RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

Since speed is constant, time taken is directly proportional to distance: Time = 60 × (105 / 35) = 60 × 3 = 180 min.

Explanation

Factorize the quadratic equation by grouping the terms:

y² - √5 y - y + √5 = 0 y(y - √5) - 1(y - √5) = 0 (y - 1)(y - √5) = 0

Setting each factor to zero gives: y - 1 = 0 ⇒ y = 1 y - √5 = 0 ⇒ y = √5

Explanation

Evaluating the terms inside the braces: 2 - 1/3 = 5/3 2/3 × 1 1/5 = 2/3 × 6/5 = 4/5 5/3 + 4/5 = (25 + 12)/15 = 37/15

Evaluating the remaining multiplication: 3/5 × 2 5/7 = 3/5 × 19/7 = 57/35

Adding both parts: 37/15 + 57/35 = (259 + 171)/105 = 430/105 = 86/21

Converting 86/21 to a mixed fraction yields 4 2/21.

Explanation

Curved surface area of a cylinder is given by 2πrh and volume by πr²h.

Given 2πrh = 660 cm² and h = 14 cm: 2 × (22/7) × r × 14 = 660 88r = 660 r = 15/2 cm

Substituting r and h into the volume formula: Volume = πr²h Volume = (22/7) × (15/2)² × 14 Volume = 22 × (225/4) × 2 Volume = 11 × 225 = 2475 cm³

Explanation

Efficiency and time taken are inversely proportional for the same total work (Work = Efficiency × Time).

Let Sundar's efficiency be 100 units/day. Since Raj is 60% more efficient than Sundar, Raj's efficiency = 160 units/day.

Total Work = Sundar's efficiency × Time taken by Sundar Total Work = 100 × 16 = 1600 units

Days taken by Raj alone = Total Work / Raj's efficiency Days taken by Raj alone = 1600 / 160 = 10 days

Explanation

Ravi's salary = Mohan's salary × (1 + Percentage Increase / 100)

Given Mohan's salary = ₹1,600 and the increase is 20%:

Ravi's salary = 1,600 × (1 + 20/100) = 1,600 × 1.20 = ₹1,920

Explanation

First, add the mixed fractions inside the bracket: 3 1/7 + 4 3/7 = (3 + 4) + (1/7 + 3/7) = 7 + 4/7 = 53/7

Next, add 7/6 to 53/7 by converting both fractions to have a common denominator of 42: 53/7 + 7/6 = (53 × 6 + 7 × 7) / 42 = (318 + 49) / 42 = 367/42

Explanation

To find the least square number divisible by 4, 5, 8, 16, and 32, we first calculate their Least Common Multiple (LCM).

Prime factorizations: 4 = 2² 5 = 5¹ 8 = 2³ 16 = 2⁴ 32 = 2⁵

LCM(4, 5, 8, 16, 32) = 2⁵ × 5¹ = 160

For a number to be a perfect square, every prime factor in its prime factorization must have an even exponent. Multiplying 160 (2⁵ × 5¹) by 2¹ × 5¹ = 10 gives 2⁶ × 5² = 1600, which is the smallest perfect square divisible by all the given numbers.

Explanation

Expressing both equations with common bases on both sides:

From 2ˣ = 4ʸ⁺¹, write 4 as 2²: 2ˣ = (2²)ʸ⁺¹ = 2²ʸ⁺² Equating the exponents: x = 2y + 2

From 3ʸ = 3ˣ⁻⁹, equating the exponents directly: y = x - 9

Substituting y = x - 9 into x = 2y + 2: x = 2(x - 9) + 2 x = 2x - 16 x = 16

Substituting x = 16 into y = x - 9: y = 16 - 9 = 7

Explanation

Cost Price = Selling Price ÷ (1 + Profit%) Cost Price = 750 ÷ 1.25 = ₹600

When the article is sold for ₹660: Profit Amount = 660 - 600 = ₹60 Profit Percentage = (Profit Amount ÷ Cost Price) × 100 = (60 ÷ 600) × 100 = 10%

Explanation

Let x be the number of rows. Since the number of trees in each row is equal to the number of rows, the total number of trees is given by:

