RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

Time = Distance / Speed

Given: Speed = 45 km/h Distance = 4/5 km

Time in hours = (4/5) / 45 = 4/225 hours

To convert hours into seconds, multiply by 3600: Time in seconds = (4/225) × 3600 = 4 × 16 = 64 seconds.

Explanation

Using the effective rate formula for 2 years at 4% per annum: Effective Rate = 4 + 4 + (4 × 4) / 100 = 8.16%

Compound Interest = Principal × Effective Rate = 7,500 × (8.16 / 100) = 75 × 8.16 = ₹612

Explanation

The diagonals of a rhombus bisect each other at right angles. Using the Pythagorean theorem, half of the second diagonal is:

√(10² - (16/2)²) = √(100 - 64) = √36 = 6 cm

So, the full length of the second diagonal (d₂) is 2 × 6 = 12 cm.

Area of the rhombus = ½ × d₁ × d₂ = ½ × 16 × 12 = 96 cm²

Explanation

For four consecutive even numbers x, x + 2, x + 4, and x + 6, the average is the midpoint of the sequence: Average = (x + x + 2 + x + 4 + x + 6) / 4 = x + 3

Given the average is 27: x + 3 = 27 x = 24

The largest of these integers is x + 6 = 24 + 6 = 30.

Explanation

Percentage Increase = [(New Premium - Original Premium) / Original Premium] × 100

Substituting the given values: Increase in premium = 1,512 - 1,440 = 72 Percentage Increase = (72 / 1,440) × 100 = (1 / 20) × 100 = 5%

Explanation

Time = Total words / Typing speed

Time = 987 / 35 = 28 7/35 minutes = 28 1/5 minutes

Convert 1/5 minute to seconds: (1/5) × 60 = 12 seconds

Combining the minutes and seconds gives 28 minutes 12 seconds.

Explanation

Rationalizing the given terms: x = (√3 + 1)/(√3 - 1) = (√3 + 1)²/(3 - 1) = (4 + 2√3)/2 = 2 + √3 y = (√3 - 1)/(√3 + 1) = (√3 - 1)²/(3 - 1) = (4 - 2√3)/2 = 2 - √3

Calculating the sum and product: x + y = (2 + √3) + (2 - √3) = 4 x × y = (2 + √3)(2 - √3) = 4 - 3 = 1

Using the algebraic identity x² + y² = (x + y)² - 2xy: x² + y² = 4² - 2(1) = 16 - 2 = 14

Explanation

1-day work of A = 1/16 1-day work of B = 1/12 1-day work of A, B, and C together = 1/4

1-day work of C = 1/4 - (1/16 + 1/12) = 1/4 - (3 + 4)/48 = 12/48 - 7/48 = 5/48

Time taken by C alone to complete the work = 1 / (5/48) = 48/5 days.

Explanation

Simplify the expression algebraically before substituting the value:

y √(y³ - y²) = y √(y²(y - 1)) = y² √(y - 1)

Substitute y = 26 into the simplified expression:

26² × √(26 - 1) = 676 × √25 = 676 × 5 = 3380

Explanation

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

Substituting the given values: Cost Price = (2600 × 100) / (100 + 30) = 260000 / 130 = ₹2,000

Explanation

Let the total work be 30 units (LCM of 15 and 30).

Efficiency of A = 30 / 15 = 2 units/day Efficiency of B = 30 / 30 = 1 unit/day

In a 2-day cycle (working on alternate days), total work completed = 2 + 1 = 3 units.

Number of 2-day cycles required to complete 30 units = 30 / 3 = 10 cycles.

Total time taken = 10 × 2 = 20 days.

Explanation

To convert the recurring decimal 5.4646... into a fraction:

Let x = 5.4646... 100x = 546.4646...

Subtracting the first equation from the second: 100x - x = 546.4646... - 5.4646... 99x = 541 x = 541/99

Using the shortcut rule: Fraction = (Complete number - Non-repeating part) / 99 = (546 - 5) / 99 = 541/99

Explanation

The first elections to provincial legislatures under the Government of India Act 1935 were held in January and February 1937 across eleven provinces of British India. This act granted provincial autonomy, abolished dyarchy at the provincial level, and expanded the electorate to about 30 million people, leading to the formation of popular provincial governments.

Explanation

Let the two perpendicular sides of the right-angled triangle be a and b, and the hypotenuse c = 26 cm.

The perimeter of the triangle is a + b + c = 60 cm, which gives: a + b = 60 - 26 = 34 cm

By Pythagoras theorem: a² + b² = c² = 26² = 676

Using the algebraic identity (a + b)² = a² + b² + 2ab: 34² = 676 + 2ab 1156 = 676 + 2ab 2ab = 1156 - 676 = 480 ab = 240

The area of a right triangle is given by ½ × a × b: Area = ½ × 240 = 120 cm²

Explanation

The mean of a dataset is calculated using the formula: Mean = (Sum of all observations) ÷ (Total number of observations).

