RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

Substitute √7 = 2.6 directly into the expression:

Numerator = 5 × √7 = 5 × 2.6 = 13

Denominator = 4 × √7 - 0.4 = (4 × 2.6) - 0.4 = 10.4 - 0.4 = 10

Value = 13 / 10 = 1.3

Explanation

Let the total work be 60 units (LCM of 10, 12, and 15).

Combined daily work of Ramu and Somu = 60 / 10 = 6 units Combined daily work of Somu and Dhamu = 60 / 12 = 5 units Combined daily work of Dhamu and Ramu = 60 / 15 = 4 units

Adding these gives 2 × (Ramu + Somu + Dhamu) = 6 + 5 + 4 = 15 units per day.

Daily work of Ramu, Somu, and Dhamu together = 15 / 2 = 7.5 units.

Time taken to complete the task together = 60 / 7.5 = 8 days.

Explanation

First, note that y is the reciprocal of x, so xy = 1.

Sum of x and y: x + y = [(√5 + 1)² + (√5 - 1)²] / [(√5 - 1)(√5 + 1)] x + y = [2((√5)² + 1²)] / [5 - 1] x + y = 2(5 + 1) / 4 = 12 / 4 = 3

Using the algebraic identity x² + y² = (x + y)² - 2xy: x² + y² = 3² - 2(1) = 9 - 2 = 7

Substitute this into the given expression: x² + y² - 4 = 7 - 4 = 3

Explanation

The sum of the interior angles of a polygon with n sides is given by the formula (n - 2) × 180°. For a quadrilateral, n = 4:

Angle sum = (4 - 2) × 180° = 2 × 180° = 360°

Alternatively, drawing a diagonal divides any quadrilateral into two triangles. Since the sum of angles in a triangle is 180°, the total angle sum of the quadrilateral is 2 × 180° = 360°.

Explanation

Using the trigonometric identity cos² θ = 1 - sin² θ, substituting sin θ = 4/5 yields:

cos² θ = 1 - (4/5)² = 1 - 16/25 = 9/25 cos θ = ±√(9/25) = ±3/5

Since sin θ is positive and θ does not lie in the first quadrant, θ must lie in the second quadrant where sine is positive and cosine is negative. Therefore, cos θ = -3/5.

Explanation

Total distance between point A and point B = Speed × Time = 104 × 7 = 728 km.

Distance covered in the first 4 hours = 118 × 4 = 472 km.

Remaining distance = 728 - 472 = 256 km.

Since the total target time is 8 hours and 4 hours have already passed, the remaining time available is 8 - 4 = 4 hours.

Required average speed for the rest of the journey = 256 / 4 = 64 km/h.

Explanation

Percentage = (Given Time / Total Time in a Day) × 100%

Since 1 day = 24 hours: Percentage = (3 / 24) × 100% = (1 / 8) × 100% = 12.5% = 12 ½ %

Explanation

HCF of fractions = (HCF of numerators) / (LCM of denominators)

For the fractions 1/2, 2/3, 3/4, and 4/5:

HCF of numerators (1, 2, 3, 4) = 1 LCM of denominators (2, 3, 4, 5) = 60

HCF = 1/60

Explanation

Let the cost price of 1 pen be C and the selling price of 1 pen be S.

Given: 12 × C = 8 × S S / C = 12 / 8 = 3 / 2 = 1.5

Gain percentage = [(S - C) / C] × 100 Gain percentage = (1.5 - 1) × 100 = 50%

Explanation

10% of x = 15% of y

(10/100) × x = (15/100) × y

10 × x = 15 × y

x / y = 15 / 10 = 3 / 2

x : y = 3 : 2

Explanation

Fraction of a pie diagram = Sector Angle / Total Angle

Substituting the given values: Fraction = 135° / 360° = 3/8

Explanation

The domain of a function is the set of all possible input values for which the function is defined. The sine function sin(x) is defined for every real angle x, whether positive, negative, or zero, because a angle of any magnitude can be represented on the unit circle. Consequently, the set of all real numbers, denoted by R, constitutes the domain of the sine function.

