RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

For the first table: Selling Price = ₹400 with a 25% profit. Cost Price = 400 ÷ 1.25 = ₹320. Profit on the first table = 400 - 320 = ₹80.

Since there is no net profit or loss overall, the gain from the first table equals the loss from the second table.

Loss on the second table = 10% of its Cost Price = ₹80. Cost Price of the second table = 80 ÷ 0.10 = ₹800.

Explanation

Amount under simple interest is given by A = P + (P × R × T / 100) = P(1 + RT / 100).

Given A₁ = ₹2,613, R = 5%, and T₁ = 6 years: 2,613 = P(1 + (5 × 6) / 100) = 1.3P P = 2,613 / 1.3 = ₹2,010

To reach an amount A₂ = ₹3,015 at the same rate: Simple Interest required = A₂ - P = 3,015 - 2,010 = ₹1,005

Using T₂ = (SI × 100) / (P × R): T₂ = (1,005 × 100) / (2,010 × 5) = 1,00,500 / 10,050 = 10 years.

Explanation

Let total sum be T. Since A receives one-third of the total amount, T = 3A.

Since total amount T = A + B + C, substituting T = 3A gives: 2A = B + C

Given that B = 3C and C = A - 1200: 2A = 3C + C 2A = 4C A = 2C

Substituting C = A - 1200 into A = 2C: A = 2(A - 1200) A = 2A - 2400 A = 2400

Explanation

Break the expression into simplified components following BODMAS rules:

First fraction: (5 + 5 × 5) / (5 × 5 + 5) = (5 + 25) / (25 + 5) = 30 / 30 = 1

Second fraction: Numerator = (1/5) ÷ (1/25) = (1/5) × 25 = 5 Denominator = (1/25) ÷ (1/5) = (1/25) × 5 = 1/5 Second fraction = 5 / (1/5) = 25

Subtrahend part: (5 - 1/5) × (10 / 2) = (24/5) × 5 = 24

Combining the simplified parts: (1 × 25) - 24 = 25 - 24 = 1

Explanation

Combined work done by John, Johnny, and Janardan in one day = 1/6 + 1/12 + 1/24 = (4 + 2 + 1) / 24 = 7/24

Total time required to finish the task working together = 24/7 = 3 3/7 days.

Explanation

Using the combined work formula, Time = (a × b) / (a + b), where a = 10 days and b = 12 days:

Total days = (10 × 12) / (10 + 12) = 120 / 22 = 60 / 11 = 5 5/11 days.

Explanation

The compound ratio of a set of ratios is found by multiplying all the antecedents (first terms) together to form the new antecedent, and all the consequents (second terms) together to form the new consequent.

Simplifying each given ratio: 45 : 75 = 3 : 5 3 : 5 = 3 : 5 51 : 68 = 3 : 4 256 : 81 = 256 : 81

Multiplying the simplified fractions: (3 / 5) × (3 / 5) × (3 / 4) × (256 / 81) = (27 / 100) × (256 / 81) = 256 / 300 = 64 / 75

This simplifies to 64 : 75.

Explanation

Since the expression (5a - 3b)/(7a + 2b) is homogeneous in a and b, we can directly substitute a = 3 and b = 4 into the given expression.

Numerator = 5a - 3b = 5(3) - 3(4) = 15 - 12 = 3 Denominator = 7a + 2b = 7(3) + 2(4) = 21 + 8 = 29

Required value = 3/29

Explanation

For an equilateral triangle of side x, the inradius r of the inscribed circle is given by: r = x / (2√3)

The diameter of this circle is: d = 2r = x / √3

A square inscribed in a circle has a diagonal equal to the diameter of the circle. Therefore, the diagonal of the square is x / √3.