Sum of observations = 1 + 1 + 1 + 2 + 4 + 5 + 5 + 7 + 8 + 12 = 46 Total number of observations = 10

Mean = 46 ÷ 10 = 4.6

Explanation

To find the number of multiples of 11 between 500 and 700:

First multiple of 11 after 500 is 11 × 46 = 506. Last multiple of 11 before 700 is 11 × 63 = 693.

Number of multiples = 63 - 46 + 1 = 18.

Alternatively, using floor division: Number of terms = floor(699 / 11) - floor(500 / 11) = 63 - 45 = 18.

Explanation

Let the present age of the son be x years. The father's present age is 3x years.

After 13 years, the father's age will be 3x + 13 and the son's age will be x + 13.

Setting up the equation based on the condition that the father will be twice as old as his son: 3x + 13 = 2(x + 13) 3x + 13 = 2x + 26 3x - 2x = 26 - 13 x = 13 years

Explanation

King Harshavardhana, who ruled northern India during the 7th century CE, was a distinguished patron of literature and a skilled playwright. He authored three major Sanskrit plays: Ratnavali, Priyadarsika, and Nagananda. 'Ratnavali' is a celebrated four-act Sanskrit drama that revolves around the romantic tale of Princess Ratnavali and King Udayana of Vatsa.

Explanation

Total Cost Price (CP) = Purchase Price + Repair Expenses = 1,200 + 300 = 1,500 Selling Price (SP) = 2,000 Profit = SP - CP = 2,000 - 1,500 = 500

Profit % = (Profit / CP) × 100 = (500 / 1,500) × 100 = (1 / 3) × 100 = 33⅓%

Explanation

The area of the triangle formed by the line ax + by + c = 0 with the coordinate axes is given by the formula: Area = c² / (2|ab|).

For the line 2x - 4y - 7 = 0, the coefficients are a = 2, b = -4, and c = -7.

Substituting these values: Area = (-7)² / (2 × |2 × (-4)|) = 49 / (2 × 8) = 49/16 unit²

Alternatively, setting y = 0 gives the x-intercept as 7/2, and setting x = 0 gives the y-intercept as -7/4. The area is ½ × |7/2| × |-7/4| = 49/16 unit².

Explanation

The product of two numbers is equal to the product of their HCF and LCM:

First Number × Second Number = HCF × LCM

Substituting the given values into the formula:

10¹² × Second Number = 10⁸ × 10¹²·⁷³ 10¹² × Second Number = 10⁽8 + 12.73) 10¹² × Second Number = 10²⁰·⁷³

Second Number = 10²⁰·⁷³ / 10¹² = 10⁽20.73 - 12) = 10⁸·⁷³

Explanation

Let the original speed of the train be v km/h.

Using the relation Time = Distance / Speed, the initial time taken is 400/v hours and the time taken at the increased speed is 400/(v + 10) hours.

Given the time difference of 2 hours: 400/v - 400/(v + 10) = 2

Dividing the equation by 2: 200/v - 200/(v + 10) = 1

Combining the terms: 2000 / [v(v + 10)] = 1 v(v + 10) = 2000 v² + 10v - 2000 = 0

Factoring the quadratic equation: (v + 50)(v - 40) = 0

Since speed cannot be negative, v = 40 km/h.

Explanation

Divide the numerator and the denominator of the given expression by cos θ to convert terms into tan θ:

(a cos θ + b sin θ) / (a cos θ - b sin θ) = (a + b tan θ) / (a - b tan θ)

Substitute tan θ = b / a into the simplified expression:

= [a + b(b / a)] / [a - b(b / a)] = (a + b² / a) / (a - b² / a) = [(a² + b²) / a] / [(a² - b²) / a] = (a² + b²) / (a² - b²)

Explanation

The given sequence is an arithmetic progression with first term a = 2 and common difference d = 3.

Using the n-th term formula of an arithmetic progression: aₙ = a + (n - 1) × d

For the 10th term (n = 10): a₁₀ = 2 + (10 - 1) × 3 a₁₀ = 2 + 9 × 3 a₁₀ = 2 + 27 = 29

Explanation

The small intestine is the site where the complete digestion of carbohydrates, proteins, and fats occurs in human beings. It receives digestive enzymes and secretions from both the liver and the pancreas, alongside producing its own intestinal juice (succus entericus). Carbohydrates are fully broken down into simple sugars like glucose, proteins are digested into amino acids, and fats are emulsified by bile and converted into fatty acids and glycerol by lipase enzymes.

