RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

Using the percentage formula: (x / 100) × 12.8 = 38.4

Solving for x: x = (38.4 × 100) / 12.8 x = (38.4 / 12.8) × 100 x = 3 × 100 x = 300

Explanation

Evaluating each term individually according to BODMAS rules:

0.077 ÷ 7 = 0.011 0.005 ÷ 5 = 0.001

Subtracting the two values: 0.011 - 0.001 = 0.01

Explanation

Relative speed = Speed of Karthik - Speed of Raju = 12 - 10 = 2 km/h

Converting relative speed into metres per minute: Relative speed = (2 × 1000) / 60 = 100/3 m/min

Distance covered by Karthik relative to Raju in 6 minutes: Distance reduced = Relative speed × Time = (100/3) × 6 = 200 m

Remaining distance between them after 6 minutes: Distance = Initial distance - Distance reduced = 300 - 200 = 100 m

Explanation

The speed ratio of the three individuals can be determined directly from the given conditions: Speed of B = 3 × Speed of C Speed of A = 2 × Speed of B = 2 × 3 × Speed of C = 6 × Speed of C

For a constant distance, time taken is inversely proportional to speed. Therefore, the ratio of time taken by A to time taken by C is 1 : 6.

Time taken by A = (Time taken by C) / 6 = 54 / 6 = 9 minutes.

Explanation

Let B be the event that the sum of the numbers appearing on two throws of a die is 10, and A be the event that the number 5 appears at least once.

The sample space of outcomes where the sum is 10 is: B = {(4, 6), (5, 5), (6, 4)} So, n(B) = 3.

The favorable outcome where the number 5 appears at least once is: A ∩ B = {(5, 5)} So, n(A ∩ B) = 1.

By conditional probability formula, P(A|B) = n(A ∩ B) / n(B) = 1/3.

Explanation

Expand both sides of the given equation:

Left side: 3(5x + 2) - 4 = 15x + 6 - 4 = 15x + 2

Right side: 2(1 - 4x) = 2 - 8x

Equating both sides gives: 15x + 2 = 2 - 8x

Rearranging terms with x to one side and constants to the other: 15x + 8x = 2 - 2 23x = 0 x = 0

Explanation

The two-point form equation of a straight line passing through (x₁, y₁) and (x₂, y₂) is:

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

Substituting the given points (-2, 5) and (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

Time required to complete a work is inversely proportional to efficiency. Since Raj is 60% more efficient than Sundar, his efficiency relative to Sundar is 160/100 = 1.6.

Days taken by Raj = Days taken by Sundar / Relative Efficiency = 16 / 1.6 = 10 days.

Explanation

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

LCM(4, 5, 8, 16, 32) = 2⁵ × 5¹ = 160

To make 160 a perfect square, all exponents in its prime factorization must be even. Multiplying by 2¹ × 5¹ gives the required least square number:

2⁶ × 5² = 64 × 25 = 1600

Explanation

The mean of a frequency distribution is calculated using the formula: Mean = Σ(f × x) / Σf

Sum of frequencies: Σf = 6 + 8 + 15 + p + 8 + 4 = 41 + p

Sum of products of values and frequencies: Σ(f × x) = (3 × 6) + (5 × 8) + (7 × 15) + (9 × p) + (11 × 8) + (13 × 4) Σ(f × x) = 18 + 40 + 105 + 9p + 88 + 52 = 303 + 9p

Given that the mean is 8: 8 = (303 + 9p) / (41 + p) 8 × (41 + p) = 303 + 9p 328 + 8p = 303 + 9p 9p - 8p = 328 - 303 p = 25

Explanation

Cost price for C = Initial cost price × (1 + Profit₁%) × (1 + Profit₂%)

Substituting the given values: Cost price for C = 4,500 × (1 + 5/100) × (1 + 20/100) = 4,500 × (21/20) × (6/5) = 4,500 × 1.26 = ₹5,670

Explanation

To find the total participation for each school, calculate the sum of students playing all five games:

School A = 34 + 21 + 12 + 9 + 98 = 174 School B = 67 + 40 + 22 + 13 + 121 = 263 School C = 23 + 26 + 19 + 7 + 78 = 153 School D = 78 + 32 + 16 + 12 + 84 = 222

School B has the highest total participation (263 students) and School C has the lowest total participation (153 students).