Explanation

By applying the distributive property of multiplication over addition, a × c + b × c = (a + b) × c:

4.6 × 13.5 + 5.4 × 13.5 = (4.6 + 5.4) × 13.5 = 10 × 13.5 = 135

Explanation

1 hour = 60 minutes × 60 seconds = 3600 seconds.

To find what decimal of an hour 1 second represents, divide 1 by 3600: 1 / 3600 = 0.0002777... ≈ 0.00027

Explanation

The product of the first n even positive integers is given by 2ⁿ × n!.

For the first six even numbers (2, 4, 6, 8, 10, 12), n = 6:

Product = 2⁶ × 6! Product = 64 × 720 = 46080

Alternatively, calculating directly: 2 × 4 × 6 × 8 × 10 × 12 = 46080

Explanation

Convert the mixed fractions into improper fractions: 5 3/9 = 5 1/3 = 16/3 3 1/4 = 13/4 7 5/16 = 117/16

Now, rewrite the given expression following BODMAS rules: (16/3) ÷ (13/4) × (117/16) + (7/8) × 16

Change division to multiplication by taking the reciprocal: = (16/3) × (4/13) × (117/16) + (7/8) × 16

Simplify the first term: = (16/16) × (117/39) × 4 = 1 × 3 × 4 = 12

Simplify the second term: = (7/8) × 16 = 14

Sum the results: = 12 + 14 = 26

Explanation

In a parallelogram, opposite angles are equal. Since A and C are opposite angles in parallelogram ABCD, ∠A = ∠C.

x + 20° = 3x - 10° 3x - x = 20° + 10° 2x = 30° x = 15°

Explanation

Given a = √4 = 2.

Evaluating each term of the expression: √9 = 3 25a = 25 × 2 = 50 √64 = 8

Substituting these values into the expression: √9 + 25a + √64 = 3 + 50 + 8 = 61

Explanation

To find the school with the maximum percentage of selection, calculate the selection percentage for each school using the formula:

Selection Percentage = (Number of Selected Students / Number of Appeared Students) × 100%

ABC: (800 / 1000) × 100 = 80% EFG: (1050 / 1200) × 100 = 87.5% IJK: (1350 / 1500) × 100 = 90% MNO: (900 / 1100) × 100 = 81.82% QRS: (750 / 900) × 100 = 83.33%

School IJK achieves the highest selection percentage at 90%.

Explanation

Height of tower = Height of observer + (Distance × tan θ)

Given: Height of observer = 1.5 m Distance from tower = 28.5 m Angle of elevation θ = 45°

Height of tower above eye level = 28.5 × tan(45°) = 28.5 × 1 = 28.5 m

Total height of tower = 28.5 m + 1.5 m = 30 m

Explanation

Expenditure on transport in the year 2013 = 60 lakh

Total expenditure on transport for all four years (2012 to 2015) = 48 + 60 + 71 + 82 = 261 lakh

Required ratio = 60 : 261 = 20 : 87

Explanation

Let the common multiplier be x, so the two numbers are 3x and 4x. Since 3 and 4 are co-prime, their Least Common Multiple (LCM) is 3 × 4 × x = 12x.

Given 12x = 180, solving for x yields x = 180 / 12 = 15.

The larger number is 4x = 4 × 15 = 60.

Explanation

For interest compounded half-yearly, the annual rate is halved and the time period in half-years is doubled: Rate per half-year, r = 10% / 2 = 5% Number of half-years, n = 1.5 × 2 = 3

Amount (A) = P × (1 + r/100)ⁿ A = 2800 × (1 + 5/100)³ A = 2800 × (21/20)³ A = 2800 × 9261 / 8000 = 3241.35

Compound Interest = Amount − Principal Compound Interest = 3241.35 − 2800 = 441.35

Explanation

A null matrix, also known as a zero matrix, is defined as a matrix in which every single element is zero. It can be of any dimension m × n, where every entry aᵢⱼ = 0 for all i and j.