Explanation

To find the greatest number, subtract the respective remainders from the given numbers and find their Highest Common Factor (HCF):

38 - 2 = 36 45 - 3 = 42 52 - 4 = 48

Required number = HCF(36, 42, 48)

Prime factorizations: 36 = 2² × 3² 42 = 2 × 3 × 7 48 = 2⁴ × 3

HCF(36, 42, 48) = 2 × 3 = 6

Explanation

The largest square that fits inside a circle is an inscribed square whose diagonal equals the diameter of the circle.

Diameter of circle = 2 × radius = 2 × 4 cm = 8 cm

Area of inscribed square = ½ × (diagonal)² = ½ × 8² = ½ × 64 = 32 cm²

Explanation

The distance d between two points (x₁, y₁) and (x₂, y₂) is given by the distance formula:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

Substituting (x₁, y₁) = (a cos α, 0) and (x₂, y₂) = (0, a sin α):

d = √[(0 - a cos α)² + (a sin α - 0)²] d = √[a² cos² α + a² sin² α] d = √[a²(cos² α + sin² α)]

Using the trigonometric identity cos² α + sin² α = 1:

d = √(a²) = |a|

Since distance is a non-negative quantity, taking the principal square root of a² yields the absolute value |a|.

Explanation

Following the BODMAS rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction), simplify the expression from the innermost brackets outwards:

Innermost parentheses: (8 - 6 + 3) = 2 + 3 = 5

Curly braces: {4 - 5} = -1

Square brackets: [6 - (-1)] = 6 + 1 = 7

Outer subtraction: 18 - 7 = 11

Explanation

The formula for calculating simple interest is SI = (P × R × T) / 100, where P is the principal amount, R is the rate of interest per annum, and T is the time period in years.

Substituting the given values: SI = (6000 × 4 × 7) / 100 = 60 × 28 = ₹1,680

Explanation

Simplify each fraction term step by step:

First term: 1/(1 + 1/3) = 1/(4/3) = 3/4

Second term: 1/(1 + 1/2) = 1/(3/2) = 2/3

Subtracting the two terms: 3/4 - 2/3 = (9 - 8)/12 = 1/12

Explanation

The general equation of a circle is x² + y² + 2gx + 2fy + c = 0, where the radius is given by the formula r = √(g² + f² - c).

For the given equation x² + y² - 6x + 8y + c = 0: 2g = -6 → g = -3 2f = 8 → f = 4

Given that the radius r = 6, substitute these values into the radius formula: 6 = √((-3)² + 4² - c) 6 = √(9 + 16 - c) 6 = √(25 - c)

Squaring both sides: 36 = 25 - c c = 25 - 36 c = -11

Explanation

The volume V of a cone with base radius r and height h is given by the formula:

V = (1/3) × π × r² × h

Substituting r = 5 cm, h = 10 cm, and π = 3.14 into the formula:

V = (1/3) × 3.14 × 5² × 10 V = (1/3) × 3.14 × 250 V = 785 / 3 V = 261.666... cm³

Rounding to two decimal places yields 261.67 cm³.

Explanation

Number of blue balls = (Ratio part of blue balls / Sum of ratio parts) × Total number of balls

Ratio of red to blue balls = 4:5 Sum of ratio parts = 4 + 5 = 9

Number of blue balls = (5 / 9) × 81 = 5 × 9 = 45

Explanation

Total weight of A, B, C, D = 4 × 56 = 224 kg Total weight of A, B, C = 3 × 52 = 156 kg Total weight of C, D = 2 × 48 = 96 kg

Adding the weight of (A + B + C) and (C + D) gives: (A + B + C) + (C + D) = 156 + 96 = 252 kg A + B + 2C + D = 252 kg

Subtracting the total weight of all four individuals (A + B + C + D = 224 kg) directly yields the weight of C: C = 252 - 224 = 28 kg

Explanation

The sum of squares of the first n natural numbers is given by the formula S = n(n + 1)(2n + 1) / 6.

Substituting n = 10: S = 10 × (10 + 1) × (2 × 10 + 1) / 6 S = (10 × 11 × 21) / 6 S = 2310 / 6 = 385

Explanation

For the first term, let θ = cos⁻¹(3/5), which gives cos θ = 3/5. Using the identity sin θ = √(1 - cos²θ), sin(cos⁻¹(3/5)) = √(1 - (3/5)²) = √(16/25) = 4/5.

For the second term, let φ = tan⁻¹(5/12), which gives tan φ = 5/12. In a right-angled triangle with opposite side 5 and adjacent side 12, the hypotenuse is √(5² + 12²) = 13. Thus, cos(tan⁻¹(5/12)) = 12/13.

