RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

The cube of 11 is 11³ = 1331 and the cube of 12 is 12³ = 1728. Thus, the required number lies strictly between 1331 and 1728.

Twice of 80 is 2 × 80 = 160. Since the number is divisible by both 160 and 6, it must be a multiple of their Least Common Multiple (LCM).

LCM(160, 6) = 480

The multiples of 480 are 480, 960, 1440, 1920, and so on.

The only multiple of 480 that lies between 1331 and 1728 is 480 × 3 = 1440.

Explanation

Cost Price of 1 watch = Selling Price / (1 + Profit%) = 1454.64 / 1.16 = 1254

Total Cost Price of a dozen (12) watches = 12 × 1254 = 15048.00

Explanation

Using the work-rate formula, Jenny's daily work rate is obtained by subtracting Manju's daily work rate from their combined daily work rate:

Jenny's rate = 1/12 - 1/16 = (4 - 3) / 48 = 1/48

Since Jenny completes 1/48 of the work in one day, she will take 48 days to complete the entire work alone.

Explanation

Using the laws of exponents, aᵐ × aⁿ = aᵐ⁺ⁿ and aᵐ ÷ aⁿ = aᵐ⁻ⁿ.

Applying these rules to the expression: 10⁹ × 10² ÷ 10³ = 10⁹⁺²⁻³ = 10¹¹⁻³ = 10⁸

Explanation

Circumference of a circle = 2πr = 22 cm.

2 × (22/7) × r = 22 r = 7/2 = 3.5 cm

Area of the semicircle = ½πr² = ½ × (22/7) × (7/2)² = ½ × (22/7) × (49/4) = 77/4 = 19.25 sq.cm

Explanation

The compound interest is given by the formula CI = P[(1 + r/100)ⁿ - 1], where P is the principal, r is the annual rate of interest, and n is the time in years.

Given: P = ₹25000, r = 12%, and n = 3 years.

CI = 25000 × [(1 + 12/100)³ - 1] CI = 25000 × [(1.12)³ - 1] CI = 25000 × [1.404928 - 1] CI = 25000 × 0.404928 CI = 10123.20

Explanation

For any two positive integers, the product of the two numbers is equal to the product of their LCM and HCF:

Number 1 × Number 2 = LCM × HCF

Substituting the given values into the formula: 24 × Number 2 = 168 × 6 Number 2 = (168 × 6) / 24 Number 2 = 168 / 4 Number 2 = 42

Explanation

The number of items actually distributed among the children is obtained by subtracting the remainder 6 from each category:

Books distributed = 78 - 6 = 72 Crayons distributed = 114 - 6 = 108 Notebooks distributed = 141 - 6 = 135

The number of children is the Highest Common Factor (HCF) of 72, 108, and 135.

72 = 9 × 8 108 = 9 × 12 135 = 9 × 15

HCF(72, 108, 135) = 9

Thus, the total number of children is 9.

Explanation

From the given equation a⁽x+y) = a⁶, equating exponents of the same base gives: x + y = 6

It is given that x is greater than y by 2, which gives: x - y = 2

Adding both equations together: (x + y) + (x - y) = 6 + 2 2x = 8 x = 4

Explanation

To solve the given expression: (12/13) × (285/96) ÷ (171/169)

Convert division into multiplication by taking the reciprocal of the fraction: = (12/13) × (285/96) × (169/171)

Simplify by dividing common terms in the numerators and denominators: 169 ÷ 13 = 13 12 ÷ 96 = 1/8 285 ÷ 171 = 5/3 (dividing both by 57)

Combine the simplified values: = (1 × 5 × 13) / (1 × 8 × 3) = 65/24

Explanation

Reading rate per day = 14 pages/hour × 3 hours/day = 42 pages/day.

Total time required to read the entire book = 378 ÷ 42 = 9 days.

Since the initial borrowing period is 1 week (7 days), the required renewal duration is 9 − 7 = 2 days.

