RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

For any sequence of consecutive odd (or even) numbers, adding k additional consecutive numbers to the sequence increases the average of the set by k. Since 2 additional consecutive odd numbers are added, the new average increases by 2 and becomes p + 2.

Algebraically, let the five consecutive odd numbers be p - 4, p - 2, p, p + 2, and p + 4, with average p. The next two consecutive odd numbers are p + 6 and p + 8. The sum of all seven numbers is 5p + (p + 6) + (p + 8) = 7p + 14. The new average is (7p + 14) / 7 = p + 2.

Explanation

Area of four walls of a room is given by the formula A = 2(L + B) × H.

Let the initial length L = 4x, breadth B = 3x, and height H = 2x. Initial wall area A₁ = 2(4x + 3x) × 2x = 28x².

After changing dimensions: New length L' = 2x New breadth B' = 6x New height H' = x

New wall area A₂ = 2(2x + 6x) × x = 16x².

Percentage decrease = [(A₁ - A₂) / A₁] × 100 = [(28x² - 16x²) / 28x²] × 100 = (12 / 28) × 100 = (3 / 7) × 100 ≈ 42.86%.

Explanation

By the division algorithm: Dividend = (Divisor × Quotient) + Remainder

Substituting the given values into the formula: 12401 = (Divisor × 76) + 13 12401 - 13 = Divisor × 76 12388 = Divisor × 76 Divisor = 12388 / 76 = 163

Explanation

The centers of three mutually touching circles of radius r = 3.5 cm form an equilateral triangle with side length s = 2r = 7 cm.

Area of the equilateral triangle = (√3/4) × s² = (1.732/4) × 7² = 21.217 cm²

The three sectors inside the triangle each subtend an angle of 60°. Total area of the 3 sectors = 3 × (60°/360°) × π × r² = ½ × (22/7) × (3.5)² = 19.25 cm²

Area of the enclosed portion = Area of triangle - Total area of 3 sectors = 21.217 - 19.25 = 1.967 cm²

Explanation

Area to be plastered = Area of 4 walls + Area of bottom = 2h(l + b) + lb = 2 × 6 × (25 + 12) + (25 × 12) = 12 × 37 + 300 = 444 + 300 = 744 m²

Rate of plastering = 75 paise = ₹0.75 per m²

Total cost = 744 × 0.75 = ₹558

Explanation

Initial quantity of milk = (27 ÷ 34) × 85 = 67.5 L Initial quantity of water = (7 ÷ 34) × 85 = 17.5 L

Since only water is added, the amount of milk remains unchanged at 67.5 L in the new mixture.

In the new milk-to-water ratio of 3 : 1, 3 parts represent 67.5 L. 1 part of water in the new mixture = 67.5 ÷ 3 = 22.5 L

Water to be added = 22.5 - 17.5 = 5 L

Explanation

The Institute of Forest Biodiversity (IFB) is located at Dulapally in Hyderabad, Telangana. Operating as a constituent research institute under the Indian Council of Forestry Research and Education (ICFRE) of the Ministry of Environment, Forest and Climate Change, it was established by upgrading the former Forest Research Centre in December 2012. The institute focuses on scientific research regarding forest biodiversity conservation, tree improvement, and eco-restoration, with particular emphasis on the Eastern Ghats and coastal ecosystems.

Explanation

Let the total work be 60 units (LCM of 12 and 10).

Efficiency of A = 60 / 12 = 5 units/day Efficiency of B = 60 / 10 = 6 units/day

In 1 cycle of 2 alternate days (A on day 1, B on day 2), work completed = 5 + 6 = 11 units. In 5 cycles (10 days), work completed = 5 × 11 = 55 units. Remaining work = 60 - 55 = 5 units.

On the 11th day, A works and completes the remaining 5 units in 5 / 5 = 1 day. Total time taken = 10 + 1 = 11 days.

Explanation

Cost Price of the mixture = Selling Price ÷ (1 + Profit %) Cost Price of the mixture = 8 ÷ (1 + 0.375) = 8 ÷ (11/8) = ₹ 64/11 per litre

Since water costs ₹ 0, the total cost of the mixture is due to milk alone. Letting M be the fraction of milk in the mixture: M × 6.40 = 64/11 M × (32/5) = 64/11 M = 10/11

Fraction of water = 1 - 10/11 = 1/11

Ratio of water to milk = (1/11) : (10/11) = 1 : 10

Explanation

The League of Nations is termed a "Child of War" because its creation was a direct outcome of the catastrophic devastation caused by World War I. Proposed as part of US President Woodrow Wilson's Fourteen Points and formally established in 1919 under the Treaty of Versailles, the organization was born out of an urgent global commitment to prevent future conflicts through collective security, diplomacy, and international cooperation.

