RRB NTPC CBT 1 2026 Mathematics Questions with Solutions

Explanation

For a period of 2 years, the difference between compound interest and simple interest is given by the formula: Difference = P × (r / 100)²

Substituting the given values: 58 = P × (4 / 100)² 58 = P × (1 / 25)² 58 = P / 625 P = 58 × 625 = 36,250

Explanation

The sum of interior angles of an n-sided polygon is given by the formula (n - 2) × 180°.

(n - 2) × 180° = 2520° n - 2 = 14 n = 16

The measure of each exterior angle of a regular n-sided polygon is given by 360° / n.

Measure of each exterior angle = 360° / 16 = 22.5°

Explanation

16% of 50 = (16 × 50) / 100 = 8

For the second part of the expression: ∛729 = 9 √25 = 5

Substituting these values into the term: (4 / 5) × 9 × 5 = 36

Now, evaluating y: y = 8 - 36 = -28

Explanation

Let the speed of the boat in still water be v km/h. Given the stream speed is 2 km/h, the downstream speed is v + 2 km/h and the upstream speed is v - 2 km/h.

Using Time = Distance / Speed: 36 / (v + 2) + 24 / (v - 2) = 6

Dividing the entire equation by 6 gives: 6 / (v + 2) + 4 / (v - 2) = 1

Simplifying: 6(v - 2) + 4(v + 2) = v² - 4 6v - 12 + 4v + 8 = v² - 4 10v = v² v = 10 km/h

Time taken downstream = 36 / (10 + 2) = 36 / 12 = 3.0 hours. Time taken upstream = 24 / (10 - 2) = 24 / 8 = 3.0 hours.

Explanation

Let total work = LCM(12, 15, 18) = 180 units.

Daily work capacities: Efficiency of A = 180 / 12 = 15 units/day Efficiency of B = 180 / 15 = 12 units/day Efficiency of C = 180 / 18 = 10 units/day

Since B works every day while C and A work on alternate days starting with C: Day 1 (B + C) = 12 + 10 = 22 units Day 2 (B + A) = 12 + 15 = 27 units

Work completed in 1 cycle (2 days) = 22 + 27 = 49 units. Work completed in 3 cycles (6 days) = 3 × 49 = 147 units. Remaining work = 180 - 147 = 33 units.

On Day 7, B and C work together and complete 22 units. Remaining work = 33 - 22 = 11 units.

On Day 8, B and A work together at a rate of 27 units/day. Time required for the remaining 11 units = 11 / 27 days.

Total time required = 7 + 11/27 = 7 11/27 days.

Explanation

Let the price at which P purchased the item be x.

Using the successive transaction relation: Price paid by R = x × (1 + Profit%/100) × (1 - Loss%/100)

Substituting the given values: 3060 = x × (153/100) × (32/100)

Dividing 3060 by 153 gives 20: 20 = x × (32/10000)

x = (20 × 10000) / 32 = 200000 / 32 = 6250

Explanation

Simplifying the given expression using basic trigonometric identities: 4 cot A sec A + 3 = 4 × (cos A / sin A) × (1 / cos A) + 3 = 4 / sin A + 3 = 4 cosec A + 3

Given cosec A = 5/4, substitute this value into the simplified expression: 4 cosec A + 3 = 4 × (5/4) + 3 = 5 + 3 = 8

Explanation

The coordinates of the centroid (geometric centre) of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) are given by ((x₁ + x₂ + x₃) / 3, (y₁ + y₂ + y₃) / 3).

Equating the x-coordinates: (9 + 4 + q) / 3 = 5 13 + q = 15 q = 2

Equating the y-coordinates: (5 + p + 4) / 3 = 5 9 + p = 15 p = 6

This gives p = 6 and q = 2.

Explanation

Using the proportion rule, if a:b::c:d, then a/b = c/d.

For 3:27::5.6:x: 3/27 = 5.6/x 1/9 = 5.6/x x = 5.6 × 9 = 50.4

For 5:35::6:y: 5/35 = 6/y 1/7 = 6/y y = 6 × 7 = 42

The ratio of x to y is: x/y = 50.4/42 = 504/420 = 6/5 = 6:5

Explanation

Present sum of ages of A and B = (24 + 8) × 2 = 64 years.

