RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

Let the required whole number be x. According to the given condition: 60% of x > 6 (60 / 100) × x > 6 0.6 × x > 6 x > 6 / 0.6 x > 10

Since x must be a whole number strictly greater than 10, the smallest whole number that satisfies this condition is 11.

Explanation

The difference between compound interest and simple interest for a period of 2 years is given by the formula D = P × (r / 100)², where D is the difference, P is the principal amount, and r is the annual rate of interest.

Substituting D = 25 and r = 5: 25 = P × (5 / 100)² 25 = P × (1 / 20)² 25 = P / 400 P = 25 × 400 = 10,000

Explanation

The work done in one day by each pair is: (X + Y)'s 1-day work = ⅓ (Y + Z)'s 1-day work = ⅙ (X + Z)'s 1-day work = ⅑

Adding these work rates: 2(X + Y + Z)'s 1-day work = ⅓ + ⅙ + ⅑ = (6 + 3 + 2)/18 = 11/18

Dividing by 2 gives the combined 1-day work rate for all three: (X + Y + Z)'s 1-day work = 11/36

Taking the reciprocal gives the total time taken to finish the task together: Time = 36/11 days

Explanation

Sum of 8 numbers = 8 × 37 = 296

When one number is excluded, 7 numbers remain and the new mean becomes 37 - 2 = 35.

Sum of remaining 7 numbers = 7 × 35 = 245

Excluded number = Sum of 8 numbers - Sum of 7 numbers = 296 - 245 = 51

Explanation

In the number 786392: Place value of 8 = 8 × 10000 = 80000 Place value of 6 = 6 × 1000 = 6000

Difference = 80000 - 6000 = 74000

Explanation

The winning party is Party D with 54 votes. The total combined votes for Party C and Others is 29 + 21 = 50 votes. Since 50 is less than 54, this group has fewer votes than Party D and cannot defeat it.

Explanation

Total combined votes of NOTA and Others = 17 + 21 = 38 votes.

Dividing these 38 votes equally gives 19 additional votes each to Party B and Party C: Party B = 37 + 19 = 56 votes Party C = 29 + 19 = 48 votes

The updated vote count across all parties becomes: Party B = 56 votes Party D = 54 votes Party C = 48 votes Party F = 39 votes

Party B wins with 56 votes. The runner-up is Party D with 54 votes.

Winning margin = 56 - 54 = 2 votes.

Explanation

For x = 2 + √5, rationalizing 1/x gives 1 / (√5 + 2) = (√5 - 2) / (5 - 4) = √5 - 2.

Subtracting 1/x from x yields: x - 1/x = (2 + √5) - (√5 - 2) = 4

Taking the cube: (x - 1/x)³ = 4³ = 64

Explanation

The product of the LCM and HCF of two numbers is equal to the product of the numbers themselves: x × y = LCM × HCF = 28

The difference between the two numbers is given as: x - y = 3

Using the algebraic identity (x + y)² = (x - y)² + 4xy: (x + y)² = 3² + 4 × 28 = 9 + 112 = 121 x + y = 11

Solving the linear equations x + y = 11 and x - y = 3 gives x = (11 + 3) / 2 = 7 and y = (11 - 3) / 2 = 4.

Explanation

By the addition theorem of probability: P(A ∩ B) = P(A) + P(B) - P(A ∪ B)

Substituting the given values: P(A ∩ B) = 1/3 + 1/4 - 1/2 = 7/12 - 6/12 = 1/12

By the definition of conditional probability: P(A/B) = P(A ∩ B) / P(B)

Substituting the calculated values: P(A/B) = (1/12) / (1/4) = 4/12 = 1/3

Explanation

Using the BODMAS rule, simplify the expression step by step:

First, evaluate the inner bracket terms: 25 ÷ (1/5) = 25 × 5 = 125 14 × 8 = 112

Subtracting these values gives the term inside the square brackets: 125 - 112 = 13

Next, perform the multiplication outside the brackets: 9 × 4 = 36

Substituting these back into the main expression: 12 + 13 - 36 = 25 - 36 = -11

Explanation

The required smallest number N is given by N = LCM(18, 21, 25, 39) + 3.