Explanation

Difference in earnings = (Friday sales in dozen - Monday sales in dozen) × 12 × Price per egg

Substituting the values from the table: Difference in earnings = (10 - 8) × 12 × 2.50 = 2 × 12 × 2.50 = 24 × 2.50 = 60 rupees.

Explanation

Combined Mean = (n₁x̄₁ + n₂x̄₂) / (n₁ + n₂)

Total marks for the first class = 12 × 67.4 = 808.8 Total marks for the second class = 15 × 72.3 = 1084.5

Sum of all marks = 808.8 + 1084.5 = 1893.3 Total number of students = 12 + 15 = 27

Combined mean = 1893.3 / 27 = 70.12

Explanation

Bill Amount = Cost Price × (1 + Tax Rate %)

Substituting the given values: Sales Tax Amount = 5% of 450 = (5 ÷ 100) × 450 = 22.50

Total Bill Amount = Cost Price + Sales Tax Amount = 450 + 22.50 = 472.50

Explanation

Speed = Distance / Time.

Walking time = 1 hr 40 min = 5/3 hours. Walking speed = 6 / (5/3) = 18/5 = 3.6 km/hr.

Running time = 1 hr 20 min = 4/3 hours. Running speed = 6 / (4/3) = 18/4 = 4.5 km/hr.

Difference in speed = 4.5 - 3.6 = 0.9 km/hr.

Explanation

By the trigonometric complementary angle identity, the sine of an angle's complement equals the cosine of the angle:

sin(90° - θ) = cos θ

In a right-angled triangle with acute angle θ, the other acute angle is 90° - θ. The side opposite to 90° - θ is the side adjacent to θ, making the sine ratio for 90° - θ identical to the cosine ratio for θ.

Explanation

Using the algebraic identity (a + b)² - (a - b)² = 4ab, expanding both terms gives:

(x + y)² - (x - y)² = (x² + 2xy + y²) - (x² - 2xy + y²) = 4xy

Explanation

Area of a rectangle = length × breadth 42 = 7 × breadth breadth = 6 cm

Perimeter of a rectangle = 2 × (length + breadth) Perimeter = 2 × (7 + 6) = 2 × 13 = 26 cm

Explanation

Let the selling price (SP) of 1 cycle be 1 unit.

Total SP of 20 cycles = 20 units Loss = SP of 2 cycles = 2 units

Cost Price (CP) = Total SP + Loss CP = 20 + 2 = 22 units

Loss % = (Loss / CP) × 100 Loss % = (2 / 22) × 100 = 100/11 % = 9 1/11 %

Explanation

Amount of sugar received by each person = Total sugar ÷ Total population

Substituting the given values: Amount per person = 285 kg ÷ 1487 = 0.19166... kg ≈ 0.191 kg

Explanation

Simple interest formula is SI = (P × R × x) / 100.

Given that SI = P / 3 and R = 6%: P / 3 = (P × 6 × x) / 100 1 / 3 = 6x / 100 x = 100 / 18 = 50 / 9 = 5 5/9 years.

Explanation

Selling price (SP) is 90% of the cost price (CP) due to a 10% loss. CP per kg = 40 / 0.9 = 400/9 per kg.

Loss per kg = CP - SP = (400/9) - 40 = 40/9 per kg.

Weight of vegetables sold = Total loss / Loss per kg = 144 / (40/9) = (144 × 9) / 40 = 32.40 kg.

Explanation

1 metric tonne is equal to 1000 kg.

Converting 1.08 tonnes into kilograms using this standard relation: 1.08 × 1000 = 1080 kg

Explanation

By the standard Pythagorean trigonometric identity, 1 + tan²θ = sec²θ. This can also be derived by dividing the fundamental identity sin²θ + cos²θ = 1 by cos²θ, giving (sin²θ / cos²θ) + (cos²θ / cos²θ) = 1 / cos²θ, which simplifies to tan²θ + 1 = sec²θ.

Explanation

Consecutive integers on the number line alternate between even and odd with a common difference of 1. Therefore, for any given even number x, the immediate next consecutive odd number is obtained by adding 1, which gives x + 1.

