RRB NTPC CBT 1 2016 Mathematics Questions with Solutions

Explanation

Speed = Distance / Time.

Speed of car A = 90 km / 2 hours = 45 km/h.

Since car B is twice as fast as car A: Speed of car B = 2 × 45 km/h = 90 km/h.

Explanation

Speed of Y = Distance ÷ Time = 420 ÷ 3 = 140 km/h.

Given the ratio of speeds of X and Y is 5:7, Speed of X = (5 ÷ 7) × Speed of Y = (5 ÷ 7) × 140 = 100 km/h.

Explanation

The mean (x̅) of the given set is (10 + 11 + 9 + 11 + 9) / 5 = 50 / 5 = 10.

The sum of squared deviations from the mean is Σ(x - x̅)² = (10 - 10)² + (11 - 10)² + (9 - 10)² + (11 - 10)² + (9 - 10)² = 0 + 1 + 1 + 1 + 1 = 4.

Using the standard deviation formula σ = √(Σ(x - x̅)² / N), where N = 5: σ = √(4 / 5) = 2 / √5.

Explanation

To count the total occurrences of the digit 5 in the numbers from 1 to 100, analyze its appearance by place value:

In the unit's place, 5 occurs 10 times: 5, 15, 25, 35, 45, 55, 65, 75, 85, and 95. In the ten's place, 5 occurs 10 times: 50, 51, 52, 53, 54, 55, 56, 57, 58, and 59.

Adding the occurrences across both positions gives 10 + 10 = 20 times. Note that the number 55 contains two 5s, contributing once to the unit's place count and once to the ten's place count.

Explanation

Two and a half kg (2.5 kg) consists of five half-kg (0.5 kg) portions, since 2.5 ÷ 0.5 = 5.

Therefore, the price of half a kg of vegetables = 120 ÷ 5 = ₹24.

Explanation

For an odd number of consecutive terms in an arithmetic progression, the middle term is equal to the average of the set.

Middle number = Sum ÷ Number of terms = 42 ÷ 3 = 14

Explanation

Expanding the given product using the distributive law:

(x + 2y)(2x - y) = x(2x - y) + 2y(2x - y) = 2x² - xy + 4xy - 2y² = 2x² + 3xy - 2y²

Combining the like terms -xy and 4xy yields 3xy.

Explanation

Using the fundamental trigonometric identities sec²θ - tan²θ = 1 and tanθ × cotθ = 1:

(sec²θ + 2tanθcotθ - tan²θ) = (sec²θ - tan²θ) + 2(tanθ × cotθ) = 1 + 2(1) = 3

Explanation

Using the inverse proportion relation M₁ × D₁ = M₂ × D₂:

Initial work capacity = 9 × 50 = 450 woman-days.

When 6 more women are added, the total workforce becomes 9 + 6 = 15 women.

Number of days required = (9 × 50) / 15 = 450 / 15 = 30 days.

Explanation

Adding the two equations eliminates y: (4x + 5y) + (x - 5y) = 14 + 16 5x = 30 x = 6

Substituting x = 6 into x - 5y = 16: 6 - 5y = 16 -5y = 10 y = -2

Explanation

To find the maximum number of gift packs that can be made such that each pack contains both types of diaries with equal distribution and none left over, calculate the Highest Common Factor (HCF) of the two quantities, 20 and 16.

Prime factorization of 20 = 2² × 5 Prime factorization of 16 = 2⁴

HCF(20, 16) = 2² = 4

Thus, 4 gift packs can be made, with each pack containing 5 big diaries and 4 small diaries.

Explanation

To find the ratio N:L, express N/L as the product of (N/M) and (M/L):

N/L = (N/M) × (M/L)

Given L:M = 3:5, M/L = 5/3. Given M:N = 2:3, N/M = 3:2.

