RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

When both dimensions of a 2D shape increase by x%, the net percentage increase in its area is given by the successive percentage change formula:

Net % increase = x + x + (x × x) / 100

Substituting x = 50:

Net % increase = 50 + 50 + (50 × 50) / 100 = 100 + 25 = 125%

Explanation

Time in hours = 2 + 15/60 = 2 + 1/4 = 9/4 hours.

Fraction of the book read = Reading rate × Time = 1/4 × 9/4 = 9/16.

Explanation

Initial total marks of 25 students = 25 × 36 = 900

Total marks of 24 students after one student leaves = 24 × 37.5 = 900 Marks of the student who left = 900 - 900 = 0

Total marks after a new student joins (25 students) = 25 × 37.2 = 930 Marks of the student who joined = 930 - 900 = 30

Average marks of the two students = (0 + 30) ÷ 2 = 15

Explanation

Exterior angle of a regular polygon = 180° - Interior angle = 180° - 140° = 40°.

The sum of all exterior angles of any convex polygon is 360°. Therefore, the number of sides n = 360° / 40° = 9.

Explanation

Dividing both the numerator and the denominator by cos α gives:

(5 sin α - 3 cos α) / (5 sin α + 2 cos α) = (5 tan α - 3) / (5 tan α + 2)

Given 5 tan α = 4, substituting this value directly into the expression yields:

(4 - 3) / (4 + 2) = 1 / 6

Explanation

Area of a rectangle is given by the formula Area = Length × Breadth (A = L × B). When the length is doubled (L' = 2L), to maintain the same area (A' = A), the new breadth B' must satisfy 2L × B' = L × B. Solving for B' yields B' = B / 2, which means the breadth must be halved.

Explanation

For interest compounded semi-annually over 1 year (2 compounding periods), the amount formula is A = P(1 + r/200)², where P is the principal and r is the annual interest rate.

72,900 = 57,600(1 + r/200)² (1 + r/200)² = 72,900 / 57,600 = 81 / 64

Taking the square root on both sides: 1 + r/200 = 9 / 8 r/200 = 1 / 8 r = 200 / 8 = 25% per annum

Explanation

Simple Interest formula is SI = (P × R × T) / 100, where P is the principal amount, R is the rate of interest per annum, and T is the time period in years.

Rearranging the formula to solve for the rate of interest: R = (SI × 100) / (P × T)

Substituting the given values into the formula: R = (20000 × 100) / (50000 × 5) R = 2000000 / 250000 = 8% per annum.

Explanation

When a journey is divided into two equal half-distances covered at speeds v₁ and v₂, the average speed for the entire journey is given by the harmonic mean formula:

Average Speed = (2 × v₁ × v₂) / (v₁ + v₂)

Substituting v₁ = 21 km/h and v₂ = 24 km/h:

Average Speed = (2 × 21 × 24) / (21 + 24) = 1008 / 45 = 112 / 5 km/h

Total Distance = Average Speed × Total Time = (112 / 5) × 5 = 112 km

Explanation

Work done by 1 man in 1 day = 1/8 Work done by 1 woman in 1 day = 1/12

Work done by 2 men in 2 days = 2 × 2 × (1/8) = ½

Remaining work = 1 - ½ = ½

Let n be the required number of women. Work done by n women in 2 days = n × 2 × (1/12) = n/6

Equating the work done by women to the remaining work: n/6 = ½ n = 3

Explanation

Using the complementary angle identity tan(90° - θ) = cot θ: tan 65° = cot 25° tan 55° = cot 35°

Since tan θ × cot θ = 1 and tan 45° = 1, regroup the given expression: 3 × (tan 25° × tan 65°) × (tan 35° × tan 55°) × tan 45° = 3 × (tan 25° × cot 25°) × (tan 35° × cot 35°) × 1 = 3 × 1 × 1 × 1 = 3

Explanation

Convert the mixed fractions into improper fractions:

2 1/5 = 11/5 5 1/5 = 26/5

Substitute these into the expression:

10 - 11/5 + 26/5

Combine the terms with the common denominator 5:

10 + (-11 + 26) / 5 = 10 + 15/5 = 10 + 3 = 13

Explanation

Using the formula for work rate when adding a person to a group: Time taken by C alone = (t₁ × t₂) / (t₁ - t₂)

Here, t₁ = 21 days (time taken by A and B together) and t₂ = 14 days (time taken by A, B, and C together).

Time taken by C alone = (21 × 14) / (21 - 14) = 294 / 7 = 42 days.

Explanation

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

Mean = S / n = n² / n = n

For the first 10 odd natural numbers (n = 10), the mean is simply 10.

Explanation

The height of the tower above the observer's eye level is: h = 26 m - 1.5 m = 24.5 m

The horizontal distance from the observer to the tower is d = 24.5 m.