Total trees = x × x = x²

Given x² = 5776, taking the square root on both sides:

x = √5776 = 76

Explanation

Using the BODMAS rule, evaluate the mathematical expression from the innermost brackets outward:

Innermost parenthesis: (8 × ½) = 4 Curly brackets: {6 - 4} = 2 Square brackets: [4 - 3 × 2] = [4 - 6] = -2 Final multiplication: 18 × (-2) = -36

Explanation

The equation of a straight line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula:

(y - y₁) = [(y₂ - y₁) / (x₂ - x₁)] × (x - x₁)

Substituting (x₁, y₁) = (-2, 5) and (x₂, y₂) = (1, 3):

(y - 5) = [(3 - 5) / (1 - (-2))] × (x - (-2))

(y - 5) = (-2 / 3) × (x + 2)

3(y - 5) = -2(x + 2)

3y - 15 = -2x - 4

2x + 3y - 11 = 0

Explanation

Convert the mixed fraction to an improper fraction: 5 44/49 = (5 × 49 + 44) / 49 = (245 + 44) / 49 = 289 / 49

Taking the square root gives: √(289 / 49) = √289 / √49 = 17 / 7

Explanation

Expand both sides of the given algebraic equation: 3(5x + 2) - 4 = 2(1 - 4x) 15x + 6 - 4 = 2 - 8x 15x + 2 = 2 - 8x

Group like terms together to isolate x: 15x + 8x = 2 - 2 23x = 0 x = 0

Explanation

For a number to be divisible by 5, its last digit S must be 0 or 5. Since the number is also divisible by 8, it must be even, which fixes S = 0.

For a number to be divisible by 8, the number formed by its last three digits, 3#S, must be divisible by 8. With S = 0, the last three digits form 3#0. Substituting # = 2 gives 320, which is divisible by 8 (320 ÷ 8 = 40). Thus, # = 2 and S = 0 satisfy both divisibility rules.

Explanation

The intercept form of the equation of a plane is x/a + y/b + z/c = 1, where a, b, and c represent the x, y, and z intercepts respectively.

Dividing both sides of the given plane equation 3x - 4y - 2z = 6 by 6: 3x/6 - 4y/6 - 2z/6 = 6/6 x/2 + y/(-3/2) + z/(-3) = 1

Comparing this with the standard intercept form yields the intercepts on the coordinate axes as 2, -3/2, and -3.

Explanation

For any two positive integers, the product of the numbers is equal to the product of their LCM and HCF:

First Number × Second Number = LCM × HCF

18 × x = 126 × 2

x = (126 × 2) / 18 = 7 × 2 = 14

Explanation

If a constant k is added to each entry xᵢ in a dataset of size n, the new arithmetic mean x̄' is given by:

x̄' = ∑(xᵢ + k) / n = (∑xᵢ + nk) / n = (∑xᵢ / n) + k = x̄ + k

Since each entry is increased by 7, k = 7. Therefore, the arithmetic mean increases by 7.

Explanation

Simple Interest = Principal × Monthly Rate × Time in months

The rate of 5 paise per rupee per month equals a monthly interest rate of 5/100 or 5%.

Simple Interest = 10,000 × (5 / 100) × 6 = 3,000

Explanation

Comparing the performance of Boys and Girls in Board X across all given years shows that Girls scored higher than Boys in 2014 (77.57% vs 75.85%), 2015 (73.80% vs 72.58%), 2017 (82.00% vs 74.45%), 2018 (78.55% vs 70.56%), and 2019 (81.48% vs 79.18%). In 2016, Boys scored 81.00% while Girls scored 77.80%, making 2016 the single year in which Boys of Board X achieved a higher score than Girls.

Explanation

The average score for Girls from Board X over the 6-year period (2014 to 2019) is calculated by taking the sum of their percentage scores and dividing by 6:

Sum = 77.57% + 73.80% + 77.80% + 82.00% + 78.55% + 81.48% = 471.20%

Average = 471.20% / 6 = 78.53%

Comparing the 6-year sum and average across all four groups: Girls from Board X: Sum = 471.20%, Average = 78.53% Girls from Board Y: Sum = 457.65%, Average = 76.28% Boys from Board X: Sum = 453.62%, Average = 75.60% Boys from Board Y: Sum = 434.35%, Average = 72.39%

Explanation

The absolute increase in percentage score for Girls of Board X in the years with positive growth from the previous year is:

Year 2016 = 77.80% - 73.80% = 4.00% Year 2017 = 82.00% - 77.80% = 4.20% Year 2019 = 81.48% - 78.55% = 2.93%

In 2015 and 2018, the percentage scores decreased. The maximum absolute increase is 4.20% in 2017.