Explanation

Volume of a rectangular tank = length × breadth × height V = 1 m × ½ m × ½ m = ¼ m³ = 0.25 m³

Since 1 m³ = 1,000 litres: Capacity in litres = 0.25 × 1,000 = 250 litres

Explanation

Let L be the total worth of the lottery.

Jane receives a share equal to L / 3. She donates 1/6 of her received share, which is given as Rs. 6000:

(1/6) × (L / 3) = 6000 L / 18 = 6000 L = 6000 × 18 = 108,000

Explanation

In the number 229301, the digit 9 is at the thousands place.

Place value of 9 = 9 × 1000 = 9000 Face value of 9 = 9

Difference = Place value - Face value = 9000 - 9 = 8991

Explanation

Let the two numbers be x and y. Any two numbers with a known sum S and product P are the roots of the quadratic equation t² - St + P = 0.

t² - 437t + 21982 = 0

Applying the quadratic formula: t = (437 ± √(437² - 4 × 21982)) / 2 t = (437 ± √(190969 - 87928)) / 2 t = (437 ± √103041) / 2 t = (437 ± 321) / 2

Solving for both roots: t₁ = (437 + 321) / 2 = 379 t₂ = (437 - 321) / 2 = 58

Explanation

Let the total amount required to start the business be T. Since the approved loan of ₹2,38,75,697 falls short by 17%, it represents 83% of the total required amount.

83% of T = ₹2,38,75,697

1% of T = 2,38,75,697 / 83 = ₹2,87,659

The additional loan amount needed by Arvind is the remaining 17% of T: 17% of T = 2,87,659 × 17 = ₹48,90,203

Explanation

The total length of the wire remains constant whether it forms a circle or a square.

Length of wire = Circumference of circle = 2 × π × r = 2 × (22/7) × 70 = 440 cm

Since the perimeter of the square equals the length of the wire: 4 × side = 440 cm side = 440 / 4 = 110 cm

Explanation

Profit percentage is given by the formula: Profit % = [(SP - CP) / CP] × 100

Given SP = (5/4) CP, the ratio of selling price to cost price is: SP / CP = 5 / 4

Substituting these values into the formula: Profit % = [(5 - 4) / 4] × 100 = (1 / 4) × 100 = 25%

Explanation

Since 90% of the students passed, the percentage of students who failed is 100% - 90% = 10%.

Let T be the total number of students who appeared for the examination.

10% of T = 600 (10 / 100) × T = 600 T = (600 × 100) / 10 = 6000

Explanation

Cost Price = Selling Price × 100 / (100 - Loss%)

Given, Selling Price = ₹17,500 and Loss = 30%.

Cost Price = 17,500 × 100 / (100 - 30) = 17,500 × 100 / 70 = 250 × 100 = ₹25,000

Explanation

Converting the boundary fractions to decimal form: 7/18 ≈ 0.389 3/5 = 0.600

For a fraction to lie between 7/18 and 3/5, its decimal value must be greater than 0.389 and less than 0.600.

Converting 1/3 to decimal form gives 1/3 ≈ 0.333. Since 0.333 < 0.389, 1/3 is less than 7/18 and lies outside the given range.

Explanation

Evaluating the expression gives (0.09)² = 0.0081. Since 0.0081 is less than 0.009, 0.8281, and 0.9919, its distance from zero is the smallest.

Explanation

Let h be the height of the building relative to the top of the houses, and d be the distance of the nearer house from the building.

For the nearer house with angle of depression 45°: tan 45° = h / d 1 = h / d d = h

For the farther house with angle of depression 30°: tan 30° = h / (d + 50) 1 / √3 = h / (h + 50)

Solving for h: h + 50 = h√3 h(√3 - 1) = 50 h = 50 / (√3 - 1)

Rationalizing the denominator: h = 50(√3 + 1) / ((√3 - 1)(√3 + 1)) h = 50(√3 + 1) / 2 h = 25(√3 + 1) m

Explanation

Using the direct proportion method:

Required value = (798.6 / 44) × 63

798.6 ÷ 44 = 18.15

18.15 × 63 = 1143.45

Explanation

In an isosceles triangle, the two base angles are equal, so ∠C = ∠B. Given that the measure of ∠A is twice that of the base angle ∠B, we have ∠A = 2∠B.