Using the trigonometric relation for the angle of elevation θ: tan θ = h / d = 24.5 / 24.5 = 1

Since tan 45° = 1, the angle of elevation θ = 45°.

Explanation

Sum of salaries = Average × 2 = 65,000 × 2 = ₹1,30,000

Total ratio parts = 15 + 11 = 26

Wife's salary = (11 / 26) × 1,30,000 = 11 × 5,000 = ₹55,000

Explanation

Probability of an event = (Number of favorable outcomes) / (Total number of possible outcomes).

The total number of letters in the English alphabet is 26, which represents the total number of possible outcomes. The vowels in the English alphabet are A, E, I, O, and U, making a total of 5 favorable outcomes.

Therefore, the probability of choosing a vowel is 5/26.

Explanation

Evaluating the numerator using the BODMAS rule (working from left to right for multiplication and division):

Numerator = 320 ÷ 8 × 8 ÷ 4 × ½ = 40 × 8 ÷ 4 × ½ = 320 ÷ 4 × ½ = 80 × ½ = 40

Evaluating the denominator:

Denominator = 180 × 5 ÷ 45 - 4 = 900 ÷ 45 - 4 = 20 - 4 = 16

Dividing the numerator by the denominator gives 40 ÷ 16 = 5/2.

Explanation

Probability = (Number of favorable outcomes) / (Total number of outcomes)

In a standard deck of 52 playing cards, the face cards consist of Jacks, Queens, and Kings. Since each of the 4 suits contains 3 face cards, the total number of face cards is 3 × 4 = 12.

Probability = 12 / 52 = 3 / 13

Explanation

The mean of a frequency distribution is calculated using the formula x̄ = Σ(f × x) / Σf.

Sum of frequencies Σf = 13 + 15 + 16 + 18 + 16 + 15 + 13 = 106

Sum of products Σ(f × x) = (19 × 13) + (21 × 15) + (23 × 16) + (25 × 18) + (27 × 16) + (29 × 15) + (31 × 13) = 247 + 315 + 368 + 450 + 432 + 435 + 403 = 2650

Mean = 2650 / 106 = 25.

By symmetry, since the values of x and their corresponding frequencies are symmetrically distributed around x = 25, the mean lies directly at the central value 25 without requiring calculation.

Explanation

Using percentage-to-fraction conversions: A / B = (100 + 25) / 100 = 125 / 100 = 5 / 4 B / C = (100 - 30) / 100 = 70 / 100 = 7 / 10

To find the ratio of A to C, multiply the two ratios: A / C = (A / B) × (B / C) = (5 / 4) × (7 / 10) = 35 / 40 = 7 / 8

Explanation

Using the mass balance equation for salt:

Initial Volume × Initial Concentration = Final Volume × Final Concentration

Initial Volume = 5 litres Initial Concentration = 3% Final Volume = 5 − 2 = 3 litres

5 × 3% = 3 × Final Concentration Final Concentration = (5 × 3%) / 3 = 5%

Explanation

Following BODMAS rules, perform multiplication and division before addition:

First term: 27 × 4.15 = 112.05 Second term: 30.8 ÷ 22 = 1.4

Now, add the results together: 112.05 + 1.4 = 113.45

Explanation

Let the two positive numbers be x and y, where x = y / 15.

Given that x × y = 0.60, substituting y = 15x gives: 15x² = 0.60 x² = 0.04 x = 0.2

Substituting x = 0.2 back into y = 15x gives y = 15 × 0.2 = 3.0.

Explanation

When individuals work together to complete a task, the total wages are shared in the ratio of their work efficiencies (work done per day).

Efficiency ratio of Imran, Tahir, and Amla = 1/11 : 1/15 : 1/13

Multiplying each term by the product of the denominators (11 × 15 × 13 = 2145): Ratio = (15 × 13) : (11 × 13) : (11 × 15) = 195 : 143 : 165

Sum of ratio terms = 195 + 143 + 165 = 503

Imran's share = (195 / 503) × 2012 = 195 × 4 = 780

Explanation

Selling Price = Cost Price × (100 + Profit %)/100

Given Cost Price = 5,340 and Profit = 5%:

Selling Price = 5,340 × (105/100) = 5,340 × 1.05 = 5,607

Alternatively, profit amount = 5% of 5,340 = 267. Selling Price = 5,340 + 267 = 5,607.

Explanation

Let the three prime numbers in descending order be p₁, p₂, and p₃, where p₁ > p₂ > p₃.

The given products are: p₁ × p₂ = 323 p₂ × p₃ = 221

Since p₂ is a common factor of both 323 and 221, p₂ = gcd(323, 221).

Using the Euclidean algorithm: 323 = 221 × 1 + 102 221 = 102 × 2 + 17 102 = 17 × 6 + 0

Thus, p₂ = 17.

