RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

Given the equation cos x + 1/cos x = 2, let cos x = t.

t + 1/t = 2 t² - 2t + 1 = 0 (t - 1)² = 0 t = 1

Thus, cos x = 1.

Substituting cos x = 1 into the required expression: cosⁿ x + 1/cosⁿ x = 1ⁿ + 1/1ⁿ = 1 + 1 = 2

Explanation

Adjacent angles of a parallelogram are supplementary, summing to 180°.

Let the smallest angle be x. The adjacent angle is given as 2x - 48°.

x + (2x - 48°) = 180° 3x = 228° x = 76°

The largest angle = 180° - 76° = 104°

Explanation

Let total capital be C.

The remaining fraction of capital = 1 - (2/3 + 1/5) = 1 - 13/15 = 2/15.

The effective overall rate of interest is: (2/3 × 6%) + (1/5 × 10%) + (2/15 × 15%) = 4% + 2% + 2% = 8%

Since total annual income is 600: 8% of C = 600 C = (600 × 100) / 8 = 7,500

Explanation

Numbers divisible by 6, 12, and 16 must be multiples of their Least Common Multiple (LCM). LCM(6, 12, 16) = 48

The multiples of 48 between 400 and 600 are: 48 × 9 = 432 48 × 10 = 480 48 × 11 = 528 48 × 12 = 576

Using the formula for the sum of an arithmetic progression: Sum = n/2 × (a + l) where n = 4 terms, first term a = 432, and last term l = 576.

Sum = 4/2 × (432 + 576) = 2 × 1008 = 2016.

Explanation

Simplifying the given terms: √36 = 6 √64 = 8

To find the LCM of 6 and 8, write their prime factorizations: 6 = 2 × 3 8 = 2³

LCM(6, 8) = 2³ × 3 = 8 × 3 = 24

Explanation

Selling Price = List Price × (100 - Discount%)/100

Since the list price remains constant, the ratio of selling prices for two different discount rates is given by: SP₂ = SP₁ × (100 - D₂) / (100 - D₁)

Substituting SP₁ = ₹36, D₁ = 10%, and D₂ = 20%: SP₂ = 36 × (100 - 20) / (100 - 10) SP₂ = 36 × 80 / 90 SP₂ = ₹32

Explanation

The combined mean x̄ of two groups with sample sizes n₁, n₂ and means x̄₁, x̄₂ is given by the weighted mean formula:

x̄ = (n₁x̄₁ + n₂x̄₂) / (n₁ + n₂)

Given: n₁ = 100, x̄₁ = 25 n₂ = 50, x̄₂ = 70

Substituting the values: x̄ = (100 × 25 + 50 × 70) / (100 + 50) x̄ = (2500 + 3500) / 150 x̄ = 6000 / 150 = 40

Explanation

If LCM(x, y) = y, then x is a factor of y.

Given that LCM(a, b) = b, a divides b. Since LCM(b, c) = c, b divides c. Since LCM(c, d) = d, c divides d.

By transitivity of divisibility, a, b, and c are all factors of d. Since d is divisible by a, b, c, and itself, the Least Common Multiple of all four numbers a, b, c, and d is d.

Explanation

Using the successive percentage change formula:

Net % change = a + b + (a × b) / 100

Substituting a = -20% (decrease) and b = +10% (increase):

Net % change = -20 + 10 + (-20 × 10) / 100 = -10 - 2 = -12%

The negative sign indicates an overall decrease, giving a net change of 12%.

Explanation

When a distance d is traveled at speed v₁ and returned at speed v₂, the total time T taken for the entire journey is given by:

T = d/v₁ + d/v₂

Substituting the given values v₁ = 20 km/h, v₂ = 5 km/h, and T = 5 h:

5 = d/20 + d/5

5 = (d + 4d) / 20

5 = 5d / 20

5 = d / 4

d = 20 km

Explanation

Given a tan α = b, we have tan α = b/a.

Using the right-triangle definition of trigonometric functions, if tan α = Perpendicular / Base = b/a, then Base = a and Perpendicular = b.

By the Pythagorean theorem: Hypotenuse = √(Base² + Perpendicular²) = √(a² + b²)

Therefore, the cosine of α is given by: cos α = Base / Hypotenuse = a / √(a² + b²)

Explanation

In 2016, the expenditure on 'Fuel etc' was ₹28.8 crores, while the total expenditure on 'Taxes' and 'Interest on loans' was ₹17.5 + ₹6.28 = ₹23.78 crores. Here, ₹23.78 crores is less than ₹28.8 crores.

