RRB NTPC CBT 1 2016 Mathematics Questions with Solutions

Explanation

Using the trigonometric identity, sec² x - 1 = tan² x.

Given sin x = 4/5, the value of cos x is: cos x = √(1 - sin² x) = √(1 - 16/25) = √(9/25) = 3/5

Now, tan x = sin x / cos x = (4/5) / (3/5) = 4/3.

Substituting this into the identity: sec² x - 1 = tan² x = (4/3)² = 16/9

Explanation

Using Euclid's division algorithm to find the Highest Common Factor (HCF):

2259 = 1757 × 1 + 502 1757 = 502 × 3 + 251 502 = 251 × 2 + 0

Since the remainder becomes zero when divided by 251, the HCF of 1757 and 2259 is 251.

Explanation

Substituting the given values into the expression w² √(z² + y²):

w² = (-2)² = 4 z² = (-½)² = ¼ y² = 0² = 0

Calculating the square root term: √(z² + y²) = √(¼ + 0) = √(¼) = ½

Multiplying the results: w² √(z² + y²) = 4 × ½ = 2

Explanation

Rearranging the linear equation x - 3 = 3x + 7 to group variable terms on one side and constant terms on the other:

3x - x = -3 - 7 2x = -10 x = -10 / 2 x = -5

Substituting x = -5 back into the original equation yields -5 - 3 = -8 on the left-hand side and 3(-5) + 7 = -8 on the right-hand side, confirming the solution.

Explanation

Following the standard left-to-right order of operations for sequential division:

33800 / 520 / 5 = (33800 / 520) / 5

First, divide 33800 by 520: 33800 / 520 = 65

Next, divide 65 by 5: 65 / 5 = 13

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 = 5 cm and w = 3 cm:

d = √(5² + 3²) = √(25 + 9) = √34 cm

As distance is always a non-negative geometric quantity, the diagonal length is strictly positive.

Explanation

The sum of all exterior angles of a regular polygon is always 360°. The number of sides n is given by:

n = 360° / Exterior Angle

Substituting the given exterior angle of 72°:

n = 360° / 72° = 5

Explanation

For a principal P at rate r% compounded annually for 2 years, the effective interest rate is 2r + r²/100 %.

Here, r = 5%, so the effective interest rate = 2(5) + 5²/100 = 10.25%.

Compound Interest = 7500 × 10.25% = 7500 × (10.25 / 100) = 75 × 10.25 = 768.75.

Explanation

Ratio of efficiency of Rakshanth and Verma = 2 : 1. Combined efficiency = 2 + 1 = 3 units/day.

Total work = Combined efficiency × Days taken together = 3 × 19 = 57 units.

Days taken by Verma alone = Total work / Verma's efficiency = 57 / 1 = 57 days.

Explanation

For two numbers in the ratio a:b (where a and b are co-prime) with an HCF of H, their LCM is given by:

LCM = a × b × H

Substituting the given values (a = 2, b = 5, and H = 18):

LCM = 2 × 5 × 18 = 180

Explanation

The sum of all exterior angles of any regular polygon is always 360°. The number of sides n is given by the formula n = 360° / exterior angle.

Substituting the given exterior angle of 40°:

n = 360° / 40° = 9

Explanation

Using the exponent rule (aᵐ)ⁿ = aᵐⁿ:

(25)⁽3/2) = (5²)⁽3/2) = 5⁽2 × 3/2) = 5³ = 125.

Explanation

By the Pythagorean theorem, the length of the diagonal d of a rectangle with length l and width w is given by the formula d = √(l² + w²). Substituting the given values l = 9 cm and w = 5 cm gives d = √(9² + 5²) = √(81 + 25) = √106 cm. Since geometric length is always a positive quantity, the length of the diagonal is √106 cm.

Explanation

Efficiency ratio of Manoj to Anuraj = 2 : 1.

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

Total work = Combined efficiency × Time taken together = 3 × 22 = 66 units.

Time taken by Anuraj alone = Total work ÷ Anuraj's efficiency = 66 ÷ 1 = 66 days.

Explanation

For two numbers in the ratio a:b (where a and b are co-prime) and having HCF h, their LCM is given by the formula:

LCM = h × a × b

Substituting h = 13, a = 4, and b = 5:

LCM = 13 × 4 × 5 = 260

Explanation

To simplify the given expression, convert each division operation into multiplication by taking the reciprocal of the dividing fraction:

5/28 ÷ 28/35 ÷ 20/112 = (5/28) × (35/28) × (112/20)

Simplifying the fractions where possible: 35/28 reduces to 5/4 by dividing numerator and denominator by 7. 112/20 reduces to 28/5 by dividing numerator and denominator by 4.

Substituting these back into the expression: (5/28) × (5/4) × (28/5)

Canceling the common terms 28 and 5 from the numerator and denominator: = 5/4

Explanation

Using unitary method, convert time into seconds: Total time = 19 minutes = 19 × 60 = 1140 seconds Given time = 15 minutes 50 seconds = (15 × 60) + 50 = 950 seconds

Cost for 950 seconds = 13 × (950 / 1140) = 13 × (5 / 6) = 65 / 6 = 10.833... ≈ 10.8 rupees.