Explanation

The Indian Statutory Commission, popularly known as the Simon Commission, was appointed by the British government in 1927 under Sir John Simon to review the functioning of the constitutional system in India and suggest further reforms. The commission was boycotted by almost all major Indian political parties, including the Indian National Congress and the Muslim League, because all seven of its members were white British parliamentarians, completely excluding Indians from deciding their own nation's constitutional future.

Explanation

Since cot θ = 1/tan θ, we have cot θ = 2 / (√3 + 1).

Substituting cot θ into the given expression: 2√3 cot θ + 1 = 2√3 × [2 / (√3 + 1)] + 1 = [4√3 / (√3 + 1)] + 1 = [4√3 + (√3 + 1)] / (√3 + 1) = (5√3 + 1) / (√3 + 1)

Explanation

Let W be the time taken to walk one way and R be the time taken to ride one way.

Given that W + R = 5 h 45 min.

Riding both ways saves 2 h, so 2R = 5 h 45 min - 2 h = 3 h 45 min.

Multiplying the first equation by 2 gives: 2W + 2R = 2 × (5 h 45 min) = 11 h 30 min

Substituting the value of 2R into this equation: 2W + 3 h 45 min = 11 h 30 min 2W = 11 h 30 min - 3 h 45 min = 7 h 45 min

Thus, the time taken to walk both ways is 7 h 45 min.

Explanation

India's first electric train was flagged off on 3 February 1925 between Bombay VT (now Chhatrapati Shivaji Maharaj Terminus) and Kurla along the Harbour Line. Operated by the Great Indian Peninsula Railway (GIPR), it ran on a 1500 V DC overhead electric traction system, marking the beginning of rail electrification in India.

Explanation

ISRO successfully launched the radar imaging earth observation satellite RISAT-2B onboard the PSLV-C46 mission on May 22, 2019, from the Satish Dhawan Space Centre in Sriharikota. Flown in its core-alone configuration without solid strap-on motors, PSLV-C46 placed the 615 kg satellite into a low Earth orbit at an altitude of 555 km with an inclination of 37 degrees. Equipped with an X-band synthetic aperture radar and a 3.6-meter radial rib antenna, the RISAT series provides all-weather, day-and-night high-resolution imaging for agriculture, forestry, and disaster management applications.

Explanation

By the principle of inclusion-exclusion, the percentage of students who passed in at least one subject is: P(E ∪ M) = P(E) + P(M) - P(E ∩ M) = 80% + 85% - 75% = 90%

The percentage of students who failed in both subjects is 100% - 90% = 10%.

Given that 10% of the total students equals 40, the total number of students is (40 ÷ 10) × 100 = 400.

Explanation

Let h be the height of the tower and x be the distance between point P and the base of the tower.

In the right-angled triangle formed at the second position, 20 m closer to the tower: tan 45° = h / (x - 20) 1 = h / (x - 20) h = x - 20

In the right-angled triangle formed at point P: tan 30° = h / x 1 / √3 = (x - 20) / x x = √3(x - 20) x = x√3 - 20√3 x(√3 - 1) = 20√3

Solving for x: x = 20√3 / (√3 - 1) x = [20√3(√3 + 1)] / [(√3 - 1)(√3 + 1)] x = [20(3 + √3)] / 2 x = 10(3 + √3) x = 30 + 10√3

Substituting √3 = 1.732: x = 30 + 10(1.732) x = 30 + 17.32 = 47.32 m

Explanation

The dataset contains n = 10 observations, which is an even number. For an even number of observations, the median is the average of the (n/2)th observation and the (n/2 + 1)th observation, which are the 5th and 6th terms.

5th term = x 6th term = x + 2

Median = (x + x + 2) / 2 = x + 1

Equating the median to 17: x + 1 = 17 x = 16

Explanation

In the right-angled triangle formed by the wall, the ground, and the ladder, the distance from the foot of the ladder to the wall represents the base, and the ladder represents the hypotenuse.

cos(θ) = Base / Hypotenuse

Given θ = 60° and Base = 15 m:

cos(60°) = 15 / Length of ladder

½ = 15 / Length of ladder

Length of ladder = 15 × 2 = 30 m

Explanation

The difference between each divisor and its corresponding remainder is constant: 20 - 15 = 5, 25 - 20 = 5, 35 - 30 = 5, and 40 - 35 = 5.

