RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

Let the total work be 36 units, so the combined rate of A and B is 1 unit per day.

In 30 days, A and B together complete 30 × 1 = 30 units of work, leaving 36 - 30 = 6 units remaining.

Since the total task takes 40 days, A works alone for the remaining 40 - 30 = 10 days to complete these 6 units.

Work rate of A = 6 / 10 = 3/5 units per day.

Time taken by A alone to complete the full 36 units of work = 36 / (3/5) = 36 × 5 / 3 = 60 days.

Explanation

Let the required number be x.

According to the given condition: x × (6/5) = 108/125

x = (108/125) × (5/6) x = (108 ÷ 6) / (125 ÷ 5) x = 18/25

Explanation

Let x be the side length of the square. The perimeter of a square is given by 4x and its area is given by x². Equating the perimeter and area gives the equation x² = 4x. Since the side length must be positive, dividing both sides by x yields x = 4 units.

Explanation

To find the greatest number dividing 58, 75, and 97 with remainders 4, 3, and 7 respectively, subtract the remainders from the corresponding numbers:

58 - 4 = 54 75 - 3 = 72 97 - 7 = 90

The required number is the Highest Common Factor (HCF) of 54, 72, and 90.

Prime factorizations: 54 = 2 × 3³ 72 = 2³ × 3² 90 = 2 × 3² × 5

HCF(54, 72, 90) = 2 × 3² = 18

Explanation

The ratio of Umesh's donation to Kapil's donation is 750 : 975.

Dividing both numbers by their highest common factor, 75: 750 ÷ 75 = 10 975 ÷ 75 = 13

The simplified ratio is 10:13.

Explanation

64.916 + 1.456 = 66.372

66.372 - 25.326 = 41.046

Explanation

To minimize the number of square slabs, the side length of each slab must be as large as possible, which equals the Highest Common Factor (HCF) of the room's dimensions.

HCF of 4.5 m and 1.5 m = 1.5 m

Minimum number of slabs = Area of the room ÷ Area of one square slab = (4.5 × 1.5) ÷ (1.5 × 1.5) = 3

Explanation

Let the efficiency of B be 1 unit/day. Since A is twice as good a workman as B, A's efficiency is 2 units/day.

Their combined efficiency is 2 + 1 = 3 units/day.

Total work = Combined efficiency × Days taken together Total work = 3 × 10 = 30 units.

Time taken by A to finish the work alone = Total work ÷ A's efficiency Time taken by A = 30 ÷ 2 = 15 days.

Explanation

Using the standard trigonometric identity: (sin θ + cos θ)² + (sin θ - cos θ)² = 2

Substitute the given value sin θ + cos θ = √5/2 into the identity: (√5/2)² + (sin θ - cos θ)² = 2 5/4 + (sin θ - cos θ)² = 2 (sin θ - cos θ)² = 2 - 5/4 (sin θ - cos θ)² = 3/4

Taking the positive square root gives: sin θ - cos θ = √(3/4) = √3/2

Explanation

The square root of 2000 lies between 44 and 45, as 44² = 1936 and 45² = 2025. The next smallest perfect square greater than 2000 is 45² = 2025. The smallest natural number that must be added to 2000 is 2025 - 2000 = 25.

Explanation

Let the required quantity to be added be x.

4/5 + x = 5/4 x = 5/4 - 4/5

Taking the LCM of the denominators 4 and 5, which is 20: x = (25 - 16) / 20 = 9/20

Explanation

Using the algebraic identity (a - b)³ = a³ - 3a²b + 3ab² - b³:

99³ = (100 - 1)³ = 100³ - 3 × 100² × 1 + 3 × 100 × 1² - 1³ = 1000000 - 30000 + 300 - 1 = 970299

Explanation

Expressing 0.000845 as a fraction gives 845 / 1,000,000. Since the denominator 1,000,000 is a perfect square (1000²), the fraction will be a perfect square if its numerator is a perfect square.

The largest perfect square less than 845 is 29² = 841.

Subtracting 841 from 845 yields 4.

Therefore, subtracting 4 / 1,000,000 = 0.000004 from 0.000845 gives 0.000841, which is (0.029)².

Explanation

Let total capacity of the tank be 70 units (LCM of 14 and 35).

