RRB NTPC CBT 1 2020 Mathematics Questions with Solutions

Explanation

Let the marked price be ₹100.

Cost Price (CP) = 75% of ₹100 = ₹75.

Selling Price (SP) = Marked Price - Discount = ₹100 - (15% of ₹100) = ₹85.

Profit Percentage = [(SP - CP) / CP] × 100

Substituting the values: Profit Percentage = [(85 - 75) / 75] × 100 = (10 / 75) × 100 = 13.33% profit.

Explanation

Fraction spent on education = 1/3 Fraction spent on food and other necessities = 2/6 = 1/3

Total fraction spent = 1/3 + 1/3 = 2/3 of the salary

Fraction saved = 1 - 2/3 = 1/3 of the salary

Amount of savings = 1/3 × ₹45000 = ₹15000

Explanation

Using the given values: tan θ = sin θ / cos θ = (3/4) / (5/4) = 3/5 cot θ = cos θ / sin θ = (5/4) / (3/4) = 5/3

Substituting these into the given expression: (1 + tan θ) / (1 - cot θ) = (1 + 3/5) / (1 - 5/3) = (8/5) / (-2/3) = (8/5) × (-3/2) = -12/5

Explanation

The curved surface area of a cylinder is given by CSA = 2πrh and the volume is given by V = πr²h.

Given h = 14 cm and CSA = 528 cm²: 2 × (22/7) × r × 14 = 528 88 × r = 528 r = 6 cm

Substituting r = 6 cm and h = 14 cm into the volume formula: V = (22/7) × 6² × 14 V = 22 × 36 × 2 = 1584 cm³

Explanation

Evaluating the given expression step by step using BODMAS:

First term: 6/5 of (1/3 + 2/4) Inside the bracket: 1/3 + 2/4 = 1/3 + 1/2 = 5/6 6/5 × 5/6 = 1

Second term: 5/6 of (3/10 + 12/20) Inside the bracket: 3/10 + 12/20 = 3/10 + 6/10 = 9/10 5/6 × 9/10 = 45/60 = 3/4

Combining all three parts: 1 - 3/4 - 1/6 = 1/4 - 1/6 = (3 - 2) / 12 = 1/12

Explanation

Using the algebraic identity a² - b² = (a + b)(a - b):

a - b = (a² - b²) / (a + b) = 25 / 5 = 5

Using the difference of cubes identity a³ - b³ = (a - b)(a² + ab + b²):

625 = 5 × (a² + ab + b²)

a² + ab + b² = 625 / 5 = 125

Explanation

The sum of all interior angles of a triangle is always equal to 180°.

Setting up the angle sum equation: 2x + (3x - 8) + (5x - 12) = 180 10x - 20 = 180 10x = 200 x = 20

Substituting x = 20 into the three angle expressions gives: First angle = 2(20) = 40° Second angle = 3(20) - 8 = 52° Third angle = 5(20) - 12 = 88°

Comparing the three values, the greatest angle is 88°.

Explanation

Let breadth = b cm and length = 2b cm.

Perimeter of rectangle = 2(length + breadth) 2(2b + b) = 48 6b = 48 b = 8 cm

Length = 2 × 8 = 16 cm

Area of rectangle = length × breadth = 16 × 8 = 128 cm²

Explanation

The ratio of total earnings is equal to the ratio of (Days Worked × Daily Wage).

Ratio of earnings (Shweta : Harish) = (20 × 5) : (30 × 6) = 100 : 180 = 5 : 9

Sum of ratio terms = 5 + 9 = 14

Shweta's share = (5 / 14) × ₹28,000 = ₹10,000

Explanation

Total number of students in the class = 50 + 70 = 120.

Number of girls attending = 40% of 50 = 20 Number of boys attending = 50% of 70 = 35 Total attendees = 20 + 35 = 55

Percentage of class attending = (55 / 120) × 100 = 45.83% ≈ 46%.