Explanation

Evaluating x and y: x = (√9 + 1) / (√9 - 1) = (3 + 1) / (3 - 1) = 4 / 2 = 2 y = (√16 + 1) / (√16 - 1) = (4 + 1) / (4 - 1) = 5 / 3

Substituting these values into x² - y²: x² - y² = 2² - (5 / 3)² = 4 - 25 / 9 = (36 - 25) / 9 = 11 / 9

Explanation

Using the BODMAS rule, simplify the inner terms first:

First, evaluate the square bracket expression: 3 - 2 = 1 6 + 8 × 1 = 14 3/7 × 14 = 6

Second, evaluate the curly bracket expression: 1/5 ÷ 7/25 = 1/5 × 25/7 = 5/7 3/7 + 4/14 = 3/7 + 2/7 = 5/7 5/7 - 5/7 = 0

Combining all terms: 1 + 6 + 0 = 7

Explanation

Number of smaller cubes along each edge (n) = Side of larger cube / Side of smaller cube = 15 / 3 = 5.

Total number of smaller cubes = n³ = 5³ = 125.

Explanation

Let the cost of a chair be C and the cost of a table be T.

From the given information: 2C + T = 880 C + 2T = 980

Adding the two equations: 3C + 3T = 1860 C + T = 620

Subtracting the first equation from the second: T - C = 100

Adding these two equations: 2T = 720 T = 360

Substituting T = 360 into C + T = 620: C = 620 - 360 = 260

The cost of each chair is 260 and each table is 360.

Explanation

When numbers are expressed in a simplified ratio a : b : c where the terms are co-prime, each actual number is obtained by multiplying its ratio term by the H.C.F.

Since 1, 2, and 3 are co-prime, the common multiplier is the H.C.F. itself, which is 15.

Multiplying each term of the ratio 1 : 2 : 3 by 15 gives: 1 × 15 = 15 2 × 15 = 30 3 × 15 = 45

Explanation

Marked Price = Cost / (1 - Commission Rate)

Substituting the given values: Marked Price = 9090 / (1 - 0.10) = 9090 / 0.90 = 10,100.00

Explanation

Converting the ratios into decimal form: 15/28 ≈ 0.536 5/18 ≈ 0.278 13/21 ≈ 0.619 19/27 ≈ 0.704

Comparing the decimal values shows that 19/27 (≈ 0.704) is the largest value among them.

Explanation

To find the median, arrange the given dataset in ascending order: 2, 2, 3, 5, 6, 7, 9, 10, 11.

The total number of observations is n = 9. Since n is odd, the median is given by the formula for the ((n + 1) / 2)-th term, which corresponds to the 5th term.

The 5th value in the ordered list is 6.

Explanation

Using the reduction formula sin(2π - θ) = -sin(θ) for an angle in the fourth quadrant:

sin(5π/3) = sin(2π - π/3) = -sin(π/3)

Since sin(π/3) = √3/2, substituting this value gives:

sin(5π/3) = -√3/2

Explanation

Squaring both sides of the equation √(5ⁿ) = 5 gives:

5ⁿ = 5²

Since the bases on both sides are equal, equating the exponents directly yields n = 2.

Explanation

Let the two numbers be 5x and 3x.

5x + 3x = 560 8x = 560 x = 70

The smaller of the two numbers corresponds to 3x: 3 × 70 = 210

Explanation

Return speed = 50 + (40% of 50) = 70 km/h

When equal distances are covered at two different speeds v₁ and v₂, the average speed is given by: Average speed = (2 × v₁ × v₂) / (v₁ + v₂)

Substituting v₁ = 50 km/h and v₂ = 70 km/h: Average speed = (2 × 50 × 70) / (50 + 70) = 7000 / 120 = 58.33 km/h

Explanation

A function f(x) is periodic with period T > 0 if T is the smallest positive real number such that f(x + T) = f(x) for all x in its domain. For the cosine function, cos(x + 2π) = cos x holds for all real numbers x, and 2π is the smallest positive value for which this equality is universally true.