Explanation

Rewriting 2ˣ = 4ʸ⁺¹ in terms of base 2 gives 2ˣ = 2²⁽ʸ⁺¹⁾, which simplifies to: x = 2y + 2

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

Substituting x = 2y + 2 into y = x - 9: y = 2y + 2 - 9 y = 7

Substituting y = 7 back into x = 2y + 2 gives: x = 2(7) + 2 = 16

Explanation

Rama's 1-day work = 1/50. Work completed by Rama in 10 days = 10 × (1/50) = 1/5. Remaining work = 1 - 1/5 = 4/5.

Ravi completes 4/5 of the work in 32 days, so Ravi's 1-day work = (4/5) ÷ 32 = 1/40.

Combined 1-day work of Rama and Ravi = 1/50 + 1/40 = (4 + 5)/200 = 9/200.

Time needed by Rama and Ravi together = 200/9 = 22 2/9 days.

Explanation

Let θ be the angle of elevation of the sun. The height of the pole and the length of its shadow form the opposite and adjacent sides of a right-angled triangle.

tan θ = Height of pole / Length of shadow tan θ = (6√3) / 6 = √3

Since tan 60° = √3, θ = 60°.

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

Comparing this with the standard intercept form yields the x, y, and z intercepts as 2, -3/2, and -3.

Explanation

Time taken = 2 min 10 s = (2 × 60 + 10) s = 130 s.

Speed in m/s = Distance / Time = 650 / 130 = 5 m/s.

To convert speed from m/s to km/hr, multiply by 18/5: Speed = 5 × (18/5) = 18 km/hr.

Explanation

The curved surface area of a cylinder is given by 2πrh and its volume is given by πr²h, where r is the radius and h is the height.

Using 2πrh = 660 with h = 14 cm: 2 × (22/7) × r × 14 = 660 88 × r = 660 r = 15/2 cm

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

Explanation

Convert the mixed fractions into improper fractions and simplify step by step:

2 - 1/3 = 5/3

(2/3) × 1 1/5 = (2/3) × (6/5) = 4/5

(3/5) × 2 5/7 = (3/5) × (19/7) = 57/35

Adding the terms inside the curly brackets: 5/3 + 4/5 = (25 + 12) / 15 = 37/15

Adding the remaining term using the LCM of 15 and 35, which is 105: 37/15 + 57/35 = (259 + 171) / 105 = 430/105 = 86/21 = 4 2/21

Explanation

The surface area S of a cube in terms of its diagonal d is given by the formula S = 2d².

Given d = 6√2 cm: d² = (6√2)² = 72 S = 2 × 72 = 144 cm²

Explanation

When smaller cubes are melted to form a single larger cube, the total volume remains conserved.

Volume of new cube = 3³ + 4³ + 5³ Volume of new cube = 27 + 64 + 125 = 216 cm³

Edge of the new cube = ∛216 = 6 cm.

Explanation

(0.2)³ = 0.2 × 0.2 × 0.2 = 0.008 (0.02)³ = 0.02 × 0.02 × 0.02 = 0.000008

Subtracting the second value from the first: 0.008000 - 0.000008 = 0.007992

Explanation

At a constant speed, time taken is directly proportional to the distance covered:

Time = (Time₁ / Distance₁) × Distance₂

Substituting the given values:

Time = (60 / 35) × 105 = 60 × 3 = 180 minutes.

Since 105 km is 3 times 35 km (105 / 35 = 3), the time taken will be 3 times 60 minutes, which equals 180 minutes.

Explanation

Using the compound interest formula A = P(1 + r/100)ⁿ, where A = 1,061, P = 1,000, and n = 2:

1,061 = 1,000 × (1 + r/100)² (1 + r/100)² = 1,061 / 1,000 = 1.061 ≈ 1.0609

Taking the square root on both sides: 1 + r/100 = √1.0609 = 1.03 r / 100 = 0.03 r = 3%

Explanation

For a quadratic polynomial f(x) = ax² + bx + c, let α and β be its zeros.

Sum of zeros, α + β = -b/a = -(-8)/1 = 8 Product of zeros, α × β = c/a = k/1 = k

Using the algebraic identity: α² + β² = (α + β)² - 2(α × β)

Given that α² + β² = 40, substitute the values into the formula: 40 = 8² - 2k 40 = 64 - 2k 2k = 64 - 40 2k = 24 k = 12

Explanation

The required number leaving a constant remainder r when divided by a set of numbers is given by LCM(numbers) + r.

Prime factorization of the given numbers: 9 = 3² 16 = 2⁴ 20 = 2² × 5 24 = 2³ × 3

LCM(9, 16, 20, 24) = 2⁴ × 3² × 5 = 16 × 9 × 5 = 720

Required least number = 720 + 3 = 723

Explanation

The required least number is given by the formula: LCM(5, 6, 8, 9, 12) + Remainder.

