RRB NTPC CBT 1 2020 Mathematics Questions with Solutions

Explanation

Area of parallelogram ABCD = base × height = BC × h, where h is the perpendicular distance between the parallel lines AD and BE.

Since BC is extended to E such that BC = CE, the base of triangle ABE is BE = BC + CE = 2 × BC.

The height of triangle ABE with respect to base BE is the same perpendicular distance h from vertex A to the line BE.

Area of triangle ABE = ½ × base × height = ½ × BE × h = ½ × (2 × BC) × h = BC × h, which is equal to the area of parallelogram ABCD.

Explanation

Cost Price = (Difference in Selling Price / Difference in Gain Percentage) × 100

Difference in Gain Percentage = 14% - 12% = 2%

Cost Price = (33 / 2) × 100 = 16.5 × 100 = ₹1650.00

Explanation

Isolating √x gives: √x = √3 - √2

Squaring both sides: x = (√3 - √2)² x = (√3)² - 2(√3)(√2) + (√2)² x = 3 - 2√6 + 2 x = 5 - 2√6

Explanation

Average AQI for Kolkata = (306 + 288 + 274 + 172) ÷ 4 = 1040 ÷ 4 = 260

Average AQI for Delhi = (248 + 246 + 246 + 236) ÷ 4 = 976 ÷ 4 = 244

Difference between the two averages = 260 - 244 = 16

Explanation

Since 34% of the total votes equals 6800, the total number of votes cast is calculated as: Total votes = 6800 ÷ 0.34 = 20000

Winning margin percentage = 48% - 34% = 14% Winning margin = 14% of 20000 = 2800 votes

Percentage of invalid votes = 100% - (48% + 34%) = 18% Number of invalid votes = 18% of 20000 = 3600 votes

Explanation

Efficiency is inversely proportional to the time taken to complete a given work: Efficiency × Time = Total Work.

If A's efficiency is taken as 1, B's efficiency is 1 - 0.20 = 0.80 = ⅘.

Time taken by B = (Time taken by A × Efficiency of A) / Efficiency of B = (12 × 1) / 0.80 = 15 days.

Explanation

Total students passing the exam in each year: 2016: 128734 + 115526 = 244260 2017: 130567 + 124313 = 254880 2018: 129209 + 122131 = 251340

Total passing students across all 3 years = 244260 + 254880 + 251340 = 750480

Average passing students per year = 750480 / 3 = 250160

Average pass percentage = (Average passing students / Total appearing students) × 100 Average pass percentage = (250160 / 354000) × 100 = 70.67% ≈ 71%

Explanation

Let the number of rows in the auditorium be x. Since the number of students in each row equals the total number of rows, the total number of students is given by x × x = x².

Given that x² = 6889, taking the square root on both sides: x = √6889

Since 80² = 6400 and 83² = (80 + 3)² = 6400 + 2(80)(3) + 3² = 6400 + 480 + 9 = 6889, the number of rows in the auditorium is 83.

Explanation

Let the total number of votes polled be 100%. Since Rana got 40% of the total votes, Ranvir got 100% - 40% = 60% of the votes.

The victory margin in percentage is 60% - 40% = 20%. Given that this 20% difference equals 60,000 votes: 1% of total votes = 60,000 ÷ 20 = 3,000

The number of votes polled in favour of Ranvir = 60% = 60 × 3,000 = 1,80,000.

Explanation

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

Solve inside the brackets: 63 - 39 = 24

Perform division and multiplication: 25 ÷ 5 = 5 7 × 24 = 168

Perform addition and subtraction from left to right: 186 - 168 + 5 = 18 + 5 = 23

Explanation

Using the empirical relationship between mean, median, and mode:

Mode = 3 × Median - 2 × Mean

Let the mean be 4x and the median be 5x. Substituting these values into the formula gives:

Mode = 3 × (5x) - 2 × (4x) = 15x - 8x = 7x

Therefore, the ratio of mean to mode is Mean : Mode = 4x : 7x = 4:7.

