RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

The coding follows the formula Code = 2n - 1, where n is the 1-based position of the letter in the English alphabet.

Applying this formula to each letter in HEAD: H (8th letter): 2 × 8 - 1 = 15 E (5th letter): 2 × 5 - 1 = 9 A (1st letter): 2 × 1 - 1 = 1 D (4th letter): 2 × 4 - 1 = 7

Combining these coded values in order yields 15917.

Explanation

Let L be the length of the train in meters and v be its speed in m/s.

The speed of the train can be found by taking the ratio of the difference in platform lengths to the difference in time taken: v = (50 - 20) / (15 - 10) = 30 / 5 = 6 m/s

When passing the 50 m platform, the total distance covered by the train is L + 50: L + 50 = v × 15 L + 50 = 6 × 15 = 90 L = 90 - 50 = 40 m

Explanation

Length of train = Speed × Time

Speed in m/s = 60 × (5/18) = 50/3 m/s

Length of train = (50/3) × 33 = 550 m

Explanation

First, calculate the product inside the parentheses: 0.032 × 0.0032 = (32 × 10⁻³) × (32 × 10⁻⁴) = 1024 × 10⁻⁷ = 0.0001024

Next, add 0.00032 to this result: 0.0001024 + 0.00032 = 0.0004224

Explanation

Evaluating the expression step-by-step:

Inside the parentheses: ½ - ¼ = ¼

Substituting this back into the bracket: ⅓ - ¼ + ½ - ¼

Combining the like terms: -¼ - ¼ = -½ -½ + ½ = 0

So the bracket simplifies to ⅓.

Finally, multiplying by the outside fraction: ½ × ⅓ = ⅙

Explanation

For any set of consecutive numbers, the average is always equal to the middle term.

Since the average of the first 3 consecutive numbers is K, the middle term is K, making the three numbers K - 1, K, and K + 1.

Adding the next two consecutive numbers extends the set to 5 terms: K - 1, K, K + 1, K + 2, and K + 3. The middle term of these 5 numbers is K + 1, which is the new average.

Explanation

When interest is compounded half-yearly, the annual interest rate is halved and the number of time periods is doubled.

Rate per half-year R = 10% / 2 = 5% Number of half-years n = 2 × 2 = 4

Amount A = P × (1 + R/100)ⁿ A = 1,800 × (1 + 5/100)⁴ A = 1,800 × (1.05)⁴ A = 1,800 × 1.21550625 = ₹2,187.91

Compound Interest = Amount - Principal Compound Interest = ₹2,187.91 - ₹1,800 = ₹387.91

Explanation

Let the required number be x.

(3/4 - 2/3) - x = -1/6

First, simplify the term in parentheses: 3/4 - 2/3 = (9 - 8) / 12 = 1/12

Substitute this back into the equation: 1/12 - x = -1/6 x = 1/12 + 1/6 x = 1/12 + 2/12 x = 3/12 x = 1/4

Explanation

Using the identity tan θ - cot θ = -2 cot 2θ:

tan 75° - cot 75° = -2 cot(2 × 75°) = -2 cot 150°

Since cot 150° = cot(180° - 30°) = -cot 30° = -√3:

tan 75° - cot 75° = -2 × (-√3) = 2√3

Alternatively, tan 75° = 2 + √3 and cot 75° = 2 - √3. Subtracting cot 75° from tan 75° gives (2 + √3) - (2 - √3) = 2√3.

Explanation

Gain = Selling Price - Cost Price = 1000 - 950 = ₹50.

Gain Percentage = (Gain / Cost Price) × 100 = (50 / 950) × 100 = 100 / 19 % = 5 5/19 %

Explanation

'Durgeshnandini' (The Chieftain's Daughter) was written by Bankim Chandra Chattopadhyay and published in 1865. It was his debut Bengali novel, which played a pioneering role in the development of Indian historical fiction and helped lay the foundation for modern Bengali prose literature.

Explanation

Vindhyachal Thermal Power Station, located in the Singrauli district of Madhya Pradesh, is a coal-fired power plant operated by the National Thermal Power Corporation (NTPC). With an installed capacity of 4,760 MW, it stands as the largest thermal power generation facility in India.

Explanation

Sum = Average × Number of terms.

From the first given condition: a + 11 + 23 + 17 = 15 × 4 a + 51 = 60 a = 9

From the second given condition: a + b + 12 + 25 = 16 × 4 9 + b + 37 = 64 b + 46 = 64 b = 18

The value of a/b = 9/18 = ½.

Explanation

The formula for permutations is ⁿPᵣ = n! / (n - r)!, which represents the product of r consecutive decreasing integers starting from n.

Expanding 5040 as a product of consecutive integers starting from 10: 10 × 9 × 8 × 7 = 5040

Since the product contains 4 terms, r = 4.