Finding the prime factorizations: 18 = 2 × 3² 21 = 3 × 7 25 = 5² 39 = 3 × 13

LCM(18, 21, 25, 39) = 2¹ × 3² × 5² × 7¹ × 13¹ = 2 × 9 × 25 × 7 × 13 = 40950

Thus, N = 40950 + 3 = 40953

Sum of the digits of N = 4 + 0 + 9 + 5 + 3 = 21

Explanation

Number of pieces = floor(Total length ÷ Length of each piece)

Number of pieces = floor(658 ÷ 34.6) = floor(6580 ÷ 346) = 19.017

Since 34.6 × 19 = 657.4 cm ≤ 658 cm, 19 complete pieces can be cut from the rod with 0.6 cm remaining.

Explanation

Using the algebraic identity (x + y)² = x² + y² + 2xy:

Substitute x + y = 9 and xy = 18: 9² = x² + y² + 2(18) 81 = x² + y² + 36 x² + y² = 81 - 36 = 45

Explanation

Using exponent rules, rewrite both sides of the equation with base 5:

√(5ⁿ) = 625 5⁽n/2) = 5⁴

Equating the exponents since the bases are equal:

n/2 = 4 n = 8

Explanation

Squaring both sides of the equation √(1 + x/121) = 12/11:

1 + x/121 = (12/11)² 1 + x/121 = 144/121

Subtracting 1 from both sides:

x/121 = 144/121 - 1 x/121 = (144 - 121) / 121 x/121 = 23/121

Multiplying both sides by 121 yields x = 23.

Alternatively, for equations of the form √(1 + x/a²) = (a + 1)/a, the value of x is given directly by (a + 1)² - a² = 12² - 11² = 144 - 121 = 23.

Explanation

Circumference of a circle = 2πr Perimeter of an equilateral triangle = 3 × side

Given radius (r) = 21 cm: Circumference = 2 × (22/7) × 21 = 132 cm

Equating the perimeter of the triangle to the circumference: 3 × side = 132 side = 132 / 3 = 44 cm

Explanation

Since both boys travel for the same duration until they meet, the ratio of their speeds is equal to the ratio of the distances they travel.

Distance covered by Rishi = 95 - 25 = 70 km Distance covered by Vamsi = 95 + 25 = 120 km

Ratio of Rishi's speed to Vamsi's speed = 70 : 120 = 7 : 12

Difference in speed ratio units = 12 - 7 = 5 units Given speed difference = 5 km/h

Since 5 units = 5 km/h, 1 unit = 1 km/h. Rishi's speed = 7 × 1 = 7 km/h.

Explanation

Let the number be x.

According to the question: 70% of x - 45% of x = 15 25% of x = 15 (25/100) × x = 15 x = (15 × 100) / 25 = 60

Explanation

For simple interest, when a principal doubles itself in T years, the interest accrued equals the principal. The annual rate of interest is given by the formula R = 100 / T %.

Substituting T = 8 years: R = 100 / 8 = 12.5% = 12½%

Explanation

Using the standard work formula: Time taken by B alone = (Time by A × Time by A and B together) / (Time by A - Time by A and B together)

Substituting the given values: Time taken by B alone = (24 × 16) / (24 - 16) = 384 / 8 = 48 days.

Explanation

Rearranging the equation x² + 25y² = 10xy gives: x² - 10xy + 25y² = 0

Using the algebraic identity (a - b)² = a² - 2ab + b², this reduces to: (x - 5y)² = 0

Taking the square root on both sides: x - 5y = 0 x = 5y

Dividing both sides by y gives: x / y = 5 / 1, which expressed as a ratio is x : y = 5 : 1.

Explanation

When two articles are sold at the same selling price, one at a gain of x% and the other at a loss of x%, the overall transaction always results in a loss given by:

Net Loss % = (x / 10)² %

Substituting x = 9:

Net Loss % = (9 / 10)² % = (81 / 100) % = 0.81% loss

Explanation

Let the two numbers be 5x and 4x, where x is their highest common factor (HCF).

