RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

Using the successive percentage change formula:

Net percentage increase = x + y + (x × y) / 100

Substituting x = 10% and y = 10%:

Net percentage increase = 10 + 10 + (10 × 10) / 100 = 21%

Assuming 100 liters of pure milk costing ₹100, adding 10% water increases the volume to 110 liters. Adding 10% more water to this mixture adds another 11 liters, making the final volume 121 liters. Since water costs nothing and the mixture is sold at the cost price of pure milk, total revenue is ₹121 for an initial cost of ₹100, giving a profit of ₹21 or 21%.

Explanation

Using the exponential property ⁿ√(a/b) = ⁿ√a / ⁿ√b:

⁵√(32/243) = ⁵√32 / ⁵√243

Since 32 = 2⁵ and 243 = 3⁵:

⁵√(2⁵) / ⁵√(3⁵) = 2/3

Explanation

When a journey is divided into two equal distances covered at speeds v₁ and v₂, the average speed for the entire journey is given by:

Average Speed = (2 × v₁ × v₂) / (v₁ + v₂)

Substituting v₁ = 40 km/h and v₂ = 50 km/h:

Average Speed = (2 × 40 × 50) / (40 + 50) = 4000 / 90 = 400/9 km/h

Total Distance = Average Speed × Total Time Total Distance = (400/9) × 18 = 800 km

Explanation

Using the standard algebraic identity: (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

Substituting the given values: 5² = 29 + 2(ab + bc + ca) 25 = 29 + 2(ab + bc + ca) 25 - 29 = 2(ab + bc + ca) -4 = 2(ab + bc + ca) ab + bc + ca = -2

Explanation

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

x² = 9 cos² A cos² B y² = 9 cos² A sin² B z² = 9 sin² A

Adding x² and y²: x² + y² = 9 cos² A (cos² B + sin² B) = 9 cos² A

Now adding z²: x² + y² + z² = 9 cos² A + 9 sin² A = 9 (cos² A + sin² A) = 9

Explanation

Given, 2(cos θ + sec θ) = 5 => cos θ + sec θ = 5/2

Squaring both sides: (cos θ + sec θ)² = (5/2)² cos²θ + sec²θ + 2(cos θ × sec θ) = 25/4

Since cos θ × sec θ = 1: cos²θ + sec²θ + 2 = 25/4 sec²θ + cos²θ = 25/4 - 2 = 17/4

Explanation

To express -40/56 as a rational number with a numerator of -5, divide both the numerator and the denominator by 8:

(-40 ÷ 8) / (56 ÷ 8) = -5/7

Explanation

In 1995, the ratio of the higher level employment rate to the total employment rate was 15/42 = 5/14. In 2010, the ratio was 15/70 = 3/14. Since 5/14 ≠ 3/14, the ratios were not equal in 1995 and 2010, making the statement false.

Explanation

To determine the least improvement, calculate the change in marks from the first term to the second term (Second Term - First Term) for each subject:

English: 67 - 65 = +2 Mathematics: 94 - 88 = +6 Science: 85 - 82 = +3 Social science: 75 - 71 = +4

In Hindi, marks decreased from 70 to 68 (68 - 70 = -2), which indicates a decline in performance rather than an improvement. Among the subjects where an actual improvement occurred, English shows the smallest increase of 2 marks.

Explanation

Expressing the decimals in scientific notation:

Numerator = 0.03 × 0.05 × 1.5 = (3 × 10⁻²) × (5 × 10⁻²) × (15 × 10⁻¹) = 225 × 10⁻⁵ = 0.00225

Denominator = 0.0225 = 225 × 10⁻⁴

Dividing the numerator by the denominator:

(225 × 10⁻⁵) / (225 × 10⁻⁴) = 10⁻⁵ / 10⁻⁴ = 10⁻¹ = 0.1

Explanation

Using the identity (a + b)⁴ - (a - b)⁴ = 8ab(a² + b²):

Let a = 2 and b = √5. Then a² = 4 and b² = 5.

Substituting these values into the formula: 8ab(a² + b²) = 8 × 2 × √5 × (4 + 5) = 16√5 × 9 = 144√5

Explanation

The length of the longest stick that can be placed inside a cuboid is equal to the length of its space diagonal, given by the formula √(l² + w² + h²).

Substituting the given dimensions l = 4 m, w = 3 m, and h = 12 m:

Diagonal = √(4² + 3² + 12²) = √(16 + 9 + 144) = √169 = 13 m

Explanation

Let L be the length of the train and S be its speed.

