RRB NTPC CBT 1 2016 Mathematics Questions with Solutions

Explanation

When interest is compounded half-yearly, the rate of interest per compounding period is halved, and the time period is doubled.

Principal (P) = Rs. 10,000 Rate per half-year (r) = 20% ÷ 2 = 10% Number of periods (n) = 2 × 2 = 4

Amount (A) = P × (1 + r/100)ⁿ A = 10000 × (1 + 10/100)⁴ A = 10000 × (1.1)⁴ A = 10000 × 1.4641 A = 14,641

Explanation

Population variance is defined as the square of the population standard deviation (σ²). Given the standard deviation σ = 4.5:

Variance = σ² = (4.5)² = 20.25

Explanation

Multiply the first equation 0.08x + 0.04y = 10 by 100 to clear the decimals: 8x + 4y = 1000

Expand and simplify the second equation 0.2(x - 1) + 0.4y = 24.8: 0.2x - 0.2 + 0.4y = 24.8 0.2x + 0.4y = 25

Multiply this equation by 10 to clear the decimals: 2x + 4y = 250

Subtract the second cleared equation from the first: (8x + 4y) - (2x + 4y) = 1000 - 250 6x = 750 x = 125

Explanation

To find the LCM of decimals, convert each decimal into a fraction in its simplest form: 0.63 = 63/100 10.5 = 21/2 2.1 = 21/10 4.20 = 21/5

The formula for the LCM of fractions is LCM of numerators divided by HCF of denominators.

LCM(63, 21, 21, 21) = 63 HCF(100, 2, 10, 5) = 1

LCM = 63 / 1 = 63

Explanation

Using modular arithmetic, 4 = 3 × 1 + 1, which means 4 ≡ 1 (mod 3).

Raising both sides to the power of 13 gives: 4¹³ ≡ 1¹³ (mod 3) = 1.

Explanation

When speed in still water is u and rate of stream is v, if the time taken to row upstream is n times the time taken to row downstream for the same distance, the relationship is:

u / v = (n + 1) / (n - 1)

Given u = 9 km/hr and n = 2:

9 / v = (2 + 1) / (2 - 1) 9 / v = 3 / 1 v = 9 / 3 = 3 km/hr

Explanation

When a value is sequentially changed by percentage amounts a and b, the net percentage change is given by the formula: Net Percentage Change = a + b + (a × b)/100

For a 60% decrease followed by a 60% increase, a = -60 and b = +60:

Net Change = -60 + 60 + (-60 × 60)/100 Net Change = 0 - 3600/100 = -36%

The negative sign indicates a net loss of 36%.

Alternatively, assuming initial earnings equal 100: Earnings after 60% decrease = 100 × 0.40 = 40 Earnings after subsequent 60% increase = 40 × 1.60 = 64 Final loss = 100 - 64 = 36%

Explanation

Distance = Speed × Time.

First, convert speed from km/h to m/s: Speed = 90 × (5/18) = 25 m/s

Now, calculate distance covered in 20 seconds: Distance = 25 × 20 = 500 m

Explanation

The cost price of the mixture is calculated using the selling price and profit percentage: Cost Price = Selling Price × 100 / (100 + Gain %) Cost Price = 408 × 100 / 120 = Rs. 340 per kg

By applying the rule of alligation: (Quantity of Darjeeling Tea) / (Quantity of Assam Tea) = (340 - 300) / (400 - 340) = 40 / 60 = 2 : 3

Explanation

When two straight lines intersect, any pair of adjacent angles forms a linear pair on a straight line, summing to 180°, which makes them supplementary. In contrast, opposite (vertically opposite) angles formed by two intersecting straight lines are equal to each other, rather than supplementary in general.

Explanation

Efficiency and time taken are inversely proportional.

Ratio of efficiency (Aaravya : Atul) = 150 : 100 = 3 : 2 Ratio of time taken (Aaravya : Atul) = 2 : 3

Given Atul's time = 3 units = 18 days 1 unit = 6 days

Aaravya's time = 2 units = 12 days

Extra time taken by Atul = 18 - 12 = 6 days.

Explanation

Factoring both quadratic expressions:

2x² + 5x - 12 = 2x² + 8x - 3x - 12 = 2x(x + 4) - 3(x + 4) = (2x - 3)(x + 4)

x² + x - 12 = x² + 4x - 3x - 12 = x(x + 4) - 3(x + 4) = (x - 3)(x + 4)

The highest common factor (H.C.F) of both expressions is (x + 4). Comparing (x + a) with (x + 4) yields a = 4.

Explanation

For a tower of height h, the relation between the change in shadow length d and the angles of elevation θ₁ and θ₂ is given by the formula d = h(cot θ₁ - cot θ₂).