The required least number is given by LCM(20, 25, 35, 40) - 5.

Prime factorization of the divisors: 20 = 2² × 5 25 = 5² 35 = 5 × 7 40 = 2³ × 5

LCM(20, 25, 35, 40) = 2³ × 5² × 7 = 1400

Required number = 1400 - 5 = 1395

Explanation

To find the HCF of decimal numbers 10.8, 28.8, and 36, convert them into integers by multiplying each number by 10:

10.8 × 10 = 108 28.8 × 10 = 288 36 × 10 = 360

Now, find the prime factorizations of these integers: 108 = 2² × 3³ 288 = 2⁵ × 3² 360 = 2³ × 3² × 5

The product of the lowest powers of common prime factors gives: HCF(108, 288, 360) = 2² × 3² = 4 × 9 = 36

Dividing this result by 10 to account for the initial scaling: HCF(10.8, 28.8, 36) = 36 / 10 = 3.6

Explanation

Standard deviation (σ) is the positive square root of variance:

σ = √(Variance)

Substituting the given value:

σ = √0.64 = 0.8

Explanation

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

For the numerator: 54.542² - 45.458² = (54.542 + 45.458) × (54.542 - 45.458) = 100 × 9.084

For the denominator: 61.738 - 52.654 = 9.084

Substituting these values into the given expression: Value = (100 × 9.084) / 9.084 = 100

Explanation

When the profit earned at one selling price equals the loss incurred at another selling price, the Cost Price (CP) is the midpoint of the two selling prices:

CP = (832 + 448) ÷ 2 = 1280 ÷ 2 = ₹640

To earn a 50% profit, the required Selling Price is:

Selling Price = CP × 1.5 = 640 × 1.5 = ₹960

Explanation

Under compound interest, if a sum becomes k times in t years, it becomes kⁿ times in n × t years.

Here, the money doubles (2¹ times) in 15 years. To become 8 times, we express 8 as a power of 2: 8 = 2³

Thus, n = 3.

Time required = 3 × 15 = 45 years.

Explanation

Cost Price (CP) = Selling Price (SP) × 100 / (100 + Gain%)

Substituting SP = ₹40.60 and Gain% = 16: CP = 40.60 × 100 / (100 + 16) CP = 4060 / 116 = ₹35

Explanation

Let Arun's initial amount be A and Sekhar's initial amount be S.

From the first condition, when Arun gives ₹30 to Sekhar: S + 30 = 2(A - 30) 2A - S = 90 --- (1)

From the second condition, when Sekhar gives ₹10 to Arun: A + 10 = 3(S - 10) A = 3S - 40 --- (2)

Substituting (2) into (1): 2(3S - 40) - S = 90 6S - 80 - S = 90 5S = 170 S = 34

Substituting S = 34 into (2) gives A = 3(34) - 40 = 62.

Explanation

Let B be Alfred's birth year. His age in 1944 was 1944 - B.

According to the question: 1944 - B = B / 80 81B = 80 × 1944 B = 155520 / 81 = 1920

Alfred was born in 1920.

His age in 1952 = 1952 - 1920 = 32 years.

Explanation

The numbers between 17 and 520 that are divisible by 6 form an arithmetic progression where: First term (a) = 18 Last term (l) = 516 Common difference (d) = 6

Number of terms (n) = ((l - a) / d) + 1 = ((516 - 18) / 6) + 1 = 83 + 1 = 84

Sum of the arithmetic progression (S) = (n / 2) × (a + l) S = (84 / 2) × (18 + 516) = 42 × 534 = 22428

Explanation

To express √(8/5) in terms of √40, multiply the numerator and denominator inside the square root by 5:

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

Substituting √40 ≈ 6.325 into the equation:

6.325 / 5 = 1.265

Explanation

The relationship between wave speed, frequency, and wavelength is given by the formula v = f × λ, where v is wave speed, f is frequency, and λ is wavelength.

Rearranging the formula to solve for wavelength gives λ = v / f.

Substituting the given values: λ = 36 / 50 = 0.72 m.