Explanation

Average expenditure = Sum of expenditure on Salary from 2013 to 2017 / Total number of years

Sum of Salary expenditure = 60.5 + 68.4 + 62.8 + 67.2 + 84.0 = 342.9 crore

Average = 342.9 / 5 = ₹68.58 crore

Explanation

Range is defined as the difference between the maximum and minimum expenditure values over the given period.

Range for Interest on loans = 10.11 - 4.88 = 5.23 ₹ crores Range for Taxes = 22.6 - 16.8 = 5.8 ₹ crores

The calculated ranges of 5.23 ₹ crores for Interest on loans and 5.8 ₹ crores for Taxes are closest to each other, representing the most similar range of expenditure.

Explanation

Percentage Increase = [(Salary in current year - Salary in previous year) / Salary in previous year] × 100

For 2017: Percentage increase = [(84.0 - 67.2) / 67.2] × 100 = (16.8 / 67.2) × 100 = 25%

Since 25% is greater than 20%, this satisfies the given condition.

Explanation

To convert 2.84 hours into hours, minutes, and seconds:

Separating the whole number part gives 2 hours.

The decimal part is converted to minutes by multiplying by 60: 0.84 × 60 = 50.4 minutes

Taking the whole number part gives 50 minutes.

The remaining decimal portion of minutes is converted to seconds by multiplying by 60: 0.4 × 60 = 24 seconds

Combining these yields 2 hours 50 minutes 24 seconds.

Explanation

By the Arithmetic Mean - Geometric Mean (AM-GM) inequality for the non-negative terms a², b², c², and d²:

(a² + b² + c² + d²) / 4 ≥ (a² × b² × c² × d²)⁽1/4)

Substituting a² + b² + c² + d² = 1:

1 / 4 ≥ (a²b²c²d²)⁽1/4)

Raising both sides to the fourth power:

(1 / 4)⁴ ≥ a²b²c²d² 1 / 256 ≥ (abcd)²

Taking the square root gives abcd ≤ 1/16, with equality holding when a = b = c = d = ½.

Explanation

Let the two quantities be 3x and 5x, where 3x is the smaller quantity.

According to the given condition: (3x - 9) / (5x - 9) = 5 / 9

Cross-multiplying to solve for x: 9(3x - 9) = 5(5x - 9) 27x - 81 = 25x - 45 2x = 36 x = 18

The smaller quantity is 3x = 3 × 18 = 54.

Explanation

To find the maximum possible daily wage, we calculate the Highest Common Factor (HCF) of the total amount offered (₹5,850) and the actual amount received (₹5,200), as the daily wage must evenly divide both amounts.

HCF(5850, 5200) = HCF(5850 - 5200, 5200) = HCF(650, 5200)

Since 5,200 ÷ 650 = 8 and 5,850 ÷ 650 = 9, the greatest common divisor of both amounts is 650.

Explanation

The smallest 6-digit number is 100000 and the largest 4-digit number is 9999.

Subtracting 9999 from 100000: 100000 - 9999 = 90001

Explanation

Profit is calculated by subtracting the Cost Price from the Selling Price: Profit = Selling Price - Cost Price Profit = ₹7,80,000 - ₹6,50,000 = ₹1,30,000

To find the percentage profit on the car, divide the total profit by the original cost price and multiply by 100: Percentage Profit = (Profit / Cost Price) × 100 Percentage Profit = (1,30,000 / 6,50,000) × 100 Percentage Profit = 1/5 × 100 = 20%

Explanation

Using trigonometric complementary angle identities and standard relations: sec 59° = cosec(90° - 59°) = cosec 31°, so sec²59° - cot²31° = cosec²31° - cot²31° = 1 sin 90° = 1 tan 56° = cot(90° - 56°) = cot 34°, so tan²56° × tan²34° = cot²34° × tan²34° = 1

Substituting these values into the given equation: 6(1) + (2/3)(1) - 3y(1) = y/3 6 + 2/3 - 3y = y/3 20/3 = 3y + y/3 20/3 = 10y/3 10y = 20 y = 2

Explanation

For quarterly compounding, the rate of interest per quarter is r = 12% / 4 = 3%, and the time duration of 6 months corresponds to n = 6 / 3 = 2 quarters.