Explanation

The numbers between 100 and 200 that are divisible by 12 form an arithmetic progression.

First term (a) = 108 Last term (l) = 192 Common difference (d) = 12

Number of terms (n) = (l - a)/d + 1 = (192 - 108)/12 + 1 = 8

Sum of an arithmetic progression = n/2 × (a + l) Sum = 8/2 × (108 + 192) = 4 × 300 = 1200

Explanation

Given √4624 = 68.

Evaluating each term by adjusting for decimal places under the square root: √46.24 = √(4624 / 100) = 68 / 10 = 6.8 √0.4624 = √(4624 / 10000) = 68 / 100 = 0.68 √0.004624 = √(4624 / 1000000) = 68 / 1000 = 0.068

Adding the terms together: 6.8 + 0.68 + 0.068 = 7.548

Explanation

Total fare collected for each mode of transport is proportional to the product of individual fare and the number of passengers.

Ratio of total fare collected (Train : Bus : Air) = (5 × 3) : (8 × 4) : (12 × 5) = 15 : 32 : 60

Sum of ratio parts = 15 + 32 + 60 = 107 parts

Value of 1 part = ₹1,07,000 ÷ 107 = ₹1,000

Total fare collected from air passengers = 60 parts = 60 × ₹1,000 = ₹60,000

Explanation

Using the identity cos θ cos 2θ cos 2²θ ... cos(2ⁿ⁻¹θ) = sin(2ⁿθ) / (2ⁿ sin θ) for θ = 20° and n = 3:

cos 20° cos 40° cos 80° = sin(2³ × 20°) / (2³ sin 20°) = sin 160° / (8 sin 20°)

Since sin 160° = sin(180° - 20°) = sin 20°:

= sin 20° / (8 sin 20°) = 1/8

Explanation

Factor out 3² from each term in the given series:

3² + 6² + 9² + ... + 42² = (3 × 1)² + (3 × 2)² + (3 × 3)² + ... + (3 × 14)² = 3² × (1² + 2² + 3² + ... + 14²) = 9 × (1² + 2² + 3² + ... + 14²)

Substituting the given value of 1015 for the sum of squares:

= 9 × 1015 = 9135

Explanation

Let the total work be 30 units (LCM of 5, 10, 15, and 30).

Work rates per day: (A + B + C) = 30 / 5 = 6 units/day (B + C + D) = 30 / 10 = 3 units/day (C + D + A) = 30 / 15 = 2 units/day (D + A + B) = 30 / 30 = 1 unit/day

Adding all four equations: 3 × (A + B + C + D) = 6 + 3 + 2 + 1 = 12 units/day A + B + C + D = 12 / 3 = 4 units/day

Time taken by A, B, C, and D working together = 30 / 4 = 7 ½ days.

Explanation

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

Perform Division: 7.2 ÷ 8 = 0.9

Perform Multiplication: 3 × 2.3 = 6.9

Substitute these values back into the expression: 24 + 0.9 - 6.9 + 5

Perform Addition and Subtraction from left to right: 24.9 - 6.9 + 5 = 18 + 5 = 23

Explanation

Let the cliff height be h = 25 m, and let x be the horizontal distance between the cliff and the tower.

From the top of the cliff, the angle of elevation θ to the top of the tower gives: tan θ = (H - 25) / x

The angle of depression θ to the foot of the cliff/tower gives: tan θ = 25 / x

Equating the two expressions for tan θ: (H - 25) / x = 25 / x H - 25 = 25 H = 50 m

Explanation

To find A : D, multiply the consecutive ratios together:

A/D = (A/B) × (B/C) × (C/D)

Substituting the given values:

A/D = (15/16) × (8/25) × (20/9)

Canceling common factors: A/D = (3 × 1 × 2) / (2 × 1 × 3) = 2/3

Explanation

Using the algebraic identity x³ + y³ = (x + y)³ - 3xy(x + y): 35 = 5³ - 3xy(5) 35 = 125 - 15xy 15xy = 90 xy = 6

Since x + y = 5 and xy = 6, the numbers are x = 3 and y = 2.

