RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

The difference between each divisor and its corresponding remainder is constant: 2 - 1 = 3 - 2 = 4 - 3 = 5 - 4 = 6 - 5 = 1.

Thus, the required number is of the form k × LCM(2, 3, 4, 5, 6) - 1 = 60k - 1.

Testing integer values of k to find a number divisible by 7: For k = 1: 60(1) - 1 = 59 (not divisible by 7) For k = 2: 60(2) - 1 = 119 (119 ÷ 7 = 17)

Explanation

Using the standard algebraic identity for the cube of a binomial: (x + y)³ = x³ + y³ + 3xy(x + y)

Substituting the given value x + y = 1 into the identity: 1³ = x³ + y³ + 3xy(1) 1 = x³ + y³ + 3xy

Alternatively, using value substitution by assuming x = 1 and y = 0: 1³ + 0³ + 3(1)(0) = 1

Explanation

To find the greatest four-digit number divisible by 5, 35, 39, and 65, first find their Least Common Multiple (LCM).

Prime factorizations: 5 = 5 35 = 5 × 7 39 = 3 × 13 65 = 5 × 13

LCM(5, 35, 39, 65) = 3 × 5 × 7 × 13 = 1365

The greatest four-digit number is 9999. Dividing 9999 by 1365 gives a quotient of 7 and a remainder of 444.

Required number = 9999 − 444 = 7 × 1365 = 9555

Explanation

To find the number by which 62557 must be divided to make it a perfect cube, factorize 62557 into its prime factors:

62557 = 47 × 1331 62557 = 47 × 11³

In this prime factorization, 11³ is already a perfect cube. Dividing 62557 by the single factor of 47 eliminates it:

62557 ÷ 47 = 1331 = 11³

This leaves 1331, which is 11³.

Explanation

Since the number of silver and gold medals won must be whole numbers, the sum of the ratio terms must divide the total number of medals without leaving a remainder.

For the ratio 9:1, the sum of the ratio terms is 9 + 1 = 10. Dividing the total 75 medals by 10 gives 75 / 10 = 7.5, which is not a whole number. Since it is impossible to win a fractional number of medals, a ratio of 9:1 cannot represent the given total.

Explanation

Let the three angles of the triangle be A, B, and C, arranged in ascending order such that A ≤ B ≤ C = 75°.

By the angle sum property of a triangle: A + B + 75° = 180° A + B = 105° A = 105° - B

To minimize the smallest angle A, the middle angle B must be maximized. Since C = 75° is the largest angle, B can be at most 75°.

Substituting B = 75° into the equation gives: A = 105° - 75° = 30°

Explanation

Let x be the original number of men in the group.

Using the work equivalence relation M₁ × D₁ = M₂ × D₂: x × 10 = (x - 5) × 12 10x = 12x - 60 2x = 60 x = 30

Explanation

The area leveled by a road roller in one complete revolution equals its curved surface area.

Curved Surface Area (CSA) = 2 × π × r × h

Given, radius (r) = 7 m, length (h) = 14 m, and number of revolutions = 100.

CSA = 2 × (22/7) × 7 × 14 = 616 m²

Total area of the field = CSA × Number of revolutions Total area = 616 × 100 = 61600 m²

Explanation

Percentage change in sales between 2014 and 2018 is given by the formula:

Percentage Change = [(Sales in 2018 - Sales in 2014) / Sales in 2014] × 100

For Fossil: Percentage Change = [(126 - 96) / 96] × 100 = (30 / 96) × 100 = 31.25%

An increase of 31.25% represents the highest percentage change among all the watch brands over this five-year period.

Explanation

Percentage Increase = [(Current Year Sales - Preceding Year Sales) / Preceding Year Sales] × 100

For Fastrack in 2017: Sales in 2016 = 175 Sales in 2017 = 312 Increase in sales = 312 - 175 = 137 Percentage increase = (137 / 175) × 100 = 78.29%

This gives the maximum percentage increase over the preceding year among all the evaluated options.