Explanation

The relationship between Marked Price (MP), Cost Price (CP), Profit percentage (P%), and Discount percentage (D%) is given by the formula:

MP / CP = (100 + P%) / (100 - D%)

Substituting P = 20 and D = 20:

MP / CP = (100 + 20) / (100 - 20) = 120 / 80 = 3 / 2 = 1.5

Percentage by which MP is above CP = ((MP - CP) / CP) × 100 = (1.5 - 1) × 100 = 50%

Explanation

A number is completely divisible by 5, 12, and 18 if it is divisible by their Least Common Multiple (LCM).

LCM(5, 12, 18) = 180

Four-digit numbers range from 1000 to 9999.

Smallest 4-digit multiple of 180 = 180 × 6 = 1080 Largest 4-digit multiple of 180 = 180 × 55 = 9900

The number of such 4-digit multiples corresponds to the range of multipliers from 6 to 55: Count = 55 - 6 + 1 = 50

Explanation

For any two positive integers, Product of two numbers = GCD × LCM.

Let the second number be x. 27 × x = 9 × 270 x = (9 × 270) / 27 x = 90

Explanation

For 18 months compounded half-yearly, the number of compounding periods n = 3 and the rate per period r = 12% / 2 = 6% = 0.06.

The formula for the difference between compound interest and simple interest for 3 periods is: Difference = P × r² × (3 + r)

Substituting P = ₹1,00,000 and r = 0.06: Difference = 1,00,000 × (0.06)² × (3 + 0.06) Difference = 1,00,000 × 0.0036 × 3.06 Difference = 360 × 3.06 = ₹1,101.6

Explanation

Convert the given decimals into fractions for simple evaluation: 0.25 = 1/4 0.625 = 5/8

Substitute these fraction values into the expression: 0.25 × (0.25 × 0.25) ÷ 0.625 = (1/4 × 1/4 × 1/4) ÷ (5/8) = (1/64) ÷ (5/8) = (1/64) × (8/5) = 1 / (8 × 5) = 1/40 = 0.025

Explanation

Let x = 2a. The given equation becomes x + 1/x = 6.

To find 64a⁶ + 1/(64a⁶), we need the value of x⁶ + 1/x⁶.

Using the algebraic identity x³ + 1/x³ = (x + 1/x)³ - 3(x + 1/x): x³ + 1/x³ = 6³ - 3(6) = 216 - 18 = 198

Next, using the identity x⁶ + 1/x⁶ = (x³ + 1/x³)² - 2: x⁶ + 1/x⁶ = 198² - 2 = 39204 - 2 = 39202

Explanation

Let the three interior angles of the triangle be A, B, and C. By the angle sum property of a triangle:

A + B + C = 180°

Given that the sum of two angles equals the third angle (A + B = C), substitute C for (A + B) in the angle sum equation:

C + C = 180° 2C = 180° C = 90°

Explanation

Mohan earns ₹60 on odd days and spends ₹50 on even days, resulting in a net gain of ₹10 every 2-day cycle.

To reach ₹200 on an earning day before spending, he needs ₹140 accumulated from full 2-day cycles, followed by the ₹60 earned on the next odd day (140 + 60 = 200).

Number of 2-day cycles required to reach ₹140 = 140 / 10 = 14 cycles. Days in 14 cycles = 14 × 2 = 28 days.

He earns ₹60 and reaches ₹200 on the very next day, which is the 29th day.

Explanation

By the division algorithm: Dividend = (Divisor × Quotient) + Remainder

Let the divisor be D. Since the divisor is one-third of the quotient, the quotient Q = 3D.

Substituting the given values: 2200 = (D × 3D) + 13 2200 - 13 = 3D² 2187 = 3D² D² = 729 D = √729 = 27

Explanation

Number of people in the 18-25 years age-group in Delhi = 80000

Number of people moving from Delhi to Goa = 25% of 80000 = (25 / 100) × 80000 = 20000

Original total population of Goa = 175000

New total population of Goa = 175000 + 20000 = 195000

Explanation

Initial population of Delhi = 250,000

January migration into Delhi: 20% of Pune's 26-40 age group (65,000) = 0.20 × 65,000 = 13,000

February migration into Delhi: 10% of Bengaluru's 18-25 age group (50,000) = 0.10 × 50,000 = 5,000

Updated 18-25 age group population in Delhi = 80,000 + 5,000 = 85,000

March migration out of Delhi to Goa: 20% of Delhi's updated 18-25 age group (85,000) = 0.20 × 85,000 = 17,000

Net change in Delhi's population = +13,000 + 5,000 - 17,000 = +1,000

Total population of Delhi in April = 250,000 + 1,000 = 251,000

Explanation

When distance is constant, time is inversely proportional to speed (Time = Distance / Speed).