Explanation

The arithmetic mean is universally considered the standard and most appropriate measure of central tendency because it incorporates every single observation in a dataset. It is rigidly defined by a clear mathematical formula, remains stable across repeated sampling, and possesses desirable algebraic properties. This makes it uniquely suited for further mathematical treatment and advanced statistical analyses, such as computing variance, standard deviation, and regression models.

Explanation

The required number is the Highest Common Factor (HCF) of (155 − 5), (260 − 10), and (315 − 15).

Subtracting the remainders from the given numbers yields: 155 − 5 = 150 260 − 10 = 250 315 − 15 = 300

Finding the HCF of 150, 250, and 300: 150 = 50 × 3 250 = 50 × 5 300 = 50 × 6

The highest common factor among 150, 250, and 300 is 50.

Explanation

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

Given Principal (P) = ₹3400, Time (T) = 4 years, and Simple Interest (SI) = ₹680:

680 = (3400 × R × 4) / 100 680 = 136 × R R = 680 / 136 = 5%

Explanation

Adjacent angles of a rhombus are supplementary, meaning their sum is equal to 180°.

Given the ratio of adjacent angles as 3 : 6, let the angles be 3x and 6x.

3x + 6x = 180° 9x = 180° x = 20°

The smallest angle of the rhombus is 3x = 3 × 20° = 60°.

Explanation

Following the BODMAS order of operations, evaluate the terms inside the parentheses first:

2³ + 3² + 1 = 8 + 9 + 1 = 18

Next, perform the division:

180 ÷ 18 = 10

Finally, perform the addition and subtraction from left to right:

10 + 37 - 20 = 27

Explanation

The smallest 5-digit number is 10000. Dividing 10000 by 7 gives a quotient of 1428 with a remainder of 4, since 10000 = 7 × 1428 + 4. To get a remainder of 6 when divided by 7, 2 must be added to 10000 (since 4 + 2 = 6). Therefore, the required number is 10000 + 2 = 10002.

Explanation

Let the common factor (HCF) of the two positive numbers be x. The two numbers are 3x and 7x.

Since 3 and 7 are co-prime, the LCM of 3x and 7x is 3 × 7 × x = 21x.

Given: 21x = 231 x = 231 / 21 = 11

The smaller number = 3x = 3 × 11 = 33.

Explanation

12% of 2400 = 12 × 24 = 288

18% of 5400 = 18 × 54 = 972

Let x be the number to be added: 288 + x = 972 x = 972 − 288 = 684

Explanation

Let X = x/2, Y = y/2, and Z = z/2.

The given equations simplify to: X + 1/Y = 1 Y + 1/Z = 1

From the first equation: 1/Y = 1 - X => Y = 1 / (1 - X)

Substitute Y into the second equation: 1/Z = 1 - 1 / (1 - X) = (1 - X - 1) / (1 - X) = -X / (1 - X)

Taking the reciprocal gives: Z = (X - 1) / X = 1 - 1/X

Rearranging the terms: Z + 1/X = 1

Substituting back X = x/2 and Z = z/2: z/2 + 2/x = 1

Explanation

Distance covered by the first train from station A in ½ hour = 50 × ½ = 25 km.

Remaining distance between the two trains = 725 - 25 = 700 km.

Relative speed of the two trains = 50 + 150 = 200 km/hr.

Time taken to meet after the second train starts = 700 ÷ 200 = 3.5 hours.

Total travel time of the first train = 0.5 + 3.5 = 4 hours.

Distance of the meeting point from station A = 50 × 4 = 200 km.

Explanation

Since the number of gents remains unchanged, the gents part in both ratios (3 : 2 and 5 : 2) is already constant at 2 units.

Increase in ladies units = 5 - 3 = 2 units.

Given that 300 ladies joined: 2 units = 300 1 unit = 150

The new number of lady members is 5 units: 5 × 150 = 750

Explanation

Using the fundamental trigonometric identity sin² x + cos² x = 1 and the complementary angle rules sin (90° - x) = cos x and cos (90° - x) = sin x:

The first term: 1 / (sin² 65° + cos² 65°) = 1 / 1 = 1

The second term: sin 15° / cos 75° = sin 15° / sin (90° - 75°) = sin 15° / sin 15° = 1

The third term: cos 65° / sin 25° = cos 65° / cos (90° - 25°) = cos 65° / cos 65° = 1

Adding the terms gives LHS = 1 + 1 + 1 = 3.