Explanation

The decrease in sales from 2017 to 2018 for each brand is calculated using: Sales decrease = Sales in 2017 - Sales in 2018

For Timex, the sales decreased from 401 thousand in 2017 to 255 thousand in 2018, giving a drop of 401 - 255 = 146 thousand.

The calculated sales decreases for the four brands are 124 thousand, 110 thousand, 146 thousand, and 20 thousand. Timex registered the highest absolute decrease of 146 thousand. In percentage terms as well, Timex experienced the highest drop of (146 / 401) × 100 = 36.4%.

Explanation

Evaluating the year-on-year sales performance from 2014 to 2017 for each brand shows that Timex increased every year (247 → 285 → 388 → 401) and Fossil increased every year (96 → 108 → 112 → 146). Meanwhile, Titan suffered a decline in 2016 (398 to 312) and Fastrack suffered declines in 2015 and 2016 (205 to 188 to 175), meaning only two brands registered a year-on-year increase throughout the entire 2014–2017 period.

Explanation

To convert 5.37 hours into hours, minutes, and seconds, keep the whole number 5 as hours.

Multiply the fractional part by 60 to convert to minutes: 0.37 × 60 = 22.2 minutes

The whole number 22 represents minutes.

Multiply the remaining fractional minute by 60 to convert to seconds: 0.2 × 60 = 12 seconds

This gives 5 hours, 22 minutes, and 12 seconds.

Explanation

Total initial age of the 11 players = 11 × 18 = 198 years.

After 4 players join, the total number of players becomes 15 and the new average age increases to 18 + 2 = 20 years.

Total age of all 15 players = 15 × 20 = 300 years.

Total age of the 4 new players = 300 - 198 = 102 years.

Average age of the new members = 102 ÷ 4 = 25.5 years.

Explanation

Using the trigonometric identity cos² A + sin² A = 1:

cos² A = 1 - sin² A cos² A = 1 - (a/b)² = 1 - (a²/b²) = (b² - a²)/b²

Taking the square root on both sides gives:

cos A = √(b² - a²)/b

Explanation

When the cost price of x articles equals the selling price of y articles, Profit % = [(x - y) / y] × 100.

Given x = 10 and y = 9: Profit % = [(10 - 9) / 9] × 100 = (1 / 9) × 100 = 100 / 9 % = 11 1/9 %

Explanation

The circumference of a circle is given by the formula C = 2πr. Substituting the given value of 30π cm:

2πr = 30π r = 15 cm

The area of a circle is given by the formula A = πr². Substituting r = 15 cm:

A = π × 15² = 225π cm²

Explanation

Each term in the numerator is 10 times the corresponding term in the denominator, as (10x)² = 100x². Factoring out 100 from the numerator gives:

Numerator = 100 × [(0.001)² + (0.022)² + (0.0333)² + (0.0444)² + (0.05555)²]

Dividing the numerator by the denominator cancels out the terms in the bracket, yielding 100.

Explanation

The smallest number after 100 divisible by 7 is 105 (7 × 15), and the largest number before 300 divisible by 7 is 294 (7 × 42).

Using the formula for the number of terms in an arithmetic progression, n = (l - a)/d + 1, where l is the last term, a is the first term, and d is the common difference:

n = (294 - 105)/7 + 1 = 189/7 + 1 = 27 + 1 = 28

Explanation

Let Hema's salary be S.

Initially, Hema decided to donate 15% of her salary, which equals 0.15 × S.

On the day of donation, she donated ₹4,500, which was 125% of her initially planned donation.

1.25 × 0.15 × S = 4,500

Converting percentages to fractions: (5/4) × (3/20) × S = 4,500 (15/80) × S = 4,500 (3/16) × S = 4,500

Solving for S: S = (4,500 × 16) / 3 S = 1,500 × 16 S = 24,000

Explanation

In a right-angled triangle formed by the kite's height, ground distance, and string length:

tan α = Perpendicular / Base = 15/8

Using the Pythagorean theorem: Hypotenuse = √(15² + 8²) = √(225 + 64) = √289 = 17 units

Given that the string length (Hypotenuse) is 102 m: 17 units = 102 m 1 unit = 6 m

Height of the kite (Perpendicular) = 15 units = 15 × 6 = 90 m.