Since 5 and 4 are coprime, the least common multiple (LCM) of 5x and 4x is: LCM = 5 × 4 × x = 20x

Given that the LCM is 120: 20x = 120 x = 6

Substituting x = 6 into the expressions for the numbers: First number = 5 × 6 = 30 Second number = 4 × 6 = 24

Explanation

The slope m of the straight line passing through the points (0, 2) and (1, -3) is: m = (-3 - 2) / (1 - 0) = -5

Using the slope-intercept form y = mx + c, where the y-intercept c = 2 since the line passes through (0, 2): y = -5x + 2

Rearranging all terms to one side gives: 5x + y - 2 = 0

Explanation

For a sum of money to double itself under simple interest, the interest earned must equal the principal amount (SI = P).

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

P = (P × 6.25 × T) / 100

1 = (6.25 × T) / 100

T = 100 / 6.25 = 16 years

Explanation

Adding the positive decimal numbers together: 42.425 + 6.526 + 0.098 = 49.049

Subtracting 15.31 from the sum: 49.049 - 15.310 = 33.739

Explanation

By the distance formula between two points (x₁, y₁) and (x₂, y₂), d = √((x₂ - x₁)² + (y₂ - y₁)²).

Given points are (x, 7) and (1, 15) with distance d = 10: 10² = (x - 1)² + (15 - 7)² 100 = (x - 1)² + 64 (x - 1)² = 36 x - 1 = ±6

Solving for x gives: x = 1 + 6 = 7 x = 1 - 6 = -5

Explanation

The area of a square of side s is given by A = s², and its perimeter is P = 4s. The ratio of the perimeters of two squares is equal to the ratio of their side lengths, which is the square root of the ratio of their areas:

P₁ / P₂ = s₁ / s₂ = √(A₁ / A₂) = √(169 / 196) = 13 / 14.

Explanation

Using the fundamental trigonometric identity cot θ = 1 / tan θ, the given equation 3 cot θ = tan θ can be rewritten as:

3 / tan θ = tan θ tan² θ = 3 tan θ = ±√3

Taking the principal value in the standard range gives tan θ = √3, which yields θ = π/3.

Explanation

Let Selling Price (SP) = 100. Then Cost Price (CP) = 92% of SP = 92.

Profit = SP - CP = 100 - 92 = 8

Percentage Profit = (Profit / CP) × 100 = (8 / 92) × 100 = (2 / 23) × 100 = 200 / 23 % ≈ 8.70%

Explanation

Amount after Year 1 = (64,000 × 1.25) - 3,200 = 80,000 - 3,200 = 76,800

Amount after Year 2 = (76,800 × 1.25) - 3,200 = 96,000 - 3,200 = 92,800

Amount after Year 3 = (92,800 × 1.25) - 3,200 = 1,16,000 - 3,200 = 1,12,800

Explanation

A point (x, y) in the Cartesian coordinate plane lies on the x-axis if and only if its y-coordinate is zero (y = 0). For the point (2, 0), the y-coordinate is 0, so it is located directly on the x-axis at a distance of 2 units from the origin along the positive x-direction.

Explanation

To find which school has the highest number of participants for the maximum number of sports, compare the participant counts across all schools for each individual sport:

Badminton: School D has the highest (78) Table Tennis: School B has the highest (40) Football: School B has the highest (22) Basketball: School B has the highest (13) Cricket: School B has the highest (121)

School B leads in 4 out of 5 sports (Table Tennis, Football, Basketball, and Cricket), making it the school with the highest number of participants in the maximum number of sports.

Explanation

Value = Scale × Height

Given scale = 20 units per cm and height = 4.7 cm:

Value = 20 × 4.7 = 94 units

Explanation

Multiply the non-zero digits first: 4 × 4 × 4 = 64.

Next, count the total number of decimal places in the three factors: 0.4 has 1 decimal place 0.04 has 2 decimal places 0.004 has 3 decimal places

Total decimal places = 1 + 2 + 3 = 6 decimal places.

Shifting the decimal point 6 positions to the left from 64 gives 0.000064.