Setting LHS = RHS: 3 = √3 tan θ tan θ = 3 / √3 = √3 θ = 60°

Explanation

For any two numbers x and y, the sum of their reciprocals is given by: (1/x) + (1/y) = (x + y) / (x × y)

Since the product of two numbers is equal to the product of their HCF and LCM (x × y = HCF × LCM): Sum of reciprocals = (x + y) / (HCF × LCM)

Substituting the given values into the formula: Sum of reciprocals = 60 / (12 × 72) = 5 / 72

Explanation

Converting the mixed fraction 5⅔ into decimal form: 5⅔ = 5 + 0.667 = 5.667

Substituting this into the expression: 5.667 + 0.73 - 3.123 = 6.397 - 3.123 = 3.274

Rounding off to two decimal places gives 3.27.

Explanation

To find the unit digit of a product of multiple numbers, multiply only the unit digits of each individual number:

4 × 8 = 32 (unit digit is 2) 2 × 4 = 8 (unit digit is 8) 8 × 6 = 48 (unit digit is 8)

Multiplying the unit digits sequentially gives 8 as the final unit digit of the product.

Explanation

Total maximum marks = 200 + 150 = 350

Minimum qualifying marks required = 50% × 350 = 175

Marks secured in the first paper = 40% × 200 = 80

Marks required in the second paper = 175 - 80 = 95

Percentage of marks needed in the second paper = (95 / 150) × 100 = 63.33%

Rounding up to the nearest whole percentage to guarantee reaching the required qualifying score gives 64%.

Explanation

Using the property of radicals √(a) × √(b) = √(a × b) and √(a) / √(b) = √(a / b):

(√(54) × √(125)) / (√(24) × √(45)) = √((54 × 125) / (24 × 45))

Simplifying the values inside the square root: 54 / 24 = 9 / 4 (dividing numerator and denominator by 6) 125 / 45 = 25 / 9 (dividing numerator and denominator by 5)

Substituting these back into the fraction inside the square root gives: (9 / 4) × (25 / 9) = 25 / 4

Taking the square root: √(25 / 4) = 5 / 2 = 2.5

Explanation

Combined daily work rate of X, Y, and Z is calculated using the relation 2(1/X + 1/Y + 1/Z) = (1/12 + 1/18 + 1/15).

Taking the LCM of 12, 18, and 15 as 180: 1/12 + 1/18 + 1/15 = (15 + 10 + 12) / 180 = 37 / 180

Thus, 1/X + 1/Y + 1/Z = 37 / 360 work per day.

Total time required for X, Y, and Z working together = 360 / 37 ≈ 9.73 days, which rounds to approximately 10 days.

Explanation

For a period of 2 years, the relationship between compound interest (CI) and simple interest (SI) at rate r% per annum is given by the formula:

CI = SI × (1 + r / 200)

Substituting SI = ₹500 and r = 10%:

CI = 500 × (1 + 10 / 200) CI = 500 × 1.05 = ₹525

Alternatively, the principal P = (SI × 100) / (r × t) = (500 × 100) / (10 × 2) = ₹2500. CI = P [(1 + r / 100)ᵗ - 1] = 2500 [(1 + 10 / 100)² - 1] = 2500 × 0.21 = ₹525.

Explanation

For any natural number n, the units digit of 6ⁿ is always 6 because 6 raised to any positive integer power ends in 6. Similarly, the units digit of 5ⁿ is always 5 because 5 raised to any positive integer power ends in 5. Therefore, the units digit of 6ⁿ - 5ⁿ is obtained by subtracting their last digits: 6 - 5 = 1.

Explanation

Let total work = LCM(20, 30) = 60 units.

