RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

Equating each expression to a common constant k: (1/3) A = (3/4) B = (1/6) C = k

Solving for A, B, and C gives: A = 3k, B = (4/3)k, and C = 6k

Multiplying the ratio 3k : (4/3)k : 6k by 3 to clear the denominator yields: A : B : C = 9 : 4 : 18

Explanation

For two similar triangles, the ratio of their perimeters is equal to the ratio of their corresponding sides:

Perimeter(ΔXYZ) / Perimeter(ΔLMN) = XY / LM

Substituting the given values:

48 / 32 = XY / 12

3 / 2 = XY / 12

XY = (12 × 3) / 2 = 18 cm

Explanation

Simple interest formula: SI = (P × R × T) / 100

For the first 4 years: SI₁ = (12,200 × 2 × 4) / 100 = 976

The total amount reinvested as the new principal: P₂ = 12,200 + 976 = 13,176

For the next 4 years: SI₂ = (13,176 × 2 × 4) / 100 = 105,408 / 100 = 1,054.08

Explanation

Substitute tan A = sin A / cos A and cot A = cos A / sin A into the expression:

sin A cos A (tan A - cot A) = sin A cos A (sin A / cos A - cos A / sin A) = sin A cos A [ (sin² A - cos² A) / (sin A cos A) ] = sin² A - cos² A

Using the fundamental Pythagorean identity cos² A = 1 - sin² A:

= sin² A - (1 - sin² A) = 2 sin² A - 1

Explanation

Ratio of shares of A : B : C : D = 3 : 4 : 5 : 7.

Given A's share = 3 units = 192 fruits. Value of 1 unit = 192 ÷ 3 = 64 fruits.

Combined share of B and C = 4 + 5 = 9 units. Number of fruits received by B and C together = 9 × 64 = 576.

Explanation

The area of a square is given by A = s², where s is the length of its side. When two multiplying factors each increase by x%, the net percentage increase is calculated using the successive percentage change formula:

Net Percentage Increase = x + y + (x × y) / 100

Since both dimensions of the square increase by 10% (x = 10, y = 10):

Percentage Increase = 10 + 10 + (10 × 10) / 100 = 20 + 1 = 21%

Alternatively, taking an initial side of 10 units gives an initial area of 10² = 100 square units. Increasing the side by 10% gives a new side of 11 units, resulting in a new area of 11² = 121 square units. The net increase in area is 121 − 100 = 21%.

Explanation

For two numbers in the co-prime ratio a:b and with HCF H, their LCM is given by the formula LCM = H × a × b.

Given: Ratio (a:b) = 4:5 HCF (H) = 3

LCM = 3 × 4 × 5 = 60

Explanation

The combined mean of two sets of observations is given by the formula:

Combined Mean = (n₁x̄₁ + n₂x̄₂) / (n₁ + n₂)

Where: n₁ = 7, x̄₁ = 10 n₂ = 3, x̄₂ = 5

Substituting the values: Combined Mean = (7 × 10 + 3 × 5) / (7 + 3) = (70 + 15) / 10 = 85 / 10 = 8.5

Explanation

The reciprocal of 5/7 is 7/5, and the reciprocal of 9/5 is 5/9.

Sum of reciprocals = 7/5 + 5/9 = (63 + 25) / 45 = 88/45

Reciprocal of the sum = 1 / (88/45) = 45/88

Explanation

Let S = √(1 + x) + √(1 - x).

Squaring both sides: S² = (1 + x) + (1 - x) + 2√((1 + x)(1 - x)) S² = 2 + 2√(1 - x²)

Given x = √3 / 2: x² = 3/4 1 - x² = 1 - 3/4 = 1/4

Substituting this value: S² = 2 + 2√(1/4) S² = 2 + 2(1/2) S² = 3

Since S > 0, taking the square root gives S = √3.

Explanation

For a circle of radius r, the area is A = πr² and the circumference is C = 2πr.

For statement 1: Squaring the circumference gives C² = (2πr)² = 4π²r². The ratio of area to squared circumference is A : C² = (πr²) / (4π²r²) = 1 / (4π) = 1 : 4π.