Explanation

When individual time (x) and combined time (y) are given, the time taken by the second person working alone is calculated using the formula: Time = (x × y) / (x - y)

Given x = 6 days and y = 4 days: Time = (6 × 4) / (6 - 4) = 24 / 2 = 12 days

Alternatively, using the LCM method: Assume total work = LCM(6, 4) = 12 units Giri's efficiency = 12 / 6 = 2 units/day Combined efficiency = 12 / 4 = 3 units/day Friend's efficiency = 3 - 2 = 1 unit/day

Time taken by friend alone = 12 / 1 = 12 days

Explanation

Number of tiles required = (Length × Width) ÷ (Side of tile)²

Convert the dimensions of the kitchen from meters to centimeters: Length = 2.4 m = 240 cm Width = 1.6 m = 160 cm

Area of the kitchen floor = 240 cm × 160 cm = 38,400 cm² Area of one square tile = 10 cm × 10 cm = 100 cm²

Number of tiles needed = 38,400 ÷ 100 = 384

Explanation

A frequency polygon is a graphical representation of quantitative data constructed by plotting the frequency of each class interval against its midpoint (or class mark) on a graph. The midpoints are marked on the horizontal axis and the corresponding frequencies are marked on the vertical axis, after which consecutive points are connected with straight line segments.

Explanation

Using the BODMAS rule, solve the operations inside the brackets in order from innermost to outermost:

Innermost parentheses: 10 × ½ = 5 Curly braces: 8 - 5 = 3 Square brackets: 6 ÷ 3 × 3 = 2 × 3 = 6 Final operation: 18 - 6 = 12

Explanation

Let the required number be x.

According to the given condition: (3/4)x - (1/3)x = 50

Subtracting the fractions on the left-hand side: (9/12 - 4/12)x = 50 (5/12)x = 50 x = (50 × 12) / 5 x = 120

Explanation

By the Basic Proportionality Theorem (Thales' Theorem), if a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio. Since PQ is parallel to AB with P on BC and Q on CA:

CP / PB = CQ / QA

Substituting CQ / QA = ⅓ and CP = 4:

4 / PB = ⅓ PB = 12

The total length of side BC is:

BC = CP + PB = 4 + 12 = 16

Explanation

The angle between two intersecting circles is defined as the angle between their tangents at the point of intersection. Two circles are said to be orthogonal when these tangents intersect at a right angle (90°). At the point of intersection of two orthogonal circles, the radius of one circle is tangent to the second circle.

Explanation

According to the BODMAS rule, the "of" operation takes precedence over division.

First, evaluate the expressions on both sides of the division operator: 860% of 50 = (860 / 100) × 50 = 430 50% of 860 = (50 / 100) × 860 = 430

Now, perform the division: 430 ÷ 430 = 1

Explanation

To find the number that must be added to 21.707 to obtain 29.053, subtract 21.707 from 29.053:

29.053 - 21.707 = 7.346

Explanation

Selling Price = Printed Price × (1 - d₁/100) × (1 - d₂/100)

Substituting the given values: Selling Price = 200 × (1 - 30/100) × (1 - 20/100) = 200 × 0.70 × 0.80 = 200 × 0.56 = ₹112

Explanation

First, calculate the inner square root: √2401 = 49

Substitute this back into the expression: √(176 + 49) = √225 = 15

Explanation

In a proportion a : b :: c : d, the product of extremes equals the product of means (a × d = b × c).

Substituting the given values: 0.75 × 8 = 5 × x 6 = 5x x = 6 ÷ 5 = 1.20

Explanation

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

Numerator = 4.2² - 1.9² = (4.2 - 1.9) × (4.2 + 1.9) = 2.3 × 6.1

Denominator = 2.3 × 6.1

Dividing the numerator by the denominator: (2.3 × 6.1) / (2.3 × 6.1) = 1