Using the formula for the area of a square in terms of its diagonal d: Area = d² / 2 = (x / √3)² / 2 = (x² / 3) / 2 = (1/6)x²

Explanation

In the right-angled triangle formed by the tower, the line of sight, and the ground:

tan θ = Height / Distance

Given θ = 45° and Height = 150 m:

tan 45° = 150 / Distance 1 = 150 / Distance Distance = 150 m

Explanation

Using the BODMAS rule, first solve the operations inside the parentheses, then perform division and multiplication, and finally subtraction:

(6 + 3) = 9 (4 + 5) = 9

Substituting these values into the expression gives:

(9 ÷ 3) - 5 × 9 = 3 - 45 = -42

Explanation

To find the Highest Common Factor (HCF), express each product in terms of its prime factors:

12 × 15 = (2² × 3) × (3 × 5) = 2² × 3² × 5 15 × 21 = (3 × 5) × (3 × 7) = 3² × 5 × 7 21 × 12 = (3 × 7) × (2² × 3) = 2² × 3² × 7

The common prime factor present in all three numbers is 3, with the lowest power being 2 in each case.

HCF = 3² = 9

Explanation

Percentage of votes obtained by the winning candidate = 100% − 37% = 63%.

Difference in vote percentage between the winning and losing candidate = 63% − 37% = 26%.

Since the difference in votes is 5,20,000: 26% of Total Votes = 5,20,000

Total number of votes cast = (5,20,000 / 26) × 100 = 20,000 × 100 = 20,00,000.

Explanation

To rationalize the denominator, multiply the numerator and the denominator by the conjugate of the denominator, which is (√2 - 1):

(4 + √2) / (√2 + 1) = [(4 + √2)(√2 - 1)] / [(√2 + 1)(√2 - 1)]

Expanding the numerator gives: (4 + √2)(√2 - 1) = 4√2 - 4 + 2 - √2 = 3√2 - 2

Expanding the denominator using the identity (a + b)(a - b) = a² - b² gives: (√2 + 1)(√2 - 1) = (√2)² - 1² = 2 - 1 = 1

Dividing the numerator by the denominator gives 3√2 - 2.

Explanation

The surface area of a sphere is proportional to the square of its radius (A ∝ r²), while its volume is proportional to the cube of its radius (V ∝ r³).

Given the ratio of surface areas A₁/A₂ = 1/4, the ratio of their radii is: r₁/r₂ = √(A₁/A₂) = √(1/4) = 1/2

Substituting the radius ratio into the volume relation gives: V₁/V₂ = (r₁/r₂)³ = (1/2)³ = 1/8

Explanation

Using the standard algebraic identity: (x + 1/x)² = x² + 1/x² + 2

Substitute x + 1/x = 2 into the identity: 2² = x² + 1/x² + 2 4 = x² + 1/x² + 2 x² + 1/x² = 4 - 2 = 2

Alternatively, x + 1/x = 2 directly implies x = 1. Substituting x = 1 into x² + 1/x² gives 1² + 1/1² = 2.

Explanation

Using √2916 = 54, the square root of a decimal number has half as many decimal places as the original number:

√29.16 = 5.4 √0.2916 = 0.54 √0.002916 = 0.054 √0.0002916 = 0.0054

Adding these values together:

5.4 + 0.54 + 0.054 + 0.0054 = 5.9994

Explanation

Let the cost price (CP) be ₹100.

To earn a profit of 10%, the selling price (SP) is: SP = 100 + 10 = ₹110

Since a discount of 5% is offered on the marked price (MP): SP = MP × (1 - 5/100) = MP × (19/20) 110 = MP × (19/20) MP = (110 × 20) / 19 = 2200 / 19 = 115 15/19

The percentage by which the marked price exceeds the cost price is: Markup % = MP - CP = 115 15/19 - 100 = 15 15/19%

Explanation

When two individuals take A and B days respectively to complete a work independently, the total time taken to complete the work together is given by the formula:

Time taken together = (A × B) / (A + B)

Substituting A = 20 and B = 30:

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

Explanation

Sum of marks of all 5 students = 5 × 39.20 = 196 Sum of marks of 3 students = 3 × 41 = 123 Sum of marks of the remaining 2 students = 196 - 123 = 73

Average of the remaining 2 students = 73 / 2 = 36.5

Explanation

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

Amount A = (8/5)P Interest SI = A - P = (8/5)P - P = (3/5)P

Substituting SI = (3/5)P and T = 5 into the formula: (3/5)P = (P × R × 5) / 100 3/5 = R / 20 R = (3 × 20) / 5 = 12% per annum

Explanation

Sachin Tendulkar scored his historic 100th international century against Bangladesh on March 16, 2012, during an Asia Cup match at the Sher-e-Bangla National Stadium in Mirpur, Dhaka. He scored 114 runs off 147 balls, becoming the first cricketer in history to reach the milestone of 100 international centuries, which comprises 51 Test centuries and 49 One Day International (ODI) centuries.

