RRB NTPC CBT 1 2016 Mathematics Questions with Solutions

Explanation

Let the three angles of the triangle be 2x, 3x, and 7x. Among these, the greatest angle is 7x and the smallest angle is 2x. The ratio of the greatest angle to the smallest angle is given by 7x : 2x = 7:2.

Explanation

Let the angle be x. Its complement is (90° - x).

According to the given condition: x = ½(90° - x) + 30° x = 45° - ½x + 30° 1.5x = 75° x = 50°

The complement angle is 90° - 50° = 40°.

The difference between the greater and smaller angles is 50° - 40° = 10°.

Explanation

By the Factor Theorem, a linear expression (x - a) is a factor of a polynomial P(x) if and only if P(a) = 0.

Simplifying the given polynomial by combining like terms: P(x) = x⁶ - x⁵ - 2x⁴ + x² - 4

To check if (x - 1) is a factor, substitute x = 1: P(1) = 1⁶ - 1⁵ - 2(1)⁴ + 1² - 4 = 1 - 1 - 2 + 1 - 4 = -5 ≠ 0

To check if (x + 1) is a factor, substitute x = -1: P(-1) = (-1)⁶ - (-1)⁵ - 2(-1)⁴ + (-1)² - 4 = 1 + 1 - 2 + 1 - 4 = -3 ≠ 0

Since neither P(1) nor P(-1) equals zero, neither (x - 1) nor (x + 1) is a factor.

Explanation

Interest on 1 rupee for 1 month = 10 paise = Rs. 0.10

Simple Interest = Principal × Rate per rupee per month × Time in months = 50 × 0.10 × 6 = Rs. 30

Explanation

Let the original numbers be represented by the ratio 5:6, so they are 5x and 6x.

Adding 6 to each number increases the ratio terms from 5 to 7 and from 6 to 8, representing a uniform increase of 2 ratio units.

2 ratio units = 6 1 ratio unit = 3

Therefore, the original numbers are 5 × 3 = 15 and 6 × 3 = 18.

Explanation

Speed of the second bus = Distance ÷ Time = 270 ÷ 15 = 18 km/hr.

The ratio of the speeds of the two buses is given as 11:9.

Speed of the first bus = (11/9) × 18 = 22 km/hr.

Explanation

Let S and A be the ages of Swati and Aparna respectively.

From the given conditions: S × A = 120 3A - S = 2

Substituting A = 120 / S into the second equation: 3(120 / S) - S = 2 360 - S² = 2S S² + 2S - 360 = 0

Factoring the quadratic equation: (S + 20)(S - 18) = 0

Since age cannot be negative, S = 18.

Explanation

Let the cost price of the water purifier be CP.

Initial selling price at 40% loss = 60% of CP New selling price at 10% profit = 110% of CP

The increase of Rs. 125 in the selling price represents the total percentage change from a 40% loss to a 10% profit:

(110% - 60%) of CP = Rs. 125 50% of CP = 125 CP = 125 × 2 = Rs. 250

Explanation

Let the selling price (SP) of 1 pen be ₹1.

Selling price of 12 pens = ₹12 Gain = Selling price of 2 pens = ₹2 Cost Price (CP) = SP - Gain = ₹12 - ₹2 = ₹10

Profit % = (Gain / CP) × 100 = (2 / 10) × 100 = 20%

Explanation

Population variance is equal to the square of the population standard deviation.

Variance = σ² = 7² = 49

Explanation

The compound interest for the nth year is given by: Interest in nth year = P × (1 + r/100)ⁿ⁻¹ × (r/100)

Given P = 2000, r = 40%, and n = 3:

Interest for the 3rd year = 2000 × (1 + 40/100)² × (40/100) = 2000 × (1.4)² × 0.4 = 2000 × 1.96 × 0.4 = 1568

Explanation

For two linear equations a₁x + b₁y = c₁ and a₂x + b₂y = c₂ to represent coincident lines, the ratio of their corresponding coefficients must be equal:

a₁/a₂ = b₁/b₂ = c₁/c₂

Substituting the values from 6x + 2y = 2 and 3x + y = C₂:

6/3 = 2/1 = 2/C₂

2 = 2/C₂

C₂ = 1

Explanation

According to the division algorithm:

Dividend = (Divisor × Quotient) + Remainder

14528 = (Divisor × 83) + 3 14525 = Divisor × 83 Divisor = 14525 ÷ 83 = 175

Explanation

Using the net percentage change formula for successive percentage changes of +20% and -20%:

Net Change = a + b + (a × b)/100 = 20 - 20 + (20 × -20)/100 = -4%

Sales in 2012 = 80 × (1 - 0.04) = 80 × 0.96 = 76.8 lakhs

Explanation

The percentage change in sales for store T from 2011 to 2012 is -8%.

Sales in 2012 = Sales in 2011 × (1 - 0.08) = 0.92 × Sales in 2011

To find the sales for 2011 as a percentage of the sales for 2012: Percentage = (Sales in 2011 / Sales in 2012) × 100 Percentage = (1 / 0.92) × 100 = 108.69% ≈ 109%

Explanation

The ratio of profit distribution is directly proportional to the product of investment amount and time duration.

Sunil's effective investment = 20,000 × 9 = 1,80,000 Gopal's effective investment = 30,000 × 12 = 3,60,000

Ratio of profit distribution (Sunil : Gopal) = 1,80,000 : 3,60,000 = 1 : 2

Sunil's share = (1 / 3) × 60,000 = Rs. 20,000

Explanation

The coefficient of variation (CV) measures the relative variability of a dataset and is calculated using the formula:

Coefficient of Variation = (Standard Deviation / Mean) × 100

Substituting the given values into the formula:

CV = (5 / 11) × 100 = 45.45%

Explanation

Sum of observations = Mean × Number of observations

Sum of initial 14 observations = 14 × 11 = 154 Sum of all 15 observations = 15 × 12 = 180

15th observation = Sum of 15 observations − Sum of 14 observations 15th observation = 180 − 154 = 26

Explanation

Let the daily efficiency of a man and a woman be 3 units and 1 unit respectively.

Total Work = 5 women × 36 days = 5 × 1 × 36 = 180 units

Daily work done by 5 men = 5 × 3 = 15 units

Number of days required = Total Work / Daily work of 5 men = 180 / 15 = 12 days

Explanation

By applying componendo and dividendo to the given equation:

(sin A + cos A + sin A - cos A) / (sin A + cos A - sin A + cos A) = (5 + 4) / (5 - 4)

2 sin A / 2 cos A = 9 / 1 tan A = 9

Squaring both sides: tan² A = 9² = 81

Substituting tan² A = 81 into the required expression:

(tan² A + 1) / (tan² A - 1) = (81 + 1) / (81 - 1) = 82 / 80 = 41 / 40

Explanation

Using the relation Work = Persons × Efficiency × Days:

Initial total work = 15 × 1 × 30 = 450 units

For 30 persons with twice the efficiency (Efficiency = 2), let the required number of days be D:

30 × 2 × D = 450 60 × D = 450 D = 450 / 60 = 7.5 days

Explanation

The ratio of the two angles is given as 2:3. Let the smaller angle be 2x and the greater angle be 3x.

The ratio of the square of the smaller angle to the square of the greater angle is calculated as: (2x)² : (3x)² = 4x² : 9x² = 4 : 9

Explanation

The sum of the first n odd natural numbers is given by the formula Sum = n².

The odd natural numbers less than 72 form a sequence starting from 1 up to 71. The number of terms n is given by (71 + 1) / 2 = 36.

Substituting n = 36 into the formula gives Sum = 36² = 1296.

Explanation

Substitute the given values into the expression 2x(w - z):

2x = 2 × 3 = 6 w - z = -2 - (-½) = -2 + ½ = -3/2

Now, multiply these results: 2x(w - z) = 6 × (-3/2) = -9

Explanation

For two numbers in the co-prime ratio a : b, their LCM is given by the formula:

LCM = HCF × a × b

Here, the ratio terms a = 4 and b = 9 are co-prime, and the HCF is 11.