Calculating x⁴ + y⁴: 3⁴ + 2⁴ = 81 + 16 = 97

Alternatively, using standard algebraic relations: x² + y² = (x + y)² - 2xy = 5² - 2(6) = 13 x⁴ + y⁴ = (x² + y²)² - 2(xy)² = 13² - 2(6)² = 169 - 72 = 97

Explanation

Correct Mean = Incorrect Mean + (Sum of Correct Values - Sum of Incorrect Values) / Total Items

Correction in sum = (38 + 46) - (83 + 76) = 84 - 159 = -75

Correction in mean = -75 / 150 = -0.5

Correct Mean = 45 - 0.5 = 44.5

Explanation

Let the side length of square ABCD be s.

Since X is the mid-point of AB, length BX = s/2. Given BY = s/3 and ∠B = 90°, the area of the right-angled triangle BXY is: Area = ½ × BX × BY = ½ × (s/2) × (s/3) = s²/12

Given that the area of ΔBXY is 300 cm²: s²/12 = 300 s² = 3600 s = 60 cm

The diagonal of a square with side s is given by AC = s√2. AC = 60√2 cm.

Explanation

Area of the room = Total cost ÷ Rate per m² Area = 3675 ÷ 12.50 = 294 m²

Let the breadth and length be 2x and 3x respectively. Area = Length × Breadth 3x × 2x = 294 6x² = 294 x² = 49 x = 7

Length = 3x = 3 × 7 = 21 m

Explanation

The greatest 3-digit number divisible by 24 is obtained by dividing 999 by 24, which gives 41 with a remainder. Thus, the greatest 3-digit multiple of 24 is 24 × 41 = 984.

The smallest 4-digit number divisible by 24 is the very next multiple of 24, which is 24 × 42 = 1008.

Since 41 and 42 are consecutive integers, they are coprime, ensuring that the highest common factor of 984 and 1008 is 24.

Sum = 984 + 1008 = 1992.

Explanation

Total work = LCM(40, 60, 80) = 240 units.

Daily work rate (efficiency): Efficiency of A = 240 / 40 = 6 units/day Efficiency of B = 240 / 60 = 4 units/day Efficiency of C = 240 / 80 = 3 units/day

To find the total duration, add the missing work of workers who left early to the total work: Work added for A leaving 11 days early = 6 × 11 = 66 units Work added for B leaving 8 days early = 4 × 8 = 32 units Adjusted total work = 240 + 66 + 32 = 338 units

Total time taken = Adjusted total work / Combined efficiency Total time taken = 338 / (6 + 4 + 3) = 338 / 13 = 26 days.

Explanation

Following the order of operations (BODMAS), solve each component of the expression separately:

First term: 62 ÷ 5 = 62/5

Second term: ((8/9) × (18/7)) × (7/5) = (16/7) × (7/5) = 16/5

Third term: (5/4) × (16/25) = 4/5

Combining all three terms over the common denominator 5: 62/5 - 16/5 + 4/5 = (62 - 16 + 4) / 5 = 50 / 5 = 10

Explanation

Using the algebraic identity (a + b)³ = a³ + b³ + 3ab(a + b):

Set a = 2x and b = 5y: (2x + 5y)³ = 8x³ + 125y³ + 30xy(2x + 5y)

Substitute the given values 2x + 5y = 12 and xy = 2 into the equation: 12³ = 8x³ + 125y³ + 30 × 2 × 12 1728 = 8x³ + 125y³ + 720 8x³ + 125y³ = 1728 - 720 = 1008

Explanation

Volume of water = Cross-sectional area × Length of water flow in given time

Cross-sectional area = Depth × Width = 2 m × 21 m = 42 m²

Speed of river flow = 5 km/h = 5000 m / 60 min = 250/3 m/min

Length of water flowing in 10 min = Speed × Time = (250/3) × 10 = 2500/3 m

Volume of water = 42 × (2500/3) = 14 × 2500 = 35,000 m³

Since 1 m³ = 1000 litres = 1 kilolitre (kl), the volume of water falling into the sea in 10 min is 35,000 kl.