Using the compound interest formula: Amount (A) = P × (1 + r/100)ⁿ A = 1200 × (1 + 3/100)² = 1200 × (1.03)² = 1200 × 1.0609 = ₹1,273.08

Compound Interest (CI) = Amount - Principal CI = ₹1,273.08 - ₹1,200 = ₹73.08

Explanation

Number of 3-year periods in 6 years = 6 ÷ 3 = 2

Total donations in 6 years = 2 × 2 = 4 times

Total blood donated = 4 × 330 ml = 1320 ml

Converting ml to litres: 1320 ml = 1320 ÷ 1000 = 1.32 L

Explanation

The area of an equilateral triangle with side length a is given by the formula:

Area = (√3 / 4) × a²

Substituting side a = 12 cm into the formula:

Area = (√3 / 4) × 12² = (√3 / 4) × 144 = 36√3 cm²

Explanation

The numbers from 1 to 26 that are divisible by 2 are the even numbers starting from 2 up to 26.

When these numbers are arranged in descending order (26, 24, 22, ..., 2), the smallest number is at the bottom. Therefore, counting from the bottom is equivalent to taking the sequence of even numbers in ascending order starting from 2.

The nth number from the bottom is given by the formula: nth term = 2 × n

For the 9th place from the bottom (n = 9): 9th term = 2 × 9 = 18

Explanation

The area of a square is related to its side length by Area = a². a = √289 = 17 cm

The length of the diagonal of a square is given by d = a√2. Substituting the side length: d = 17√2 cm

Explanation

The smallest five-digit number is 10000. Dividing 10000 by 472 yields a quotient of 21 and a remainder of 88 ( since 10000 = 472 × 21 + 88 ). To find the smallest five-digit multiple of 472, we take the next integer quotient 22 and compute 472 × 22 = 10384. Alternatively, adding the difference between the divisor and the remainder (472 - 88 = 384) to 10000 gives 10384.

Explanation

To find the maximum possible area of the square designs, the side length of each square must be equal to the Highest Common Factor (HCF) of the dimensions of the room.

HCF(12, 8) = 4 m

Number of square designs required = (Length ÷ Side) × (Width ÷ Side) = (12 ÷ 4) × (8 ÷ 4) = 3 × 2 = 6

Explanation

The measure of an interior angle of a regular polygon with n sides is given by the formula ((n - 2) × 180°) / n.

For a regular pentagon (n = 5): Interior angle = ((5 - 2) × 180°) / 5 = (3 × 180°) / 5 = 108°

For a regular octagon (n = 8): Interior angle = ((8 - 2) × 180°) / 8 = (6 × 180°) / 8 = 135°

Ratio of the interior angle of a pentagon to an octagon = 108 : 135 = 4 : 5.

Explanation

Circumference of the wheel = π × diameter = (22/7) × 88 cm = 1936/7 cm = 1936/700 m

Distance covered in one revolution equals the circumference of the wheel.

Number of revolutions = Total distance ÷ Circumference = 8712 ÷ (1936 / 700) = (8712 × 700) ÷ 1936 = 4.5 × 700 = 3150

Explanation

Daily earnings equal the product of working hours per day and wages per hour.

Using the net percentage change formula for the product of two quantities: Net % change = a + b + (a × b) / 100

Given a = +35% and b = -25%: Net % change = 35 - 25 + (35 × (-25)) / 100 = 10 - 875 / 100 = 10 - 8.75 = 1.25%

Explanation

Simplify the numerator and denominator separately:

Numerator = 6.25 + (5/7) × 28 − 5 = 6.25 + 20 − 5 = 21.25

Denominator = (3/4) × (15.8 − 3.4) + 5 × 2.39 = (3/4) × 12.4 + 11.95 = 9.3 + 11.95 = 21.25

Dividing the numerator by the denominator: 21.25 / 21.25 = 1

Explanation

Distance = Speed × Time

Given: Speed = 50 km/h Time = 2½ h = 2.5 h

Distance = 50 × 2.5 = 125 km

Explanation

Relative Speed = Speed of dog - Speed of cat Relative Speed = 24 km/h - 10 km/h = 14 km/h

Converting relative speed to meters per minute: Relative Speed = (14 × 1000) / 60 = 700/3 m/min

Time = Distance / Relative Speed Time = 280 / (700/3) = (280 × 3) / 700 = 1.2 minutes

Explanation

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

For n = 12: S = 12 × (12 + 1) × (2 × 12 + 1) / 6 S = 12 × 13 × 25 / 6 S = 2 × 13 × 25 = 650

Explanation

The LCM of a set of fractions is given by the formula: LCM of fractions = LCM of numerators / HCF of denominators

For the given fractions 1/3, 7/6, 5/9, 4/27, and 8/15:

Numerators = 1, 7, 5, 4, 8 LCM(1, 7, 5, 4, 8) = 280

Denominators = 3, 6, 9, 27, 15 HCF(3, 6, 9, 27, 15) = 3

Substituting these values into the formula yields 280 / 3.