For statement 2: The ratio of area to circumference is A : C = (πr²) / (2πr) = r / 2 = r : 2.

Both statements are mathematically correct.

Explanation

Taking the tangent on both sides of A + B = 45°:

tan(A + B) = tan 45° = 1

(tan A + tan B) / (1 - tan A tan B) = 1

tan A + tan B = 1 - tan A tan B

tan A + tan B + tan A tan B = 1

Adding 1 to both sides:

1 + tan A + tan B + tan A tan B = 2

(1 + tan A)(1 + tan B) = 2

Alternatively, using quick angle substitution, let A = 0° and B = 45°:

(1 + tan 0°)(1 + tan 45°) = (1 + 0)(1 + 1) = 2

Explanation

Squaring both sides of the given equation: √(0.03 × 0.3 × p) = 0.3 × 0.03 × √q

0.03 × 0.3 × p = (0.3)² × (0.03)² × q

Dividing both sides by (0.03 × 0.3 × q): p/q = [(0.3)² × (0.03)²] / (0.03 × 0.3) p/q = 0.3 × 0.03 = 0.009

Explanation

The Least Common Multiple (LCM) of a set of numbers must be divisible by each individual number in that set. Since 4 is one of the numbers, the LCM must be divisible by 4. Dividing 18 by 4 gives 4.5, which is not an integer, so 18 is not a multiple of 4 and cannot be the LCM of 4, 6, and x.

Explanation

Using the simple interest formula, SI = (P × R × T) / 100:

Given that the monthly interest SI = ₹1,500, time T = 1/12 year, and interest rate R = 12% per annum:

1,500 = (P × 12 × 1/12) / 100 1,500 = P / 100 P = 1,500 × 100 = ₹1,50,000 = ₹1.5 lakhs

Alternatively, an annual rate of 12% corresponds to a monthly rate of 1%. Since 1% of the required capital is ₹1,500, the full capital (100%) is 1,500 × 100 = ₹1,50,000, which equals ₹1.5 lakhs.

Explanation

Using the ratio method for 3 years at 5% (1/20) per annum, let the principal sum be 20³ = 8,000 units.

The compound interest components for 3 years in ratio 3 : 3 : 1 are: First level interest = 8,000 × 1/20 = 400 units Second level interest = 400 × 1/20 = 20 units Third level interest = 20 × 1/20 = 1 unit

Total CI = (3 × 400) + (3 × 20) + (1 × 1) = 1,261 units

Given CI = ₹3,783, so 1 unit = 3,783 / 1,261 = ₹3.

The simple interest (SI) for 3 years consists only of the primary interest component: SI = 3 × 400 = 1,200 units

SI = 1,200 × 3 = ₹3,600.

Explanation

In geometry, the fundamental properties of triangle concurrence points establish that the three medians of a triangle always meet at a single point called the centroid; the three altitudes drawn from the vertices to the opposite sides intersect at a single point called the orthocenter; and the three internal angle bisectors concur at a single point called the in-center, which serves as the center of the inscribed circle. Since all three statements accurately describe these fundamental definitions of triangle centers, all statements 1, 2, and 3 are correct.

Explanation

Using the algebraic identity (a + b + c)² = a² + b² + c² + 2(ab + bc + ca):

3² = 6 + 2(ab + bc + ca) 9 = 6 + 2(ab + bc + ca) 2(ab + bc + ca) = 3 ab + bc + ca = 3/2

The given relation 1/a + 1/b + 1/c = 1 simplifies to: (ab + bc + ca) / abc = 1 ab + bc + ca = abc

Substituting ab + bc + ca = 3/2 yields abc = 3/2.

Explanation

Let total work = 60 units (LCM of 12, 15, and 5).