Here, d = 30 m, θ₁ = 30°, and θ₂ = 60°.

Substituting the values: 30 = h(cot 30° - cot 60°) 30 = h(√3 - 1/√3) 30 = h(2/√3) h = (30 × √3) / 2 = 15√3 m

Explanation

Profit percentage when cheating in weight/quantity is given by the formula:

Profit % = (Weight Saved / Actual Weight Given) × 100

For a standard order of 100 units, Deepak gives 20% less, which is 80 units.

Profit % = (20 / 80) × 100 = 25%

Explanation

A number is divisible by 3 if the sum of its digits is divisible by 3. Summing the digits of 267Y2 gives 2 + 6 + 7 + Y + 2 = 17 + Y. For this sum to be a multiple of 3, the smallest non-negative integer value for Y is 1, as 17 + 1 = 18, which is divisible by 3.

Explanation

Let the first discount percentage be d%.

The relationship between marked price, selling price, and successive discounts is given by the formula: Selling Price = Marked Price × (1 - d/100) × (1 - Second Discount/100)

Substituting the given values: 160 = 400 × (1 - d/100) × (1 - 20/100) 160 = 400 × (1 - d/100) × 0.8 160 = 320 × (1 - d/100) 1 - d/100 = 160 ÷ 320 1 - d/100 = 0.5 d/100 = 0.5 d = 50%

Alternatively, working backward from the selling price of Rs. 160 after the 20% second discount gives the intermediate price as 160 ÷ 0.8 = Rs. 200. The discount from the initial Rs. 400 list price to Rs. 200 is 200 ÷ 400 × 100 = 50%.

Explanation

To calculate the profit percentage, equalize the number of bananas by taking the LCM of 5 and 4, which is 20.

Cost Price (CP) of 20 bananas = 4 × 4 = Rs. 16 Selling Price (SP) of 20 bananas = 5 × 5 = Rs. 25

Profit = SP - CP = 25 - 16 = Rs. 9

Profit % = (Profit / CP) × 100 = (9 / 16) × 100 = 56.25%

Explanation

Let total work = LCM(14, 21) = 42 units.

Anil's efficiency = 42 ÷ 14 = 3 units/day Rohit's efficiency = 42 ÷ 21 = 2 units/day

Work done by Rohit alone in the last 3 days = 3 × 2 = 6 units Remaining work done together = 42 - 6 = 36 units Time taken working together = 36 ÷ (3 + 2) = 36/5 days

Total number of days = 36/5 + 3 = 51/5 days.

Explanation

To find the combined ratio A : B : C, equalize the term for B in both ratios by taking the LCM of 4 and 6, which is 12.

A : B = (3 × 3) : (4 × 3) = 9 : 12 B : C = (6 × 2) : (5 × 2) = 12 : 10

Thus, A : B : C = 9 : 12 : 10.

Taking A = 9 and C = 10: A + C = 9 + 10 = 19

A : (A + C) = 9 : 19

Explanation

Complementary angles sum to 90°.

Sum of the ratio terms = 11 + 7 = 18 parts. 18 parts = 90° 1 part = 90° / 18 = 5°

The smaller angle corresponds to 7 parts in the ratio: 7 × 5° = 35°

Explanation

Let h be the height of the vertical pole and θ be the angle of elevation of the sun.

The length of the shadow on the ground is √3 h.

Using the right-triangle trigonometric relation: tan θ = height of pole / length of shadow tan θ = h / (√3 h) = 1 / √3

Since tan 30° = 1 / √3, the angle of elevation θ = 30°.

Explanation

The sum of all interior angles in a triangle is 180°.

Given ratio of angles = 2:5:8 Total ratio parts = 2 + 5 + 8 = 15

Largest angle = (8 / 15) × 180° = 8 × 12° = 96°

Explanation

The unit digit of any base ending in 1 raised to any positive integer power is always 1.

Unit digit of (4211)¹⁰² = 1 Unit digit of (361)⁵² = 1

Unit digit of product = 1 × 1 = 1

Explanation

The traffic lights will change simultaneously at intervals equal to the Least Common Multiple (LCM) of 15, 18, 27, and 30 seconds.

Prime factorization of each number: 15 = 3 × 5 18 = 2 × 3² 27 = 3³ 30 = 2 × 3 × 5

LCM(15, 18, 27, 30) = 2¹ × 3³ × 5¹ = 270 seconds

Converting 270 seconds into minutes: 270 seconds = 4 minutes and 30 seconds

Adding this duration to the initial time of 6:10:00 hours: 6:10:00 hours + 0:04:30 hours = 6:14:30 hours

Explanation

The Coefficient of Variation (CV) measures relative variability and is calculated using the formula:

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

Substituting the given values: CV = (7 / 21) × 100% = (1 / 3) × 100% = 33.33%

Explanation

Time taken = 2 hours + 20/60 hours + 24/3600 hours = 2 + 1/3 + 1/150 = 117/50 hours.