Explanation

In a computer interface, a menu is a graphical control element that displays a list of executable commands, functions, or options to the user. Selecting a menu item triggers a specific action or command within the software application, such as saving a file, editing text, or printing a document.

Explanation

To find the factors of N = 2² × 3¹ × 5² × 7¹ that are divisible by 50 (2¹ × 5²) but not by 100 (2² × 5²), we express each factor in prime factorization form 2ᵃ × 3ᵇ × 5ᶜ × 7ᵈ.

For a factor to be divisible by 50, the exponent of 5 must be at least 2 (c = 2) and the exponent of 2 must be at least 1. To ensure the factor is not divisible by 100, the exponent of 2 cannot be 2 or higher, which forces a = 1.

The available choices for each prime factor's exponent are: For prime 2: a = 1 (1 possibility) For prime 3: b = 0 or 1 (2 possibilities) For prime 5: c = 2 (1 possibility) For prime 7: d = 0 or 1 (2 possibilities)

Total required factors = 1 × 2 × 1 × 2 = 4.

Explanation

The area of a path of width x running along the inner boundary of a rectangular field of length L and breadth B is given by: Area of path = 2x(L + B - 2x)

Given L = 37 m, B = 30 m, and Area = 570 m²: 570 = 2x(37 + 30 - 2x) 570 = 2x(67 - 2x) 285 = x(67 - 2x) 2x² - 67x + 285 = 0

Factoring the quadratic equation: (x - 5)(2x - 57) = 0

Since x = 57/2 = 28.5 m would reduce the inner dimensions below zero, the width of the path is x = 5 m.

Explanation

Let Narendra's present age be x years, making his mother's present age 4x years.

Four years ago, Narendra was (x - 4) years old and his mother was (4x - 4) years old.

According to the given condition: 4x - 4 = 6(x - 4) 4x - 4 = 6x - 24 2x = 20 x = 10

Narendra's present age is 10 years, and his mother's present age is 40 years (4 × 10).

Explanation

Prime factorization of 120 is 2³ × 3¹ × 5¹.

The number of odd divisors is found by taking only the powers of the odd prime factors 3¹ and 5¹: Odd divisors = (1 + 1) × (1 + 1) = 4

The total number of divisors is calculated using all prime factors: Total divisors = (3 + 1) × (1 + 1) × (1 + 1) = 16

Subtracting the odd divisors from the total number of divisors gives the even divisors: Even divisors = 16 − 4 = 12

Explanation

Since ABC is an equilateral triangle, each of its interior angles is 60°, so ∠ABC = 60°. In a cyclic quadrilateral, the sum of opposite angles is 180°. Therefore, ∠ABC + ∠CDA = 180°, which gives ∠CDA = 180° - 60° = 120°.

Explanation

Net percentage change = a + b + (a × b)/100

Here, markup percentage a = +20 and discount percentage b = -20.

Net change = 20 - 20 + (20 × -20)/100 = -400/100 = -4%

The negative result indicates a loss of 4%.