Rate of each inlet pipe = 70 / 14 = +5 units/hr Rate of each drainpipe = 70 / 35 = -2 units/hr

When all 28 pipes are opened together, the tank fills in 2.5 hours, giving a net filling rate of: Net rate = 70 / 2.5 = 28 units/hr

Let the number of drainpipes be d. Then the number of inlet pipes is (28 - d).

5(28 - d) - 2d = 28 140 - 7d = 28 7d = 112 d = 16

Explanation

Let the cost price of the item for A be ₹x.

Using the successive transaction formula: x × (1 + 25/100) × (1 - 12/100) = 231 x × (125/100) × (88/100) = 231 x × (5/4) × (22/25) = 231 x × (11/10) = 231 x = (231 × 10) / 11 = 210

Explanation

Volume of sphere = ⁴/₃ π r³ Volume of cone = ⅓ π r² h

Given total volume = ⁷/₃ π r³

⁴/₃ π r³ + ⅓ π r² h = ⁷/₃ π r³

⅓ π r² h = ⁷/₃ π r³ - ⁴/₃ π r³ ⅓ π r² h = π r³

Canceling π r² from both sides: ⅓ h = r h = 3r

Therefore, h/r = 3.

Explanation

The prime factorizations of the given numbers are: 57 = 3 × 19 67 = 67 (a prime number) 77 = 7 × 11 87 = 3 × 29

Since 67 is a prime number and does not share any prime factor with the other numbers, there is no common prime factor among all four numbers. Therefore, the highest common factor (HCF) is 1.

Explanation

Mean of the observations x̄ = (6 + 7 + 10 + 12 + 13 + 8 + 14) / 7 = 70 / 7 = 10.

The variance formula is σ² = Σ(xᵢ - x̄)² / n.

Calculating the squared deviations from the mean: (6 - 10)² = 16 (7 - 10)² = 9 (10 - 10)² = 0 (12 - 10)² = 4 (13 - 10)² = 9 (8 - 10)² = 4 (14 - 10)² = 16

Sum of squared deviations = 16 + 9 + 0 + 4 + 9 + 4 + 16 = 58.

Variance σ² = 58 / 7 = 8.2857... ≈ 8.29.

Explanation

The sub-duplicate ratio of a ratio a : b is defined as √a : √b.

Substituting the given numbers into the formula: Sub-duplicate ratio = √625 : √64 = 25 : 8

Explanation

Rate of 1 paisa per rupee per month equals 1% per month, which is 12% per annum.

Using the simple interest formula SI = (P × R × T) / 100:

SI = (4000 × 12 × 5) / 100 = ₹2,400.

Explanation

HCF of fractions = HCF of numerators ÷ LCM of denominators

HCF of numerators (15, 10, 35) = 5 LCM of denominators (4, 3, 6) = 12

HCF of given fractions = 5/12

Explanation

The product of numbers with decimals can be found by counting the total number of decimal places in the factors.

In 2.31, there are 2 decimal places, and in 0.326, there are 3 decimal places. The total number of decimal places in the product will be 2 + 3 = 5 decimal places.

Since 231 × 326 = 75306, shifting the decimal point 5 places to the left gives: 2.31 × 0.326 = 75306 / 100000 = 0.75306

Explanation

Substitute x = 4 into the given linear equation 2x + 3y = 23:

2(4) + 3y = 23 8 + 3y = 23 3y = 23 - 8 3y = 15 y = 5

Explanation

The difference between a gain of 10% and a loss of 10% is equal to 10% + 10% = 20% of the cost price.

Given that this difference is ₹150: 20% of Cost Price = 150 Cost Price = (150 × 100) / 20 = 750

Explanation

Let the three positive numbers be x, 3x, and 4x.

Sum of their squares = x² + (3x)² + (4x)² = 26x²

Given that 26x² = 2106: x² = 2106 / 26 = 81 x = 9

The largest number is 4x = 4 × 9 = 36.

Explanation

In a right isosceles triangle with equal legs of length a, the hypotenuse is given by h = a√2.

