RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

When a point (x, y) is reflected across the x-axis, its x-coordinate remains unchanged while its y-coordinate changes sign, according to the transformation rule (x, y) → (x, -y).

Applying this rule to the point (7, 8) gives the image point (7, -8).

Explanation

Distance = Speed × Time

Given: Speed = 5 km/h Time = 15 minutes = 15/60 hour = 1/4 hour

Length of the bridge = 5 × (1/4) km = 1.25 km = 1.25 × 1000 metres = 1250 metres.

Explanation

The simple interest is calculated using the formula SI = (P × R × T) / 100, where Principal (P) = ₹16,800, Rate (R) = 6¼% = 25/4% per annum, and Time (T) = 9 months = 9/12 years = ¾ years.

Substituting the given values into the formula: SI = (16,800 × 25 × 3) / (100 × 4 × 4) SI = (168 × 75) / 16 SI = 12,600 / 16 = ₹787.50

Explanation

Converting the fractions to decimal form gives 5/7 ≈ 0.714, 7/9 ≈ 0.778, 4/5 = 0.800, and 7/6 ≈ 1.167. Comparing these values shows that 0.714 is the least, making 5/7 the smallest fraction.

Explanation

Net percentage change = x + y + (x × y)/100

Here, markup percentage x = +10% and discount percentage y = -10%:

Net change = 10 - 10 + (10 × (-10))/100 = -1%

The negative result indicates a loss of 1%.

Explanation

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

Given: 6 × CP = 7 × SP

CP / SP = 7 / 6

Since the cost price (7 units) is greater than the selling price (6 units), it results in a loss.

Loss percentage = ((CP - SP) / CP) × 100 = ((7 - 6) / 7) × 100 = (1 / 7) × 100 = 14.28% loss

Explanation

The present worth P of an amount A due in n years at rate r% per annum compound interest is given by the formula P = A / (1 + r/100)ⁿ.

Substituting A = 338, r = 4, and n = 2: P = 338 / (1 + 4/100)² P = 338 / (26/25)² P = (338 × 625) / 676 P = 625 / 2 = ₹312.50

Explanation

Let the required number be x.

Using the percentage formula: (40 / 100) × x = 800 x = (800 × 100) / 40 x = 2000

Explanation

Let the number of students in the school be x. Since the fee paid by each student is equal to the total number of students, the total fee collected is given by x × x = x².

x² = 2601 x = √2601 x = 51

There were 51 students in the school.

Explanation

Substitute y = 5 directly into the expression:

10y = 10 × 5 = 50

y³ - y² = 5³ - 5² = 125 - 25 = 100

√(y³ - y²) = √100 = 10

Multiplying the components gives: 10y × √(y³ - y²) = 50 × 10 = 500

Explanation

Let the number to be subtracted be x.

(15 - x) / (19 - x) = ¾

Cross-multiplying: 4 × (15 - x) = 3 × (19 - x) 60 - 4x = 57 - 3x x = 60 - 57 = 3

Verification: (15 - 3) / (19 - 3) = 12 / 16 = ¾.

Explanation

The mean of a dataset is given by the formula: Mean = (Sum of all observations) ÷ (Total number of observations).

Sum of the observations = 2 + 5 + 4 + 1 + 8 = 20 Total number of observations = 5

Mean = 20 ÷ 5 = 4

Explanation

For any natural number n, n and n + 1 are two consecutive integers. Let d be the Highest Common Factor (HCF) of n and n + 1. Since d divides both n and n + 1, d must also divide their difference:

(n + 1) - n = 1

The only positive integer that divides 1 is 1. Therefore, HCF(n, n + 1) = 1 for all natural numbers n, which means consecutive natural numbers are always co-prime.

Explanation

The least number that is exactly divisible by a given set of numbers is their Least Common Multiple (LCM).

Finding the prime factorization of each number: 12 = 2² × 3 15 = 3 × 5 18 = 2 × 3²

Taking the highest power of each involved prime factor: LCM = 2² × 3² × 5 = 4 × 9 × 5 = 180.