Explanation

Total number of undergraduate students (B.B.A + B.C.A) = 182 + 91 = 273

Total number of postgraduate students (M.B.A + M.C.A) = 455 + 182 = 637

Ratio of undergraduate students to postgraduate students = 273 : 637

Dividing both terms by their greatest common divisor 91: 273 ÷ 91 = 3 637 ÷ 91 = 7

Ratio = 3 : 7

Explanation

Radius of the inner circle, r = 6 cm. Radius of the outer circle, R = 6 + 50% of 6 = 9 cm.

The area of the annulus (ring between two concentric circles) is given by the formula: Area = π(R² - r²)

Using π = 22/7 and substituting the radius values: Area = (22/7) × (9² - 6²) Area = (22/7) × (81 - 36) Area = (22/7) × 45 Area = 990/7 cm²

Explanation

A rational number in its simplest form, p/q, has a terminating decimal representation if and only if the prime factorization of the denominator q contains no prime factors other than 2 and 5 (that is, q = 2ᵃ × 5ᵇ, where a and b are non-negative integers). For 2/5, the denominator is 5 = 2⁰ × 5¹, which satisfies this condition and yields the terminating decimal 0.4.

Explanation

Using the trigonometric identity: cosec²A - cot²A = 1

Factoring as a difference of squares: (cosecA - cotA)(cosecA + cotA) = 1

Substituting cosecA - cotA = 3 into the equation: 3 × (cosecA + cotA) = 1 cosecA + cotA = 1/3

Explanation

Let a/b = b/c = c/d = k.

Expressing a, b, and c in terms of d and k: c = dk, b = dk², and a = dk³.

Substituting these values into the numerator and denominator: b³ + c³ + d³ = d³k⁶ + d³k³ + d³ = d³(k⁶ + k³ + 1) a³ + b³ + c³ = d³k⁹ + d³k⁶ + d³k³ = d³k³(k⁶ + k³ + 1)

Dividing the numerator by the denominator: (b³ + c³ + d³) / (a³ + b³ + c³) = [d³(k⁶ + k³ + 1)] / [d³k³(k⁶ + k³ + 1)] = 1 / k³

Since a/d = (dk³)/d = k³, taking the reciprocal gives 1/k³ = d/a.

Explanation

The net percentage change in area when two dimensions are increased by a% and b% is given by the formula:

Net percentage increase = a + b + (a × b)/100 %

Substituting a = 25 and b = 25:

Net percentage increase = 25 + 25 + (25 × 25)/100 % = 50 + 625/100 % = 50 + 6.25% = 56.25% = 56 ¼ %

Explanation

Using modular arithmetic properties:

11 ≡ 1 (mod 10)

Since 11 leaves a remainder of 1 when divided by 10, any positive integer power of 11 will also leave a remainder of 1 when divided by 10: 11⁹ ≡ 1⁹ ≡ 1 (mod 10)

Adding 7 to both sides of the congruence: (11⁹ + 7) ≡ (1 + 7) (mod 10) (11⁹ + 7) ≡ 8 (mod 10)

Dividing 8 by 10 leaves a remainder of 8.

Explanation

The prime factorizations of the given numbers are:

156 = 2² × 3 × 13 455 = 5 × 7 × 13 1183 = 7 × 13²

The highest common prime factor present in all three numbers is 13.

Therefore, HCF(156, 455, 1183) = 13.

Explanation

To convert the recurring decimal x = 0.1246464... into p/q form, multiply x by powers of 10 to isolate the repeating portion:

100x = 12.464646... 10000x = 1246.464646...