Efficiency of A = 60 ÷ 12 = 5 units/day Efficiency of B = 60 ÷ 15 = 4 units/day Combined efficiency of A, B, and C = 60 ÷ 5 = 12 units/day

Efficiency of C = 12 - (5 + 4) = 3 units/day

Since all of them work together for the entire duration, the money is distributed in the ratio of their efficiencies: Share of C = (3 ÷ 12) × ₹2,880 = ¼ × ₹2,880 = ₹720

Explanation

According to the BODMAS rule, the operations inside the brackets are solved first, following the order of Division, Multiplication, Addition, and Subtraction:

Inside the brackets: 12 ÷ 4 × 3 = 3 × 3 = 9 2 × 2 = 4 20 + 9 - 4 = 25

Substituting this value back into the original expression: 36 - 2(25) + 10 = 36 - 50 + 10 = -14 + 10 = -4

Explanation

Since a : b = 3 : 2, substitute a = 3 and b = 2 into the given expression:

7a + 9b = 7(3) + 9(2) = 21 + 18 = 39

5a + 7b = 5(3) + 7(2) = 15 + 14 = 29

Therefore, (7a + 9b) : (5a + 7b) = 39 : 29.

Explanation

Using the identity x³ + y³ = (x + y)³ - 3xy(x + y):

36 = 6³ - 3xy(6) 36 = 216 - 18xy 18xy = 180 xy = 10

Evaluating the required expression: 1/x + 1/y = (x + y) / xy = 6 / 10 = 3/5

Explanation

In a right-angled triangle, the median drawn from the vertex containing the right angle to the hypotenuse is equal to half the length of the hypotenuse (BD = ½ × AC).

Given AC = 4√2 units: BD = ½ × 4√2 = 2√2 units.

Explanation

The two-digit numbers divisible by 4 are 12, 16, 20, ..., 96, which can be expressed as 4 × k for k = 3, 4, 5, ..., 24.

The sum of their squares is given by: S = (4 × 3)² + (4 × 4)² + ... + (4 × 24)² S = 16 × (3² + 4² + ... + 24²)

Using the standard sum of squares formula, Σ k² = n(n + 1)(2n + 1) / 6: Σₖ₌₁²⁴ k² = (24 × 25 × 49) / 6 = 4900

Subtracting 1² + 2² = 5 to find the sum from k = 3 to 24: 3² + 4² + ... + 24² = 4900 - 5 = 4895

Multiplying by 16: S = 16 × 4895 = 78320

Explanation

Using the algebraic identity a⁴ - b⁴ = (a² - b²)(a² + b²):

(0.3⁴ - 0.2⁴) / (0.3² + 0.2²) = [(0.3² - 0.2²)(0.3² + 0.2²)] / (0.3² + 0.2²)

Canceling (0.3² + 0.2²) from the numerator and denominator:

= 0.3² - 0.2² = 0.09 - 0.04 = 0.05

Explanation

Using the algebraic identity (a + b)² = a² + b² + 2ab:

(x² + x⁻²)² = x⁴ + x⁻⁴ + 2 (x² + x⁻²)² = 1154 + 2 = 1156 x² + x⁻² = √1156 = 34

Next, applying the same identity to x + x⁻¹:

(x + x⁻¹)² = x² + x⁻² + 2 (x + x⁻¹)² = 34 + 2 = 36 x + x⁻¹ = √36 = 6

Explanation

Product of two numbers = HCF × LCM

Let the required number be x. x × 210 = 30 × 2310 x = (30 × 2310) ÷ 210 x = 2310 ÷ 7 x = 330

Explanation

Ratio of efficiency of Raju to Vinod = 3 : 1 Combined efficiency = 3 + 1 = 4 units per day

Total work = Combined efficiency × Number of days = 4 × 21 = 84 units

Time taken by Vinod alone = Total work / Efficiency of Vinod = 84 / 1 = 84 days

Explanation

Distance travelled by B = 60 + 12 = 72 km. Distance travelled by A = 60 - 12 = 48 km.

Since both start simultaneously and meet at the same time, the ratio of their speeds equals the ratio of the distances travelled:

Speed of A / Speed of B = 48 / 72 = 2 / 3

Let Speed of A = 2x and Speed of B = 3x. Difference in speed = 3x - 2x = x = 4 km/h.

Speed of A = 2x = 2 × 4 = 8 km/h.

Explanation

Let the marked price of the cooker be ₹100.