Explanation

When two pipes empty a tank individually in time T₁ and T₂, the combined time T required to empty the full tank when working together is given by the formula:

T = (T₁ × T₂) / (T₁ + T₂)

Substituting the given values (T₁ = 6 h and T₂ = 18 h):

T = (6 × 18) / (6 + 18) = 108 / 24 = 4.5 h

Explanation

Arranging the given 10 observations in ascending order: 2, 2, 3, 3, 3, 5, 5, 7, 7, 9

Since the number of observations n = 10 is even, the median is the average of the 5th and 6th terms: Median = (3 + 5) ÷ 2 = 4

The mode is the observation with the highest frequency. In this dataset, 3 appears 3 times, which is more frequent than any other value, so Mode = 3.

Explanation

To convert the recurring decimal 3.127 (where 7 is recurring) into fraction form, let x = 3.12777...

Multiplying x by 100 and 1000 gives: 100x = 312.777... 1000x = 3127.777...

Subtracting the first equation from the second: 1000x - 100x = 3127.777... - 312.777... 900x = 2815 x = 2815 / 900

Dividing the numerator and denominator by 5: x = 563 / 180

Explanation

Simplifying the right side of the given equation: √(169/289) = √169 / √289 = 13/17

Substituting this back into the equation: 52/x = 13/17

Cross-multiplying to solve for x: x = (52 × 17) / 13 = 4 × 17 = 68

Explanation

Rationalizing the denominator by multiplying the numerator and denominator by (√2 - 1):

x = 1 / (√2 + 1) = (√2 - 1) / [(√2 + 1)(√2 - 1)] = (√2 - 1) / (2 - 1) = √2 - 1

Now, adding 1 to both sides gives:

x + 1 = (√2 - 1) + 1 = √2

Explanation

Area of square = side² = 12 × 12 = 144 cm²

Area of triangle = ½ × base × altitude

Since the area of the triangle equals the area of the square: ½ × 12 × altitude = 144 6 × altitude = 144 altitude = 144 / 6 = 24 cm

Explanation

Raj's total marks obtained = 18 + 17 + 16 + 15 = 66 out of a total maximum of 80 marks. Raj's aggregate percentage = (66 / 80) × 100 = 82.5%.

Ramesh's total marks obtained = 85 + 70 + 60 + 75 = 290 out of a total maximum of 400 marks. Ramesh's aggregate percentage = (290 / 400) × 100 = 72.5%.

Difference between their aggregate percentages = 82.5% - 72.5% = 10%.

Explanation

Let the two numbers be x and y.

Given: x - y = 14 x² - y² = 56

Using the algebraic identity x² - y² = (x - y)(x + y): 56 = 14 × (x + y) x + y = 4

Adding and subtracting the two equations: 2x = 4 + 14 = 18 ⇒ x = 9 2y = 4 - 14 = -10 ⇒ y = -5

The two numbers are 9 and -5.

Explanation

Let the two numbers be x and y.

From the first given condition: x/3 + y/2 = 8 Multiplying by 6 gives: 2x + 3y = 48

From the second given condition: x/5 + y/6 = 4 Multiplying by 30 gives: 6x + 5y = 120

Multiplying 2x + 3y = 48 by 3 yields: 6x + 9y = 144

Subtracting 6x + 5y = 120 from 6x + 9y = 144: 4y = 24 y = 6

Substituting y = 6 back into 2x + 3y = 48: 2x + 18 = 48 2x = 30 x = 15

The two numbers are 15 and 6, making 15 the largest of the two numbers.

Explanation

Let x be the required number to be subtracted from both the numerator and the denominator.

(42 - x) / (45 - x) = 5 / 6

Cross-multiplying both sides: 6 × (42 - x) = 5 × (45 - x) 252 - 6x = 225 - 5x 252 - 225 = 6x - 5x x = 27

Subtracting 27 from the numerator yields 42 - 27 = 15, and from the denominator yields 45 - 27 = 18, which simplifies to 15/18 = 5/6.

Explanation

Ratio of votes polled for the two candidates is 2 : 9, so the successful candidate received 9 parts of the total votes.

Given that 9 parts = 984321 1 part = 984321 / 9 = 109369

Total votes polled = 2 + 9 = 11 parts Total votes = 11 × 109369 = 1203059

Explanation

From the given relation, 15% of A = 18% of B. Simplifying gives A = (18 / 15) B = 1.2 B.

Substituting A = 1.2 B into 20% of A: 20% of A = 20% of (1.2 B) = 24% of B.