The biggest prime number p₁ = 323 ÷ 17 = 19.

Explanation

Using the exponent law aᵐ × aⁿ = aᵐ⁺ⁿ:

From (3ˣ)(3ʸ) = 9: 3ˣ⁺ʸ = 3² x + y = 2 --- (Equation 1)

From (25ˣ)(5ʸ) = 625: (5²)ˣ × 5ʸ = 5⁴ 5²ˣ⁺ʸ = 5⁴ 2x + y = 4 --- (Equation 2)

Subtracting Equation 1 from Equation 2: (2x + y) - (x + y) = 4 - 2 x = 2

Substituting x = 2 into Equation 1: 2 + y = 2 y = 0

This gives (x, y) = (2, 0).

Explanation

Sum of the ratio parts = 4 + 5 + 6 + 7 = 22 parts

Value of 1 ratio part = 176 / 22 = 8 cm

The longest side corresponds to the largest ratio part, which is 7 parts.

Length of the longest side = 7 × 8 = 56 cm

Explanation

Let the two numbers be 31x and 31y, where x and y are co-prime positive integers with gcd(x, y) = 1.

Given that the sum of the numbers is 434: 31x + 31y = 434 31(x + y) = 434 x + y = 14

The co-prime pairs (x, y) of positive integers with x < y that satisfy x + y = 14 are: (1, 13) (3, 11) (5, 9)

Pairs like (2, 12), (4, 10), (6, 8), and (7, 7) are excluded because they share common factors greater than 1, which would change the H.C.F.

Thus, there are 3 such pairs of numbers: (31, 403), (93, 341), and (155, 279).

Explanation

The increase in simple interest depends directly on the change in rate of interest, principal, and time:

Increase in Interest = (Principal × Increase in Rate × Time) / 100

Given Principal = 480, Time = 5 years, and Increase in Rate = 2%:

Increase in Interest = (480 × 2 × 5) / 100 = 4800 / 100 = 48

Explanation

Using the algebraic identity (a + b)² = a² + b² + 2ab, we have (x² + 1/x²)² = x⁴ + 1/x⁴ + 2. Substituting x⁴ + 1/x⁴ = 322 gives (x² + 1/x²)² = 322 + 2 = 324, so x² + 1/x² = √324 = 18.

Next, using (a - b)² = a² + b² - 2ab, we have (x - 1/x)² = x² + 1/x² - 2. Substituting x² + 1/x² = 18 gives (x - 1/x)² = 18 - 2 = 16. Since x > 1, taking the positive square root yields x - 1/x = √16 = 4.

Explanation

To find the square root of a number ending in .25, use the identity for numbers ending in 0.5: (n + 0.5)² = n × (n + 1) + 0.25

Here, the integer part is 182, which factors into the product of two consecutive integers: 13 × 14 = 182

Substituting n = 13: (13.5)² = 13 × 14 + 0.25 = 182.25

Taking the square root on both sides gives √182.25 = 13.5.

Explanation

Using standard trigonometric values, sin 60° = √3 / 2 and cos 60° = ½.

Multiplying these values directly: sin 60° cos 60° = (√3 / 2) × (½) = √3 / 4

Explanation

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

15 × 30 = 9 × D₂ 450 = 9 × D₂ D₂ = 450 ÷ 9 = 50 days.

Explanation

Squaring both sides of the given equation x + 1/x = 2 cos θ:

(x + 1/x)² = (2 cos θ)² x² + 1/x² + 2 = 4 cos² θ x² + 1/x² = 4 cos² θ - 2 x² + 1/x² = 2(2 cos² θ - 1)

Using the trigonometric identity 2 cos² θ - 1 = cos 2θ:

x² + 1/x² = 2 cos 2θ

Explanation

The total number of possible outcomes when tossing three coins is 2³ = 8. Using the complement rule, the probability of getting at least one head is given by:

P(at least one head) = 1 − P(no heads)

Since there is only 1 outcome with no heads (TTT), P(no heads) = 1/8.

P(at least one head) = 1 − 1/8 = 7/8.

Explanation

The given product can be calculated by multiplying the non-zero digits and adjusting for the net power of 10:

7 × 7 × 7 × 7 × 7 = 7⁵ = 16807

Counting the net decimal shift: 1 (from 0.7) + 2 (from 0.07) + 3 (from 0.007) - 1 (from 70) = 5 decimal places to the left.