When the train crosses a man on the platform: Speed S = L / 10

When the train crosses a 260 m long platform: Speed S = (L + 260) / 20

Equating the two expressions for speed: L / 10 = (L + 260) / 20 2L = L + 260 L = 260 m

Alternatively, the extra time taken to cross the platform compared to the man is 20 s - 10 s = 10 s. This extra 10 s is required solely to cover the platform's length of 260 m. Since the train takes 10 s to cross its own length as well, the length of the train is 260 m.

Explanation

To find the ratio of HCF to LCM for the numbers 10, 25, and 35, determine their prime factorizations: 10 = 2 × 5 25 = 5² 35 = 5 × 7

HCF is the product of the lowest powers of common prime factors: HCF = 5

LCM is the product of the highest powers of all prime factors involved: LCM = 2¹ × 5² × 7¹ = 2 × 25 × 7 = 350

Ratio of HCF to LCM = 5 : 350 = 1 : 70.

Explanation

The mean of a set of observations is the sum of all observations divided by the total number of observations.

Sum of observations = (x + 1) + (x + 3) + (x + 5) + (x + 7) + (x + 9) = 5x + 25

Number of observations = 5

Mean = (5x + 25) / 5 = x + 5

Given that the mean is 6: x + 5 = 6 x = 6 − 5 = 1

Explanation

A line perpendicular to the line Ax + By = C is represented in the form Bx - Ay = k.

For the given line x - y = 5, any perpendicular line has the general equation x + y = k.

Since this line passes through the point (1, 5), substitute x = 1 and y = 5 into the equation:

1 + 5 = k k = 6

Substituting k = 6 back into the general equation gives x + y = 6.

Explanation

By definition, the arithmetic mean x̄ of n observations x₁, x₂, ..., xₙ is defined as x̄ = (∑ᵢ₌₁ⁿ xᵢ) / n, which gives ∑ᵢ₌₁ⁿ xᵢ = n x̄.

The sum of the deviations of all observations from their arithmetic mean is: ∑ᵢ₌₁ⁿ (xᵢ - x̄) = ∑ᵢ₌₁ⁿ xᵢ - ∑ᵢ₌₁ⁿ x̄ = n x̄ - n x̄ = 0

The algebraic sum of deviations of any set of observations from their arithmetic mean is always zero.

Explanation

To find the range for the cube root of 250, compare 250 with the cubes of consecutive integers:

6³ = 216 7³ = 343

Since 216 < 250 < 343, taking the cube root across the inequality gives 6 < ³√250 < 7. Therefore, the cube root of 250 lies between 6 and 7.

Explanation

The slope of a line ax + by + c = 0 is given by m = -a/b.

For the line x + y = 2, the slope is m₁ = -1. For the line x - y = 7, the slope is m₂ = 1.

The product of their slopes is m₁ × m₂ = (-1) × 1 = -1. When the product of the slopes of two lines is -1, the lines are perpendicular to each other, which means the angle between them is 90°.

Explanation

Simplifying the right-hand side of the equation: (2/2)⁽x-1) = 1⁽x-1) = 1

Substituting this back into the original equation: (2/3)⁽x+1) = 1

Since any non-zero number raised to the power of 0 equals 1, set the exponent to 0: x + 1 = 0 x = -1

Explanation

Let the total work be LCM(12, 15) = 60 units.

Ram's efficiency = 60 / 12 = 5 units per day. Shyam's efficiency = 60 / 15 = 4 units per day. Combined efficiency of Ram and Shyam = 5 + 4 = 9 units per day.

Work completed together in 5 days = 5 × 9 = 45 units. Remaining work = 60 - 45 = 15 units.

Time required by Ram to finish the remaining work = 15 / 5 = 3 days.

Explanation

Let the cost price of the watch be 100%.

Selling price at 10% loss = 90% of CP Selling price at 10% gain = 110% of CP

Difference in selling price = 110% - 90% = 20% of CP

Given that 20% of CP = ₹120: Cost Price = (120 / 20) × 100 = ₹600

Explanation

Using logarithmic properties, express each term on the right-hand side in base 3 using log(aⁿ) = n log(a):

log(9) = log(3²) = 2 log(3) log(27) = log(3³) = 3 log(3) log(81) = log(3⁴) = 4 log(3)

Summing these terms gives: log(9) + log(27) + log(81) = (2 + 3 + 4) log(3) = 9 log(3)

Substituting this back into the equation: 9 log(x) = 9 log(3) log(x) = log(3) x = 3

Explanation

To find the relative speed, convert the speed of the second train from m/s to km/h: Speed of second train = 25 × (18/5) = 90 km/h.