Given h = 28√2 cm: a√2 = 28√2 a = 28 cm

The area of the right isosceles triangle is given by the formula: Area = ½ × a² Area = ½ × 28 × 28 = ½ × 784 = 392 cm²

Explanation

For a time period of 1½ years (18 months) with annual compounding, the amount is calculated using: Amount = Principal × (1 + R/100) × (1 + (R/2)/100)

Substituting Principal = 1400 and Rate = 20%: Amount = 1400 × (1 + 20/100) × (1 + 10/100) Amount = 1400 × 1.2 × 1.1 = 1848

Compound Interest = Amount - Principal = 1848 - 1400 = 448

Explanation

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

Given: 10 × CP = 9 × SP SP / CP = 10 / 9

Using the profit percentage formula: Profit % = [(SP - CP) / CP] × 100 = [(SP/CP) - 1] × 100

Substitute SP / CP = 10 / 9: Profit % = [(10 / 9) - 1] × 100 = (1 / 9) × 100 = 100 / 9 % = 11 1/9 % ≈ 11.11 %

Rounding to the nearest integer yields 11 %.

Explanation

Circumference of a circle C = 2πr 2πr = 30π r = 15 cm

Area of a circle A = πr² A = π × 15² = 225π cm²

Explanation

Each term in the denominator is obtained by dividing the corresponding term in the numerator by 10. That is, 0.001 = 0.01/10, 0.022 = 0.22/10, 0.0333 = 0.333/10, and so on.

Squaring each term in the denominator gives: (x/10)² = x²/100

Factoring out 1/100 from every term in the denominator gives: Denominator = (1/100) × [(0.01)² + (0.22)² + (0.333)² + (0.4444)² + (0.5555)²] = Numerator / 100

Substituting this back into the fraction gives: Numerator / (Numerator / 100) = 100

Explanation

The number of multiples of 7 up to a given number N is given by the floor division ⌊N / 7⌋.

The total multiples of 7 up to 299 are ⌊299 / 7⌋ = 42. The total multiples of 7 up to 100 are ⌊100 / 7⌋ = 14.

Subtracting these gives the number of multiples of 7 between 100 and 300: 42 - 14 = 28.

Alternatively, the first multiple of 7 after 100 is 105 (7 × 15) and the last multiple before 300 is 294 (7 × 42). Using the arithmetic progression formula for the number of terms, n = ((last - first) / difference) + 1 = ((294 - 105) / 7) + 1 = (189 / 7) + 1 = 28.

Explanation

Let Hema's salary be S. The initially planned donation was 15% of S, or (15/100) S.

Since the actual donation made was 125% of the planned donation: Actual donation = 125% × 15% of S = (5/4) × (3/20) S = (3/16) S

Given that the actual donation was ₹4,500: (3/16) S = 4,500 S = (4,500 × 16) / 3 = 1,500 × 16 = ₹24,000

Explanation

Using the right-angled triangle formed by the string, the height of the kite, and the level ground:

tan α = perpendicular / base = 15 / 8

By the Pythagorean theorem, the hypotenuse ratio is: Hypotenuse = √(15² + 8²) = √(225 + 64) = √289 = 17

Therefore: sin α = perpendicular / hypotenuse = 15 / 17

The height of the kite is: Height = String length × sin α Height = 102 × (15 / 17) Height = 6 × 15 = 90 m

Explanation

To find the required fraction, separate the whole numbers and the fractional parts of the given expression:

Sum of whole numbers = 3 + 6 + 4 + 5 + 7 = 25

Sum of fractional parts = 2/3 + 7/12 + 9/36 + 1/12

Converting all fractions to a common denominator of 12: 2/3 = 8/12 7/12 = 7/12 9/36 = 3/12 1/12 = 1/12

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

Total sum = 25 + 1 + 7/12 = 26 + 7/12

To make the total sum a whole number, the fractional part 7/12 must be completed to 1: Required fraction = 1 - 7/12 = 5/12

Explanation

Summing the units digits gives 9 + 7 + 8 = 24, which leaves 4 in the units digit and carries over 2 to the tens column.

Summing the hundreds digits together with the carryover from the tens column yields 5 + 3 + 2 + Carryover = 11. This implies the carryover from the tens column to the hundreds column must be 1.

Thus, the tens column equation is P + R + Q + 2 = 11, which simplifies to P + R + Q = 9.

To maximize Q, set the digits P and R to their minimum possible value of 0, yielding Q = 9.

Explanation

According to the VBODMAS rule, operations under the vinculum (bar) are performed first, followed by brackets, division, multiplication, addition, and subtraction.