LCM = 11 × 4 × 9 = 396

Explanation

Using the combined work formula, time taken together T = (t₁ × t₂) / (t₁ + t₂), where t₁ = 15 minutes and t₂ = 20 minutes:

T = (15 × 20) / (15 + 20) = 300 / 35 = 60 / 7 = 8 4/7 minutes.

Explanation

By the Pythagorean theorem, the length of the diagonal d of a rectangle with length l and width w is given by d = √(l² + w²).

Substituting l = 9 cm and w = 6 cm: d = √(9² + 6²) = √(81 + 36) = √117 = √(9 × 13) = 3√13 cm

Since geometric length is always a positive quantity, the length of the diagonal is 3√13 cm.

Explanation

Convert the time durations into seconds: 3 minutes 30 seconds = (3 × 60) + 30 = 210 seconds 4 minutes 10 seconds = (4 × 60) + 10 = 250 seconds

Using the unitary method: Cost for 210 seconds = ₹10 Cost for 250 seconds = (10 / 210) × 250 = 250 / 21 = 11.904... rupees

Rounding to one decimal place gives 11.9 rupees.

Explanation

For a chain of division operations, perform the divisions sequentially from left to right:

35968 ÷ 562 ÷ 8 = (35968 ÷ 562) ÷ 8

Dividing 35968 by 562 gives 64.

Dividing 64 by 8 gives 8.

Alternatively, combining the divisors: 35968 ÷ (562 × 8) = 35968 ÷ 4496 = 8

Explanation

Let W be the time taken to walk one way and R be the time taken to ride one way.

Time for walking and riding (W + R) = 9 hrs 15 min Time for walking both ways (2W) = 11 hrs 12 min

The time taken to ride both ways (2R) is given by:

2R = 2 × (W + R) − 2W 2R = 2 × (9 hrs 15 min) − (11 hrs 12 min) 2R = 18 hrs 30 min − 11 hrs 12 min = 7 hrs 18 min

Explanation

Average Speed = Total Distance ÷ Total Time

From the table, the distance travelled by Truck Q on Friday is 302 km. Given time taken = 8 hours.

Average speed = 302 ÷ 8 = 37.75 km/hr

Explanation

Average distance = Total distance travelled / Total number of days

Total distance travelled by Truck S across the 6 days: 325 + 314 + 312 + 278 + 292 + 274 = 1795 km

Number of days = 6

Average distance = 1795/6 km

Explanation

The total distance travelled by all six trucks on Saturday is calculated by summing the distances covered by trucks P, Q, R, S, T, and U on that day:

Total distance = 292 + 284 + 260 + 274 + 280 + 242 = 1632 km.

Explanation

Target Selling Price = Given SP × (100 + Desired Profit %) ÷ (100 - Given Loss %)

Target Selling Price = 3400 × (100 + 15) ÷ (100 - 15) = 3400 × (115 ÷ 85) = 3400 × (23 ÷ 17) = 200 × 23 = 4600

Explanation

Sum of the ratio terms = 1 + 2 + 4 + 5 = 12 parts.

Value of each ratio part = 156 / 12 = 13.

Multiplying each term of the ratio by 13 gives the respective shares: 1 × 13 = 13 2 × 13 = 26 4 × 13 = 52 5 × 13 = 65

Explanation

Converting the fractions to decimal values:

2/7 ≈ 0.286 0.3 = 0.300 3/7 ≈ 0.429

Comparing the decimal values from smallest to largest gives 0.286 < 0.300 < 0.429, which corresponds to the ascending order 2/7, 0.3, 3/7.

Explanation

Effective compound interest rate for 2 years at 4% per annum is calculated using the successive percentage change formula: Effective Rate = R + R + (R × R)/100 = 4 + 4 + (4 × 4)/100 = 8.16%

Compound Interest = Principal × Effective Rate / 100 = 3500 × 8.16 / 100 = 35 × 8.16 = Rs. 285.6

Alternatively, using the standard formula: Amount = P(1 + R/100)ⁿ = 3500(1 + 4/100)² = 3500 × (1.04)² = 3500 × 1.0816 = Rs. 3785.6

Compound Interest = Amount - Principal = 3785.6 - 3500 = Rs. 285.6

Explanation

The formula for compound amount when interest is compounded quarterly is A = P(1 + r/100)ⁿ, where P is the principal, r is the quarterly rate of interest, and n is the number of quarters.