Explanation

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

Cost Price = (390 × 100) / (100 + 20) = (390 × 100) / 120 = ₹325

Explanation

Using the BODMAS rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction), evaluate the nested brackets from inside out:

Innermost round brackets: 8 - 6 + 3 = 2 + 3 = 5

Curly braces: 4 - 5 = -1

Square brackets: 15 - (-1) = 15 + 1 = 16

Outer expression: 20 - 16 = 4

Explanation

The height h of an object when moving a distance d towards it, changing the angle of elevation from α to β, is given by the formula:

h = d / (cot α - cot β)

Given d = 30 m, α = 30°, and β = 60°:

cot 30° = √3 cot 60° = 1/√3

cot 30° - cot 60° = √3 - 1/√3 = 2/√3

Substituting these values into the formula:

h = 30 / (2/√3) = (30 × √3) / 2 = 15√3 m

Explanation

Arranging the given real numbers in increasing order from smallest to largest:

Negative numbers are smaller than positive numbers, and among negative numbers, the number with the larger absolute value is smaller. Therefore, -0.96 is smaller than -0.64.

Ordering the positive numbers by magnitude gives 0.24 < 0.58 < 0.83.

Combining all the numbers in ascending order: -0.96 (1) < -0.64 (4) < 0.24 (3) < 0.58 (5) < 0.83 (2)

This yields the sequence 1, 4, 3, 5, 2.

Explanation

Let the current age of the daughter be d years. Mother's current age = 3d years.

Five years ago: Mother's age = 3d - 5 Daughter's age = d - 5

According to the problem statement: 3d - 5 = 4(d - 5) 3d - 5 = 4d - 20 4d - 3d = 20 - 5 d = 15

Explanation

Let x be the total amount in ₹.

(60/100) × (40/100) × (32/100) × x = 432

Converting fractions to simplest form: (3/5) × (2/5) × (8/25) × x = 432 (48 / 625) × x = 432

Solving for x: x = (432 × 625) / 48 x = 9 × 625 x = 5,625

Explanation

Let y = √(x/(1 - x)). The given equation simplifies to:

y + 1/y = 25/12 (y² + 1)/y = 25/12 12y² - 25y + 12 = 0

Factoring the quadratic equation gives: (4y - 3)(3y - 4) = 0 y = 3/4 or y = 4/3

Case 1: When y = 3/4 √(x/(1 - x)) = 3/4 x/(1 - x) = 9/16 16x = 9 - 9x 25x = 9 x = 9/25

Case 2: When y = 4/3 √(x/(1 - x)) = 4/3 x/(1 - x) = 16/9 9x = 16 - 16x 25x = 16 x = 16/25

Explanation

Taking the common denominator xyz, the given expression simplifies to:

(x² / yz) + (y² / zx) + (z² / xy) = (x³ + y³ + z³) / xyz

According to the algebraic identity, x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx).

Since x + y + z = 0, the right-hand side becomes 0, which gives x³ + y³ + z³ = 3xyz.

Substituting this into the expression:

3xyz / xyz = 3

Explanation

Let the two adjacent sides of the rectangle be 3x and 4x.

Perimeter = 2 × (3x + 4x) = 14x 14x = 42 cm x = 3 cm

Using the Pythagorean theorem, the length of the diagonal is: Diagonal = √((3x)² + (4x)²) = √(25x²) = 5x

Substituting x = 3 cm: Diagonal = 5 × 3 = 15 cm

Explanation

The difference between compound interest and simple interest for 2 years is given by the formula:

Difference = P × (R / 100)²

Substituting P = 16,000 and Difference = 1,000 into the formula:

1,000 = 16,000 × (R / 100)² (R / 100)² = 1,000 / 16,000 (R / 100)² = 1 / 16

Taking the square root on both sides:

R / 100 = 1 / 4 R = 25%

Explanation

Expanding the given equation: (1 + tan A)(1 + tan B) = 2 1 + tan A + tan B + tan A tan B = 2 tan A + tan B = 1 - tan A tan B

Dividing both sides by (1 - tan A tan B): (tan A + tan B) / (1 - tan A tan B) = 1

Using the compound angle identity tan(A + B) = (tan A + tan B) / (1 - tan A tan B): tan(A + B) = 1

Explanation

In a symmetric distribution, data values are evenly and balancedly distributed around the center. As a result, the highest peak representing the most frequent value (mode), the central value dividing the dataset into two equal halves (median), and the arithmetic average (mean) all coincide at the exact same central point. Therefore, Mean = Median = Mode.

Explanation

The time after which all three will meet again at the starting point is equal to the Least Common Multiple (LCM) of their individual revolution times: LCM(42, 63, 84).

