RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

The radius of the hemisphere r = diameter / 2 = 7 / 2 cm.

The total surface area of a hemisphere is given by the formula: Total Surface Area = 3πr²

Substituting r = 7 / 2 cm and π = 22 / 7: Total Surface Area = 3 × (22 / 7) × (7 / 2)² = 3 × (22 / 7) × (49 / 4) = (3 × 22 × 7) / 4 = 462 / 4 = 115.50 cm²

Explanation

A pie diagram represents a complete dataset using a full circle divided into central sectors. Since the angle subtended at the center by a complete circle is 360°, the sum of the central angles of all the individual sectors in a pie diagram always equals 360°.

Explanation

Selling Price = Cost Price × (100 + Gain%) / 100

Selling Price = 800 × (120 / 100) = 800 × 1.2 = ₹960.

Explanation

When two individuals take 'a' and 'b' days respectively to complete a task, the time taken by both of them working together is given by the formula:

Time = (a × b) / (a + b)

Substituting a = 20 and b = 30:

Time = (20 × 30) / (20 + 30) = 600 / 50 = 12 days.

Explanation

Total expenditure on raw materials over the 5 years = 452 + 569 + 659 + 706 + 783 = 3169 million.

Total expenditure on salary over the 5 years = 340 + 398 + 409 + 456 + 479 = 2082 million.

Difference = 3169 - 2082 = 1087 million.

Explanation

For positive fractions where the difference between the denominator and numerator is constant (here, denominator - numerator = 1), the fraction with the largest numerator and denominator has the highest value. Writing each fraction in the form 1 - (1/d), where d is the denominator, 15/16 equals 1 - 1/16. Since 1/16 is the smallest value being subtracted from 1, 15/16 yields the greatest value.

Explanation

Substitute p = 3 into the quadratic expression p² - 7p + 12:

p² - 7p + 12 = (3)² - 7(3) + 12 = 9 - 21 + 12 = 21 - 21 = 0

Explanation

Let the missing number be x.

0.008 ÷ x = 0.8

x = 0.008 ÷ 0.8 = 8/1000 ÷ 8/10 = 1/100 = 0.01

Explanation

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

Substituting SP = ₹34.80 and Loss% = 25%:

Cost Price = 34.80 × 100 / 75 = 34.80 × 4 / 3 = 11.60 × 4 = ₹46.40

Explanation

Taking the square root of 1296 gives: √1296 = 36

Given that √1296 = x², we have: x² = 36 x = √36 = 6

Explanation

To find the percentage, convert both quantities to the same unit (grams): 7.2 kg = 7.2 × 1000 g = 7200 g

Required percentage = (18 / 7200) × 100 = 18 / 72 = 0.25%

Explanation

Relative speed in the same direction = 4 km/hr - 3.5 km/hr = 0.5 km/hr.

Time = Distance / Relative Speed = 1.5 km / 0.5 km/hr = 3 hours.

Explanation

Using the trigonometric identity cos(A - B) = cos A cos B + sin A sin B:

cos 100° cos 40° + sin 100° sin 40° = cos(100° - 40°) = cos 60° = ½

Explanation

When a person covers a certain distance at speed v₁ and returns the same distance at speed v₂, the distance d is given by the formula:

d = (v₁ × v₂) / (v₁ + v₂) × Total Time

Substituting v₁ = 2 km/h, v₂ = 3 km/h, and Total Time = 5 hours:

d = (2 × 3) / (2 + 3) × 5 = (6 / 5) × 5 = 6 km

Explanation

Points that divide a line segment into three equal parts divide it internally in the ratios of 1:2 and 2:1 relative to the endpoints. Consequently, these two points are called the points of trisection.

Explanation

1 litre equals 1000 ml. To express 180 ml as a decimal fraction of a litre, divide 180 by 1000:

180 ÷ 1000 = 0.18

Explanation

Let x be the required value to be subtracted.

