RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

Ammonia has the chemical formula NH₃, containing one nitrogen atom and three hydrogen atoms. The atomic mass of nitrogen is 14 u and that of hydrogen is 1 u. Therefore, the total mass of hydrogen in ammonia is 3 × 1 u = 3 u. The ratio of nitrogen to hydrogen by mass is 14 : 3.

Explanation

Increase in simple interest rate = 12½% - 8% = 4.5% = 9/2%

For a period of 1 year, the increase in interest income is given by the formula: Increase in Interest = (Principal × Increase in Rate × Time) / 100

Substituting the given values: 369 = (Principal × 4.5 × 1) / 100 Principal = (369 × 100) / 4.5 = 41 × 200 = 8,200

Explanation

Given that 19 × CP = 25 × SP, the ratio of Cost Price to Selling Price is CP / SP = 25 / 19.

Taking CP = 25 and SP = 19: Loss = CP - SP = 25 - 19 = 6

Percentage loss = (Loss / CP) × 100 = (6 / 25) × 100 = 24%

Explanation

Road density is defined as the length of the road network per unit area of land (typically measured per 1,000 square kilometers). Kerala has the highest road density among Indian states, exceeding 5,000 km per 1,000 square km, which is well above the national average. This extensive network is largely driven by the state's high population density, unique urban-rural continuum, and dispersed settlement patterns, necessitating widespread road connectivity across all rural and urban areas.

Explanation

The series increases by consecutive prime numbers:

1 + 2 = 3 3 + 3 = 6 6 + 5 = 11 11 + 7 = 18 18 + 11 = 29 29 + 13 = 42

The prime number following 13 is 17. Adding 17 to 42 gives 42 + 17 = 59.

Explanation

The Bhabha Atomic Research Centre (BARC), established by Dr. Homi Jehangir Bhabha as the Atomic Energy Establishment, Trombay (AEET) in January 1954, is India's premier multidisciplinary nuclear research facility. BARC is regarded as the mother institution of nuclear and accelerator technology R | D facilities in India, as it provided the foundational infrastructure, expertise, and trained scientific workforce that led to the establishment of several specialized Department of Atomic Energy research centres across the country.

Explanation

Given the equation 2.5x = 0.05y, we can express y in terms of x by dividing both sides by 0.05:

y / x = 2.5 / 0.05 = 250 / 5 = 50

Thus, y = 50x.

Substituting y = 50x into the given expression:

(y - x) / (y + x) = (50x - x) / (50x + x) = 49x / 51x = 49/51

Explanation

For any two prime numbers x and y, their LCM is equal to their product x × y.

Given x × y = 119, factorizing 119 gives 7 × 17. Since x > y, x = 17 and y = 7.

Substituting these values into the required expression: 3y - x = 3(7) - 17 = 21 - 17 = 4

Explanation

Tangents drawn from an external point to a circle are equal in length, so SA = SB = 12 cm. In triangle SAB, since SA = SB, the angles opposite to these sides are equal, making angle SAB = angle SBA.

The sum of angles in a triangle is 180°. Given angle ASB = 60°, each of the base angles is calculated as (180° - 60°) / 2 = 60°.

Since all three angles are equal to 60°, triangle SAB is an equilateral triangle. Therefore, the length of chord AB is equal to SA and SB, which is 12 cm.

Explanation

Convert the mixed fractions and decimals: 5½ = 5.5, 7½ = 7.5, and (0.2)² = 0.04.

Simplify the expression 4.25 ÷ (0.2)²: 4.25 ÷ 0.04 = 106.25

Simplify inside the curly braces: 5.5 - (7.5 + 2.8 ÷ x) = 5.5 - 7.5 - 2.8 ÷ x = -2 - 2.8 ÷ x

Substitute these back into the given equation: 8.5 - (-2 - 2.8 ÷ x) × 106.25 = 306 8.5 + (2 + 2.8 ÷ x) × 106.25 = 306

Subtract 8.5 from both sides: (2 + 2.8 ÷ x) × 106.25 = 297.5

Divide both sides by 106.25: 2 + 2.8 ÷ x = 297.5 ÷ 106.25 = 2.8

Subtract 2 from both sides: 2.8 ÷ x = 0.8 x = 2.8 ÷ 0.8 = 3.5

Explanation

Time taken by train R to cover the distance = 7 hours (5 a.m. to 12 p.m.). Time taken by train S to cover the distance = 10.5 hours or 21/2 hours (7 a.m. to 5:30 p.m.).