Given speed ratio = 5 : 4 : 6

Ratio of time taken = (1/5) : (1/4) : (1/6)

To clear the fractions, multiply each term by the LCM of denominators (LCM of 5, 4, 6 = 60): (1/5 × 60) : (1/4 × 60) : (1/6 × 60) = 12 : 15 : 10

Explanation

Using the percentage formula, Percentage = (Rate / 100) × Base Value:

First, calculate 25% of ₹900: 25% of ₹900 = (25 / 100) × 900 = 225

Next, calculate 6% of ₹225: 6% of ₹225 = (6 / 100) × 225 = 13.50

Alternatively, multiplying the percentages directly: (6 / 100) × (25 / 100) × 900 = (150 × 900) / 10000 = 13.50

Explanation

Let x be the required value to be added.

(½ + ⅓ + ⅕) + x = 3

Taking the LCM of the denominators 2, 3, and 5, which is 30: ½ + ⅓ + ⅕ = 15/30 + 10/30 + 6/30 = 31/30

Now, substituting the sum into the equation: 31/30 + x = 3 x = 3 - 31/30 x = (90 - 31) / 30 x = 59/30

Explanation

In square ABCD, AB = BC and ∠ABC = 90°. Since △QAB is equilateral, QB = AB and ∠QBA = 60°.

Since QB = AB and BC = AB, it follows that QB = BC, making △QBC an isosceles triangle.

The angle ∠QBC = ∠ABC - ∠QBA = 90° - 60° = 30°.

In △QBC, the angles opposite to equal sides are equal, so ∠QCB = ∠BQC. Using the angle sum property of a triangle: ∠QBC + ∠QCB + ∠BQC = 180° 30° + 2 × ∠QCB = 180° 2 × ∠QCB = 150° ∠QCB = 75°

Explanation

Using the formula for combinations: ⁿC₄ = n(n - 1)(n - 2)(n - 3) / 24

Given that ⁿC₄ = 70: n(n - 1)(n - 2)(n - 3) / 24 = 70 n(n - 1)(n - 2)(n - 3) = 70 × 24 = 1680

Expressing 1680 as the product of four consecutive decreasing integers: 1680 = 8 × 7 × 6 × 5

Comparing the highest factors on both sides gives n = 8.

Explanation

When a number leaves remainders r₁ and r₂ when divided by divisors d₁ and d₂ such that the difference (d₁ - r₁) = (d₂ - r₂) = k is constant, the required least number is given by LCM(d₁, d₂) - k.

Here, 18 - 11 = 7 and 24 - 17 = 7, so k = 7.

LCM(18, 24) = 72

Required least number = 72 - 7 = 65

Explanation

Arranging the positions in descending order of height: 1 (tallest) > 2 > 3 > 4 > 5 (shortest)

Since only two people are shorter than D, D must be in position 3: _ > _ > D > _ > _

The given conditions state that A is taller than E, and E is taller than F, yielding the relative order A > E > F. Since A is not the tallest, A cannot be in position 1. Therefore, A must occupy position 2, while E and F occupy positions 4 and 5 respectively: _ > A > D > E > F

The remaining person, C, must occupy position 1 as the tallest individual. Consequently, zero people are taller than C.

Explanation

Using the order of operations (BODMAS):

Evaluate the innermost parentheses: 5 - 2 = - 7

Substitute this back into the curly brackets: 5 - (- 7) - 5 = - 5 + 7 - 5 = - 3

Multiply by the outer term: 5 × (- 3) = 15

Explanation

The prime factorisation of 1024 is 2¹⁰. Since the LCM of two numbers is 1024, all prime factors of both numbers must be factors of 1024. The only prime factor of 1024 is 2, which means the prime number among the two given numbers is 2. As 2 is a factor of 1024, the greatest common divisor (GCD) of 2 and a number whose LCM with 2 is 1024 is 2.

Explanation

Let the two numbers be 3x and 5x.

3x + 5x = 80 8x = 80 x = 10

The greater number = 5x = 5 × 10 = 50.

Explanation

Let the length, breadth, and height of the cuboid be l, b, and h respectively.

Given condition: l² + b² + h² = ½ × 2(lb + bh + hl) l² + b² + h² - lb - bh - hl = 0

Multiplying both sides by 2 gives: (l - b)² + (b - h)² + (h - l)² = 0

Since the sum of non-negative squares is zero, l = b = h. Thus, the cuboid is a cube.

Given volume l³ = 729 cm³, which gives side length l = 9 cm.

Lateral surface area of the cube = 4l² = 4 × 9² = 4 × 81 = 324 cm².

Explanation

Let the cost price of 1 book be CP and the selling price of 1 book be SP.