When two objects move in opposite directions, their relative speed is the sum of their individual speeds. A man sitting in the first train moves at the speed of the first train (60 km/h).

Relative speed = Speed of first train + Speed of second train = 60 km/h + 90 km/h = 150 km/h.

Explanation

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

Substituting n = 50 into the formula: S = 50 × (50 + 1) / 2 S = 50 × 51 / 2 S = 25 × 51 = 1275

Explanation

To find the sum of 1 5/7 and 1 2/5, convert the mixed fractions to improper fractions:

1 5/7 = 12/7 1 2/5 = 7/5

Now, add the two improper fractions by finding the least common multiple (LCM) of the denominators 7 and 5, which is 35:

12/7 + 7/5 = (12 × 5 + 7 × 7) / 35 = (60 + 49) / 35 = 109 / 35

Converting 109/35 back into a mixed fraction gives 3 with a remainder of 4, which is written as 3 4/35.

Explanation

Volume of cylinder = πR²H = π × 6² × 5 = 180π m³ Volume of a sphere = (4/3)πr³ = (4/3) × π × 3³ = 36π m³

Number of spheres = Volume of cylinder / Volume of one sphere = 180π / 36π = 5

Explanation

When a train crosses a stationary person, the distance covered equals the length of the train.

Speed = Distance / Time = 250 / 10 = 25 m/s

Converting m/s into km/h: Speed = 25 × (18/5) = 90 km/h

Explanation

Using the BODMAS rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction), evaluate from the innermost brackets outward:

Innermost parenthesis: 12 - 20 ÷ 2 = 12 - 10 = 2 Curly braces: {25 ÷ 5 - 2 × 2} = {5 - 4} = 1 Square brackets: [16 ÷ 1] = 16 Final subtraction: 20 - 16 = 4

Explanation

From the given equation a + b = ab: b = ab - a b = a(b - 1) a = b / (b - 1)

Dividing both sides by b: a / b = 1 / (b - 1)

Thus, the ratio a : b is equal to 1 : (b - 1).

Explanation

Adding the given equations x + y = 40 and x - y = 20 yields 2x = 60, which gives x = 30. Subtracting the second equation from the first yields 2y = 20, which gives y = 10. The factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30, and the factors of 10 are 1, 2, 5, 10. The highest common factor (HCF) of 30 and 10 is 10, as it is the largest integer that divides both numbers completely.

Explanation

Using the identity sec²θ - tan²θ = 1: (secθ + tanθ)(secθ - tanθ) = 1

Given secθ + tanθ = 2 + √3: secθ - tanθ = 1 / (2 + √3) = 2 - √3

Adding both equations: 2 secθ = (2 + √3) + (2 - √3) = 4 secθ = 2

Since cosθ = 1 / secθ: cosθ = ½

Explanation

Evaluate each term separately:

√0.0121 = √(121 / 10000) = 11 / 100 = 0.11

∛0.008 = ∛(8 / 1000) = 2 / 10 = 0.2

Adding the two simplified values together: 0.11 + 0.2 = 0.31

Explanation

Speed of the train = (Difference in bridge lengths) ÷ (Difference in time) Speed = (600 - 200) ÷ (80 - 40) = 400 ÷ 40 = 10 m/s

Length of the train = (Speed × Time taken for second bridge) - Length of second bridge Length = (10 × 40) - 200 = 400 - 200 = 200 m

Explanation

Using the algebraic identity (x + y)³ = x³ + y³ + 3xy(x + y):

Substituting x + y = -2 into the identity: (-2)³ = x³ + y³ + 3xy(-2) -8 = x³ + y³ - 6xy

Alternatively, setting y = 0 gives x = -2: x³ + y³ - 6xy = (-2)³ + 0³ - 6(-2)(0) = -8

Explanation

To convert any fraction to a percentage, multiply the fraction by 100%.

Percentage = (2 / 5) × 100% = 2 × 20% = 40%

Explanation

In a proportion a : b :: c : d, the product of extremes equals the product of means (a × d = b × c).

x² × x = 18 × 12 x³ = 216 x = ∛216 = 6

Explanation

tan(15°) can be evaluated using the trigonometric angle subtraction identity tan(A - B) = (tan A - tan B) / (1 + tan A tan B).

