter = 2 × (7x + 6x) = 26x

Substituting the value of x: Perimeter = 26 × 35 = 910 m

Explanation

Converting each fraction into its decimal value: 2/3 ≈ 0.667 3/7 ≈ 0.429 1/5 = 0.200 1/6 ≈ 0.167

Arranging these decimal values from largest to smallest gives 0.667 > 0.429 > 0.200 > 0.167. Replacing the decimals with their original fractional forms gives 2/3, 3/7, 1/5, 1/6.

Explanation

When two objects move in opposite directions, their relative speed is the sum of their individual speeds: Relative Speed = 9 km/h + 11 km/h = 20 km/h

The time taken to meet on a circular track of length 2 km is given by: Time = Distance / Relative Speed = 2 km / 20 km/h = 1/10 hour

Converting hours to minutes: Time = (1/10) × 60 minutes = 6 minutes

Explanation

Using the complementary angle identities, cosec 50° = sec 40° and sec 50° = cosec 40°.

First term: tan 40° / cosec 50° = tan 40° / sec 40° = (sin 40° / cos 40°) / (1 / cos 40°) = sin 40°

Second term: cot 40° / sec 50° = cot 40° / cosec 40° = (cos 40° / sin 40°) / (1 / sin 40°) = cos 40°

Squaring both terms and adding them gives: (sin 40°)² + (cos 40°)² = sin² 40° + cos² 40° = 1

Explanation

1 man's 1-day work = 1/20 1 woman's 1-day work = 1/45

1-day work of 2 men and 3 women = 2(1/20) + 3(1/45) = 1/10 + 1/15 = (3 + 2)/30 = 5/30 = 1/6

Total time required = 1 ÷ (1/6) = 6 days.

Explanation

By the Remainder Theorem, when a polynomial P(x) is divided by (x - 1), the remainder is P(1).

For the first polynomial P(x) = 4x³ + ax² - 3x + 1: P(1) = 4(1)³ + a(1)² - 3(1) + 1 = 4 + a - 3 + 1 = a + 2

For the second polynomial Q(x) = x⁴ + x³ - x² + 6: Q(1) = 1⁴ + 1³ - 1² + 6 = 1 + 1 - 1 + 6 = 7

Since both polynomials leave the same remainder when divided by (x - 1): P(1) = Q(1) a + 2 = 7 a = 5

Explanation

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

Using the given percentages: 40% of x = 12 ⇒ x = (12 × 100) / 40 = 30 50% of y = 24 ⇒ y = (24 × 100) / 50 = 48

The ratio of the first number to the second number is: x : y = 30 : 48 = 5 : 8

Explanation

Time taken to double, T₁ = 4 years 4 months = 4 + 4/12 = 13/3 years.

At simple interest, the interest earned is directly proportional to time.

For the sum to double, the interest earned equals the principal in 13/3 years. For the sum to triple, the interest earned must be twice the principal.

Time required = 2 × (13/3) = 26/3 = 8⅔ years.

Explanation

The sum of all interior angles in a triangle is 180°.

Given the ratio of the angles is 4 : 5 : 6, total ratio parts = 4 + 5 + 6 = 15.

Value of 1 part = 180° ÷ 15 = 12°

Smallest angle = 4 parts = 4 × 12° = 48°

Explanation

Pressure is defined as the force applied perpendicular to the surface of an object per unit area (P = F/A). In the International System of Units (SI), force is measured in newtons (N) and area in square meters (m²). Therefore, the SI unit of pressure is newton per square meter (N/m²), which is officially named the Pascal (Pa) in honor of the scientist Blaise Pascal (1 Pa = 1 N/m²).

Explanation

Distance covered by a rotating wheel in N revolutions is given by: Distance = N × 2πr

Converting distance into metres: 66 km = 66000 m.

Substituting the given values: 66000 = 21000 × 2 × (22/7) × r 66000 = 3000 × 44 × r 66000 = 132000 × r r = 66000 / 132000 = ½ m

Explanation

NITI Aayog's Atal Innovation Mission (AIM) signed a Statement of Intent (SOI) with Adobe to drive digital literacy and foster creative skills among school children in India. Under this collaboration, Adobe implements its Digital Disha Programme across schools equipped with Atal Tinkering Labs (ATLs), providing students and educators with free access to creative learning tools such as Adobe Spark Premium to enhance technology-based learning.

Explanation

The formula for the mean of the squares of the first n natural numbers is:

Mean = (n + 1)(2n + 1) / 6

For the first 10 natural numbers (n = 10):

Mean = (10 + 1)(2 × 10 + 1) / 6 = (11 × 21) / 6 = 77 / 2

Explanation

A line parallel to the y-axis has a constant x-coordinate for all values of y, giving it an equation of the form x = k, where k is a constant.

Simplifying x + 3 = 4 - 2x: 3x = 1 x = ⅓

Since this simplifies to x = ⅓, it represents a vertical line parallel to the y-axis.

Explanation

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

9 = (5 + 7 + 9 + α + 11 + 12) / 6 9 = (44 + α) / 6 54 = 44 + α α = 54 - 44 = 10

Explanation

To find the Least Common Multiple (LCM) of 14, 21, 25, and 70, write each number in its prime factorized form: 14 = 2 × 7 21 = 3 × 7 25 = 5² 70 = 2 × 5 × 7

Taking the highest power of every prime factor involved: LCM = 2¹ × 3¹ × 5² × 7¹ = 2 × 3 × 25 × 7 = 1050

Explanation

Following the BODMAS rule (Brackets, Orders, Division/Multiplication, Addition/Subtraction):

Evaluate the expression inside the bracket: (225 ÷ 25 - 10) = 9 - 10 = -1

Perform multiplication: (-1) × 5 + 2 - 7 × 3 = -5 + 2 - 21

Perform addition and subtraction from left to right: -5 + 2 - 21 = -3 - 21 = -24

Explanation

Given the exponential equations: a⁽2x) = b b⁽2y) = c c⁽2z) = a

Substituting the expression for b into the second equation: (a⁽2x))⁽2y) = c a⁽4xy) = c

Substituting this expression for c into the third equation: (a⁽4xy))⁽2z) = a a⁽8xyz) = a¹

Equating the exponents on both sides: 8xyz = 1 xyz = 1/8

Explanation

Simplify each square root and fraction in the given expression:

√196 = 14 √441 = 21 √225 = 15

Substitute these values into the expression:

(14 ÷ 7) × (21 ÷ 7) × (120 ÷ 15) = 2 × 3 × 8 = 48