Explanation

To find the required fraction, sum the fractional parts of the given expression:

Fractional sum = 2/3 + 7/12 + 9/36 + 1/12

Simplifying 9/36 to 1/4 and taking 12 as the common denominator:

2/3 = 8/12 7/12 = 7/12 9/36 = 1/4 = 3/12 1/12 = 1/12

Sum of fractions = (8 + 7 + 3 + 1) / 12 = 19/12 = 1 + 7/12

Adding the whole numbers (3 + 6 + 4 + 5 + 7) gives 25, so the total sum is 25 + 1 + 7/12 = 26 + 7/12.

To make the total sum a whole number (27), the fraction that must be added is 1 - 7/12 = 5/12.

Explanation

Expanding the three-digit numbers into place-value form gives: (500 + 10P + 9) + (300 + 10R + 7) + (200 + 10Q + 8) = 1114

Summing the constants: 1024 + 10(P + Q + R) = 1114 10(P + Q + R) = 90 P + Q + R = 9

To maximize Q, the single-digit variables P and R must take their minimum possible value of 0. Substituting P = 0 and R = 0 yields Q = 9.

Explanation

According to the BODMAS rule, operations inside brackets and under a vinculum are simplified first.

Evaluating the term under the vinculum gives 5 + 3 = 8.

Solving inside the innermost parenthesis yields 6 − 8 = −2.

Simplifying inside the next set of brackets gives 9 − (−2) = 9 + 2 = 11.

Performing the multiplication within the outer square brackets gives 8 × 11 = 88.

Subtracting from 50 yields 50 − 88 = −38.

Explanation

Let the original sum be S.

Using the formula for successive percentage deductions: Remaining Amount = S × (1 - d₁/100) × (1 - d₂/100)

Substituting the given values: 3600 = S × (1 - 10/100) × (1 - 20/100) 3600 = S × 0.90 × 0.80 3600 = S × 0.72 S = 3600 / 0.72 = 5000

Explanation

Using the algebraic identity: x³ + y³ + z³ - 3xyz = (x + y + z)[(x + y + z)² - 3(xy + yz + zx)]

Substitute the given values x + y + z = 16 and xy + yz + zx = 78: x³ + y³ + z³ - 3xyz = 16 × (16² - 3 × 78) = 16 × (256 - 234) = 16 × 22 = 352

Explanation

Given √(x) + 1/√(x) = 3.

Squaring both sides: (√(x) + 1/√(x))² = 3² x + 1/x + 2 = 9 x + 1/x = 7

Squaring both sides again: (x + 1/x)² = 7² x² + 1/x² + 2 = 49 x² + 1/x² = 47

Explanation

Let x be the required number.

According to the question, the difference between multiplying the number by 6/5 and 5/6 is equal to 572:

(6/5)x - (5/6)x = 572

Taking the common denominator 30:

(36/30)x - (25/30)x = 572 (11/30)x = 572 x = (572 × 30) / 11 x = 52 × 30 x = 1560

Explanation

Let the speed of the train be T km/h.

According to the given speed ratios: Speed of horse = T / 3 km/h Speed of steamer = T / 1.5 = 2T / 3 km/h

Using the relation Time = Distance / Speed, the total travel time of 13 h 30 min (13.5 hours) is the sum of the individual travel times:

(120 / (2T/3)) + (450 / T) + (60 / (T/3)) = 13.5 (180 / T) + (450 / T) + (180 / T) = 13.5 810 / T = 13.5 T = 810 / 13.5 = 60 km/h

Explanation

In compound interest, the amount grows by a constant multiplication factor over equal intervals of time.

Growth factor for 4 years = Amount after 4 years / Principal = 6,000 / 4,800 = 5/4

Since 12 years consists of 3 consecutive 4-year periods, the amount multiplies by (5/4) thrice:

Amount after 12 years = 4,800 × (5/4)³ = 4,800 × (125 / 64) = 75 × 125 = 9,375

Explanation

The total volume remains conserved when the three metal cubes are melted into a single cube:

Volume of new cube = s₁³ + s₂³ + s₃³ Volume = 3³ + 4³ + 5³ = 27 + 64 + 125 = 216 cm³

Side of the new cube = ∛216 = 6 cm.