Explanation

Numerator of the main fraction: 3¼ - (4/5 × 5/6) = 13/4 - 2/3 = (39 - 8)/12 = 31/12

Denominator of the main fraction: 4⅓ ÷ 1/5 - (3/10 + 21⅕) = 13/3 × 5 - (3/10 + 106/5) = 65/3 - 215/10 = 65/3 - 43/2 = (130 - 129)/6 = 1/6

Main fraction value: (31/12) ÷ (1/6) = (31/12) × 6 = 31/2

Second expression to subtract: 1⅔ × 1½ = 5/3 × 3/2 = 5/2

Subtracting the second expression from the main fraction: 31/2 - 5/2 = 26/2 = 13

Explanation

Using the algebraic identity: (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

Substituting the given values: 10² = a² + b² + c² + 2(31) 100 = a² + b² + c² + 62 a² + b² + c² = 100 - 62 = 38

Explanation

When the cost price of x items equals the selling price of y items, the percentage gain is calculated using the formula:

Gain % = [(x - y) / y] × 100

Substituting x = 15 and y = 12:

Gain % = [(15 - 12) / 12] × 100 = (3 / 12) × 100 = 25%

Explanation

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

Given the ratio of angles as 2 : 3 : 4, the total ratio parts equal 2 + 3 + 4 = 9 parts.

The difference between the greatest angle (4 parts) and the smallest angle (2 parts) is 4 - 2 = 2 parts.

Difference = (2 ÷ 9) × 180° = 40°

Explanation

Area of road = Outer area - Inner area

Length of the inner square = 18 m Area of the inner square = 18² = 324 m²

Since the road of width 3 m is constructed outside around the park, the side length increases by 3 m on both opposite sides: Length of the outer square = 18 + (2 × 3) = 24 m Area of the outer square = 24² = 576 m²

Area of the road = 576 - 324 = 252 m²

Explanation

Total distance to be covered = Length of train + Length of platform = 450 m + 150 m = 600 m.

Speed of the train = 80 km/h = 80 × (5/18) m/s = 200/9 m/s.

Time taken = Distance ÷ Speed = 600 ÷ (200/9) = (600 × 9) ÷ 200 = 27 seconds.

Explanation

Net rate of filling per hour = 1/4 - 1/6 = 1/12 of the tank.

Time required to fill the complete tank = 1 / (1/12) = 12 hours.

Time required to make the tank half-full = 12 × ½ = 6 hours.

Explanation

Using the value substitution method, let a = 1, b = 1, and c = -2 so that a + b + c = 0.

Numerator: (a² + b² + c²)² = (1² + 1² + (-2)²)² = (1 + 1 + 4)² = 6² = 36

Denominator: a²b² + b²c² + c²a² = (1² × 1²) + (1² × (-2)²) + ((-2)² × 1²) = 1 + 4 + 4 = 9

Evaluating the expression: 36 / 9 = 4

Alternatively, squaring both sides of (a + b + c)² = 0 gives a² + b² + c² = -2(ab + bc + ca). Squaring again gives (a² + b² + c²)² = 4(ab + bc + ca)² = 4(a²b² + b²c² + c²a² + 2abc(a + b + c)). Since a + b + c = 0, (a² + b² + c²)² = 4(a²b² + b²c² + c²a²), which yields 4.

Explanation

When 3 identical cubes of side a are joined side by side in a line, 2 internal joints are formed, covering 4 cube faces in total. The total surface area of the resulting cuboid is therefore the surface area of 3 individual cubes minus the area of the 4 hidden faces:

Surface Area = 3 × 6a² - 4a² = 14a²

Given side a = 16 cm: Surface Area = 14 × 16² = 14 × 256 = 3584 cm²

Explanation

To find the greatest length that can measure the given dimensions exactly, we compute the Highest Common Factor (HCF) of the three lengths expressed in centimeters.

Converting all lengths to centimeters: 7 m = 700 cm 3 m 85 cm = 385 cm 12 m 95 cm = 1295 cm

Prime factorizing each number: 700 = 2² × 5² × 7 385 = 5 × 7 × 11 1295 = 5 × 7 × 37

The common prime factors are 5 and 7. HCF(700, 385, 1295) = 5 × 7 = 35 cm

Explanation

By the divisibility rule of 11, a number is divisible by 11 if the difference between the sum of its digits at odd positions and the sum of its digits at even positions is 0 or a multiple of 11.