Explanation

The secant function is periodic with a period of 360°. Dividing 2100° by 360° gives a quotient of 5 and a remainder of 300°:

2100° = 5 × 360° + 300°

Therefore, sec(2100°) = sec(300°).

Since 300° is in the fourth quadrant, where the secant function is positive:

sec(300°) = sec(360° - 60°) = sec(60°) = 1 / cos(60°) = 1 / (½) = 2

Explanation

Simplifying each term by taking the square roots:

√(225 / 729) = 15 / 27 = 5 / 9

√(25 / 144) = 5 / 12

√(16 / 81) = 4 / 9

Substitute these values back into the expression:

(5 / 9) - (5 / 12) - (4 / 9)

Group the terms with common denominators:

(5 / 9 - 4 / 9) - (5 / 12) = (1 / 9) - (5 / 12)

Taking the LCM of 9 and 12, which is 36:

(4 - 15) / 36 = -11 / 36

Explanation

To find the total dress material bought, add the lengths purchased for both daughters:

Total material = 6.35 m + 5.65 m = 12.00 m = 12 m

Explanation

Since a/b = 4/3, substitute a = 4 and b = 3 directly into the given expression:

Numerator: 6a + 2b = 6(4) + 2(3) = 24 + 6 = 30 Denominator: 4a - 2b = 4(4) - 2(3) = 16 - 6 = 10

Value = 30 / 10 = 3

Explanation

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

Here, a = √5 and b = √11.

(√5 + √11)(√5 - √11) = (√5)² - (√11)² = 5 - 11 = -6

Explanation

Time taken = 1 h + 40/60 h + 48/3600 h = 1 + 2/3 + 1/75 = 42/25 h.

Reduced speed = Distance / Time = 42 / (42/25) = 25 km/h.

Since Reduced Speed = (5/7) × Actual Speed: Actual Speed = 25 × (7/5) = 35 km/h.

Explanation

The mean proportional between two positive numbers a and b is given by the formula √(a × b).

Substituting a = 169 and b = 121: Mean proportional = √(169 × 121) = √169 × √121 = 13 × 11 = 143

Explanation

Using the double-angle identity: 2 sin α cos α = sin 2α

Substituting α = 30°: 2 sin 30° cos 30° = sin(2 × 30°) = sin 60° = √3/2

Alternatively, substituting the values directly: 2 sin 30° cos 30° = 2 × ½ × (√3/2) = √3/2

Explanation

Using the percentage relationship: 50% × x = 25% × y ½ × x = ¼ × y x / y = ¼ / ½ = 1 / 2

So, x : y = 1 : 2.

Explanation

To find the value of the given expression:

Numerator = (5 × 1.6) - (2 × 1.4) = 8.0 - 2.8 = 5.2

Denominator = 1.3

Dividing the numerator by the denominator: 5.2 ÷ 1.3 = 4

Explanation

For a sum of money to double itself under simple interest, the interest earned (SI) must equal the principal (P).

Using the time formula for simple interest, T = (SI × 100) / (P × R): T = (100 / 18) = 50 / 9 = 5 5/9 years

Converting 5/9 of a year into months: (5 / 9) × 12 = 20 / 3 = 6.67 months, which rounds to 7 months.

Explanation

Using the percentage relation: (x / 100) × 160 = 8

Solving for x: x = (8 × 100) / 160 x = 800 / 160 x = 5

Explanation

Let the original ratio be 2 : 3 and the new ratio be 3 : 4.

Since both numbers are increased by equal amounts (20), the change in ratio units for both numbers is (3 - 2) = (4 - 3) = 1 unit.

1 unit = 20

Therefore, the original numbers are: First number = 2 × 20 = 40 Second number = 3 × 20 = 60

Explanation

Area of the square = side² = 8² = 64 cm².

Since the area of the rectangle is equal to the area of the square, the area of the rectangle is also 64 cm².

Let the width of the rectangle be x cm and its length be 4x cm. Area = length × width = 4x × x = 4x² = 64 x² = 16 x = 4 cm

Length = 16 cm and width = 4 cm. Perimeter of the rectangle = 2 × (length + width) = 2 × (16 + 4) = 40 cm.