Explanation

First, divide 4082 by 157: 4082 ÷ 157 = 26

Next, perform the subtraction: 26 - 23 = 3

Explanation

To solve the linear equation for x:

1000x - 5 = -6

Add 5 to both sides: 1000x = -6 + 5 1000x = -1

Divide both sides by 1000: x = -1/1000

Explanation

For any two positive integers, the fundamental relation between their product, Highest Common Factor (HCF), and Least Common Multiple (LCM) is given by:

Product of two numbers = HCF × LCM

Substituting the given values into the formula: 2522 = HCF × 97

HCF = 2522 ÷ 97 = 26

Explanation

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

Given that W + R = 9 hours 20 minutes, doubling this gives 2(W + R) = 18 hours 40 minutes, which represents the combined time to walk both ways and ride both ways.

The time taken to walk both ways is 2W = 11 hours 15 minutes.

Subtracting 2W from 2(W + R) gives the total time taken to ride back both ways (2R): 2R = 18 hours 40 minutes - 11 hours 15 minutes = 7 hours 25 minutes.

Explanation

The arithmetic mean is calculated using the formula: Mean = (Sum of all observations) / (Total number of observations)

Sum of observations = 2 + 9 + 9 + 3 + 6 + 9 + 4 = 42 Total number of observations = 7

Mean = 42 / 7 = 6

Explanation

Using the direct selling price formula:

New Selling Price = Given Selling Price × (100 + Profit%) / (100 - Loss%)

Substituting the given values:

New Selling Price = 20,400 × (100 + 15) / (100 - 15) = 20,400 × 115 / 85 = 20,400 × 23 / 17 = 1,200 × 23 = Rs. 27,600

Explanation

Effective compound interest rate for 2 years at 4% per annum = 4 + 4 + (4 × 4) / 100 = 8.16%.

Compound Interest = Principal × Effective Rate = 4500 × 8.16 / 100 = 45 × 8.16 = Rs. 367.2

Explanation

Sum of the ratio terms = 1 + 3 + 6 + 7 = 17 parts.

Value per ratio part = 221 / 17 = 13.

Multiplying each term of the ratio by 13 gives the respective amounts: 1 × 13 = 13 3 × 13 = 39 6 × 13 = 78 7 × 13 = 91

Explanation

Using the trigonometric identity: tan(A - B) = (tan A - tan B) / (1 + tan A × tan B)

Substitute tan A = 15/8 and tan B = 7/24 into the formula:

tan A - tan B = 15/8 - 7/24 = 45/24 - 7/24 = 38/24

1 + tan A × tan B = 1 + (15/8 × 7/24) = 1 + 105/192 = 297/192

tan(A - B) = (38/24) ÷ (297/192) = (38/24) × (192/297) = 38 × 8 / 297 = 304/297

Explanation

To find the product of 2.31 and 0.34, multiply the numbers ignoring the decimal points first:

231 × 34 = 7854

Count the total number of decimal places in both factors: 2.31 has two decimal places and 0.34 has two decimal places, giving a total of four decimal places. Placing the decimal point four places from the right yields 0.7854.

Explanation

To convert a decimal number to a fraction, place the digits after the decimal point over a power of 10 corresponding to the number of decimal digits. Since 0.018 has three decimal places, the denominator is 10³ = 1000, giving 18/1000.

Explanation

Converting each fraction into its decimal form:

1/5 = 0.20 1/3 = 0.33... 2/5 = 0.40

Arranging these decimal values in increasing order gives 0.20 < 0.33 < 0.40, which corresponds to 1/5, 1/3, 2/5.

Explanation

Expanding the given algebraic expression using the distributive property: 5x - 2x(x - 1) = 5x - 2x² + 2x

Combining the like terms: (5x + 2x) - 2x² = 7x - 2x²

Explanation

In a right-angled triangle with 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, cos x = 3/5, which gives sec x = 1/cos x = 5/3.

Evaluating the given expression: sec x / sin x = (5/3) / (4/5) = (5/3) × (5/4) = 25/12

Explanation

Simple Interest SI = (P × R × T) / 100 Total Amount = Principal + SI

Given Principal P = Rs. 650, Rate R = 6%, and Time T = 4 years: SI = (650 × 6 × 4) / 100 = Rs. 156

Total Amount = 650 + 156 = Rs. 806

Explanation

Rotational symmetry refers to the number of times a geometric figure matches its original orientation during a complete rotation of 360°. A trapezium aligns with its original shape only after completing a full turn of 360°. Consequently, the order of rotational symmetry of a trapezium is 1.

Explanation

To find the median, arrange the given 10 observations in ascending order: -5, -3, -2, 0, 1, 4, 4, 5, 7, 10

Since the total number of observations n = 10 is even, the median is the mean of the (n/2)th and ((n/2) + 1)th terms, which correspond to the 5th and 6th values in the sorted list.