Using the angle sum property of a triangle: ∠A + ∠B + ∠C = 180° 2∠B + ∠B + ∠B = 180° 4∠B = 180° ∠B = 45°

Since ∠C = ∠B, the measure of ∠C is 45°.

Explanation

Let the breadth of the rectangle be b. Since the length l is twice its breadth, l = 2b.

The perimeter of a rectangle is given by the formula P = 2(l + b).

Substituting the given values: 2(2b + b) = 120 6b = 120 b = 20 m

Thus, length l = 2 × 20 = 40 m.

Area of the rectangle = l × b = 40 × 20 = 800 m².

Explanation

The required least number is given by LCM(9, 8, 10, 12) + 3.

Prime factorization of the given divisors: 9 = 3² 8 = 2³ 10 = 2 × 5 12 = 2² × 3

LCM(9, 8, 10, 12) = 2³ × 3² × 5 = 8 × 9 × 5 = 360.

Required number = 360 + 3 = 363.

Explanation

Number of cubes = Volume of cuboid / Volume of a cube

Volume of cuboid = length × breadth × height = 9 × 6 × 6 = 324 cm³ Volume of a cube = side³ = 3³ = 27 cm³

Number of cubes = 324 / 27 = 12

Explanation

In a parallelogram, opposite sides are parallel. When two parallel lines are cut by a transversal, the interior angles on the same side of the transversal (consecutive angles) are supplementary. Therefore, the sum of any two consecutive angles of a parallelogram is equal to 180°.

Explanation

Let L and H be the LCM and HCF of the two numbers.

L + H = 682 L - H = 638

Solving these equations: 2L = 1320 ⇒ L = 660 2H = 44 ⇒ H = 22

Let the two numbers be 22a and 22b, where a and b are co-prime integers. Given sum = 22(a + b) = 286 ⇒ a + b = 13. Product of co-prime factors a × b = LCM / HCF = 660 / 22 = 30.

The pair of co-prime integers satisfying a + b = 13 and a × b = 30 is 10 and 3. The two numbers are 22 × 10 = 220 and 22 × 3 = 66.

Explanation

Cost Price (CP) of 1 banana = ₹20 / 5 = ₹4 Selling Price (SP) of 1 banana = ₹15 / 3 = ₹5

Gain Percentage = [(SP - CP) / CP] × 100 = [(5 - 4) / 4] × 100 = (1 / 4) × 100 = 25%

Explanation

By the Pythagorean theorem, for a right-angled triangle formed by the wall, the ground, and the ladder:

Length of ladder² = (Distance from wall)² + (Height of window)²

Length of ladder² = (2.5)² + 6² Length of ladder² = 6.25 + 36 = 42.25 Length of ladder = √42.25 = 6.5 m

Explanation

Percentage increase = [r / (100 - r)] × 100%

Substituting r = 40: Percentage increase = [40 / (100 - 40)] × 100 = (40 / 60) × 100 = 200/3 % ≈ 66.67%

Alternatively, assuming Y's income is 100, X's income becomes 60. Y's income is 40 more than X's income, making it (40 / 60) × 100 = 66.67% more than X's income.

Explanation

To find the median, arrange the given dataset of 10 numbers in ascending order:

22, 34, 39, 45, 54, 54, 56, 68, 78, 84

Since the number of observations n = 10 is even, the median is the average of the 5th and 6th terms:

Median = (5th term + 6th term) / 2 = (54 + 54) / 2 = 54

Explanation

Length of the rectangle = p Width of the rectangle = p/3

Area of a rectangle = Length × Width = p × (p/3) = p²/3

Explanation

Mean = (Sum of observations) ÷ (Total number of observations)

Sum of the 5 observations = (x + 77) + (x + 7) + (x + 5) + (x + 3) + (x - 2) = 5x + 90

Mean = (5x + 90) ÷ 5 = x + 18