Product = 0.527 × 2.013 = 1.060851

According to the given condition: 1.060851 - x = 1 x = 1.060851 - 1 = 0.060851

Explanation

To find the state with the lowest difference between total RTR and total AT, subtract total AT from total RTR for each state using the total values provided in the table:

UP: 45158969 - 44098473 = 1060496 HP: 9359022 - 8428032 = 930990 Punjab: 14860518 - 13722869 = 1137649 Haryana: 10358582 - 9151587 = 1206995

Comparing these differences, HP has the lowest value of 930990.

Explanation

Total maximum marks for all 6 subjects combined = 100 + 100 + 100 + 100 + 50 + 50 = 500.

70% of 500 = 0.70 × 500 = 350 marks. 80% of 500 = 0.80 × 500 = 400 marks.

The question asks for students with total marks strictly between 350 and 400. Pundir scored 355 marks (71%) and Devesh scored 360 marks (72%), giving a total count of 2 students.

Explanation

To find the Highest Common Factor (HCF) of algebraic expressions, multiply the HCF of their numerical coefficients by each common variable raised to its lowest power:

HCF(28, 70) = 14 Lowest exponent of p between p⁵ and p³ = p³ Lowest exponent of q between q² and q⁴ = q²

Combining these gives 14p³q².

Explanation

For the printer model L-1100, the quantities sold each year from 2014 to 2018 are: 2014: 523 2015: 673 2016: 567 2017: 590 2018: 623

The maximum number of L-1100 printers sold in a single year is 673 (in 2015), and the minimum number sold is 523 (in 2014).

Difference = Maximum - Minimum = 673 - 523 = 150

Explanation

When 1000 is divided by 15, the quotient is 66 with a remainder of 10, since 15 × 66 = 990.

The least number that must be added to 1000 to make it divisible by 15 is equal to the divisor minus the remainder: 15 - 10 = 5

Adding 5 to 1000 gives 1005, which is exactly divisible by 15 (1005 ÷ 15 = 67).

Explanation

Time = Distance / Speed

Converting speed to metres per minute: Speed = 30 × (1000 / 60) = 500 m/min

Time taken = 600 / 500 = 6/5 = 1 1/5 minutes

Explanation

Let the initial quantities of alcohol and water in the mixture be 5x litres and 4x litres, respectively.

When 9 litres of water is added to the mixture, the new quantity of water becomes (4x + 9) litres while the quantity of alcohol remains 5x litres.

Given the new ratio of alcohol to water is 4 : 5: 5x / (4x + 9) = 4 / 5

Cross-multiplying the equation: 25x = 16x + 36 9x = 36 x = 4

The quantity of alcohol in the mixture is 5 × 4 = 20 litres.

Explanation

Using the BODMAS rule for order of operations:

Innermost parenthesis: 11 + 4 - 2 = 13 Curly brackets: 2 × 13 = 26 Square brackets: 2 + 26 = 28 Multiplication and subtraction: (2/7) × 28 - 2 = 2 × 4 - 2 = 8 - 2 = 6

Explanation

Curved surface area (CSA) of a cylinder = 2πrh Total surface area (TSA) of a cylinder = 2πr(h + r)

Given, CSA / TSA = 3 / 7

(2πrh) / [2πr(h + r)] = 3 / 7 h / (h + r) = 3 / 7 7h = 3h + 3r 4h = 3r h / r = 3 / 4

Explanation

0.36 expressed as a fraction is 36/100.

To convert this fraction into its simplest form, divide both the numerator and the denominator by their greatest common divisor, which is 4:

Numerator = 36 ÷ 4 = 9 Denominator = 100 ÷ 4 = 25

Thus, the simplest fractional form is 9/25.

The sum of the numerator and the denominator is 9 + 25 = 34.

Explanation

Simple Interest (SI) = (P × R × T) / 100

where P is the principal sum, R is the interest rate per annum, and T is the time in years.