Explanation

To find the median, first arrange the given data points in ascending order: 2, 14, 16, 18, 20, 20, 28, 30, 32, 48

Since the total number of observations is n = 10 (an even number), the median is calculated using the formula: Median = ½ × [(n/2)-th term + (n/2 + 1)-th term]

The 5th term is 20 and the 6th term is 20.

Median = (20 + 20) / 2 = 20

Explanation

The altitude (h) of an equilateral triangle with side length 'a' is given by the formula:

h = (√3 / 2) × a

Given side length a = 10 cm, substituting the value into the formula gives:

h = (√3 / 2) × 10 = 5√3 cm

Alternatively, dropping a perpendicular from a vertex to the opposite side bisects the base into two equal segments of 5 cm each. By applying the Pythagorean theorem on one of the resulting right-angled triangles:

h = √(10² - 5²) = √(100 - 25) = √75 = 5√3 cm

Explanation

The horizontal distance between the poles is 24 m, and the difference in their heights is 38 - 20 = 18 m.

Connecting the top of the shorter pole horizontally to the taller pole forms a right-angled triangle where: Base = 24 m Perpendicular = 18 m Hypotenuse = Distance between their tops

Using the Pythagoras theorem: Distance² = Base² + Perpendicular² Distance² = 24² + 18² Distance² = 576 + 324 = 900 Distance = √900 = 30 m

Explanation

When two runners cover the same distance D at speeds v₁ and v₂, the distance is given by the formula:

D = (v₁ × v₂ / |v₁ - v₂|) × Δt

Given v₁ = 15 km/h, v₂ = 16 km/h, and the time difference Δt = 16 minutes = 16/60 hours.

Substituting these values into the formula:

D = (15 × 16 / |15 - 16|) × (16 / 60) D = 240 × (16 / 60) D = 4 × 16 = 64 km

Alternatively, since speed and time are inversely proportional for a constant distance, the speed ratio of 15 : 16 gives a time ratio of 16 : 15. The difference of 1 unit in the time ratio represents 16 minutes, so 15 units equal 15 × 16 = 240 minutes = 4 hours. Multiplying the speed of 16 km/h by 4 hours yields 64 km.

Explanation

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

First Number × Second Number = HCF × LCM

Substituting the given values: 16 × Second Number = 2 × 144 Second Number = (2 × 144) / 16 = 18

Explanation

The interest earned in 1.5 years (from 2 years to 3.5 years) is ₹873 - ₹756 = ₹117.

Simple interest for 1 year = 117 / 1.5 = ₹78

Simple interest for 2 years = 78 × 2 = ₹156

Sum (Principal) = Amount after 2 years - Interest for 2 years Principal = 756 - 156 = ₹600

Explanation

The common difference between each divisor and its corresponding remainder is 1 (since 2 - 1 = 3 - 2 = 4 - 3 = 5 - 4 = 6 - 5 = 1).

Thus, the required number is of the form N = LCM(2, 3, 4, 5, 6) × k - 1.

LCM(2, 3, 4, 5, 6) = 60, giving N = 60k - 1.

To make N a multiple of 7, 60k - 1 must be divisible by 7: 60k - 1 = (56k + 4k - 1)

Since 56k is divisible by 7, (4k - 1) must be divisible by 7. For k = 1: 4(1) - 1 = 3 (not divisible by 7) For k = 2: 4(2) - 1 = 7 (divisible by 7)

Substituting k = 2: N = 60 × 2 - 1 = 119

Explanation

Using the algebraic identity (x + y)³ = x³ + y³ + 3xy(x + y):

Substitute x + y = 1 into the identity: (1)³ = x³ + y³ + 3xy(1) 1 = x³ + y³ + 3xy

Alternatively, choosing simple values such as x = 1 and y = 0 satisfying x + y = 1: 1³ + 0³ + 3(1)(0) = 1

Explanation

Prime factorization of the given numbers: 5 = 5 35 = 5 × 7 39 = 3 × 13 65 = 5 × 13