Efficiency of A = 60 ÷ 20 = 3 units/day Efficiency of B = 60 ÷ 30 = 2 units/day

In the last 5 days, B worked alone and completed: Work by B = 5 days × 2 units/day = 10 units

Remaining work to be done jointly by A and B: 60 units - 10 units = 50 units

Time taken by A and B together: 50 units ÷ (3 + 2) units/day = 10 days

Total time taken to complete the work = 10 days + 5 days = 15 days.

Explanation

Let the fraction be x/y.

According to the given conditions: Numerator x is 2 less than denominator y, so x = y - 2.

When the numerator is multiplied by 2 and the denominator by 3, the new fraction is (2x)/(3y) = 2/9.

Simplifying the equation: (2x)/(3y) = 2/9 x/y = (2/9) × (3/2) x/y = 1/3

Checking the condition: for x = 1 and y = 3, the numerator 1 is 2 less than the denominator 3 (3 - 1 = 2). Thus, the original fraction is 1/3.

Explanation

For two coprime numbers, LCM is simply their product. The sum of 1489 and 11 is 1489 + 11 = 1500. Since 1489 and 11 have no common factors other than 1, their HCF is 1. LCM = (1489 × 11) / 1 = 16379.

Explanation

When a quantity is increased by x% and subsequently decreased by y%, the net percentage change is given by the formula:

Net % change = x + y + (x × y) / 100

Substituting x = 40 and y = -40:

Net % change = 40 - 40 + (40 × (-40)) / 100 = -1600 / 100 = -16%

The negative sign indicates a net decrease of 16%. Alternatively, when a number is increased and then decreased by the same percentage x%, there is always a net decrease equal to (x / 10)²% = (40 / 10)²% = 16% decrease.

Explanation

Total Work = Efficiency × Time

Given the ratio of working capacity (efficiency) of A to B is 3 : 4, let A's efficiency be 3 units per day and B's efficiency be 4 units per day.

Total work = Efficiency of A × Time taken by A = 3 × 12 = 36 units

Time taken by B = Total work / Efficiency of B = 36 / 4 = 9 days

Explanation

From the given equations: x = a sin t ⇒ a/x = cosec t y = b tan t ⇒ b/y = cot t

Substituting these into the expression: a²/x² - b²/y² = cosec² t - cot² t

Using the trigonometric identity cosec² t - cot² t = 1, the result simplifies directly to 1.

Explanation

Using the co-function identity sin θ = cos(90° - θ), the equation cos 2θ = sin θ is rewritten as:

cos 2θ = cos(90° - θ)

Equating the angles for θ between 0° and 90°: 2θ = 90° - θ 3θ = 90° θ = 30°

This satisfies the equation since cos(2 × 30°) = cos 60° = ½, and sin 30° = ½.

Explanation

Area of an equilateral triangle with side length a is given by the formula (√3 / 4) × a². For side length a = 4 cm, the area of triangle ABC = (√3 / 4) × 4² = 4√3 cm².

Joining the midpoints of the three sides of a triangle divides the triangle into four congruent triangles of equal area. Therefore, the area of mid-point triangle PQR is one-fourth of the area of triangle ABC:

Area of triangle PQR = (1/4) × 4√3 = √3 cm².

Explanation

Using the algebraic identity (a - b)² = a² - 2ab + b², the expression for P simplifies as follows:

P = (0.3)² + (0.03)² - 2 × (0.3) × (0.03) = (0.3 - 0.03)² = (0.27)²

Dividing P by Q:

P/Q = (0.27 × 0.27) / 0.54 = 0.27 / 2 = 0.135

Explanation

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

Since the simple interest generated by both parts is equal: P₁ × 10 × 5 = P₂ × 9 × 6 50 × P₁ = 54 × P₂ P₁ / P₂ = 54 / 50 = 27 / 25

The total sum of ₹2600 is divided in the ratio 27 : 25. Sum of ratio parts = 27 + 25 = 52 parts

Value of 1 part = 2600 / 52 = ₹50

The part lent out at 10% per annum (P₁) = 27 × 50 = ₹1350.00

Explanation

Using the algebraic identity: if a + b + c = 0, then a³ + b³ + c³ = 3abc.

Let a = p², b = q², and c = -r². Given that p² + q² - r² = 0, we have a + b + c = 0.