Setting A = 45° and B = 30°: tan(15°) = (tan 45° - tan 30°) / (1 + tan 45° × tan 30°)

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

Rationalizing the expression by multiplying numerator and denominator by (√3 - 1): tan(15°) = (√3 - 1)² / (3 - 1) = (3 + 1 - 2√3) / 2 = 2 - √3

Explanation

Population growth follows the compound interest formula P = P₀(1 + r/100)ᵗ, where P₀ is the initial population, r is the growth rate per second, and t is time in seconds.

Substituting P₀ = 400 cells, r = 5, and t = 2 seconds: P = 400 × (1 + 5/100)² P = 400 × (21/20)² P = 400 × (441/400) P = 441 cells

Explanation

The nth term of an Arithmetic Progression (AP) is given by the formula:

aₙ = a + (n - 1)d

Here, the first term a = 2, the common difference d = 7 - 2 = 5, and the term number n = 10.

Substituting these values into the formula:

a₁₀ = 2 + (10 - 1) × 5 = 2 + 9 × 5 = 2 + 45 = 47

Explanation

Using the compound interest formula:

Amount = P(1 + R/100)ᵀ

Given: Principal P = ₹8,000, Rate R = 21%, Time T = 1½ = 3/2 years

Amount = 8000 × (1 + 21/100)⁽3/2) Amount = 8000 × (121/100)⁽3/2) Amount = 8000 × (√(121/100))³ Amount = 8000 × (11/10)³ Amount = 8000 × (1331 / 1000) Amount = 8 × 1331 = ₹10,648

Compound Interest = Amount − Principal CI = 10,648 − 8,000 = ₹2,648

Explanation

Let the Cost Price (CP) of the dress be ₹x.

Selling price with 10% profit = 110% of x Selling price with 5% loss = 95% of x

The difference between a 10% profit and a 5% loss is 15% of the Cost Price.

15% of x = 150 (15 / 100) × x = 150 x = (150 × 100) / 15 = 1000

Thus, the cost price of the dress is ₹1000.

Explanation

Minimum passing mark per question = 70% of 10 = 7 marks.

For Mery: Marks obtained in Q1, Q2, Q3, Q4, Q5 are 10, 9, 8, 6, and 8 respectively. Marks in Q4 (6) is less than 7, so Mery did not pass 1 question.

For Therry: Marks obtained in Q1, Q2, Q3, Q4, Q5 are 5, 7, 10, 9, and 9 respectively. Marks in Q1 (5) is less than 7, so Therry did not pass 1 question.

Explanation

Let the present age of A be x years. Since A is 6 years older than B, the present age of B is x - 6 years.

10 years ago, A's age was x - 10 years and B's age was x - 16 years.

According to the given condition: x - 16 = ¾(x - 10) 4(x - 16) = 3(x - 10) 4x - 64 = 3x - 30 x = 34

Thus, the present age of A is 34 years.

Explanation

The first 100 natural numbers are 1, 2, 3, ..., 100. The highest common factor (HCF) of a set of numbers is the greatest positive integer that divides every number in the set without leaving a remainder. Since the set includes the number 1, whose only positive factor is 1, the only common factor for all 100 numbers is 1.

Explanation

Using the percentage multiplication property, x% of y% of Z = (x × y × Z) / 10000.

For 19% of 16% of 1480: Product of percentages = 19 × 16 = 304

For 38% of 8% of 1480: Product of percentages = 38 × 8 = (19 × 2) × 8 = 19 × 16 = 304

Since both percentage products equal 304, the two expressions yield the exact same value.

Explanation

Total work done is constant, given by Work = Men × Days.

Let M be the initial number of men. 50M = (M + 5) × (50 − 10) 50M = 40M + 200 10M = 200 M = 20

Explanation

Let the shorter diagonal be d₁ = d, then the longer diagonal d₂ = 2d.

Area of a rhombus = ½ × d₁ × d₂ = ½ × d × 2d = d² Since the area is given as K, d² = K.

The diagonals of a rhombus bisect each other at right angles. By Pythagoras theorem, the side of the rhombus s = √[(d₁/2)² + (d₂/2)²] = √[(d/2)² + d²] = √(5d²/4).

Substituting d² = K gives s = √(5K/4) cm.

Explanation

Total percentage difference between a 12.5% profit and a 10.5% loss = 12.5% + 10.5% = 23% of the Cost Price (CP).

Given, 23% of CP = ₹161 1% of CP = 161 / 23 = ₹7

Selling price required for a 19% profit = 119% of CP = 119 × 7 = ₹833

Explanation

The amount saved represents the 25% discount on the original price of the sweater.

Let the original price be x. 25% of x = 200 (25 / 100) × x = 200 x = (200 × 100) / 25 x = 800