Subtracting the first equation from the second: 10000x - 100x = 1246.464646... - 12.464646... 9900x = 1234 x = 1234 / 9900

Simplifying by dividing the numerator and denominator by 2: x = 617 / 4950

Alternatively, using the standard formula for mixed recurring decimals: Numerator = (Entire digits up to repeating block) - (Non-repeating digits) = 12464 - 124 = 12340 Denominator = Two 9s (for repeating digits 6, 4) followed by three 0s (for non-repeating digits 1, 2, 4) = 99000

p/q = 12340 / 99000 = 617 / 4950

Explanation

Given sin A = 4/sqrt(17) for an acute angle A, cos A = 1/sqrt(17) and tan A = 4. Applying the double-angle identity tan 2A = (2 tan A) / (1 - tan² A) gives tan 2A = 2(4) / (1 - 16) = -8/15.

Explanation

Let the required number be x.

According to the given condition: x/6 - x/9 = 100

Taking the LCM of 6 and 9, which is 18: (3x - 2x) / 18 = 100 x / 18 = 100 x = 100 × 18 = 1800

Explanation

Let the common multiplier for the numbers be x. The two numbers are 3x and 5x.

Since 3 and 5 are co-prime, the LCM of 3x and 5x is 3 × 5 × x = 15x.

Given that the LCM is 105: 15x = 105 x = 7

The difference between the two numbers is 5x − 3x = 2x.

Substituting x = 7: Difference = 2 × 7 = 14

Explanation

Daily earnings of 2 men and 3 women = 49 / 7 = 7 Daily earnings of 3 men and 6 women = 96 / 8 = 12

Let m and w be the daily earnings of 1 man and 1 woman respectively: 2m + 3w = 7 3m + 6w = 12 ⇒ m + 2w = 4

Multiplying the second equation by 2 gives 2m + 4w = 8. Subtracting 2m + 3w = 7 from 2m + 4w = 8 yields w = 1. Substituting w = 1 into m + 2w = 4 gives m = 2.

Daily earnings of 1 man and 1 woman = m + w = 2 + 1 = 3. Number of days required to earn 27 = 27 / 3 = 9 days.

Explanation

Mean = (Sum of observations) / (Total number of observations)

Sum = 5.2 + 13.3 + 17.5 + 19.5 + 21.4 = 76.9

Number of observations = 5

Mean = 76.9 / 5 = 15.38

Explanation

Let the principal amount be P. Since the amount doubles in 5 years, the simple interest earned in 5 years is equal to P. Therefore, the simple interest earned per year is P/5.

In 8 years, the total simple interest earned is 8 × (P/5) = 8P/5.

The total amount after 8 years is P + 8P/5 = 13P/5 = 2 3/5 times the original amount.

Explanation

Using the work-rate relation, Rama's packing rate is 36 boxes/h and her sister's rate is 36 / 3 = 12 boxes/h.

Combined packing rate = 36 + 12 = 48 boxes/h.

Time taken together = Total boxes / Combined rate = 36 / 48 h = ¾ h = ¾ × 60 min = 45 min.

Alternatively, using the combined time formula T = (t₁ × t₂) / (t₁ + t₂), where t₁ = 1 h and t₂ = 3 h: T = (1 × 3) / (1 + 3) = ¾ h = 45 min.

Explanation

For two similar polygons, the ratio of their areas is equal to the square of the ratio of their corresponding side lengths:

(Area of larger polygon) / (Area of smaller polygon) = (Side ratio)²

Given the side ratio is 7 : 5, the area ratio is (7/5)² = 49 / 25.

1127 / (Area of smaller polygon) = 49 / 25

Area of smaller polygon = (1127 × 25) / 49 = 23 × 25 = 575 cm²

Explanation

To find the greatest 4-digit number divisible by 18, 25, 45, and 55, first calculate their Least Common Multiple (LCM).

Prime factorization of each number: 18 = 2 × 3² 25 = 5² 45 = 3² × 5 55 = 5 × 11

LCM(18, 25, 45, 55) = 2¹ × 3² × 5² × 11¹ = 2 × 9 × 25 × 11 = 4950

The required number is the largest multiple of 4950 that is a 4-digit number.

4950 × 2 = 9900 4950 × 3 = 14850

Thus, 9900 is the greatest 4-digit number divisible by 18, 25, 45, and 55.