Explanation

Let the two numbers be 19a and 19b, where a and b are co-prime factors of the LCM. The problem states that the other two factors of their LCM are 11 and 13, so a = 11 and b = 13.

The larger of the two numbers is obtained by multiplying the HCF by the larger co-prime factor:

Larger number = 19 × 13 = 247

Explanation

Using the complementary angle identity sin(90° - θ) = cos θ, sin 75° can be written as sin(90° - 15°) = cos 15°.

Substituting sin 75° = cos 15° into the given expression (cos 15° - sin 75°) / (cos 15° + sin 75°) gives:

Numerator = cos 15° - cos 15° = 0

Denominator = cos 15° + cos 15° = 2 cos 15°

Since the numerator evaluates to 0 and the denominator is non-zero, the value of the entire expression is 0 / (2 cos 15°) = 0.

Explanation

Let the two numbers be x and y.

The sum of two numbers is given as x + y = 84.

The product of two numbers is equal to the product of their HCF and LCM: x × y = HCF × LCM = 3 × 124 = 372

The sum of the reciprocals of the two numbers is: 1/x + 1/y = (x + y) / (x × y)

Substituting the values: 1/x + 1/y = 84 / 372 = 7 / 31

Explanation

The area of an isosceles right-angled triangle with hypotenuse h is given by the formula Area = h² / 4.

Substituting the given area of 81 cm² into the formula: 81 = h² / 4 h² = 324 h = √324 = 18 cm

Explanation

For an n-sided polygon, the sum of interior angles is given by (n - 2) × 180° and the sum of exterior angles is always 360°.

According to the problem: (n - 2) × 180° = 3 × 360° (n - 2) × 180° = 1080° n - 2 = 6 n = 8

Explanation

Let the first number be x and the second number be y.

35% of x = 30% of y 35x = 30y x/y = 30/35 = 6/7

Thus, x : y = 6 : 7.

Explanation

To convert an angle from degrees to radians, multiply the degree measure by π / 180°.

Radians = 54° × (π / 180°) = 54π / 180

Dividing both the numerator and the denominator by 18 yields 3π/10.

Explanation

Let the two numbers be x and y. Given, x + y = 80 and x - y = 8.

Adding the two equations gives: 2x = 88 ⇒ x = 44

Subtracting the two equations gives: 2y = 72 ⇒ y = 36

The required ratio of the first number to the second number is: x : y = 44 : 36 = 11 : 9

Explanation

Let CP be the cost price of one item and SP be the selling price of one item.

Given, 40 × SP = 35 × CP SP / CP = 35 / 40 = 7 / 8

Since SP is less than CP, it is a case of loss.

Loss percentage = [(CP - SP) / CP] × 100 Loss percentage = [(8 - 7) / 8] × 100 = (1 / 8) × 100 = 12½%

Explanation

Average speed for two equal halves of a journey is given by the formula 2 × v₁ × v₂ / (v₁ + v₂).

Substituting v₁ = 25 km/h and v₂ = 30 km/h: Average speed = (2 × 25 × 30) / (25 + 30) = 1500 / 55 = 300 / 11 km/h

Total distance = Average speed × Total time Total distance = (300 / 11) × 11 = 300 km

Explanation

To convert a mixed recurring decimal 0.53̄2̄ (where 32 repeats indefinitely) into a fraction, use the standard conversion formula:

Fraction = (Complete number after decimal - Non-recurring part) / (Number of 9s equal to repeating digits followed by 0s equal to non-repeating digits)

For 0.53̄2̄: Numerator = 532 - 5 = 527 Denominator = 990 (two 9s for '32' and one 0 for '5')

Algebraically: Let x = 0.5323232... 10x = 5.323232... 1000x = 532.323232... 1000x - 10x = 532 - 5 990x = 527 x = 527/990

Explanation

Average marks in Science = Sum of marks scored by all students in Science / Number of students

Sum of Science marks = 69 + 80 + 82 + 80 + 75 = 386 Number of students = 5

Average = 386 / 5 = 77.2

Explanation

Justin scored the highest marks in English (88), Maths (85), and Geography (90). Having secured the top score in three subjects, Justin is the student who scored the highest marks in at least 2 subjects.