Shifting 16807 left by 5 decimal places: 16807 × 10⁻⁵ = 0.16807

Explanation

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

Given Principal (P) = ₹2,500 Rate of Interest (R) = 8¼% = 33/4% per annum

The period from 1st March 2018 to 12th May 2018 consists of: March = 31 days April = 30 days May = 12 days Total time = 31 + 30 + 12 = 73 days

Converting days into years: T = 73 / 365 = 1/5 year

Substituting the values into the formula: SI = (2500 × 33/4 × 1/5) / 100 SI = (25 × 33) / 20 SI = 165 / 4 = 41.25

Explanation

Let the marked price (MP) of 1 pen be ₹1.

Cost price (CP) of 20 pens = MP of 15 pens = ₹15.

Selling price (SP) of 20 pens after a 1% discount on the marked price = ₹20 × (1 - 0.01) = ₹19.80.

Profit = SP - CP = ₹19.80 - ₹15 = ₹4.80.

Profit percent = (Profit / CP) × 100 = (4.80 / 15) × 100 = 32%.

Explanation

The perpendicular distance d from the origin (0, 0) to a line Ax + By + C = 0 is given by the formula:

d = |C| / √(A² + B²)

For the line 4x + 3y + 6 = 0, we have A = 4, B = 3, and C = 6. Substituting these values:

d = |6| / √(4² + 3²) = 6 / √(16 + 9) = 6 / √25 = 6/5

Explanation

Let 2a = 3b = 4c = k. Then, a = k/2, b = k/3, and c = k/4.

Given that a + b + c = 13√2: k/2 + k/3 + k/4 = 13√2 (6k + 4k + 3k) / 12 = 13√2 13k / 12 = 13√2 k = 12√2

Substituting k back gives: a = (12√2) / 2 = 6√2 b = (12√2) / 3 = 4√2 c = (12√2) / 4 = 3√2

Now, calculating a² + b² + c²: a² = (6√2)² = 72 b² = (4√2)² = 32 c² = (3√2)² = 18

a² + b² + c² = 72 + 32 + 18 = 122

Explanation

For the standard equation of a circle x² + y² + 2gx + 2fy + c = 0, the radius is given by the formula r = √(g² + f² - c).

Comparing x² + y² + 7x + 4y + 9 = 0 with the standard equation gives: 2g = 7 → g = 7/2 2f = 4 → f = 2 c = 9

Substituting these values into the radius formula: r = √((7/2)² + 2² - 9) r = √(49/4 + 4 - 9) r = √(49/4 - 5) r = √((49 - 20)/4) r = √(29/4) = √29/2

Explanation

The ratio of covered distance to remaining distance is 5 : 7. Therefore, out of 12 total parts, the covered distance represents 5 parts.

Distance covered = (5 ÷ 12) × 32 = 40/3 km.

Time taken = 1 hour 20 minutes = 1 + 20/60 = 4/3 hours.

Speed = Distance ÷ Time = (40/3) ÷ (4/3) = 10 km/hr.

Explanation

For a square of side length a, the perimeter is P = 4a and the diagonal length is d = a√2.

Given perimeter P = 160 cm: a = 160 / 4 = 40 cm

Substituting the side length into the diagonal formula: d = 40√2 cm

Explanation

For any numbers in the ratio a : b : c with a highest common factor of HCF, the least common multiple is given by the formula:

LCM = HCF × LCM(a, b, c)

Given the ratio 1 : 2 : 3 and HCF = 8:

LCM(1, 2, 3) = 6

LCM = 8 × 6 = 48

Explanation

Let CP_A and CP_B be the cost prices of cars A and B respectively. Since the selling price of both cars is the same, 120% of CP_A equals 75% of CP_B:

1.20 × CP_A = 0.75 × CP_B

CP_A / CP_B = 0.75 / 1.20 = 5 / 8

Dividing the total cost of ₹6,50,000 in the ratio 5:8 gives:

CP_A = (5 / 13) × 6,50,000 = ₹2,50,000 CP_B = (8 / 13) × 6,50,000 = ₹4,00,000

Explanation

Using the BODMAS rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction), evaluate the nested brackets from the innermost outward:

Innermost parentheses: 11 + 5 + 6 = 22 Curly braces: 3 + 22 = 25 Square brackets: 3 + 25 = 28 Outer multiplication: (3/5) × 28 = 84/5 = 16 4/5

Explanation

Let F be the current age of the father and S be the current age of the son.

According to the given conditions: F - S = 24 S + 2 = ½ × (F + 2)

Simplifying the second equation: 2S + 4 = F + 2 F - 2S = 2

Subtracting this from the first equation (F - S = 24): (F - S) - (F - 2S) = 24 - 2 S = 22

Substituting S = 22 back into F = S + 24 yields F = 22 + 24 = 46 years.

Explanation

Under compound interest, if a sum of money becomes k times itself in t years, it becomes kⁿ times itself in n × t years. Here, the money doubles (k = 2) in 12 years (t = 12). To become 4 times itself, which is 2² (n = 2), the required time is n × t = 2 × 12 = 24 years.