Prime factorization of the numbers: 5 = 5 6 = 2 × 3 8 = 2³ 9 = 3² 12 = 2² × 3

LCM(5, 6, 8, 9, 12) = 2³ × 3² × 5 = 8 × 9 × 5 = 360

Required least number = 360 + 4 = 364

Explanation

The compound interest formula is A = P(1 + r/100)ⁿ, where A is the final amount, P is the principal sum, r is the annual interest rate, and n is the time period in years.

Given: P = ₹4,000, A = ₹4,840, and r = 10%.

Substituting these values into the formula: 4840 = 4000(1 + 10/100)ⁿ 4840 / 4000 = (11/10)ⁿ 121 / 100 = (11/10)ⁿ (11/10)² = (11/10)ⁿ

Equating the powers on both sides yields n = 2 years.

Explanation

Factorizing both algebraic expressions into their irreducible prime factors:

a³ + b³ = (a + b)(a² - ab + b²) a⁴ - b⁴ = (a² + b²)(a² - b²) = (a² + b²)(a + b)(a - b)

To find the Least Common Multiple (LCM), take the product of all distinct factors raised to their highest powers:

LCM = (a + b)(a² - ab + b²)(a² + b²)(a - b)

Since (a + b)(a² - ab + b²) = a³ + b³, simplifying yields:

LCM = (a³ + b³)(a² + b²)(a - b)

Explanation

The volume of a cylinder is given by the formula V = π × r² × h, where r is the radius and h is the height.

Substituting r = 28 cm and h = 40 cm (using π = 22/7): V = (22/7) × 28 × 28 × 40 V = 22 × 4 × 28 × 40 V = 98560 cm³

Explanation

Using the trigonometric identity tan(A - B) = (tan A - tan B) / (1 + tan A tan B), substitute A = 85° and B = 25°:

(tan 85° - tan 25°) / (1 + tan 85° tan 25°) = tan(85° - 25°) = tan 60° = √3

Explanation

For compounding half-yearly, the annual interest rate is halved and the time period is doubled: Principal (P) = ₹20,000 Half-yearly rate (r) = 4% / 2 = 2% Number of half-year periods (n) = 2 × 2 = 4

Amount (A) = P × (1 + r/100)ⁿ A = 20,000 × (1 + 2/100)⁴ A = 20,000 × (1.02)⁴ A = 20,000 × 1.08243216 = ₹21,648.64

Compound Interest = Amount − Principal Compound Interest = ₹21,648.64 − ₹20,000 = ₹1,648.64

Explanation

Selling price at a 20% loss corresponds to 80% of the cost price (CP).

80% of CP = ₹1,600

CP = 1,600 / 0.80 = ₹2,000

To achieve a 20% profit, the selling price must be 120% of the cost price:

Required Selling Price = 120% of CP = 1.20 × 2,000 = ₹2,400

Explanation

Profit = ⅖ × Selling Price = ⅖ × 125 = ₹50

Cost Price = Selling Price - Profit = 125 - 50 = ₹75

Explanation

The surface area of a sphere is given by the formula 4πr², where r is the radius.

Given r = 21 m and π = 22/7: Surface area = 4 × (22/7) × 21 × 21 = 4 × 22 × 3 × 21 = 5544 m²

Explanation

To simplify the given ratio 1/3 : 4/9 : 3/4, multiply each term by the LCM of the denominators (3, 9, 4), which is 36.

Simplified ratio = (1/3 × 36) : (4/9 × 36) : (3/4 × 36) = 12 : 16 : 27

Sum of the ratio parts = 12 + 16 + 27 = 55

First part = (12 ÷ 55) × 825 = 12 × 15 = 180

Explanation

To find the greatest three-digit number divisible by 6, 8, 10, and 16, first find the Least Common Multiple (LCM) of the numbers.

Prime factorization: 6 = 2 × 3 8 = 2³ 10 = 2 × 5 16 = 2⁴

LCM(6, 8, 10, 16) = 2⁴ × 3 × 5 = 240

Any number divisible by 6, 8, 10, and 16 must be a multiple of 240. Dividing the largest three-digit number (999) by 240 gives a quotient of 4 and a remainder of 39.

Subtracting the remainder from 999 gives 999 − 39 = 960, which is 240 × 4.

Explanation

First, calculate the product of 0.27 and 3.54: 0.27 × 3.54 = 0.9558

To find how much this product is less than 5.743, subtract 0.9558 from 5.743: 5.7430 - 0.9558 = 4.7872

Explanation

Simplifying the fraction 7/77 gives 1/11.