Given: Principal (P) = ₹10,000 Quarterly rate (r) = 20% / 4 = 5% Number of quarters (n) = 6 months / 3 months = 2

Substituting the values into the formula: A = 10000 × (1 + 5/100)² A = 10000 × (1.05)² A = 10000 × 1.1025 = ₹11,025

Explanation

Standard deviation is defined as the positive square root of variance:

Standard Deviation = √(Variance)

Substituting the given variance:

Standard Deviation = √144 = 12

Because standard deviation is a measure of dispersion and distance from the mean, it is always a non-negative quantity.

Explanation

When a trader cheats by x% while buying and x% while selling using tampered weights, the total percentage gain is given by the successive percentage change formula:

Net Gain % = x + x + (x × x)/100

Substituting x = 9:

Net Gain % = 9 + 9 + (9 × 9)/100 = 18 + 0.81 = 18.81%

Explanation

tan x / cot x = tan x / (1 / tan x) = tan² x

Given sin x = 4/5, the opposite side is 4 and the hypotenuse is 5. Using the Pythagorean theorem, the adjacent side is √(5² - 4²) = 3.

Therefore, tan x = 4/3.

Substituting tan x gives tan² x = (4/3)² = 16/9.

Explanation

The sum of all exterior angles of any convex polygon is always 360°. For a regular polygon with n sides, all exterior angles are equal, so each exterior angle is given by the formula:

Exterior Angle = 360° / n

Substituting the given exterior angle of 45°:

45° = 360° / n n = 360° / 45° = 8

Explanation

Let the cost price of the article be C.

First selling price (SP₁) at 2% profit = 102% of C Second selling price (SP₂) at 14% profit = 114% of C

Ratio of the two selling prices = SP₁ : SP₂ = 102 : 114

Dividing both terms by 6 gives 17 : 19. The given cash difference of Rs. 3 is extra information and is not required to calculate the ratio.

Explanation

Using the quotient rule of exponents, aᵐ ÷ aⁿ = aᵐ⁻ⁿ:

2⁴ ÷ 2⁻¹ = 2⁴⁻⁽⁻¹⁾ = 2⁴⁺¹ = 2⁵ = 32

Explanation

For a mixed recurring decimal where 36 repeats endlessly (0.023636...):

Numerator = (Entire number) - (Non-recurring part) = 236 - 2 = 234 Denominator = Number of 9s equal to recurring digits followed by 0s equal to non-recurring decimal digits = 9900

Converting to simplest fraction form: 234 / 9900 = 13 / 550

Explanation

The mode of a dataset is defined as the value that appears most frequently. In the given set of data (-5, -3, -2, 0, 1, 4, 4, 5, 7, 10), the number 4 appears twice, while all other numbers appear only once. Since 4 has the highest frequency of occurrence, it is the mode of the dataset.

Explanation

To multiply a number by 9999, use the identity 9999 = (10000 - 1):

54367 × 9999 = 54367 × (10000 - 1) = 543670000 - 54367 = 543615633

Explanation

Expand the given expression using the distributive property: 7x + 3x(x - 4) = 7x + 3x² - 12x

Group and combine the like terms (7x and -12x): 3x² + (7x - 12x) = 3x² - 5x

Explanation

For any two numbers, the relation between their product, HCF, and LCM is:

Product of two numbers = HCF × LCM

Substituting the given values into the formula: 3192 = HCF × 56 HCF = 3192 ÷ 56 = 57

Explanation

Ratio of efficiency of Prashanth to Tarun = 2 : 1.

Combined efficiency = 2 + 1 = 3 units per day.

Total work = Combined efficiency × Days taken together = 3 × 21 = 63 units.

Time taken by Tarun alone = Total work / Efficiency of Tarun = 63 / 1 = 63 days.