Let the total distance between Chennai and Vijayawada be D. Speed of train R = D/7 Speed of train S = 2D/21

In the 2 hours before train S starts (from 5 a.m. to 7 a.m.), train R covers a distance of 2 × (D/7) = 2D/7. Remaining distance at 7 a.m. = D - 2D/7 = 5D/7.

Relative speed of both trains moving in opposite directions = D/7 + 2D/21 = 5D/21.

Time taken to cross each other after 7 a.m. = (5D/7) ÷ (5D/21) = 3 hours.

Crossing time = 7 a.m. + 3 hours = 10 a.m.

Explanation

Time taken by Reena alone = (T₁ × T₂) / (T₂ - T₁)

Here, T₁ = 6 days (combined time of Reena and Jaya) and T₂ = 8 days (time taken by Jaya alone).

Substituting the values into the formula: Time = (6 × 8) / (8 - 6) = 48 / 2 = 24 days

Alternatively, using 1-day work rates: Reena's 1-day work = (1/6) - (1/8) = (4 - 3) / 24 = 1/24 work per day, which means Reena alone requires 24 days to complete the entire work.

Explanation

Translating the word problem into a mathematical expression:

Value = 3 × (3/10) × (3/100) × (3/1000) × 30

Multiplying the numerators together: Numerator = 3 × 3 × 3 × 3 × 30 = 2430

Multiplying the denominators together: Denominator = 10 × 100 × 1000 = 1,000,000

Dividing the numerator by the denominator: Value = 2430 / 1,000,000 = 0.00243

Explanation

Total ratio units = 35 + 28 + 20 = 83. Value of 1 ratio unit = 1162 / 83 = 14.

Share of A = 35 × 14 = 490 Share of B = 28 × 14 = 392 Share of C = 20 × 14 = 280

Explanation

A number is divisible by 9 if the sum of its digits is divisible by 9.

Sum of digits of 7x5462 = 7 + x + 5 + 4 + 6 + 2 = 24 + x

The next multiple of 9 after 24 is 27. Therefore, 24 + x = 27, which gives x = 3.

Explanation

Ratio of Gold to Copper in Alloy A = 7 : 2 (sum = 9 parts) Ratio of Gold to Copper in Alloy B = 7 : 11 (sum = 18 parts)

To take equal quantities of both alloys, make the total parts equal by multiplying Alloy A's ratio by 2: Alloy A = 14 : 4 Alloy B = 7 : 11

When mixed in equal quantities to form C: Total Gold in C = 14 + 7 = 21 parts Total Copper in C = 4 + 11 = 15 parts

Ratio of Gold to Copper in C = 21 : 15 = 7 : 5

Since 5 parts of Copper correspond to 10 kg, 1 part = 2 kg. Amount of Gold in C = 7 parts = 7 × 2 kg = 14 kg.

Explanation

In any triangle ABC, the sum of all interior angles is 180°, so ∠A + ∠B + ∠C = 180°. Given that ∠A + ∠B = ∠C, substituting this into the angle sum property gives ∠C + ∠C = 180°, which simplifies to 2∠C = 180°, yielding ∠C = 90°.

Explanation

Using the algebraic identity for conjugate sums, (a + b)/(a - b) + (a - b)/(a + b) = 2(a² + b²)/(a² - b²):

x + y = 2((√3)² + 1²) / ((√3)² - 1²) = 2(3 + 1) / (3 - 1) = 8 / 2 = 4

Since y is the reciprocal of x, xy = 1.

Applying the identity x² + y² = (x + y)² - 2xy:

x² + y² = 4² - 2(1) = 16 - 2 = 14

Explanation

Let Madhan's earnings be 100.

Since Ajay earned 30% less than Madhan, Ajay's earnings = 100 - 30 = 70.

Since Ravi earned 50% more than Ajay, Ravi's earnings = 70 + (50% of 70) = 70 + 35 = 105.

Percentage by which Ravi earned more than Madhan = ((105 - 100) / 100) × 100 = 5%.