Summing the two values: 4/5 + 12/13 = (52 + 60) / 65 = 112/65.

Explanation

Total production of bananas from all three states in 2016 = 3 + 5 + 4.1 = 12.1 tons

Total production of bananas from all three states in 2018 = 5.3 + 4 + 5 = 14.3 tons

Difference in total production = 14.3 - 12.1 = 2.2 tons

Explanation

Selection percentage = (Number of students selected / Number of students appeared) × 100

For K.V., the percentage of students selected relative to those who appeared is: (800 / 1000) × 100 = 80%

This represents the lowest selection ratio among the schools.

Explanation

The general equation of a straight line is Ax + By + C = 0, with slope m = -A/B.

For the line x + y - 3 = 0, slope m₁ = -1/1 = -1. For the line x + y + 2 = 0, slope m₂ = -1/1 = -1. For the line 3x + 3y - 7 = 0, slope m₃ = -3/3 = -1.

Since m₁ = m₂ = m₃ = -1 and all three lines have different constant terms (which give distinct intercepts), the lines are parallel to each other.

Explanation

According to the order of operations (BODMAS), division inside the parentheses is performed first, followed by addition:

1250 ÷ 50 = 25

7500 + 25 = 7525

Explanation

To determine the position of a point (x₁, y₁) relative to a circle S ≡ x² + y² + 2gx + 2fy + c = 0, substitute the point coordinates into the equation to calculate S₁ = x₁² + y₁² + 2gx₁ + 2fy₁ + c. If S₁ < 0, the point lies inside the circle; if S₁ = 0, it lies on the circle; and if S₁ > 0, it lies outside the circle.

For the point (1, 2) and the circle x² + y² - 3x - 4y + 1 = 0:

S₁ = 1² + 2² - 3(1) - 4(2) + 1 S₁ = 1 + 4 - 3 - 8 + 1 = -5

Since S₁ < 0, the point lies inside the circle.

Explanation

The n-th term of a geometric progression is given by the formula Tₙ = a × rⁿ⁻¹, where a is the first term and r is the common ratio. Substituting a = 2, r = 3, and n = 5 yields T₅ = 2 × 3⁵⁻¹ = 2 × 3⁴ = 2 × 81 = 162.

Explanation

Let the two numbers be 9x and 5x, where 9x is the greater number and 5x is the smaller number.

According to the given condition: 9x + 9 = 3 × (5x - 5) 9x + 9 = 15x - 15 15x - 9x = 9 + 15 6x = 24 x = 4

The two numbers are 9 × 4 = 36 and 5 × 4 = 20.

Explanation

Let the number of boys in the class be x. The number of girls is ³⁄₅x.

Total students = x + ³⁄₅x = ⁸⁄₅x = 40

Solving for x: x = (40 × 5) / 8 = 25

Explanation

For π < x < 3π/2, x lies in the third quadrant.

Since cos x = -½, we have: sin² x = 1 - cos² x = 1 - (-½)² = ¾ tan² x = sin² x / cos² x = (¾) / (¼) = 3 cosec² x = 1 / sin² x = 4/3

Substituting these values into the expression: 4 tan² x - 3 cosec² x = 4(3) - 3(4/3) = 12 - 4 = 8

Explanation

Using the work formula M₁ × D₁ × H₁ = M₂ × D₂ × H₂:

24 × 15 × 8 = M₂ × 10 × 6 2880 = 60 × M₂ M₂ = 2880 / 60 = 48

Explanation

The area of the four walls of a room is given by the formula: Area of walls = Perimeter of the floor × Height

Substituting the given values: Area of walls = 18 m × 3 m = 54 m²

Explanation

A quadratic equation with roots α and β is given by the formula x² - (α + β)x + (α × β) = 0.

For the given roots α = -2 and β = 3: Sum of roots = -2 + 3 = 1 Product of roots = -2 × 3 = -6

Substituting these values into the standard formula yields: x² - (1)x + (-6) = 0 x² - x - 6 = 0

Explanation

Using the identity sin²θ = 1 - cos²θ = (1 - cosθ)(1 + cosθ):

Simplify the third term: sin²θ / (1 + cosθ) = (1 - cosθ)(1 + cosθ) / (1 + cosθ) = 1 - cosθ

Combine the second and fourth terms: (1 + cosθ) / sinθ - sinθ / (1 - cosθ) = [(1 + cosθ)(1 - cosθ) - sin²θ] / [sinθ(1 - cosθ)] = (1 - cos²θ - sin²θ) / [sinθ(1 - cosθ)] = (1 - 1) / [sinθ(1 - cosθ)] = 0

Substituting these into the given expression gives: 1 + 0 - (1 - cosθ) = 1 - 1 + cosθ = cosθ