New speed = Distance / Time = 39 / (117/50) = (39 × 50) / 117 = 50/3 km/hr.

Since the running speed is 10/9 of the actual speed: Actual speed = (9/10) × (50/3) = 15 km/hr.

Explanation

Let Sonam's present age be S and her father's present age be F.

8 years ago, the ratio of their ages was 1:3: (S - 8) / (F - 8) = 1/3 3S - 24 = F - 8 F = 3S - 16

8 years from now, Sonam's age will be half of her father's age: S + 8 = ½(F + 8) 2S + 16 = F + 8 F = 2S + 8

Equating the two expressions for F: 3S - 16 = 2S + 8 S = 24

Substituting S = 24 into F = 2S + 8: F = 2(24) + 8 = 56 years.

Explanation

Using the standard relationship between Cost Price (CP) and Marked/Printed Price (MP):

CP / MP = (100 - Discount%) / (100 + Profit%)

Substituting Discount = 15% and Profit = 20%:

CP / MP = (100 - 15) / (100 + 20) = 85 / 120

Simplifying by dividing both numerator and denominator by 5:

CP / MP = 17 / 24

Explanation

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

Here, the average of 5 consecutive even numbers is given as 40, which means the middle (3rd) number is 40.

The smallest (1st) number = 40 - 2 × 2 = 36.

Explanation

Under simple interest, the interest earned per unit time remains constant.

When a principal P doubles in 10 years, the simple interest earned in 10 years is 2P - P = P.

In 20 years, the total simple interest earned is 2 × P = 2P.

The total amount after 20 years is Principal + Interest = P + 2P = 3P.

Explanation

Using the algebraic identity x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx):

Let x = a, y = c, and z = 1.

The expression becomes: a³ + c³ + 1³ - 3(a)(c)(1) = (a + c + 1)(a² + c² + 1 - ac - c - a)

Since (a + c + 1) = 0, substituting this value gives: a³ + c³ + 1 - 3ac = 0 × (a² + c² + 1 - ac - c - a) = 0

Explanation

Using the algebraic identity for difference of cubes: x³ - y³ = (x - y)(x² + xy + y²)

Dividing both sides by (x - y): (x³ - y³)/(x - y) = x² + xy + y²

Expanding the given condition: (x + y)² - xy = 0 x² + 2xy + y² - xy = 0 x² + xy + y² = 0

Substituting x² + xy + y² = 0 yields (x³ - y³)/(x - y) = 0.

Explanation

Let L be the length of the ladder and x be the initial distance of the base of the ladder from the wall.

By Pythagoras' theorem for the initial right-angled triangle formed by the wall, floor, and ladder: L² = x² + 9² L² = x² + 81

When the base of the ladder is pulled 3 m further away, the top of the ladder reaches the bottom of the wall (height = 0), meaning the ladder lies flat on the ground: L = x + 3, which gives x = L - 3

Substituting x = L - 3 into the Pythagorean equation: L² = (L - 3)² + 81 L² = L² - 6L + 9 + 81 6L = 90 L = 15 m

Explanation

Total domestic sales of cars of all five manufacturers in 2006 = 480 + 410 + 390 + 380 + 440 = 2100 thousand.

Domestic sales of car manufacturer M in 2006 = 410 thousand.

Percentage share of manufacturer M = (410 / 2100) × 100 ≈ 19.52% ≈ 19.5%.

Explanation

Percentage increase = [(Sales in 2010 - Sales in 2009) / Sales in 2009] × 100

For manufacturer N, sales increased from 420 thousand in 2009 to 495 thousand in 2010. Percentage increase for N = [(495 - 420) / 420] × 100 = (75 / 420) × 100 ≈ 17.86%.

This growth rate of 17.86% is the highest percentage increase recorded in 2010 over the previous year.

Explanation

From the given table, manufacturer L recorded domestic car sales of 520 thousand units in the year 2009. Comparing the sales values for the given combinations, 520 thousand is the highest sales figure recorded across the entire dataset.

Explanation

Using the standard work-time formula, if two people together take T days to complete a task and one person alone takes B days, the time taken by the other person A working alone is given by:

A = (T × B) / (B - T)

Given: Together time (T) = 5 days Binay's time (B) = 8 days

Substituting the values: A = (5 × 8) / (8 - 5) = 40/3 days

Alternatively, calculating 1 day's work: Combined 1 day work = 1/5 Binay's 1 day work = 1/8 Atul's 1 day work = 1/5 - 1/8 = (8 - 5) / 40 = 3/40

Taking the reciprocal, Atul alone takes 40/3 days to complete the work.