Explanation

Let the two parts of the sum be P₁ and P₂.

Using the simple interest formula SI = (P × R × T) / 100, set the interest from both parts equal: (P₁ × 15 × 4) / 100 = (P₂ × 10 × 6) / 100 60 P₁ = 60 P₂ P₁ = P₂

Since the total sum P₁ + P₂ = ₹2,700, each part is half of the total: P₁ = 2,700 / 2 = ₹1,350

Explanation

To find the Least Common Multiple (LCM) of numbers given in prime-factorized form, take the highest power of each prime factor present across all the numbers.

Comparing the exponents for each prime factor: For 2: max(2, 3, 1) = 3 → 2³ For 3: max(3, 2, 1) = 3 → 3³ For 5: max(1, 2, 3) = 3 → 5³ For 7: max(2, 4, 1) = 4 → 7⁴ For 11: max(0, 0, 1) = 1 → 11

Multiplying these highest powers gives the LCM = 2³ × 3³ × 5³ × 7⁴ × 11.

Explanation

Distance d = (v₁ × v₂ / |v₁ - v₂|) × Δt

Given v₁ = 6 km/h, v₂ = 7.5 km/h, and the time difference Δt = 1 h 4 min = 64/60 hours = 16/15 hours.

Substituting the values: d = (6 × 7.5 / 1.5) × (16/15) d = 30 × (16/15) d = 32 km

Explanation

Since x and y are positive acute angles, x + y lies between 0° and 180°.

Given cos(x + y) = ½, taking the inverse cosine gives: x + y = 60°

Given sin(x - y) = 0 and x ≥ y, taking the inverse sine gives: x - y = 0°, which means x = y

Substituting x = y into x + y = 60° gives: 2x = 60° x = 30°

Since x = y, y = 30°.

Explanation

Let the initial number of apples be x.

After selling ⅓ of the apples to the first customer, the remaining number of apples is (⅔)x.

The second customer gets ½ of the remaining apples, which is ½ × (⅔)x = (⅓)x. The remaining apples after this sale equal (⅔)x - (⅓)x = (⅓)x.

The third customer gets ⅓ of (⅓)x plus 5 apples, which equals (1/9)x + 5.

The remaining apples after the third customer are: (⅓)x - ((1/9)x + 5) = (2/9)x - 5

Given that the man is left with 3 apples: (2/9)x - 5 = 3 (2/9)x = 8 x = 36

Explanation

Squaring both given equations and adding them together:

x² = (sin A + sin B)² = sin²A + sin²B + 2 sin A sin B y² = (cos A + cos B)² = cos²A + cos²B + 2 cos A cos B

Adding x² and y²: x² + y² = (sin²A + cos²A) + (sin²B + cos²B) + 2 (cos A cos B + sin A sin B)

Using the identities sin²θ + cos²θ = 1 and cos A cos B + sin A sin B = cos(A - B): x² + y² = 1 + 1 + 2 cos(A - B) x² + y² = 2 + 2 cos(A - B)

Rearranging the terms: 2 cos(A - B) = x² + y² - 2 cos(A - B) = (x² + y² - 2) / 2

Explanation

For any two distinct prime numbers x and y, their LCM is simply their product, x × y.

Factoring 319 into prime numbers gives 319 = 29 × 11.

Since x > y, x = 29 and y = 11.

Substituting these values into the given expression: 2x - 5y = 2(29) - 5(11) = 58 - 55 = 3.

Explanation

The total surface area of a cuboid is given by the formula A = 2 × (lb + bh + hl), where l is length, b is breadth, and h is height.

Given l = 16 m, b = 14 m, and h = 7 m:

Surface Area = 2 × (16 × 14 + 14 × 7 + 7 × 16) = 2 × (224 + 98 + 112) = 2 × 434 = 868 m²

Explanation

Calculate the square roots: √625 = 25 √25 = 5 √196 = 14

Substitute these values into the given expression: (√625 / 11) × (14 / √25) × (11 / √196) = (25 / 11) × (14 / 5) × (11 / 14)

Canceling the common terms 11 and 14 from both numerator and denominator simplifies the expression to: 25 / 5 = 5

Explanation

By modular arithmetic, 17 ≡ -1 (mod 18). Raising both sides to the power of 200 gives 17²⁰⁰ ≡ (-1)²⁰⁰ (mod 18). Since the exponent 200 is an even number, (-1)²⁰⁰ = 1.

