RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

The radius drawn to the point of contact of a tangent is perpendicular to the tangent, forming a right-angled triangle where the distance from the center to point P is the hypotenuse.

By Pythagoras theorem: r² + (length of tangent)² = (distance from center)² r² + 12² = 13² r² + 144 = 169 r² = 25 r = 5 cm

Explanation

To find a number between 3 and 5 in radical form, convert the boundaries into square roots: 3 = √9 and 5 = √25

Any real number x satisfying 9 < x < 25 will have its square root √x lying strictly between 3 and 5. Since 17 satisfies 9 < 17 < 25 and is not a perfect square, √17 is an irrational number between 3 and 5.

Explanation

Let N be the total number of enrolled voters.

Voters who cast their votes = 75% of N = (3/4) × N

Valid votes = 98% of votes cast = (49/50) × (3/4) × N

Candidate's votes = 75% of valid votes = (3/4) × (49/50) × (3/4) × N = 9261

N × (441 / 800) = 9261

N = 9261 × (800 / 441)

N = 21 × 800 = 16800

Explanation

Percentage increase = [(Candidates in 2016 - Candidates in 2012) / Candidates in 2012] × 100

Calculating the percentage increase for the states: Bihar = (500 - 410) / 410 × 100 = 90 / 410 × 100 ≈ 21.95% Punjab = (350 - 280) / 280 × 100 = 70 / 280 × 100 = 25% UP = (430 - 300) / 300 × 100 = 130 / 300 × 100 ≈ 43.33% MP = (540 - 250) / 250 × 100 = 290 / 250 × 100 = 116%

The lowest percentage increase is recorded by Bihar at 21.95%.

Explanation

Given sin θ / cos θ = 5/12, which means tan θ = 5/12.

In a right-angled triangle, tan θ is the ratio of the Opposite side to the Adjacent side. Therefore, Opposite = 5 and Adjacent = 12.

Using the Pythagorean theorem: Hypotenuse = √(Opposite² + Adjacent²) = √(5² + 12²) = √(25 + 144) = √169 = 13

The sine of the angle is given by sin θ = Opposite / Hypotenuse = 5/13.

Explanation

Given the ratio x : y = 3 : 5, substitute x = 3 and y = 5 into the expression:

Numerator = 5x - 3y = 5(3) - 3(5) = 15 - 15 = 0 Denominator = 10x + 6y = 10(3) + 6(5) = 30 + 30 = 60

Evaluating the fraction yields 0 / 60 = 0.

Explanation

The x-coordinate of the centroid of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) is given by the formula:

x-centroid = (x₁ + x₂ + x₃) / 3

Substituting the x-coordinates of the given points (3, 5), (-7, 4), and (10, -6):

k = (3 + (-7) + 10) / 3 = 6 / 3 = 2

Explanation

To find the greatest number that leaves a remainder of 3 when dividing 39 and 84, subtract the remainder 3 from both numbers and calculate their Highest Common Factor (HCF):

39 - 3 = 36 84 - 3 = 81

HCF(36, 81): 36 = 9 × 4 81 = 9 × 9

HCF(36, 81) = 9

Explanation

The total wet surface area of a cistern filled with water comprises the base area and the area of the four vertical walls up to the height of the water level.

Formula: Total wet surface area = (l × b) + 2 × (l + b) × h

Given parameters: Length (l) = 8 m Width (b) = 4 m Water depth (h) = 1 m 22 cm = 1.22 m

Base area = 8 × 4 = 32 m² Area of four walls = 2 × (8 + 4) × 1.22 = 24 × 1.22 = 29.28 m²

Total wet surface area = 32 + 29.28 = 61.28 m²

Explanation

Time taken is inversely proportional to efficiency (Time = Total Work ÷ Efficiency).

Let Kamal's efficiency be 100 units per day. Since Vimal is 40% more efficient, Vimal's efficiency = 140 units per day.

Total Work = Kamal's Efficiency × Kamal's Time = 100 × 14 = 1400 units.

Time taken by Vimal = Total Work ÷ Vimal's Efficiency = 1400 ÷ 140 = 10 days.

Explanation

Total number of persons = 4 men + 3 women + 2 boys = 9.

The total number of ways to select 3 persons out of 9 is: ⁹C₃ = (9 × 8 × 7) / (3 × 2 × 1) = 84

To have 2 men selected, the group of 3 must consist of exactly 2 men and 1 non-man (selected from the 5 women and boys).