Explanation

Let the salaries of A and B be A and B respectively. Since A spends 95% of his salary, he saves 5%. Since B spends 80% of his salary, he saves 20%.

Given that their savings are equal: 5% of A = 20% of B A = 4B

Substituting A = 4B into the total salary equation: A + B = 4300 4B + B = 4300 5B = 4300 B = 860

Thus, A's salary = 4 × 860 = 3440.

Explanation

Mean (x̅) = (3 + 4 + 5 + 6 + 7) / 5 = 25 / 5 = 5

Variance (σ²) = Σ(x - x̅)² / n = [(-2)² + (-1)² + 0² + 1² + 2²] / 5 = (4 + 1 + 0 + 1 + 4) / 5 = 10 / 5 = 2

Standard deviation (σ) = √(Variance) = √2

Explanation

In a parallelogram, adjacent angles are supplementary, so their sum is 180°. Let x be the smallest angle. The adjacent larger angle is given as 2x - 12°.

x + (2x - 12°) = 180° 3x - 12° = 180° 3x = 192° x = 64°

The largest angle is 2(64°) - 12° = 116°.

Explanation

The prime factorizations of 85 and 153 are: 85 = 5 × 17 153 = 3² × 17

The Highest Common Factor (HCF) of 85 and 153 is 17.

Equating the HCF to the given expression: 85x - 153 = 17 85x = 17 + 153 85x = 170 x = 170 / 85 = 2

Explanation

Let the total capital be C.

The remaining fraction of the capital is 1 - ¼ - ⅓ = 5/12.

The overall effective annual simple interest rate is: Effective Rate = (¼ × 3½%) + (⅓ × 7½%) + (5/12 × 8%) = (7/8 + 5/2 + 10/3)% = (21 + 60 + 80) / 24 % = 161 / 24 %

Given that the yearly simple interest is ₹2,576: (161 / 2400) × C = 2576 C = (2576 × 2400) / 161 C = 16 × 2400 C = ₹38,400

Explanation

Following the BODMAS/PEMDAS order of operations, multiplication and division are performed from left to right before subtraction:

Multiplication: 0.5 × 0.5 = 0.25 Consecutive divisions (left to right): (0.5 ÷ 5) ÷ 0.5 = 0.1 ÷ 0.5 = 0.2 Subtraction: 0.25 - 0.2 = 0.05

Explanation

For a prime factorization of the form 2ᵃ × 3ᵇ × 5ᶜ × 7ᵈ, the number of even factors is given by the formula a × (b + 1) × (c + 1) × (d + 1), since an even factor must contain at least one factor of 2 (giving a choices: 2¹, 2², ..., 2ᵃ).

Given N = 2⁷ × 3⁴ × 5³ × 7¹: a = 7, b = 4, c = 3, d = 1

Number of even factors = 7 × (4 + 1) × (3 + 1) × (1 + 1) = 7 × 5 × 4 × 2 = 280.

Explanation

The given expression is: [(1 - x⁴) / (1 + x)] ÷ [(1 + x²) / x] × [1 / (x(x - 1))]

Converting division into multiplication by taking the reciprocal: = [(1 - x⁴) / (1 + x)] × [x / (1 + x²)] × [1 / (x(x - 1))]

Factoring 1 - x⁴ as (1 - x²)(1 + x²) = (1 - x)(1 + x)(1 + x²): = [(1 - x)(1 + x)(1 + x²) / (1 + x)] × [x / (1 + x²)] × [1 / (x(x - 1))]

Canceling the common terms (1 + x), (1 + x²), and x from the numerator and denominator: = (1 - x) / (x - 1) = -(x - 1) / (x - 1) = -1

Explanation

Using the standard algebraic identity for the cube of a binomial: (x - y)³ = x³ - y³ - 3xy(x - y)

Substituting the given value x - y = 1 into the identity: (1)³ = x³ - y³ - 3xy(1) 1 = x³ - y³ - 3xy

Alternatively, setting simple values that satisfy x - y = 1, such as x = 1 and y = 0: 1³ - 0³ - 3(1)(0) = 1 - 0 - 0 = 1