Explanation

Volume of a sphere is proportional to the cube of its radius (V ∝ R³). When a large sphere of radius R is melted to form n smaller identical spheres of radius r, the total volume remains conserved:

n = (R / r)³

Given R = 10 cm and r = 2.5 cm:

n = (10 / 2.5)³ = 4³ = 64

Explanation

The area of a circle is directly proportional to the square of its radius (A ∝ r²). When a linear dimension is reduced by x%, the percentage reduction in area is given by the formula:

Percentage reduction in area = 2x - x²/100 %

Substituting x = 15: Percentage reduction = 2(15) - (15)²/100 = 30 - 225/100 = 30 - 2.25 = 27.75%

Explanation

The curved surface area (CSA) of a cone is given by the formula CSA = πrl, where r is the base radius and l is the slant height.

Since the base diameters are equal, the radii of both cones are equal (r₁ = r₂ = r). The ratio of their curved surface areas is:

CSA₁ / CSA₂ = (π × r × l₁) / (π × r × l₂) = l₁ / l₂

Given that the ratio of their slant heights l₁ : l₂ is 3 : 4, the ratio of their curved surface areas is also 3 : 4.

Explanation

Let total work = LCM(18, 24, 36) = 72 units.

Daily work done by each pair: A + B = 72 / 18 = 4 units B + C = 72 / 24 = 3 units A + C = 72 / 36 = 2 units

Adding all three equations gives 2(A + B + C) = 9 units, which means A + B + C = 4.5 units/day.

A's daily work = (A + B + C) - (B + C) = 4.5 - 3 = 1.5 units/day Time taken by A = 72 / 1.5 = 48 days

B's daily work = (A + B + C) - (A + C) = 4.5 - 2 = 2.5 units/day Time taken by B = 72 / 2.5 = 28 4/5 days

C's daily work = (A + B + C) - (A + B) = 4.5 - 4 = 0.5 units/day Time taken by C = 72 / 0.5 = 144 days

Explanation

Let the total length of the pole be L meters.

Sum of colored fractions = 1/10 + 1/20 + 1/30 + 1/40

Taking the LCM of the denominators (120): Sum = (12 + 6 + 4 + 3) / 120 = 25/120 = 5/24

Fraction of the green portion = 1 - 5/24 = 19/24

Given that the green portion measures 4.75 m: (19/24) × L = 4.75 L = (4.75 × 24) / 19 L = 114 / 19 = 6 m

Explanation

Let C be the cost price of the chair and T be the cost price of the table.

From the given conditions: -0.05C + 0.15T = 300 0.05C + 0.05T = 180

Multiplying both equations by 20 simplifies them to: -C + 3T = 6000 C + T = 3600

Adding the two linear equations eliminates C: 4T = 9600 T = 2400

The actual cost price of the table is ₹2,400.

Explanation

Since savings = income - expenditure, the ratio unit difference for both A and B is the same: For A: 5 - 3 = 2 units For B: 4 - 2 = 2 units

Thus, 2 ratio units represent the savings of ₹1,600.

1 ratio unit = 1600 / 2 = ₹800

Income of A = 5 × ₹800 = ₹4,000

Explanation

The maximum production in a single week occurred in the fifth week, which is 1400 cycles.

If 1400 cycles were produced in each of the 5 weeks, total production would be: 5 × 1400 = 7000 cycles

Actual total cycles produced over the 5 weeks: 800 + 1360 + 1000 + 900 + 1400 = 5460 cycles

Number of additional cycles that would have been produced: 7000 − 5460 = 1540 cycles

Explanation

Number of days from 5 Feb 2005 to 18 April 2005 (both dates included): February (non-leap year) = 24 days March = 31 days April = 18 days Total days = 24 + 31 + 18 = 73 days = 73 / 365 = 1/5 year

Using the simple interest formula: Simple Interest = (P × R × T) / 100 = (3000 × 25/4 × 1/5) / 100 = (3000 × 5) / 400 = 37.50

Explanation

The number of pairs of non-adjacent persons among n people standing in a circle is given by the formula for the number of diagonals in an n-sided polygon: Non-adjacent pairs = n(n - 3)/2

Since each non-adjacent pair sings a 2-minute song, the total time for all pairs is: Total time = [n(n - 3)/2] × 2 = n(n - 3) minutes

Given that the total time taken is 40 minutes: n(n - 3) = 40 n² - 3n - 40 = 0 (n - 8)(n + 5) = 0

Since the number of persons n must be a positive integer, n = 8.