Ways to select 2 men from 4 men = ⁴C₂ = (4 × 3) / (2 × 1) = 6 Ways to select 1 person from 5 non-men = ⁵C₁ = 5

Number of favorable outcomes = ⁴C₂ × ⁵C₁ = 6 × 5 = 30

Probability = Favorable outcomes / Total outcomes = 30 / 84 = 5 / 14

Explanation

Amount = P × (1 + r₁/100) × (1 + r₂/100) × (1 + r₃/100)

Substituting P = ₹25,000, r₁ = 4%, r₂ = 5%, and r₃ = 6%: Amount = 25000 × (104/100) × (105/100) × (106/100) Amount = 25000 × (26/25) × (21/20) × (53/50) Amount = 26 × 21 × 53 = ₹28,938

Compound Interest = Amount - Principal Compound Interest = 28,938 - 25,000 = ₹3,938

Explanation

To combine the ratios, make the term b common in both: a : b = 3 : 2 = 9 : 6 b : c = 3 : 5 = 6 : 10

This gives a : b : c = 9 : 6 : 10.

Substituting these values into the required expression: (a + b) / (b + c) = (9 + 6) / (6 + 10) = 15 / 16

Explanation

The total number of balls in the urn is 10 + 5 = 15.

Using combinations, the probability of drawing 2 black balls out of 10 without replacement from a total of 15 balls is:

P(both black) = ¹⁰C₂ / ¹⁵C₂ = (10 × 9 / 2) / (15 × 14 / 2) = 45 / 105 = 3/7.

Alternatively, multiplying sequential probabilities yields (10/15) × (9/14) = (2/3) × (9/14) = 3/7.

Explanation

Given that 35% of A's income is equal to 40% of B's income:

35% × A = 40% × B

35 × A = 40 × B

A / B = 40 / 35

Simplifying the fraction by dividing both numerator and denominator by 5 gives A / B = 8 / 7, yielding the ratio A : B = 8 : 7.

Explanation

Total population increase over the decade = 2,22,600 - 1,85,500 = 37,100.

Percentage increase over 10 years = (37,100 / 1,85,500) × 100 = 20%.

Average percentage increase per year = 20% / 10 = 2%.

Explanation

Following the BODMAS rule, perform division before addition.

First, convert the mixed fractions into improper fractions: 5 1/10 = 51/10 3 2/5 = 17/5

Now, evaluate the division: (51/10) ÷ (17/5) = (51/10) × (5/17) = 3/2

Finally, add 56/5 to 3/2: 56/5 + 3/2 = (112 + 15)/10 = 127/10 = 12 7/10

Explanation

For any two numbers, the product of the two numbers is equal to the product of their HCF and LCM:

First number × Second number = HCF × LCM

Substituting the given values into the formula: 148 × Second number = 37 × 740 Second number = (37 × 740) / 148 = 740 / 4 = 185

Explanation

LCM of fractions = LCM of numerators / HCF of denominators

LCM of numerators (2, 4, 7, 3) = 84 HCF of denominators (3, 9, 12, 5) = 1

LCM of given fractions = 84 / 1 = 84

Explanation

Given a/b = 0.75 = 3/4, we can set a = 3 and b = 4.

Substituting these values into the first term: (2a + b) / (2a - b) = (2 × 3 + 4) / (2 × 3 - 4) = 10 / 2 = 5

Adding 4/9 to this result: 5 + 4/9 = (45 + 4) / 9 = 49/9

Explanation

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

25 × SP = 20 × CP SP / CP = 20 / 25 = 4 / 5

Loss % = [(CP - SP) / CP] × 100 Loss % = [(5 - 4) / 5] × 100 = (1 / 5) × 100 = 20%

Explanation

Evaluating the expression step-by-step from the innermost square root:

√196 = 14 √(50 + 14) = √64 = 8 √(217 + 8) = √225 = 15

Explanation

For a right-angled triangle with tan θ = 12/5, opposite = 12 and adjacent = 5. Hypotenuse = √(12² + 5²) = √(144 + 25) = √169 = 13

Therefore, sin θ = 12/13.

Substituting this into the given expression: (1 + sin θ) / (1 - sin θ) = (1 + 12/13) / (1 - 12/13) = (25/13) / (1/13) = 25

Explanation

When one value is r% more than another, the percentage by which the second value is less than the first is given by the formula: [r / (100 + r)] × 100%.

Substituting r = 25%: Percentage less = [25 / (100 + 25)] × 100% = (25 / 125) × 100% = 20%.

Alternatively, assuming Rahim earns ₹100, Sita earns ₹125. Rahim earns ₹25 less than Sita, so as a percentage of Sita's income, this is (25 / 125) × 100 = 20%.