Cost Price (CP) = 9/10 × 100 = ₹90 Selling Price (SP) = 100 + 8 = ₹108

Gain = SP - CP = 108 - 90 = ₹18

Gain Percentage = (Gain / CP) × 100 = (18 / 90) × 100 = 20%

Explanation

Cross-multiplying the given ratio:

(3a² - 8b²) / (a² + 4b²) = 2 / 1

3a² - 8b² = 2(a² + 4b²) 3a² - 8b² = 2a² + 8b² 3a² - 2a² = 8b² + 8b² a² = 16b²

Taking the square root on both sides yields a = ±4b. Since a and b are of the same sign, a and b must both be positive or both be negative, so a = 4b. Therefore, a : b = 4 : 1.

Explanation

To convert 8.17 hours into hours, minutes, and seconds:

The whole number part represents 8 hours.

Convert the decimal part (0.17 hours) to minutes: 0.17 × 60 = 10.2 minutes

The whole number part represents 10 minutes.

Convert the remaining decimal part (0.2 minutes) to seconds: 0.2 × 60 = 12 seconds

Combining these gives 8 hours, 10 minutes, and 12 seconds.

Explanation

Let the total number of pages in the book be x.

On the first day, after reading 3/8 of the book, the fraction of unread pages remaining is 1 - 3/8 = 5/8.

On the second day, the boy reads 4/5 of the remainder, leaving 1 - 4/5 = 1/5 of that remainder unread.

Fraction of the total book remaining unread = (5/8) × (1/5) = 1/8.

Given that 1/8 of the book equals 45 pages: x = 45 × 8 = 360 pages.

Explanation

The altitude (h) of an equilateral triangle with side length 'a' is given by the formula:

h = (√3 / 2) × a

Substituting a = 3√3 cm into the formula:

h = (√3 / 2) × 3√3 h = (3 × 3) / 2 h = 9 / 2 h = 4.5 cm

Explanation

For a sum of money to double under simple interest, the interest earned must equal the principal, giving the relationship R × T = 100, where R is the rate of interest and T is the time period in years.

In statement 1, R × T = 16 × 6 = 96 ≠ 100. In statement 2, R × T = 18 × 5 = 90 ≠ 100.

Since neither condition yields 100, both statements are incorrect.

Explanation

The HCF of fractions is given by the formula: HCF of fractions = (HCF of Numerators) / (LCM of Denominators)

For the given fractions 3/2, 9/8, 81/16, and 27/10: HCF of numerators (3, 9, 81, 27) = 3 LCM of denominators (2, 8, 16, 10) = 80

Substituting these values into the formula: HCF = 3/80

Explanation

The effective compound interest rate for 2 years at a rate of r% per annum is given by (2r + r²/100)%.

At 10% per annum, the effective rate for 2 years = 2(10) + (10²/100) = 21%. At 20% per annum, the effective rate for 2 years = 2(20) + (20²/100) = 44%.

Difference in effective rate = 44% - 21% = 23%.

Additional interest received = 23% of ₹200 = (23 / 100) × 200 = ₹46.

Explanation

Using the complementary angle identity: tan 48° = tan(90° - 42°) = cot 42° = cos 42° / sin 42°

Since cos 42° = p, by the Pythagorean trigonometric identity: sin 42° = √(1 - cos² 42°) = √(1 - p²)

Substituting these into the expression for cot 42°: tan 48° = p / √(1 - p²)

Explanation

Using the complementary angle trigonometric identity: cos θ = sin(90° - θ)

Substituting θ = 75°: cos 75° = sin(90° - 75°) = sin 15°

Equating this to the given relation: sin 2A = sin 15°

For the smallest positive value of A: 2A = 15° A = 7.5°

Explanation

Total maximum marks for each student = 100 + 100 + 100 + 50 = 350

Total marks obtained by P = 50 + 85 + 55 + 40 = 230 Aggregate percentage of P = (230 / 350) × 100 = 65.71%

Total marks obtained by Q = 65 + 90 + 50 + 45 = 250 Aggregate percentage of Q = (250 / 350) × 100 = 71.43%

Difference in aggregate percentage = 71.43% - 65.71% = 5.72% ≈ 6%

Explanation

Expenditure on taxes in the year 2014 = 2 million Total expenditure on transport across all years (2012 to 2015) = 48 + 60 + 71 + 82 = 261 million

Required ratio = 2 : 261 = 1 : (261 ÷ 2) = 1 : 130.5

Explanation

Let total work = LCM(8, 7) = 56 units.