Substituting these values into the identity: (p²)³ + (q²)³ + (-r²)³ = 3(p²)(q²)(-r²) p⁶ + q⁶ - r⁶ = -3p²q²r²

Dividing both sides by p²q²r²: (p⁶ + q⁶ - r⁶) ÷ (p²q²r²) = -3

Explanation

Let the total capital invested be C.

Remaining fraction of capital = 1 - (½ + ⅓) = ⅙

Total interest earned in one year = Simple Interest 1 + Simple Interest 2 + Simple Interest 3 = (C × ½ × 5/100) + (C × ⅓ × 8/100) + (C × ⅙ × 10/100) = C × (5/200 + 8/300 + 10/600) = C × (15/600 + 16/600 + 10/600) = 41C / 600

Equating the total interest to ₹820.00: 41C / 600 = 820 41C = 820 × 600 C = 20 × 600 = 12000

Explanation

The length of the longest pole that can fit in a cuboidal room is equal to the length of its space diagonal, given by the formula √(l² + b² + h²).

Substituting the given dimensions l = 10 m, b = 8 m, and h = 6 m:

Diagonal = √(10² + 8² + 6²) = √(100 + 64 + 36) = √200 = 10√2 m

Explanation

To find the median and mode, arrange the 20 given marks in ascending order: 7, 9, 12, 12, 12, 13, 13, 14, 14, 14, 14, 15, 16, 17, 18, 18, 18, 19, 20, 20

Since the total number of observations N = 20 is even, the median is the average of the 10th and 11th terms. 10th term = 14 11th term = 14 Median = (14 + 14) / 2 = 14

The mode is the value that occurs most frequently in the dataset. Here, 14 appears 4 times, which is the highest frequency, so Mode = 14.

Evaluating the given expression: 2 × median - mode = 2 × 14 - 14 = 28 - 14 = 14

Explanation

The circumradius R of an equilateral triangle with side length a is given by the formula:

R = a / √3

Given the side length a = √3 units:

R = √3 / √3 = 1 unit

Explanation

The smallest four-digit number is 1000.

To find the largest perfect square less than 1000, calculate √1000 ≈ 31.62. Taking the nearest integer less than 31.62 gives 31.

Since 31² = 961, the smallest number that must be subtracted from 1000 to obtain a perfect square is 1000 - 961 = 39.

Explanation

Using the BODMAS rule:

First, evaluate the first product: ½ × ⅔ = ⅓

Next, simplify the expression inside the parentheses: (½ × ⅓) + 5/6 = 1/6 + 5/6 = 6/6 = 1

Now, evaluate the second product: ¾ × 1 × 4/21 = 3/21 = 1/7

Finally, subtract the second term from the first term: ⅓ - 1/7 = (7 - 3) / 21 = 4/21

Explanation

Squaring both sides of the equation √(19 - x√12) = √4 - √3:

19 - x√12 = (2 - √3)² 19 - x√12 = 4 + 3 - 4√3 19 - x√12 = 7 - 4√3

Rearranging the terms to isolate x√12:

x√12 = 19 - (7 - 4√3) x√12 = 12 + 4√3

Dividing both sides by √12 (which equals 2√3):

x = (12 + 4√3) / (2√3) x = 12 / (2√3) + (4√3) / (2√3) x = 2√3 + 2 = 2 + 2√3

Explanation

The first eight composite natural numbers are 4, 6, 8, 9, 10, 12, 14, and 15. Sum = 4 + 6 + 8 + 9 + 10 + 12 + 14 + 15 = 78 Mean = 78 / 8

The first eight prime numbers are 2, 3, 5, 7, 11, 13, 17, and 19. Sum = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 = 77 Mean = 77 / 8

Difference = (78 / 8) - (77 / 8) = 1/8

Explanation

By the rule of alligation, the ratio of the number of boys to girls is calculated as:

(Average age of girls - Combined average age) : (Combined average age - Average age of boys)

Ratio = (18.2 - 16.8) : (16.8 - 15.4) = 1.4 : 1.4 = 1 : 1