Given SI = ₹650, R = 13%, and T = 5 years:

650 = (P × 13 × 5) / 100 650 = 65P / 100 P = (650 × 100) / 65 = ₹1,000

Explanation

To find the median, arrange the given weights in ascending order: 61, 62, 63, 64, 65, 66, 67

Since the total number of observations n = 7 is odd, the median is the ((n + 1) / 2)th term, which is the 4th term. The 4th term in the ordered data is 64 kg.

Explanation

75% of 75% = (75 ÷ 100) × (75 ÷ 100) = 0.75 × 0.75 = 0.5625.

Explanation

Multiplying 3.14 by 10⁶ is equivalent to moving the decimal point six places to the right:

3.14 × 10⁶ = 3.14 × 1,000,000 = 3,140,000

Explanation

Let the three numbers in ascending order be x, y, and z.

The sum of the numbers is: x + y + z = 53 × 3 = 159

Using the given range and difference conditions: x = z - 28 y = x + 8 = z - 20

Substituting x and y into the sum equation: (z - 28) + (z - 20) + z = 159 3z - 48 = 159 3z = 207 z = 69

Explanation

Given equation: √(x) ÷ √(6.25) = 2

Since √(6.25) = 2.5, substituting this into the equation gives: √(x) ÷ 2.5 = 2 √(x) = 2 × 2.5 = 5

Squaring both sides: x = 5² = 25

Explanation

The standard equation of a circle with center (h, k) and radius r is (x - h)² + (y - k)² = r².

Substituting h = -2, k = 3, and r = 4: (x - (-2))² + (y - 3)² = 4² (x + 2)² + (y - 3)² = 16 x² + 4x + 4 + y² - 6y + 9 = 16 x² + y² + 4x - 6y + 13 = 16 x² + y² + 4x - 6y - 3 = 0

Explanation

Let the original cost price of the car be C.

The selling price after a 20% loss = C × (1 - 0.20) = 0.80 × C

The final selling price after the friend sells it at a 30% profit = 0.80 × C × (1 + 0.30) = 1.04 × C

Given that the final selling price is ₹58,000: 1.04 × C = 58,000 C = 58,000 / 1.04 = 55,769.23

Rounding to the nearest integral value gives ₹55,769.

Explanation

Cost Price (CP) of 20 articles = ₹16 Selling Price (SP) of 20 articles = 20 × ₹1.50 = ₹30

Gain = SP - CP = ₹30 - ₹16 = ₹14

Gain Percentage = (Gain / CP) × 100 = (14 / 16) × 100 = 87.5%

Explanation

The standard equation of a circle x² + y² + 2gx + 2fy + c = 0 has center (-g, -f) and radius √(g² + f² - c).

For the first circle x² + y² - 12x - 6y + 41 = 0: Center C₁ = (6, 3) Radius r₁ = √(36 + 9 - 41) = √4 = 2

For the second circle x² + y² + kx + 6y - 59 = 0: Center C₂ = (-k/2, -3) Radius r₂ = √((k²/4) + 9 + 59) = √((k²/4) + 68)

The square of the distance between centers d²: d² = (6 + k/2)² + (3 + 3)² = 36 + 6k + (k²/4) + 36 = (k²/4) + 6k + 72

The angle θ between two circles satisfies: cos θ = |r₁² + r₂² - d²| / (2 r₁ r₂)

Since θ = 45°, cos 45° = 1/√2: r₁² + r₂² - d² = 4 + ((k²/4) + 68) - ((k²/4) + 6k + 72) = -6k

1/√2 = |-6k| / (2 × 2 × √((k²/4) + 68)) 1/√2 = 3|k| / √(k² + 272)

Squaring both sides: 1/2 = 9k² / (k² + 272) 18k² = k² + 272 17k² = 272 k² = 16 k = ± 4

Explanation

Using the trigonometric identity sin²θ + cos²θ = 1, the expression can be rewritten as:

9 sin²θ + 10 cos²θ = 9(sin²θ + cos²θ) + cos²θ = 9 + cos²θ

Since the minimum value of cos²θ for any real angle θ is 0, the minimum value of 9 + cos²θ is 9 + 0 = 9.