Explanation

The sum of n observations is given by Sum = Mean × n.

Sum of initial 5 observations = 5 × 20 = 100

With the addition of the 6th observation, the new mean increases by 2 to become 20 + 2 = 22 for 6 observations.

Sum of 6 observations = 6 × 22 = 132

Value of the 6th observation = Sum of 6 observations - Sum of 5 observations = 132 - 100 = 32

Explanation

Number of college teachers = 1500 Number of school girls = 3000

Ratio of college teachers to school girls = 1500 : 3000 = 1 : 2

Explanation

Average sale per customer for Cakes and Pastries = 210 / 14 = ₹15 Average sale per customer for Biscuits = 120 / 12 = ₹10

To maximize the increase in sales, each customer buying bread must purchase the item with the higher value, which is Cakes and Pastries at ₹15 per customer.

Total customers buying any type of bread = 40 + 16 = 56

Maximum increase in sales = 56 × ₹15 = ₹840

Explanation

From the given equation a - 5 = b, rearranging terms gives a - b = 5 and b - a = -5.

Taking the absolute values: |a - b| = |5| = 5 |b - a| = |-5| = 5

Adding the two absolute values yields: |a - b| + |b - a| = 5 + 5 = 10.

Explanation

Speed = Distance ÷ Time

Distance = 600 m = 0.6 km Time = 5 minutes = 5/60 hour = 1/12 hour

Substituting these values into the formula: Speed = 0.6 ÷ (1/12) = 0.6 × 12 = 7.2 km/h

Alternatively, calculating speed in meters per second first: Speed = 600 m ÷ (5 × 60 s) = 600 ÷ 300 = 2 m/s

Converting from m/s to km/h by multiplying by 18/5: Speed = 2 × (18/5) = 36/5 = 7.2 km/h

Explanation

Using the standard algebraic identity x² - 2xy + y² = (x - y)², the expression simplifies to:

√(x² - 2xy + y²) = √((x - y)²) = |x - y|

Substituting x = 25 and y = 13:

|25 - 13| = 12

Explanation

When a circle circumscribes a square, the diameter of the circumscribed circle is equal to the diagonal of the square.

Given radius r = 8 cm, the diameter of the circle is 2 × 8 = 16 cm.

For a square of side length a, the length of the diagonal is given by a√2.

Setting the diagonal equal to the diameter: a√2 = 16 a = 16 / √2 = 8√2 cm

Explanation

The difference (D) between compound interest and simple interest on a principal sum (P) for 2 years at an annual interest rate of R% is given by the formula:

D = P × (R / 100)²

Substituting D = 16.32 and R = 5:

16.32 = P × (5 / 100)² 16.32 = P × (1 / 20)² 16.32 = P / 400 P = 16.32 × 400 = 6,528

Explanation

Total students across all institutions = (3000 + 2500) + (9000 + 11000) + (15000 + 16000) = 56500

Total teachers in college and university = 1500 + 2000 = 3500

Ratio = 56500 : 3500 = 113 : 7

Explanation

Distance = Speed × Time = 3 km/h × 8 hours = 24 km

Time taken at 16 km/h = Distance / Speed = 24 km / 16 km/h = 1.5 hours

Explanation

For compound interest, the amount after t years is given by P(1 + r/100)ᵗ.

Equating the amounts of A and B: A(1 + 3/100)⁷ = B(1 + 3/100)⁹

Dividing both sides by (1 + 3/100)⁷: A/B = (1 + 3/100)² = (103/100)² = 10609/10000

The ratio of A to B is 10609 : 10000. Since the sum of the ratio units is 10609 + 10000 = 20609, which equals the total amount ₹20,609: A = ₹10,609 B = ₹10,000