Explanation

Adding the given decimal numbers step by step:

0.56 + 0.43 = 0.99 0.99 + 0.89 = 1.88

Explanation

Let the length of the train be L meters and its speed be v km/hr.

When the train overtakes a boy moving in the same direction, the relative speed is (v - boy's speed) km/hr. Converting to m/s by multiplying by 5/18, the distance covered to pass each boy equals the length of the train L.

For the first boy (speed = 5 km/hr, time = 6 s): L = (v - 5) × (5/18) × 6 = (5/3)(v - 5)

For the second boy (speed = 7 km/hr, time = 9 s): L = (v - 7) × (5/18) × 9 = (5/2)(v - 7)

Equating the two expressions for L: (5/3)(v - 5) = (5/2)(v - 7) 2(v - 5) = 3(v - 7) 2v - 10 = 3v - 21 v = 11 km/hr

Substituting v = 11 km/hr back into the formula for L: L = (5/3) × (11 - 5) = (5/3) × 6 = 10 m

Explanation

Area of an incircle is given by π × r² = 462. Using π = 22/7: (22/7) × r² = 462 r² = (462 × 7) / 22 = 147 r = √147 = 7√3 cm

For an equilateral triangle of side length a, the inradius r = a / (2√3). a = 2√3 × r = 2√3 × 7√3 = 42 cm

Perimeter = 3a = 3 × 42 = 126 cm

Explanation

Selling Price = Cost Price × (100 - Loss%) / 100

Selling Price = 2400 × (100 - 10) / 100 = 2400 × 0.9 = ₹2,160

Explanation

Using the algebraic identity (a + b)² = a² + b² + 2ab: (a + b)² = 82 + 2(9) = 100 a + b = 10

Now, applying the sum of cubes formula: a³ + b³ = (a + b)(a² + b² - ab) a³ + b³ = 10 × (82 - 9) = 10 × 73 = 730

Explanation

Converting the fractions to decimal form yields 7/10 = 0.70, 5/7 ≈ 0.714, 3/4 = 0.75, and 7/8 = 0.875. Since 0.70 is the least decimal value, 7/10 is the smallest fraction. Alternatively, comparing 7/10 with 5/7 by cross-multiplication gives 7 × 7 = 49 and 10 × 5 = 50; since 49 < 50, 7/10 is smaller than 5/7 as well as all other given options.

Explanation

The area of a triangle is given by the formula Area = ½ × base × altitude.

The ratio of the areas of two triangles is: A₁ / A₂ = (b₁ × h₁) / (b₂ × h₂)

Substituting the given values: 8 / 15 = (4 / 5) × (h₁ / h₂)

h₁ / h₂ = (8 / 15) × (5 / 4) = 40 / 60 = 2 / 3

Explanation

The exterior angle of a regular polygon is 180° - interior angle = 180° - 120° = 60°.

The number of sides n = 360° / exterior angle = 360° / 60° = 6.

The number of diagonals of an n-sided polygon is given by the formula n(n - 3) / 2.

Substituting n = 6: Number of diagonals = 6 × (6 - 3) / 2 = 6 × 3 / 2 = 9.

Explanation

Simplify each term inside the curly brackets by eliminating the decimal points:

2.4 / 0.9 = 24 / 9 = 8 / 3 0.27 / 0.08 = 27 / 8 0.50 / 0.05 = 50 / 5 = 10

Multiplying these terms together inside the brackets: (8 / 3) × (27 / 8) × 10 = (27 / 3) × 10 = 9 × 10 = 90

Multiply the result by the outer factor 4: 4 × 90 = 360

Explanation

The expression 4⁷ − 4 can be factored as 4 × (4⁶ − 1). Since 4⁶ is even, 4⁶ − 1 is an odd number. Therefore, the prime factorization of 4⁷ − 4 contains 2² = 4 as its highest power of 2. Since 8 = 2³, the expression lacks a third factor of 2 and is not a multiple of 8.

Explanation

Let the required number be x.

According to the question: 30% of x + 38 = 50 0.3x = 50 - 38 0.3x = 12 x = 12 ÷ 0.3 = 40

Explanation

For a set of 7 distinct observations arranged in ascending order x₁ < x₂ < x₃ < x₄ < x₅ < x₆ < x₇, the median is the 4th observation, x₄ = 21.5.