N/L = (3/2) × (5/3) = 5/2

Explanation

Total remaining dozen of each fruit: Remaining apples = 10 × (1 - ½) = 5 dozen Remaining mangoes = 15 × (1 - ⅓) = 10 dozen Remaining oranges = 20 × (1 - ¼) = 15 dozen

Total remaining fruits in dozen = 5 + 10 + 15 = 30 dozen Total remaining fruits = 30 × 12 = 360

Explanation

Using the trigonometric identity tan(-θ) = -tan(θ) and the periodicity property tan(360° + θ) = tan(θ):

tan(-405°) = -tan(405°) = -tan(360° + 45°) = -tan(45°) = -1

Explanation

To find the highest common factor (HCF) of 360 and 450, express both numbers in terms of their prime factorizations:

360 = 2³ × 3² × 5¹ 450 = 2¹ × 3² × 5²

Taking the product of the lowest powers of all common prime factors: HCF = 2¹ × 3² × 5¹ = 2 × 9 × 5 = 90

Alternatively, using Euclid's division algorithm: 450 = 360 × 1 + 90 360 = 90 × 4 + 0

Since the remainder becomes 0, the divisor 90 is the HCF.

Explanation

For two years at a simple interest rate of 10% per annum, the total simple interest rate is 20%. Since the simple interest is Rs. 2,000, the principal P = 2000 × (100 / 20) = Rs. 10,000.

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

D = P × (R / 100)²

Substituting P = 10,000 and R = 10:

D = 10,000 × (10 / 100)² = 10,000 × (1 / 100) = Rs. 100

Explanation

By definition, a prime number is a natural number greater than 1 that has exactly two distinct positive integer divisors: 1 and the number itself. For example, numbers like 2, 3, 5, 7, and 11 can only be divided evenly by 1 and themselves.

Explanation

Simple Interest formula is SI = (P × R × T) / 100, where P is principal, R is rate of interest per annum, and T is time in years.

Interest from Shiva = (6000 × R × 3) / 100 = 180R Interest from Krishna = (8000 × R × 5) / 100 = 400R

Total Interest = 180R + 400R = 580R

Given total interest is Rs. 5,220: 580R = 5220 R = 5220 / 580 = 9%

Explanation

Substitute Q = 10 into the given equation: P + 10 = 2(P - 10)

Expanding the right side: P + 10 = 2P - 20

Rearranging terms to solve for P: 20 + 10 = 2P - P P = 30

Explanation

The area of a square with diagonal d is given by the formula Area = d² / 2.

Substituting the given diagonal d = 13√2:

Area = (13√2)² / 2 = (169 × 2) / 2 = 169 sq. units.

Explanation

Converting the given fractions into their decimal equivalents: 6/5 = 1.20 5/4 = 1.25 4/3 = 1.33 3/2 = 1.50

Comparing these decimal values, 1.20 is smaller than 1.25, 1.33, and 1.50, which makes 6/5 the smallest fraction.

Explanation

Cost Price = Selling Price / (1 + Profit%/100)

Substituting the given values: Cost Price = 1300 / (1 + 30/100) = 1300 / 1.3 = 1000

Explanation

Total pens = 2 dozen = 2 × 12 = 24 pens. Cost price (CP) = ₹ 480.

Number of packets made = 24 / 4 = 6 packets. Total selling price (SP) = 6 × ₹ 100 = ₹ 600.

Profit = SP - CP = 600 - 480 = ₹ 120.

Profit percentage = (Profit / CP) × 100 = (120 / 480) × 100 = 25%.

Explanation

Range = Maximum value - Minimum value

From the given data set: Maximum value = 19 Minimum value = 9

Range = 19 - 9 = 10

Explanation

Using the standard algebraic identity (a + b)³ = a³ + 3a²b + 3ab² + b³:

Substitute a = s and b = 2: (s + 2)³ = s³ + 3(s²)(2) + 3(s)(2²) + 2³ = s³ + 6s² + 12s + 8

Explanation

Fraction of work done = (Number of days worked) / (Total days required)

Substituting the given values: Work done = 20 / 50 = 2/5

Explanation

Using the property of similar triangles, the ratio of the height of the tree to the height of the electric pole is equal to the ratio of the shadow length to the total distance from the pole to the tip of the shadow.

Let s be the length of the shadow of the tree.