Now, divide 7777077 by 11 by breaking the number into multiples of 77:

7777077 = 7700000 + 77000 + 77

Dividing each part by 11:

(7700000 + 77000 + 77) ÷ 11 = 700000 + 7000 + 7 = 707007

Explanation

Profit is shared in proportion to the product of investment and time.

For a half-yearly period of 6 months: A's effective investment = 20,000 × 6 = 120,000 B's effective investment = 16,000 × 6 = 96,000 C's effective investment = 15,000 × (6 - 3) = 45,000

Profit sharing ratio (A : B : C) = 120,000 : 96,000 : 45,000 = 40 : 32 : 15

Sum of ratio terms = 40 + 32 + 15 = 87

Value per unit ratio = 4,350 ÷ 87 = 50 B's share = 32 × 50 = ₹1,600

Explanation

To find the median, first arrange the 8 observations in ascending order: 7, 8, 11, 12, 13, 14, 15, 20

Since the number of observations n = 8 is even, the median is the average of the (n/2)-th term and the (n/2 + 1)-th term, corresponding to the 4th and 5th terms:

Median = (4th term + 5th term) / 2 = (12 + 13) / 2 = 25 / 2 = 12.5

Explanation

Speed = Distance / Time

When a train passes a point object like an electric post, the distance covered is equal to the length of the train.

Distance = 100 m Time = 6 s

Speed in m/s = 100 / 6 = 50 / 3 m/s

To convert m/s to km/h, multiply by 18 / 5:

Speed = (50 / 3) × (18 / 5) = 60 km/h

Explanation

In a rhombus, the diagonals bisect each other at right angles (90°). Therefore, the side of the rhombus forms the hypotenuse of a right-angled triangle whose perpendicular sides are half the lengths of the diagonals.

Side of a rhombus a = √((d₁/2)² + (d₂/2)²)

Given d₁ = 32 cm and d₂ = 24 cm: d₁/2 = 16 cm d₂/2 = 12 cm

a = √(16² + 12²) = √(256 + 144) = √400 = 20 cm

Explanation

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

[(28.4)² - (24.4)²] / (28.4 - 24.4) = (28.4 - 24.4)(28.4 + 24.4) / (28.4 - 24.4)

Canceling out the common term (28.4 - 24.4) from the numerator and denominator leaves:

28.4 + 24.4 = 52.8

Explanation

In right-angled triangle ABC, right-angled at C, the cosine of angle A is the ratio of the adjacent side (AC) to the hypotenuse (AB):

cos A = AC / AB

Substituting A = 60° and AC = 20√3 units:

cos 60° = (20√3) / AB

Since cos 60° = ½:

½ = (20√3) / AB AB = 2 × 20√3 = 40√3 units

Explanation

Using the algebraic identity (x + 1/x)² = x² + 1/x² + 2, we can express x² + 1/x² as (x + 1/x)² - 2.

Substituting x + 1/x = 26: x² + 1/x² = 26² - 2 x² + 1/x² = 676 - 2 = 674

Explanation

Mean = Sum of observations / Total number of observations

Sum of the 11 observations = 22 + 35 + 15 + 75 + 66 + 50 + 52 + 36 + 70 + 86 + 10 = 517

Total number of observations = 11

Mean = 517 / 11 = 47

Explanation

The Highest Common Factor (HCF) of fractions is given by the formula:

HCF of fractions = (HCF of numerators) / (LCM of denominators)

For the given fractions, the numerators are 7, 9, and 11: HCF(7, 9, 11) = 1

The denominators are 16, 24, and 64: 16 = 2⁴ 24 = 2³ × 3 64 = 2⁶

LCM(16, 24, 64) = 2⁶ × 3 = 64 × 3 = 192

Substituting these values into the formula: HCF = 1 / 192

Explanation

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

Given SI = ₹4,500, R = 9% per annum, and T = 5 years: 4,500 = (P × 9 × 5) / 100 4,500 = 45P / 100 P = (4,500 × 100) / 45 = 10,000

Explanation

Using the standard formula for OR/AND time and work problems:

Time taken = Total Days / [(Men asked / Men given) + (Children asked / Children given)]

Time taken = 92 / [(20 / 5) + (10 / 9)] = 92 / [4 + 10/9] = 92 / (46 / 9) = (92 × 9) / 46 = 18 days.

Alternatively, since 5 men = 9 children, 1 man = 9/5 children. Thus, 20 men + 10 children = 20 × (9/5) + 10 = 36 + 10 = 46 children. Since 9 children complete the work in 92 days, 46 children will take (9 × 92) / 46 = 18 days.