Increasing the largest 3 observations (x₅, x₆, and x₇) by 4 affects only the 5th, 6th, and 7th positions in the ordered sequence. The 4th observation remains unchanged at x₄ = 21.5, so the median of the new set remains the same as that of the original set.

Explanation

Dimensions of the floor in centimeters: Length = 3 m = 300 cm Breadth = 1 m 50 cm = 150 cm

To pave the floor completely with the largest possible square slabs, the side length of each square slab must be equal to the Highest Common Factor (HCF) of the length and breadth: Side of square slab = HCF(300, 150) = 150 cm

Number of slabs required = (Area of floor) / (Area of one slab) Number of slabs = (300 × 150) / (150 × 150) = 2

Explanation

√529 = 23.

Substituting this into the given equation: √x + 10 = 23 √x = 23 - 10 √x = 13

Squaring both sides: x = 13² = 169

Explanation

Using componendo and dividendo on the given equation:

(sec θ + tan θ + sec θ - tan θ) / (sec θ + tan θ - sec θ + tan θ) = (5 + 3) / (5 - 3)

2 sec θ / 2 tan θ = 8 / 2

sec θ / tan θ = 4

Since sec θ = 1 / cos θ and tan θ = sin θ / cos θ, simplifying the left side gives:

(1 / cos θ) / (sin θ / cos θ) = 1 / sin θ = 4

Taking the reciprocal yields sin θ = 1/4.

Explanation

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

Taking the square root of each term: √225 : √144 = 15 : 12

Explanation

Using the algebraic identity (a + b)² + (a - b)² = 2(a² + b²), let a = 74 and b = 47.

Substituting these values into the given expression: Numerator = (74 + 47)² + (74 - 47)² = 2(74² + 47²)

Dividing by the denominator: 2(74² + 47²) / (74² + 47²) = 2

Explanation

The angle swept by the minute hand in 60 minutes is 360°, so in 5 minutes it sweeps an angle θ = (5 / 60) × 360° = 30°.

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The area of the sector formed is calculated using the formula: Area = (θ / 360°) × πr²

Substituting radius r = 6 cm, θ = 30°, and π = 22/7: Area = (30° / 360°) × (22 / 7) × 6² = (1 / 12) × (22 / 7) × 36 = 3 × (22 / 7) = 66 / 7 ≈ 9.43 cm²

Explanation

Let the two positive numbers be x and y.

Given that √x = 7, squaring both sides gives x = 49. The square of the first number is x² = 49² = 2401.

Using the relation x² + y² = 2437: 2401 + y² = 2437 y² = 2437 - 2401 y² = 36 y = √36 = 6

Explanation

The sum of the first n odd natural numbers is given by the formula S = n². Substituting n = 25 gives S = 25² = 625.

Explanation

Let total work = LCM(4, 6, 8) = 24 units.

Work done per hour (efficiencies): A + B = 24 / 4 = 6 units/hour B + C = 24 / 6 = 4 units/hour A + C = 24 / 8 = 3 units/hour

Adding all three equations: 2(A + B + C) = 6 + 4 + 3 = 13 units/hour A + B + C = 13 / 2 = 6.5 units/hour

Efficiency of C = (A + B + C) - (A + B) = 6.5 - 6 = 0.5 units/hour

Time taken by C alone = Total Work / Efficiency of C = 24 / 0.5 = 48 hours.

Explanation

To convert 9.23 hours into hours, minutes, and seconds:

Whole hours = 9 hours Fractional hours = 0.23 × 60 = 13.8 minutes Whole minutes = 13 minutes Fractional minutes = 0.8 × 60 = 48 seconds

Combining these gives 9 hours, 13 minutes, and 48 seconds.

Explanation

Squaring both sides of the equation √(15 × 27 × x) = 180 gives:

15 × 27 × x = 180²

405 × x = 32400

x = 32400 / 405 = 80

Explanation

The total time taken by a person to complete the entire work is given by the formula: Total time = Days taken / Fraction of work done

Time taken by A = 12 / (½) = 24 days Time taken by B = 10 / (20/100) = 50 days Time taken by C = 8 / (½) = 16 days Time taken by D = 12 / (⅕) = 60 days

Since C takes the least amount of time (16 days) to finish the complete work individually, C will be the first to complete it.