Explanation

Given 4A = 5B, the ratio A/B = 5/4. Given 6B = 7C, the ratio B/C = 7/6.

Multiplying the two ratios yields A:C: A/C = (A/B) × (B/C) = (5/4) × (7/6) = 35/24

Explanation

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

(√5 + √7)² = (√5)² + (√7)² + 2(√5)(√7) = 5 + 7 + 2√35 = 12 + 2√35

Since 35 is not a perfect square, √35 is an irrational number. The sum of a rational number (12) and an irrational number (2√35) is an irrational number.

Explanation

Dividing the given equation 2xy cos θ + (x² - y²) sin θ = x² + y² by cos θ gives: 2xy + (x² - y²) tan θ = (x² + y²) sec θ

Squaring both sides and substituting sec² θ = 1 + tan² θ: [(x² - y²) tan θ + 2xy]² = (x² + y²)² (1 + tan² θ)

Expanding and collecting like terms: [(x² + y²)² - (x² - y²)²] tan² θ - 4xy(x² - y²) tan θ + [(x² + y²)² - 4x²y²] = 0

Using the algebraic identities (x² + y²)² - (x² - y²)² = 4x²y² and (x² + y²)² - 4x²y² = (x² - y²)²: 4x²y² tan² θ - 4xy(x² - y²) tan θ + (x² - y²)² = 0

This simplifies to a perfect square: [2xy tan θ - (x² - y²)]² = 0

Taking the square root: 2xy tan θ = x² - y² tan θ = (x² - y²) / (2xy)

Explanation

Evaluating the square roots in the expression: √144 = 12 √196 = 14

Substituting these values into the given expression gives: (12/7) × (14/12) × (7/14)

Canceling out the common terms in the numerator and denominator: (12 × 14 × 7) / (7 × 12 × 14) = 1

Explanation

Converting each fraction to its decimal value:

2/9 ≈ 0.222 3/8 = 0.375 4/9 ≈ 0.444 5/9 ≈ 0.555 8/11 ≈ 0.727

Arranging these values from smallest to largest gives 0.222 < 0.375 < 0.444 < 0.555 < 0.727, which corresponds to 2/9 < 3/8 < 4/9 < 5/9 < 8/11.

Explanation

Let the two numbers be 2x and 3x, where x is their highest common factor (HCF).

Since 2 and 3 are co-prime, the LCM of 2x and 3x is given by: LCM = 2 × 3 × x = 6x

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

The smallest of the two numbers is 2x = 2 × 20 = 40.

Explanation

Using the work-rate formula: 1 / Raju's time = (1 / Combined time) - (1 / Ranga's time)

Substituting the given values: 1 / Raju's time = (1 / 6) - (1 / 18) = (3 - 1) / 18 = 2 / 18 = 1 / 9

Taking the reciprocal gives Raju's time as 9 days.

Explanation

Dividing 45 by 0.09 can be solved by converting the decimal 0.09 into a fraction:

45 ÷ 0.09 = 45 ÷ (9/100) = 45 × (100/9) = 5 × 100 = 500

Explanation

Speed = Distance ÷ Time = 15.5 km ÷ 5 hours = 3.1 km/h

Distance covered in 6 hours = Speed × Time = 3.1 km/h × 6 hours = 18.6 km

Explanation

The place value of a digit in a number is calculated by multiplying the digit by its position value. In the number 76897, counting from right to left, the digit 8 occupies the hundreds place:

Place Value = Digit × Position Value = 8 × 100 = 800

Explanation

The standard trigonometric values are: sin 90° = 1 tan 45° = 1 cosec 90° = 1

Substituting these values into the given expression: sin 90° + tan 45° + cosec 90° = 1 + 1 + 1 = 3

Explanation

To find the smallest fraction, convert each fraction into its approximate decimal form:

11/12 ≈ 0.916 9/11 ≈ 0.818 10/14 ≈ 0.714 8/13 ≈ 0.615

Comparing the decimal values shows that 0.615 is the lowest value, making 8/13 the smallest fraction.