Height of tree / Height of pole = s / (10 + s) 6 / 18 = s / (10 + s) 1 / 3 = s / (10 + s) 10 + s = 3s 2s = 10 s = 5 feet

Explanation

Standard deviation measures the spread or dispersion of data points relative to their mean. It is calculated using the formula σ = √(Σ(xᵢ - μ)² / N), where μ is the mean and N is the total number of observations.

For the given set {10, 10, 10, 10, 10}: Mean μ = (10 + 10 + 10 + 10 + 10) / 5 = 10 Variance σ² = ((10 - 10)² + (10 - 10)² + (10 - 10)² + (10 - 10)² + (10 - 10)²) / 5 = 0 Standard deviation σ = √0 = 0.

Explanation

Cost price of 12 pens = ₹120.

Since there are 4 packs of 3 pens each in a dozen (12 ÷ 3 = 4), and each pack is sold for ₹35: Total selling price = 4 × 35 = ₹140.

Gain = Selling Price - Cost Price = 140 - 120 = ₹20. Gain percentage = (20 / 120) × 100 = 16.67%.

Explanation

Since efficiency is inversely proportional to time taken, the ratio of efficiencies of G and S is 2 : 1, which means the ratio of time taken by G and S is 1 : 2.

The difference in the time ratio is 2 - 1 = 1 unit, which corresponds to 30 days. Time taken by G = 1 × 30 = 30 days Time taken by S = 2 × 30 = 60 days

Total work = Efficiency × Time = 2 × 30 = 60 units. Combined efficiency of G and S = 2 + 1 = 3 units per day.

Time taken working together = Total work / Combined efficiency = 60 / 3 = 20 days.

Explanation

Simple Interest SI = (P × R × T) / 100

Since the principal amount P and interest rate R = 12% are the same in both cases, simple interest is directly proportional to time period T.

Ratio of Simple Interest = T₁ : T₂ = 9 : 12 = 3 : 4

Explanation

Range is calculated as the difference between the maximum and minimum values in the dataset: Range = Maximum value - Minimum value = 29 - 16 = 13.

Mode is the value that appears most frequently in the dataset. Frequency of each number: 16 appears 1 time 17 appears 2 times 18 appears 3 times 19 appears 1 time 28 appears 1 time 29 appears 1 time

Since 18 has the highest frequency (3 times), the mode is 18.

Explanation

Gain Percentage = [(Selling Price - Cost Price) / Cost Price] × 100

Gain = 13,500 - 11,000 = 2,500

Gain Percentage = (2,500 / 11,000) × 100 = 250 / 11 = 22.727...% ≈ 22.7%

Explanation

Using the algebraic expansion (a + b)² = a² + b² + 2ab:

(sin θ + cos θ)² = sin²θ + cos²θ + 2 sin θ cos θ

Substituting the fundamental trigonometric identity sin²θ + cos²θ = 1 yields:

(sin θ + cos θ)² = 1 + 2 cos θ sin θ

Explanation

In every set of 10 consecutive integers, each digit from 0 to 9 appears exactly once in the unit's place. The range from 1 to 100 contains 10 complete sets of 10 numbers (1–10, 11–20, ..., 91–100).

Therefore, the digit 3 occurs in the unit's place exactly once in each set, giving a total of 10 times: 3, 13, 23, 33, 43, 53, 63, 73, 83, and 93.

Explanation

Prime factorization of the given numbers: 20 = 2² × 5 28 = 2² × 7

The Greatest Common Factor (G.C.F.) is the product of the lowest powers of common prime factors: G.C.F.(20, 28) = 2² = 4

The Least Common Multiple (L.C.M.) is the product of the highest powers of all prime factors involved: L.C.M.(20, 28) = 2² × 5 × 7 = 140

Thus, the G.C.F. and L.C.M. of 20 and 28 are 4 and 140 respectively.

Explanation

Total distance = 300 m Total time = 2 minutes 30 seconds = 150 seconds

Speed (in m/s) = Distance / Time = 300 / 150 = 2 m/s

To convert speed from m/s to km/h, multiply by 18/5: Speed = 2 × (18/5) = 36/5 = 7.2 km/h

Explanation

Let the other number be y.