For the seven-digit number 67843A2: Sum of digits at odd positions = 6 + 8 + 3 + 2 = 19 Sum of digits at even positions = 7 + 4 + A = 11 + A

Difference = 19 - (11 + A) = 8 - A

Since A is a single-digit whole number, 8 - A must be equal to 0 for the difference to be divisible by 11. 8 - A = 0 A = 8

Explanation

The smallest multiple of 7 greater than 1000 is 1001 (143 × 7) and the largest multiple of 7 less than 3000 is 2996 (428 × 7).

Using the arithmetic progression formula l = a + (n - 1)d, where first term a = 1001, last term l = 2996, and common difference d = 7:

2996 = 1001 + (n - 1) × 7 1995 = (n - 1) × 7 n - 1 = 285 n = 286

Alternatively, using floor division, the count is ⌊2999/7⌋ - ⌊1000/7⌋ = 428 - 142 = 286.

Explanation

Using the complementary angle identity tan(90° - θ) = cot θ and the product identity tan θ × cot θ = 1, the given expression can be paired from the outer terms inward:

(tan 1° × tan 89°) × (tan 2° × tan 88°) × ... × (tan 44° × tan 46°) × tan 45°

Since tan(90° - θ) = cot θ, tan 89° = cot 1°, tan 88° = cot 2°, and so on up to tan 46° = cot 44°.

Each pair simplifies to tan θ × cot θ = 1. The middle term tan 45° is also equal to 1.

Thus, the entire product evaluates to 1 × 1 × ... × 1 = 1.

Explanation

The notation DDD represents a three-digit number where each digit is D, which can be written as 100D + 10D + D = 111D.

For DDD to be divisible by 8, the term 111D must be divisible by 8. Since 111 and 8 are coprime (gcd(111, 8) = 1), 8 must divide the single digit D.

The only non-zero single-digit integer that is a multiple of 8 is 8. Substituting D = 8 yields the three-digit number 888, which is divisible by 8 (888 ÷ 8 = 111).

Explanation

Rationalize the denominator by multiplying both the numerator and the denominator by (2√2 + √7):

(2√2 + √7) / (2√2 - √7) = (2√2 + √7)² / ((2√2)² - (√7)²)

Expanding the numerator using the algebraic identity (a + b)² = a² + b² + 2ab: (2√2 + √7)² = (2√2)² + (√7)² + 2(2√2)(√7) = 8 + 7 + 4√14 = 15 + 4√14

Simplifying the denominator using (a - b)(a + b) = a² - b²: (2√2)² - (√7)² = 8 - 7 = 1

Thus, the expression simplifies to 15 + 4√14.

Comparing 15 + 4√14 with x + y√14 yields x = 15 and y = 4.

Explanation

Using the algebraic identity a² - b² = (a - b)(a + b), simplify the term in the brackets: 75² - 55² = (75 - 55) × (75 + 55) = 20 × 130 = 2600

Substitute this back into the original expression: (4/11) × (121/16) × 24 × 2600 × (1/100)

Simplify step-by-step: (4/11) × (121/16) = 11/4 (11/4) × 24 = 66 2600 × (1/100) = 26

Multiply the remaining terms: 66 × 26 = 1716

Explanation

Rearrange the expression by pairing each decimal in the numerator with its corresponding term in the denominator:

Expression = 9 × (0.9 / 0.3) × (0.09 / 0.03) × (0.009 / 0.003)

Simplifying each term in the brackets: 0.9 / 0.3 = 3 0.09 / 0.03 = 3 0.009 / 0.003 = 3

Substituting these values back gives: Expression = 9 × 3 × 3 × 3 = 9 × 27 = 243

Explanation

The height of the kite from the ground is given by the formula: Total Height = L × sin(θ) + h

Where L is the length of the thread (296 m), θ is the angle of elevation (30°), and h is the height of the person's hand from the ground (2 m).

Substituting the given values: Total Height = 296 × sin(30°) + 2 Total Height = 296 × ½ + 2 Total Height = 148 + 2 = 150 m

Explanation

To find the least number to be added to 3500, first calculate the Least Common Multiple (LCM) of 42, 49, 56, and 63:

42 = 2 × 3 × 7 49 = 7² 56 = 2³ × 7 63 = 3² × 7

LCM(42, 49, 56, 63) = 2³ × 3² × 7² = 8 × 9 × 49 = 3528

The smallest multiple of 3528 greater than 3500 is 3528 itself.

Number to be added = 3528 - 3500 = 28