LCM(5, 35, 39, 65) = 3 × 5 × 7 × 13 = 1365

To find the greatest four-digit number divisible by these numbers, divide the largest four-digit number (9999) by the LCM (1365): 9999 ÷ 1365 = 7 with a remainder of 444

Subtracting the remainder from 9999: 9999 - 444 = 9555

Explanation

To determine the required divisor, find the prime factorization of 62557:

62557 = 47 × 11 × 11 × 11 = 47 × 11³

In the prime factorization, 11 forms a complete triplet (11³), whereas 47 appears only once. Dividing 62557 by 47 eliminates the extra factor, leaving 1331 (11³), which is a perfect cube.

Explanation

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

According to the given condition: (1/5) × (40/100) × x = (2/3) × (30/100) × y

Simplifying both sides: (2/25) x = (1/5) y

Rearranging to find the ratio x : y: x / y = (1/5) ÷ (2/25) x / y = (1/5) × (25/2) x / y = 5/2

Thus, the ratio of the first number to the second number is 5:2.

Explanation

Since medals are discrete items that cannot be fractional, the total number of medals (75) must be divisible by the sum of the ratio terms. For the ratio 9:1, the sum of the terms is 9 + 1 = 10. Since 75 ÷ 10 = 7.5 is not a whole number, it is impossible to divide 75 medals into this ratio.

Explanation

Let the angles of the triangle be A, B, and C such that A ≤ B ≤ C = 75°.

The sum of angles in a triangle is 180°: A + B + 75° = 180° A + B = 105° A = 105° - B

To minimize the smallest angle A, the intermediate angle B must be maximized. Since 75° is the largest angle in the triangle, B can be at most 75°.

Substituting B = 75°: A = 105° - 75° = 30°

Explanation

Let x be the original number of men in the group.

Using Total Work = Number of Men × Number of Days: 10 × x = 12 × (x - 5) 10x = 12x - 60 2x = 60 x = 30

Explanation

Curved surface area of the cylindrical road roller = 2 × π × r × h

Substituting r = 7 m, h = 14 m, and π = 22/7: Area covered in 1 revolution = 2 × (22/7) × 7 × 14 = 616 m²

Total area of the field leveled in 100 revolutions = 616 × 100 = 61600 m²

Explanation

The percentage change in sales between 2014 and 2018 is calculated using the formula:

Percentage Change = (|Sales in 2018 - Sales in 2014| / Sales in 2014) × 100

For Fossil, sales increased from 96 thousand in 2014 to 126 thousand in 2018:

Percentage Change = ((126 - 96) / 96) × 100 = (30 / 96) × 100 = 31.25%

This increase of 31.25% represents the highest percentage change in sales between 2014 and 2018 across all the given brands.

Explanation

Percentage increase = [(Sales in given year - Sales in preceding year) / Sales in preceding year] × 100%

For Fastrack in 2017: Sales in 2016 = 175 Sales in 2017 = 312 Percentage increase = [(312 - 175) / 175] × 100% = (137 / 175) × 100% = 78.29%

This represents the maximum percentage increase over the preceding year among all sales figures in the given data.

Explanation

To find the brand with the highest decrease in sales in 2018 compared to 2017, we calculate the reduction in sales for Timex:

Sales of Timex in 2017 = 401 thousand Sales of Timex in 2018 = 255 thousand Decrease in sales = 401 - 255 = 146 thousand

Percentage decrease = (146 / 401) × 100 = 36.41%

Timex registered the maximum decrease in sales between 2017 and 2018, both in absolute terms (146 thousand) and in percentage terms (36.41%).

Explanation

A continuous year-on-year increase during 2014–2017 requires sales to rise every year from 2014 through 2017.

Timex sales grew continuously each year: 247 in 2014, 285 in 2015, 388 in 2016, and 401 in 2017.

Fossil sales also grew continuously each year: 96 in 2014, 108 in 2015, 112 in 2016, and 146 in 2017.

Titan sales dropped in 2016 (from 398 to 312), and Fastrack sales dropped in 2015 (from 205 to 188). Consequently, exactly two brands—Timex and Fossil—showed a year-on-year increase in sales during this period.