Explanation

For any two similar geometric figures, such as equilateral triangles, the ratio of their areas is equal to the square of the ratio of their corresponding side lengths.

Area(PQR) / Area(ABC) = (Side(PQR) / Side(ABC))²

Given that Area(PQR) = 16 × Area(ABC) and Side(ABC) = 28 cm:

16 = (Side(PQR) / 28)²

Taking the square root on both sides:

4 = Side(PQR) / 28

Side(PQR) = 4 × 28 = 112 cm

Explanation

Simplify each term by factoring out perfect squares under the radical:

√150 = √(25 × 6) = 5√6 √54 = √(9 × 6) = 3√6 √24 = √(4 × 6) = 2√6

Substitute these values into the given expression: √150 - √54 - √24 = 5√6 - 3√6 - 2√6 = (5 - 3 - 2)√6 = 0 × √6 = 0

Explanation

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

Given: 8 × CP = 5 × SP SP / CP = 8 / 5

Gain % = [(SP - CP) / CP] × 100 = [(8 - 5) / 5] × 100 = (3 / 5) × 100 = 60%

Explanation

Distance covered by the first train in 1 hour = 80 km/h × 1 h = 80 km.

Relative speed of the second train relative to the first train = 100 km/h - 80 km/h = 20 km/h.

Time taken by the second train to overtake the first train = Distance / Relative Speed = 80 / 20 = 4 hours.

Explanation

Let L and H be the LCM and HCF of the two numbers respectively.

Given: L + H = 369 L - H = 351

Adding and subtracting the two equations: L = (369 + 351) / 2 = 360 H = (369 - 351) / 2 = 9

Using the formula: Product of two numbers = LCM × HCF 72 × Second Number = 360 × 9 Second Number = (360 × 9) / 72 = 45

Explanation

By the Principle of Inclusion-Exclusion, the percentage of candidates who passed in at least one subject is: P(E ∪ M) = P(E) + P(M) - P(E ∩ M) P(E ∪ M) = 55% + 35% - 20% = 70%

nearby The percentage of candidates who failed in both subjects is: 100% - 70% = 30%

Since 30% of the total candidates equals 1200, the total number of candidates who took the exam is: (1200 / 30) × 100 = 4000

Explanation

In compound interest, the amount at the end of any year acts as the principal for the next year. Therefore, the interest earned in the 3rd year is the interest accrued on the 2nd year's amount over one year.

Interest earned in the 3rd year = ₹13,310 - ₹12,100 = ₹1,210

Rate of interest r = (Interest for the year / Principal for that year) × 100 r = (1,210 / 12,100) × 100 = 10%

Explanation

Overall interest rate per annum = (Total Interest × 100) / (Principal × Time) Overall interest rate = (1,750 × 100) / (5,000 × 5) = 7%

Using the rule of alligation: Ratio of principal lent at 5% to principal lent at 10% = (10 - 7) : (7 - 5) = 3 : 2

Amount lent at 10% = [2 / (3 + 2)] × 5,000 = (2 / 5) × 5,000 = ₹2,000

Explanation

Total cost price of all five goods = 173 + 219 + 350 + 481 + 227 = 1450

Total selling price by Shopkeeper X = 180 + 220 + 360 + 490 + 250 = 1500 Total profit for Shopkeeper X = 1500 - 1450 = 50 Percentage profit for Shopkeeper X = (50 / 1450) × 100 = 3.45%

Total selling price by Shopkeeper Y = 175 + 225 + 357 + 500 + 235 = 1492 Total profit for Shopkeeper Y = 1492 - 1450 = 42

Since 50 > 42, Shopkeeper X made more profit in total, with a percentage profit of 3.45%.

Explanation

Since 3 men do the work of 6 boys, 1 man is equivalent to 2 boys.