Explanation

Using the order of operations (BODMAS):

3 + [3 × {3 - (3 + 3) ÷ 6}] = 3 + [3 × {3 - 6 ÷ 6}] = 3 + [3 × {3 - 1}] = 3 + [3 × 2] = 3 + 6 = 9

Explanation

By the section formula, a point (x, y) dividing the line segment joining (x₁, y₁) and (x₂, y₂) in the ratio m:n satisfies:

x = (m·x₂ + n·x₁) / (m + n)

Substituting x = 4, x₁ = 2, and x₂ = 7:

4 = (7m + 2n) / (m + n) 4m + 4n = 7m + 2n 3m = 2n m : n = 2 : 3

Explanation

The Reserve Bank of India (RBI) was established on 1st April 1935 in accordance with the provisions of the Reserve Bank of India Act, 1934. It was set up based on the recommendations of the Hilton Young Commission (Royal Commission on Indian Currency and Finance) of 1926 to respond to the post-World War I economic difficulties and to manage currency and credit in the country. Initially established as a shareholder's bank with a paid-up capital of ₹5 crore, it commenced operations in Kolkata before relocating its central office to Mumbai in 1937.

Explanation

A number is divisible by 4 if the number formed by its last two digits is divisible by 4. For the number 2564S4, the last two digits are S4.

Substituting 5 for S gives the two-digit number 54. Dividing 54 by 4 yields 13 with a remainder of 2, meaning 54 is not divisible by 4. Thus, 5 cannot be in the place of S.

Explanation

India's first marine national park was created in the Gulf of Kutch in 1982, after the area was designated as a marine sanctuary in 1980. Located along the southern shoreline in Gujarat, this protected area spans an archipelago of 42 islands surrounded by coral reefs, intertidal mudflats, and extensive mangrove ecosystems that harbor rich marine biodiversity including sea turtles, dolphins, and dugongs.

Explanation

Squaring p gives p² = (sinθ + cosθ)² = sin²θ + cos²θ + 2 sinθ cosθ = 1 + 2 sinθ cosθ. Rearranging gives p² - 1 = 2 sinθ cosθ.

Expressing q in terms of sinθ and cosθ gives q = secθ + cosecθ = 1/cosθ + 1/sinθ = (sinθ + cosθ) / (sinθ cosθ) = p / (sinθ cosθ).

Substituting these into the given expression yields q(p² - 1) = [p / (sinθ cosθ)] × [2 sinθ cosθ] = 2p.

Explanation

The perimeter of a rectangle is given by the formula P = 2(l + b), where l is the length and b is the breadth.

Given the ratio l : b = 6 : 5, let l = 6x and b = 5x.

Substituting these into the perimeter formula: 2(6x + 5x) = 110 22x = 110 x = 5

Therefore, the breadth b = 5x = 5 × 5 = 25 m.

Explanation

Total work = LCM(6, 4, 10) = 60 units.

Efficiency of A = 60 ÷ 6 = 10 units/day Efficiency of B = 60 ÷ 4 = 15 units/day Efficiency of C = 60 ÷ 10 = 6 units/day

Combined efficiency of A, B, and C = 10 + 15 + 6 = 31 units/day.

Time taken working together = Total work ÷ Combined efficiency = 60 ÷ 31 = 1 29/31 days.

Explanation

Stoppage time per hour = (Speed excluding stoppages - Speed including stoppages) / Speed excluding stoppages × 60 minutes

Substituting the given values: Stoppage time per hour = (80 - 52) / 80 × 60 = (28 / 80) × 60 = 21 minutes.

Explanation

Let the total number of valid votes be V. Since the winning candidate received 58% of the valid votes, the losing candidate received 100% − 58% = 42% of the valid votes.

The difference in percentage between the two candidates is: 58% − 42% = 16%

Given that the winning margin is 256 votes: 16% of V = 256 V = (256 × 100) / 16 = 1600

Total votes polled = Valid votes + Invalid votes Total votes polled = 1600 + 86 = 1686

Explanation

Using the property √(x / 10²ⁿ) = (√x) / 10ⁿ and the given value √225 = 15:

√2.25 = √(225 / 100) = 15 / 10 = 1.5 √0.0225 = √(225 / 10000) = 15 / 100 = 0.15 √0.000225 = √(225 / 1000000) = 15 / 1000 = 0.015

Sum = 1.5 + 0.15 + 0.015 = 1.665

Explanation

The formula for the total amount under simple interest is A = P(1 + rt/100), where P is the principal amount, r is the rate of interest per annum, and t is the time period in years.

Substituting A = 230, r = 5, and t = 3:

230 = P(1 + (5 × 3)/100) 230 = P(1 + 15/100) 230 = P(115/100) P = (230 × 100) / 115 = 200

Explanation

Given the equation 2160 / √x = 240:

√x = 2160 / 240 √x = 9

Squaring both sides gives: x = 9² = 81

Explanation

Percentage = (Number of Sciences Girls in Low category / Total student sample) × 100

From the table, the number of girl students in the Sciences group belonging to the Low category of addiction is 23.