Evaluate the expression under the vinculum: 5 + 3 = 8 Solve inside the parenthesis: 6 - 8 = -2 Solve inside the curly braces: 9 - (-2) = 9 + 2 = 11 Multiply within the square brackets: 8 × 11 = 88 Perform the final subtraction: 50 - 88 = -38

Explanation

From the given equations, express x and z in terms of y: x = y / 2 z = 6y

Substituting these into the ratio x : y : z gives: x : y : z = (y / 2) : y : 6y

Dividing through by y gives: x : y : z = ½ : 1 : 6

Multiplying the ratio throughout by 2 yields 1 : 2 : 12.

Explanation

Let the original sum be P. After a 10% deduction, the remaining amount is 90% of P. Deducting 20% from this remainder leaves 80% of the remaining amount, which equals 72% of the original sum (0.90 × 0.80 = 0.72).

Given that the final remaining sum is ₹3,600: 0.72 × P = 3,600 P = 3,600 / 0.72 = 5,000

Explanation

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

Substituting 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

Using the algebraic identity (a + b)² = a² + b² + 2ab:

First, square both sides of √x + 1/√x = 3: (√x + 1/√x)² = 3² x + 1/x + 2 = 9 x + 1/x = 7

Next, square both sides of x + 1/x = 7: (x + 1/x)² = 7² x² + 1/x² + 2 = 49 x² + 1/x² = 47

Shortcut: If √x + 1/√x = k, then x + 1/x = k² - 2, and x² + 1/x² = (k² - 2)² - 2. Substituting k = 3 gives (3² - 2)² - 2 = 7² - 2 = 47.

Explanation

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

Based on the given conditions, the speed of the steamer is T / 1.5 = 2T / 3 km/h and the speed of the horse is T / 3 km/h.

Using the relation Time = Distance / Speed, the total time of 13 hours 30 minutes (13.5 hours) is: 13.5 = 120 / (2T / 3) + 450 / T + 60 / (T / 3) 13.5 = 180 / T + 450 / T + 180 / T 13.5 = 810 / T T = 810 / 13.5 = 60 km/h

Explanation

In compound interest, the principal grows by a constant multiplying factor over equal time intervals.

For a 4-year period, the growth factor k is: k = Amount after 4 years / Principal = 6,000 / 4,800 = 5/4

Since 12 years consists of three 4-year intervals (12 = 4 × 3), the amount grows by a factor of k³ over 12 years: Amount after 12 years = Principal × k³ Amount = 4,800 × (5/4)³ Amount = 4,800 × (125 / 64) Amount = 75 × 125 = 9,375

Explanation

Let the uncovered side of the rectangular field be 30 ft.

Area = length × width 690 = 30 × width width = 690 / 30 = 23 ft

The field is fenced on three sides, leaving one 30 ft side uncovered. Thus, the required fencing covers the remaining 30 ft side and both 23 ft sides.

Fencing required = 30 + 23 + 23 = 76 ft

Explanation

When three cubes are melted to form a single cube without any loss of metal, the total volume remains conserved.

Volume of the new cube = 3³ + 4³ + 5³ Volume = 27 + 64 + 125 = 216 cm³

Side of the new cube = ∛216 = 6 cm

Explanation

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

Since the total number of observations (n) is 10 (an even number), the median is the average of the 5th and 6th observations:

5th observation = 20 6th observation = 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.

Substituting a = 10 cm: h = (√3/2) × 10 = 5√3 cm.

Explanation

The two poles form a right-angled triangle with the horizontal ground line. Horizontal distance (base) = 24 m Difference in heights (perpendicular) = 38 m - 20 m = 18 m

Using the Pythagorean theorem: Distance between tops = √(base² + perpendicular²) = √(24² + 18²) = √(576 + 324) = √900 = 30 m

Explanation

Distance D = (S₁ × S₂ / |S₁ - S₂|) × Δt

Given S₁ = 15 km/h, S₂ = 16 km/h, and time difference Δt = 16 minutes = 16/60 hours:

D = (15 × 16 / 1) × (16 / 60) = 240 × (4 / 15) = 64 km.

Explanation

For any two numbers, the product of the numbers is equal to the product of their HCF and LCM: First Number × Second Number = HCF × LCM

Given, HCF = 2, LCM = 144, and First Number = 16.

16 × Second Number = 2 × 144 16 × Second Number = 288 Second Number = 288 ÷ 16 = 18

Explanation

Simple Interest for 1.5 years = ₹873 - ₹756 = ₹117

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

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

Principal sum = Amount after 2 years - Simple Interest for 2 years Principal sum = ₹756 - ₹156 = ₹600