Prime factorisation of the given times: 42 = 2 × 3 × 7 63 = 3² × 7 84 = 2² × 3 × 7

LCM(42, 63, 84) = 2² × 3² × 7 = 4 × 9 × 7 = 252 s.

Explanation

Using the standard algebraic identity: x³ + 1/x³ = (x + 1/x)³ - 3(x + 1/x)

Let k = x + 1/x. Substituting x³ + 1/x³ = 18 into the identity gives: k³ - 3k = 18 k³ - 3k - 18 = 0

Substituting k = 3 into the equation: 3³ - 3(3) = 27 - 9 = 18

Explanation

Let the discount offered be D. Selling Price (SP) = 3D Marked Price (MP) = SP + Discount = 3D + D = 4D

Discount percentage = (Discount / MP) × 100 = (D / 4D) × 100 = 25%

Since profit percentage equals discount percentage, Profit = 25%. Cost Price (CP) = SP / (1 + Profit/100) = 3D / 1.25 = 12D / 5

Ratio of discount offered to cost price = D : (12D / 5) = 5 : 12

Explanation

Using the algebraic identity a³ - b³ = (a - b)(a² + ab + b²), let a = 5.4 and b = 0.4 (since 0.4³ = 0.064).

The given expression is in the form: [(5.4)³ - (0.4)³] / [(5.4)² + (5.4 × 0.4) + (0.4)²]

Canceling the common term (a² + ab + b²) from the numerator and denominator leaves: a - b = 5.4 - 0.4 = 5

Explanation

Simplifying each factor in the product gives:

1 + 1/x = (x + 1)/x 1 + 1/(x + 1) = (x + 2)/(x + 1) 1 + 1/(x + 2) = (x + 3)/(x + 2) 1 + 1/(x + 3) = (x + 4)/(x + 3)

Multiplying these terms together:

[(x + 1)/x] × [(x + 2)/(x + 1)] × [(x + 3)/(x + 2)] × [(x + 4)/(x + 3)]

All intermediate numerators and denominators cancel out in a telescoping manner, leaving:

(x + 4) / x

Explanation

Side of the square field = 28 m Radius of each tethered quadrant, r = 28 ÷ 2 = 14 m

Grazable area = 4 × (¼ × π × r²) = π × r² = (22/7) × 14 × 14 = 616 m²

Total area of the square field = 28 × 28 = 784 m²

Area beyond reach = Total area - Grazable area = 784 - 616 = 168 m²

Daily consumption rate = Grazable area ÷ Days = 616 ÷ 22 = 28 m²/day

Days for remaining grass = Ungrazable area ÷ Daily consumption rate = 168 ÷ 28 = 6 days

Explanation

Using the compound angle formula tan(A - B) = (tan A - tan B) / (1 + tan A tan B):

Setting A = 45° and B = 30°, the given expression reduces to tan(45° - 30°) = tan 15°.

Substituting tan 45° = 1 and tan 30° = 1/√3 into the expression: tan 15° = (1 - 1/√3) / (1 + 1/√3) = (√3 - 1) / (√3 + 1)

Multiplying the numerator and denominator by (√3 - 1) to rationalize the denominator: tan 15° = (√3 - 1)² / ((√3 + 1)(√3 - 1)) = (3 + 1 - 2√3) / (3 - 1) = (4 - 2√3) / 2 = 2 - √3

Explanation

Time is inversely proportional to speed when the distance traveled is constant (Time = Distance / Speed).

Given the ratio of the speeds of the three trains is 2 : 3 : 5, the ratio of the time taken to cover the same distance is: 1/2 : 1/3 : 1/5

Explanation

Total Cost Price = Purchase Price + Repair Cost Total Cost Price = ₹4,000 + ₹400 = ₹4,400

Gain = Selling Price - Total Cost Price Gain = ₹5,000 - ₹4,400 = ₹600

Gain % = (Gain / Total Cost Price) × 100 Gain % = (600 / 4,400) × 100 = 150 / 11 = 13 7/11%