5th term = 1 6th term = 4

Median = (1 + 4) / 2 = 5 / 2 = 2.5

Explanation

Let each equal distance be d km. The total distance travelled is 3d km.

Time = Distance / Speed

Total time in hours = d/3 + d/4 + d/8 = d × (8 + 6 + 3) / 24 = 17d/24 hours

Converting total time from minutes to hours: 42.5 minutes = 42.5 / 60 = 17 / 24 hours

Equating both: 17d/24 = 17/24 d = 1 km

Total distance = 3 × 1 = 3 km

Explanation

Using the work-equivalence formula M₁ × D₁ = M₂ × D₂:

4 × 12 = 6 × D₂ 48 = 6 × D₂ D₂ = 48 / 6 = 8 days

Explanation

Let the cost price of the article be 100 units.

First selling price at 2% profit = 100 + 2 = 102 units Second selling price at 12% profit = 100 + 12 = 112 units

Ratio of the two selling prices = 102 : 112 = 51 : 56

Explanation

When a trader cheats by x% while buying and y% while selling using faulty weights, the net percentage profit is calculated using the successive percentage formula:

Net Gain % = x + y + (x × y)/100

Given x = 8% and y = 8%:

Net Gain % = 8 + 8 + (8 × 8)/100 = 16 + 64/100 = 16 + 0.64 = 16.64%

Explanation

Substitute the given values into the expression: z/w + x = (-½) / (-2) + 3

Dividing -½ by -2 gives ¼.

Adding 3 to ¼ gives: ¼ + 3 = 3¼

Explanation

Using the standard order of operations for sequential division, operations are evaluated from left to right:

28854 ÷ 458 ÷ 9 = (28854 ÷ 458) ÷ 9 = 28854 ÷ (458 × 9)

Multiplying the divisors: 458 × 9 = 4122

Dividing the dividend by 4122: 28854 ÷ 4122 = 7

Explanation

Profit ratio in a partnership is equal to the ratio of the products of their respective investments and time periods: Profit ratio = (Capital₁ × Time₁) / (Capital₂ × Time₂)

Let Mohan's time be 12 months and Sohan's time in the business be t months. Substituting the given values:

3 / 1 = (100000 × 12) / (40000 × t) 3 / 1 = 30 / t 3t = 30 t = 10 months

Sohan was active in the business for 10 months, which means he joined after 12 - 10 = 2 months.

Explanation

Complementary angles add up to 90°. For two angles in the ratio 4:5, the total number of parts is 4 + 5 = 9.

Value of one part = 90° / 9 = 10°

The greater angle corresponds to 5 parts = 5 × 10° = 50°.

Explanation

Average = (Total pages printed in 4th month) ÷ (Number of printers)

Sum of pages printed by printers P, Q, R, S, T, and U in the 4th month: = 256 + 563 + 347 + 651 + 412 + 321 = 2550

Average = 2550 ÷ 6 = 425

Explanation

The total number of pages printed by Printer S across all 5 months is calculated by summing its monthly output: 552 + 438 + 527 + 651 + 582 = 2750 pages. Comparing this with the total pages printed by the other printers over the same period (P = 2646, Q = 2665, R = 2595, T = 2731, U = 2712), Printer S has the highest overall total.

Explanation

Number of pages printed by Printer Q in the 2nd month = 441 Number of pages printed by Printer U in the 5th month = 693

Required Ratio = 441 : 693

Dividing both terms by 63: 441 ÷ 63 = 7 693 ÷ 63 = 11

The simplified ratio is 7:11.

Explanation

Rearranging the given equation into a sum of two perfect squares: 10x² - 6xy + y² - 4x + 4 = 0 (9x² - 6xy + y²) + (x² - 4x + 4) = 0 (3x - y)² + (x - 2)² = 0

Since the sum of two squared real terms is zero, each term must individually equal zero: x - 2 = 0 ⇒ x = 2 3x - y = 0 ⇒ y = 3x = 6

Substituting x = 2 and y = 6 into (x + 2y): x + 2y = 2 + 2(6) = 2 + 12 = 14

Explanation

The Coefficient of Variation (CV) measures the relative variability of a dataset and is given by the formula:

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

Substituting the given values: Coefficient of Variation = (4 / 20) × 100% = 0.20 × 100% = 20%

Explanation

Sum of interior angles of a triangle is always 180°.

Sum of ratio terms = 2 + 5 + 8 = 15

Smallest angle = (2 / 15) × 180° = 2 × 12° = 24°

Explanation

The decrease in the selling price is 360 - 345 = Rs. 15. Since loss increases by 4% of the cost price (CP) due to this price drop, 4% of CP = 15. Therefore, CP = (15 × 100) / 4 = Rs. 375.

Explanation

The average of the first n multiples of a number k is given by the formula:

Average = k × (n + 1) / 2

Substituting k = 9 and n = 30:

Average = 9 × (30 + 1) / 2 = 9 × 31 / 2 = 279 / 2 = 139.5

Alternatively, since the multiples form an arithmetic progression, the average is simply the mid-point of the first term (9) and the last term (270):

Average = (9 + 270) / 2 = 139.5