Given that the selling price of 65 books equals the cost price of 100 books: 65 × SP = 100 × CP SP / CP = 100 / 65 = 20 / 13

Profit percentage = [(SP - CP) / CP] × 100 = [(20 - 13) / 13] × 100 = (7 / 13) × 100 = 700 / 13 % = 53 11/13 %

Explanation

1-day work of Ram and Shyam = 1/30 1-day work of Shyam and Bharti = 1/40 1-day work of Bharti and Ram = 1/60

Adding these three pairs gives 2 × (1-day work of Ram + Shyam + Bharti) = 1/30 + 1/40 + 1/60 = (4 + 3 + 2)/120 = 9/120 = 3/40.

Therefore, 1-day work of Ram, Shyam, and Bharti together = 3/(40 × 2) = 3/80.

Total time taken by all three together = 80/3 = 26 2/3 days.

Explanation

The highest common factor (HCF) of any two consecutive natural numbers n and n + 1 is always 1, because gcd(n, n + 1) = gcd(n, 1) = 1. Since the set of natural numbers from 200 to 478 includes consecutive integers such as 200 and 201, the only factor common to all the numbers in the range is 1.

Explanation

For an arithmetic progression, the average of the terms is equal to the average of the first and last terms:

Average = (First term + Last term) / 2

Substituting the first term (7) and the last term (77):

Average = (7 + 77) / 2 = 84 / 2 = 42

Explanation

Given x : y = 2 : 3 and y : z = 2 : 3. To make the common term y equal in both ratios, multiply x : y by 2 and y : z by 3.

x : y = (2 × 2) : (3 × 2) = 4 : 6 y : z = (2 × 3) : (3 × 3) = 6 : 9

Combining both ratios yields x : y : z = 4 : 6 : 9.

Explanation

Let N be the total number of women visiting the hotel.

Each woman spent N/4 rupees.

Total bill = Number of women × Amount spent by each woman 20,449 = N × (N/4) N² = 20,449 × 4 N² = 81,796 N = √81,796 = 286

Explanation

Janata Dal (United) is a regional political party primarily based in Bihar and affiliated with the National Democratic Alliance. It has no presence or political coalition with the Maha Vikas Aghadi in Maharashtra.

Explanation

Cancelling the common factor 0.9 in the numerator and denominator yields:

(0.9 × 0.9 × 0.9) / 0.9 = 0.9 × 0.9 = 0.81

Explanation

The Tomb of Sheikh Salim Chishti is located within the courtyard of the Jama Masjid at Fatehpur Sikri, Uttar Pradesh. It was constructed by Mughal Emperor Akbar between 1580 and 1581 to honor the Sufi saint Sheikh Salim Chishti, who had predicted the birth of Akbar's son, Prince Salim (Emperor Jahangir). Built with white marble and featuring exquisite serpent brackets and intricately carved jali screens, it is considered one of the finest examples of Mughal architecture.

Explanation

Solving the given equations: ∠A + ∠B = 110° ∠B - ∠A = 10°

Adding both equations gives 2∠B = 120°, which yields ∠B = 60°. Substituting this into ∠A + 60° = 110° gives ∠A = 50°.

Using the angle sum property of a triangle (∠A + ∠B + ∠C = 180°): ∠C = 180° - 110° = 70°

Comparing the interior angles gives ∠A (50°) < ∠B (60°) < ∠C (70°).

In any triangle, the side opposite the largest angle is the longest side, and the side opposite the smallest angle is the shortest side. The side opposite to largest angle ∠C (70°) is AB, and the side opposite to smallest angle ∠A (50°) is BC. Therefore, AB is the longest side and BC is the shortest side.

Explanation

Distance covered in 3 hours = 10⅕ km = 51/5 km

Speed = Distance ÷ Time = (51/5) ÷ 3 = 17/5 km/h

Distance covered in 5 hours = Speed × Time = (17/5) × 5 = 17 km

Explanation

Cement Corporation of India Limited (CCI) was incorporated in 1965 as a Central Public Sector Enterprise under the administrative control of the Ministry of Heavy Industries. It is categorized as a Miniratna enterprise and does not hold Maharatna status, which is granted to larger public sector units with extensive financial autonomy.

Explanation

Television remotes use infrared light to transmit signals to electronic devices. A light-emitting diode (LED) at the front of the remote pulses invisible infrared light at a specific frequency, typically around 940 nanometers, to send binary-coded signals. An infrared receiver on the device detects these light pulses and converts them back into electrical signals that execute specific commands, such as changing channels or adjusting the volume.

Explanation

The smallest perfect cube greater than 2712 is 14³, which equals 2744 (since 13³ = 2197 and 14³ = 2744).

To find the required positive integer to be added: 2744 - 2712 = 32