According to the given condition, twice the difference between the two numbers is equal to their sum: 2(15 - y) = 15 + y 30 - 2y = 15 + y 3y = 15 y = 5

Explanation

Let the three consecutive numbers be x − 1, x, and x + 1.

Sum = (x − 1) + x + (x + 1) = 3x = 126 x = 126 ÷ 3 = 42

The three numbers are 41, 42, and 43. The highest number among them is 43.

Explanation

Let Surya's present age be S and his son's present age be s.

According to the given conditions: S - s = 25 In 5 years, S + 5 = 2(s + 5), which simplifies to S - 2s = 5.

Subtracting the second equation from the first: (S - s) - (S - 2s) = 25 - 5 s = 20

Surya's present age S = 20 + 25 = 45 years. Surya's age after 3 years = 45 + 3 = 48 years.

Explanation

The exterior angle of a regular polygon is E = 180° - 108° = 72°. The number of sides n is given by n = 360° / E = 360° / 72° = 5. A polygon with 5 sides is a pentagon.

Explanation

For sin θ = 1 / √2, θ = 45°.

Therefore, cos θ = 1 / √2 and tan θ = 1.

Substituting these values into the expression: tan θ + cos θ = 1 + 1 / √2 = (1 + √2) / √2

Explanation

Mean is calculated as the sum of all values divided by the number of observations: Mean = (7 + 5 + 5 + 2 + 7 + 6 + 5 + 3 + 7 + 6) / 10 = 53 / 10 = 5.3

To find the median, arrange the 10 values in ascending order: 2, 3, 5, 5, 5, 6, 6, 7, 7, 7

Since n = 10 is even, the median is the average of the 5th and 6th values: Median = (5 + 6) / 2 = 5.5

Explanation

To find the maximum number of sets that can be formed without any items leftover, calculate the Highest Common Factor (HCF) of 32, 24, and 16.

32 = 8 × 4 24 = 8 × 3 16 = 8 × 2

HCF(32, 24, 16) = 8

Thus, 8 sets can be made, with each set containing 4 pens, 3 pencils, and 2 erasers.

Explanation

Using the algebraic expansion formula (a - b)² = a² - 2ab + b²:

Substituting a = w and b = 9: (w - 9)² = w² - 2(w)(9) + 9² = w² - 18w + 81

Explanation

The effective annual rate of interest with half-yearly compounding is calculated using the formula:

Effective Rate = (1 + r/n)ⁿ - 1

Here, nominal annual rate r = 10% = 0.10, and compounding frequency n = 2 per year.

Rate per half-year = 10% / 2 = 5% = 0.05

Effective Rate = (1 + 0.05)² - 1 = 1.05² - 1 = 1.1025 - 1 = 0.1025 = 10.25%

Explanation

Total rate per unit = Base rate + Margin = 67.89 + 1.11 = Rs. 69.00

Cost of 150 units = 150 × 69 = Rs. 10,350

Explanation

The area of a circle is given by the formula Area = πr², where r is the radius.

Given Area = 9π sq. cm: πr² = 9π r² = 9 r = 3 cm

The circumference of a circle is given by the formula Circumference = 2πr.

Substituting r = 3 cm: Circumference = 2 × π × 3 = 6π cm

Explanation

Let the number of Rs. 5, Rs. 10, and Rs. 20 currency notes be 3x, 2x, and 1x respectively.

Total value = (5 × 3x) + (10 × 2x) + (20 × 1x) = 15x + 20x + 20x = 55x

Given total value = 165 55x = 165 x = 3

Number of Rs. 20 notes = 1x = 3 notes. Value of currency notes of Rs. 20 denomination = 3 × 20 = Rs. 60.

Explanation

Converting each fraction to decimal form: 3/2 = 1.50 4/3 = 1.33... 5/4 = 1.25 6/5 = 1.20

Comparing these values, 1.50 is the largest decimal.