Father's 1-day work = 56 / 8 = 7 units. Son's 1-day work = 56 / 7 = 8 units.

In a 2-day cycle (Father on day 1, Son on day 2), work completed = 7 + 8 = 15 units.

In 3 complete cycles (6 days), work completed = 3 × 15 = 45 units. Remaining work = 56 - 45 = 11 units.

On the 7th day, the father works and completes 7 units. Remaining work = 11 - 7 = 4 units.

On the 8th day, it is the son's turn. Time taken to finish remaining 4 units = 4 / 8 = ½ day.

Total days required = 6 + 1 + ½ = 7½ days.

Explanation

Using the algebraic identity: a³ + b³ + c³ - 3abc = (a + b + c)[(a + b + c)² - 3(ab + bc + ca)]

Substituting the given values: a³ + b³ + c³ - 3(168) = 17 × [17² - 3(94)] a³ + b³ + c³ - 504 = 17 × (289 - 282) a³ + b³ + c³ - 504 = 17 × 7 a³ + b³ + c³ - 504 = 119 a³ + b³ + c³ = 119 + 504 = 623

Explanation

Using the formula for two numbers: First Number × Second Number = HCF × LCM

Substituting the given values: 435 × Second Number = 15 × 6090 Second Number = (15 × 6090) / 435

Dividing 435 by 15 gives 29. Second Number = 6090 / 29 = 210

Explanation

A median of a triangle is defined as a line segment connecting any vertex to the midpoint of its opposite side. The centroid is the point of concurrency of the three medians of a triangle. By the centroid theorem, the centroid divides each median into two parts such that the distance from the vertex to the centroid is twice the distance from the centroid to the midpoint of the opposite side, establishing a 2:1 ratio.

Explanation

Let L be the total length of the pillar.

First part = L / 4 Second part = (4 / 8) × (L / 4) = L / 8

The sum of the first two parts is: L / 4 + L / 8 = 3L / 8

The third part represents the remaining length: L - 3L / 8 = 5L / 8

Given that the third part is 10 m: 5L / 8 = 10 L = 10 × 8 / 5 = 16 m

Explanation

Simple Interest = (Principal × Rate × Time) / 100

Since the simple interest obtained from each of the three schemes is equal, let the principal investments be P₁, P₂, and P₃:

(P₁ × 10 × 6) / 100 = (P₂ × 12 × 10) / 100 = (P₃ × 15 × 12) / 100

60 P₁ = 120 P₂ = 180 P₃

Dividing throughout by 60: 1 P₁ = 2 P₂ = 3 P₃

To find the ratio P₁ : P₂ : P₃, multiply the reciprocals of the coefficients by their LCM (6): P₁ : P₂ : P₃ = 1/1 : 1/2 : 1/3 = 6 : 3 : 2

Explanation

Using the successive percentage formula: Total Gain = x + y + (x × y) / 100, where x and y are the percentage gains of the first and second persons respectively.

38 = 15 + y + (15 × y) / 100 23 = y + 0.15y 23 = 1.15y y = 23 / 1.15 = 20%

Explanation

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

The odd numbers between 0 and 52 are 1, 3, 5, ..., 51. The total number of terms n = (51 - 1)/2 + 1 = 26.

Substituting n = 26 into the formula: Sum = 26² = 676

Explanation

The distance covered in each hour forms an arithmetic progression with first term a = 40 km and common difference d = 20 km.

Sum of distance covered in 5 hours S₅ = (5 / 2) × [2(40) + (5 - 1)20] = 2.5 × [80 + 80] = 400 km.

The remaining distance to cover = 470 - 400 = 70 km.

In the 6th hour, the speed of the train becomes 40 + 5 × 20 = 140 km/h.

Time taken to cover the remaining 70 km = 70 / 140 = ½ hour = 30 minutes.

Total time required = 5 hours + 30 minutes = 5 h 30 min.