Total student sample = 500

Percentage = (23 / 500) × 100 = 4.6%

Explanation

Total boys in the High Addiction group = 35 (Commerce) + 18 (Sciences) + 22 (Social Sciences) = 75

Total girls in the High Addiction group = 22 (Commerce) + 20 (Sciences) + 26 (Social Sciences) = 68

Total students in the High Addiction group = 75 + 68 = 143

Percentage of boys in the High Addiction group = (75 / 143) × 100 ≈ 52.45% ≈ 52%

Explanation

Percentage fluctuation = |Current Year - Previous Year| / Previous Year × 100

For 2005-06: Scooters fluctuation = (1316 - 904) / 904 × 100 = 45.58% Motorcycles fluctuation = (2019 - 1654) / 1654 × 100 = 22.07% Together fluctuation = (3335 - 2558) / 2558 × 100 = 30.38% None of the fluctuations in 2005-06 exceeded 50%.

For 2006-07: Scooters fluctuation = (2017 - 1316) / 1316 × 100 = 53.27% Motorcycles fluctuation = (2019 - 722) / 2019 × 100 = 64.24% Together fluctuation = (3335 - 2739) / 3335 × 100 = 17.87% Although scooters and motorcycles separately registered fluctuations of more than 50%, their combined registration fluctuation of 17.87% did not exceed 50%.

Neither year satisfies the condition for scooters, motorcycles, and total registrations all simultaneously registering a fluctuation of more than 50%.

Explanation

By the Law of Cosines, cos A = (b² + c² - a²) / (2bc).

Substituting a = 25, b = 45, and c = 30 into the formula: cos A = (45² + 30² - 25²) / (2 × 45 × 30) cos A = (2025 + 900 - 625) / 2700 cos A = 2300 / 2700 cos A = 23 / 27

Explanation

Calculating the cubes directly: (0.2)³ = 0.008 (0.02)³ = 0.000008

Subtracting the two values: 0.008 - 0.000008 = 0.007992

Explanation

Arithmetic Mean = Sum of observations / Total number of observations

Total number of observations (n) = 6, and the given arithmetic mean = 17.

Sum of 6 observations = Mean × n = 17 × 6 = 102

Sum of the known observations + p = 102 (16 + 18 + 15 + 17 + 14) + p = 102 80 + p = 102 p = 102 - 80 = 22

Explanation

The diagonal of a cube with edge length a is given by d = a√3.

Given d = 6√2 cm: a√3 = 6√2 a² = (6√2 / √3)² = 72 / 3 = 24 cm²

The total surface area of a cube is 6a²: Surface area = 6 × 24 = 144 cm²

Explanation

Using the compound interest formula: A = P(1 + r/100)ⁿ

Substituting the given values P = 1,500, A = 1,591.35, and n = 2: 1,591.35 = 1,500 × (1 + r/100)² (1 + r/100)² = 1,591.35 / 1,500 = 1.0609

Taking the square root on both sides: 1 + r/100 = √1.0609 1 + r/100 = 1.03 r/100 = 0.03 r = 3%

Explanation

Let total work be 1 unit.

Rama's 1-day work = 1/50 Work done by Rama in 10 days = 10 × (1/50) = 1/5 Remaining work = 1 - (1/5) = 4/5

Ravi completes 4/5 of the work in 32 days. Ravi's 1-day work = (4/5) ÷ 32 = 1/40

Combined 1-day work of Rama and Ravi = (1/50) + (1/40) = (4 + 5)/200 = 9/200

Time taken by both working together = 200/9 = 22 2/9 days.

Explanation

Time taken = 2 min 10 s = (2 × 60 + 10) s = 130 s.

Speed = Distance / Time = 650 / 130 = 5 m/s.

Converting speed from m/s to km/h by multiplying by 18/5: Speed = 5 × (18 / 5) = 18 km/h.

Explanation

The relationship between the height of an object, the length of its shadow, and the angle of elevation θ of the sun is given by the trigonometric ratio:

tan θ = height / shadow length

Substituting the given values into the formula: tan θ = (6√3) / 6 = √3

Since tan 60° = √3, the angle of elevation θ is 60°.

Explanation

Total number of balls in the box = 2 + 6 + 4 = 12. The number of ways to select 2 balls from 12 is 12C2 = (12 × 11) / (2 × 1) = 66. The number of ways to select 2 green balls from 6 green balls is 6C2 = (6 × 5) / (2 × 1) = 15.

Probability = Favorable outcomes / Total outcomes = 15 / 66 = 5 / 22.

Explanation

Using the successive percentage change formula, the amount paid by C is obtained by sequentially applying the profit margins:

Amount paid by C = 4500 × (105/100) × (120/100) = 4500 × 1.05 × 1.2 = 4500 × 1.26 = 5,670