Total work = 6 boys × 20 days = 120 boy-days.

Converting the target group of 6 men and 8 boys into boys: 6 men + 8 boys = (6 × 2) boys + 8 boys = 20 boys.

Number of days required = Total work / Combined capacity = 120 / 20 = 6 days.

Explanation

In an isosceles triangle, the altitude from the vertex opposite to the base bisects the base into two equal segments of 8 ÷ 2 = 4 cm each.

By the Pythagorean theorem in the right-angled triangle formed by the altitude, half the base, and an equal side:

Height = √((Equal side)² - (Half base)²) = √(5² - 4²) = √(25 - 16) = √9 = 3 cm

Explanation

Let x = 0.232323...

Multiplying both sides by 100 gives: 100x = 23.232323...

Subtracting x from 100x: 100x - x = 23.232323... - 0.232323... 99x = 23 x = 23/99

Using the standard shortcut for a purely recurring decimal, the numerator is the repeating digits (23) and the denominator contains as many 9s as there are repeating digits (2 digits, hence 99), giving 23/99.

Explanation

Target SP = Given SP × (100 + Gain%) / (100 - Loss%)

Substituting the given values: Target SP = 138 × (100 + 4) / (100 - 8) = 138 × 104 / 92 = 138 × 26 / 23 = 6 × 26 = ₹156

Explanation

When two fair six-sided dice are thrown simultaneously, the total number of possible outcomes is 6 × 6 = 36. The sum of the numbers appearing on the dice is 12 only when both dice show a 6, which corresponds to a single favorable outcome, (6, 6). Therefore, the probability that the sum is 12 is 1 / 36.

Explanation

The expression is a product of cosine values for integer angles from 1° to 179°. One of the terms in this sequence is cos 90°. Since cos 90° = 0, multiplying any sequence of real numbers by zero results in a product of 0.

Explanation

Percentage reduction = [(Initial value - Final value) / Initial value] × 100

Given: Initial visitors on day 1 = 84,500 Visitors remaining on day 4 = 16,900

Reduction in visitors = 84,500 - 16,900 = 67,600

Percentage reduction = (67,600 / 84,500) × 100 = (4 / 5) × 100 = 80%

Explanation

When two objects move in opposite directions, their relative speed is the sum of their individual speeds: Relative speed = 30 km/h + 3.5 km/h = 33.5 km/h

Distance = Relative speed × Time Distance = 33.5 km/h × 2.5 h = 83.75 km

Explanation

The sum of the given numbers is 1 + 3 + 7 + 9 + 11 + 13 = 44.

The smallest perfect square greater than 44 is 7² = 49.

The number that must be added is 49 - 44 = 5.

Explanation

Applying the BODMAS rule to evaluate the expression step-by-step:

First part: (4 + 2 - 16 ÷ 4 + 3) = (4 + 2 - 4 + 3) = 5

Second part: [(1 + 8 × 7) ÷ 19] = (1 + 56) ÷ 19 = 57 ÷ 19 = 3

Third part: [(3 + 5 - 4) + (17 - 9 × 4)] = 4 + (17 - 36) = 4 + (-19) = -15

Combining the evaluated parts: 5 + 3 × (-15) = 5 - 45 = -40

Explanation

Using the trigonometric identity cos(180° - θ) = -cos θ:

cos 168° = cos(180° - 12°) = -cos 12° cos 96° = cos(180° - 84°) = -cos 84°

Substituting these values into the given expression: cos 12° + cos 84° + cos 168° + cos 96° = cos 12° + cos 84° - cos 12° - cos 84° = 0

Explanation

The required number increased by 3 must be equal to the Least Common Multiple (LCM) of 27, 35, 25, and 21.

Prime factorization of the numbers: 27 = 3³ 35 = 5 × 7 25 = 5² 21 = 3 × 7

LCM(27, 35, 25, 21) = 3³ × 5² × 7 = 27 × 25 × 7 = 4725

Required smallest number = LCM - 3 = 4725 - 3 = 4722