Explanation

Dividend = (Divisor × Quotient) + Remainder

Substituting the given polynomial values into the formula: Dividend = (x + 2)(4x - 5) + 12 = 4x² - 5x + 8x - 10 + 12 = 4x² + 3x + 2

Explanation

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

First, calculate the interest produced by Rs. 800 in 3 years at (9/2)% per annum: SI = (800 × 9 × 3) / (2 × 100) = Rs. 108

Equating this to the simple interest produced by the required principal sum P in 9 years at 8% per annum: (P × 8 × 9) / 100 = 108 72P / 100 = 108 P = (108 × 100) / 72 = Rs. 150

Explanation

Using the rule of alligation for the gain/loss percentages:

Ratio of Cost Price of Cylinder to Stove = (4 - (-20)) : (25 - 4) = 24 : 21 = 8 : 7

Sum of ratio terms = 8 + 7 = 15

Cost price of the gas cylinder = (8 / 15) × 4500 = Rs. 2400

Explanation

Percentage reduction in consumption = [r / (100 + r)] × 100

Substituting r = 35%: Percentage reduction = [35 / (100 + 35)] × 100 = (35 / 135) × 100 = 700 / 27 = 25.92% ≈ 26%

Explanation

Sum of initial 10 observations = 10 × 13 = 130

With 2 additional observations, total observations become 12. Sum of total 12 observations = 12 × 14 = 168

Sum of the 2 new observations = 168 - 130 = 38 Mean of the 2 new observations = 38 / 2 = 19

Explanation

Population variance (σ²) is equal to the square of the population standard deviation (σ).

Variance = (Standard Deviation)² = 6.5² = 42.25

Explanation

The smallest natural number greater than 15 divisible by 7 is 21 (7 × 3), and the largest natural number less than 85 divisible by 7 is 84 (7 × 12).

The total number of multiples is given by: Count = 12 - 3 + 1 = 10

Explanation

Approximating the given expression by rounding off the terms: 5.168 ≈ 5.2 3.194 ≈ 3.2 67.999 ≈ 68 4453 / 4224.017 ≈ 1.054

Evaluating the simplified expression: (5.2 × 3.2 × 1.054) / 68 ≈ 17.54 / 68 ≈ 0.258

Evaluating the difference between 0.258 and the options shows that 0.2 has the smallest absolute difference.

Explanation

When the difference between the denominator and the numerator is constant for a set of proper fractions, the fraction with the largest numerator and denominator has the greatest value.

For each ratio, the difference between the denominator and the numerator is 8: 15 - 7 = 8 23 - 15 = 8 25 - 17 = 8 29 - 21 = 8

A proper fraction with numerator n and denominator d can be written as n/d = 1 - (8/d). As the denominator d increases, the term 8/d decreases, making 1 - (8/d) larger. Since 21:29 has the largest denominator (29), its value 21/29 ≈ 0.724 is the greatest.

Explanation

From the given equation 3.5x = 0.07y, the ratio of x to y is: x / y = 0.07 / 3.5 = 7 / 350 = 1 / 50

This gives y = 50x. Substituting y = 50x into the expression (y - x) / (y + x): (50x - x) / (50x + x) = 49x / 51x = 49 / 51

Explanation

The difference between each divisor and its corresponding remainder is constant: 6 - 2 = 7 - 3 = 8 - 4 = 9 - 5 = 12 - 8 = 4.

The least number X is given by: X = LCM(6, 7, 8, 9, 12) - 4

Calculating the LCM of 6, 7, 8, 9, and 12: LCM = 2³ × 3² × 7 = 8 × 9 × 7 = 504

Thus, X = 504 - 4 = 500.

Now, calculating 150% of X: 150% of 500 = (150 / 100) × 500 = 750

Explanation

Multiplying both sides of the equation a² + b² + c² = ab + bc + ca by 2 and rearranging gives:

(a - b)² + (b - c)² + (c - a)² = 0

Since a, b, and c are real numbers, the sum of squares equals zero if and only if each term is zero, which implies a = b = c.

Substituting b = a and c = a into the expression:

(a + b + c)² = (a + a + a)² = (3a)² = 9a²

Explanation

Percentage less = [r / (100 + r)] × 100%

Here, r = 75/4%.

Percentage boyfriend earns less = [(75/4) / (100 + 75/4)] × 100 = [(75/4) / (475/4)] × 100 = (75 / 475) × 100 = (3 / 19) × 100 = 300 / 19 = 15.79% ≈ 16%