Present sum of ages of A, B, and C = 32 × 3 = 96 years.

Present age of C = 96 - 64 = 32 years.

Age of C after 16 years = 32 + 16 = 48 years.

Explanation

The sum of the first n odd positive integers is given by the formula n². The odd numbers from 1 to 100 are 1, 3, 5, ..., 99, which gives n = 50 terms. Thus, the sum is 50² = 2500.

The sum of an arithmetic progression is given by S = (N / 2) × (a + l), where N is the number of terms, a is the first term, and l is the last term. For even numbers from 150 to 200, a = 150, l = 200, and common difference d = 2. N = (200 - 150) / 2 + 1 = 26. Sum = (26 / 2) × (150 + 200) = 13 × 350 = 4550.

The required ratio is 2500 : 4550 = 50 : 91.

Explanation

Let the monthly incomes of Ajay and Amit be 6x and 7x respectively.

Since expenditure = income − savings and each saves ₹63,000 per month, the ratio of their expenditures is given by: (6x − 63000) / (7x − 63000) = 2 / 4

Simplifying 2 / 4 to 1 / 2 and cross-multiplying: 2 × (6x − 63000) = 1 × (7x − 63000) 12x − 126000 = 7x − 63000 5x = 63000 x = 12600

Monthly income of Ajay = 6x = 6 × 12600 = ₹75,600.

Explanation

Using the algebraic identity (x + y)² = x² + 2xy + y²:

(2a + 3b)² = (2a)² + (3b)² + 2(2a)(3b) (2a + 3b)² = 4a² + 9b² + 12ab

Substituting 4a² + 9b² = 49 and ab = 6 into the expansion:

(2a + 3b)² = 49 + 12(6) (2a + 3b)² = 49 + 72 (2a + 3b)² = 121

Taking the positive square root on both sides gives 2a + 3b = √121 = 11.

Explanation

Using BODMAS / PEMDAS rules, perform division and multiplication first, followed by addition and subtraction from left to right:

16 + 20 ÷ 4 - 3 × 3 = 16 + 5 - 9 = 21 - 9 = 12

Explanation

The mode of a dataset is defined as the value that appears with the highest frequency. In the given dataset of 25 observations, the number 7 appears 4 times, which is a higher frequency than any other number in the set.

Explanation

The total surface area (TSA) of a right circular cone is given by the formula TSA = πr(r + l), where r is the radius and l is the slant height.

Given: Radius (r) = 7 cm Slant height (l) = 25 cm π = 22 / 7

Substituting the values: TSA = (22 / 7) × 7 × (7 + 25) TSA = 22 × 32 TSA = 704 cm²

Explanation

Ratio of investments of Meenakshi and Meena = 12 : 32 = 3 : 8.

Total ratio units = 3 + 8 = 11. Difference in ratio units = 8 - 3 = 5.

Value of 1 ratio unit = 2904 / 11 = ₹264.

Difference between their profits = 5 × 264 = ₹1,320.

Explanation

The volume of the newly formed cube equals the sum of the volumes of the three melted cubes: V = 1³ + 6³ + 8³ = 1 + 216 + 512 = 729 cm³

If the edge length of the new cube is a, then: a³ = 729 a = 9 cm

The length of the body diagonal of a cube of side a is given by the formula d = a√3. Substituting a = 9 gives the diagonal as 9√3 cm.

Explanation

Rate of filling by the first tap = +1/4 per hour Rate of filling by the second tap = +1/26 per hour Rate of emptying by the third tap = -1/26 per hour

Net filling rate when all three taps are opened together = 1/4 + 1/26 - 1/26 = 1/4 of the cistern per hour.

Time required to fill 1/4 of the cistern = (1/4) / (1/4) = 1 hour.

Explanation

Prime factorization of the given numbers: 64 = 2⁶ 128 = 2⁷ 4480 = 2⁷ × 5¹ × 7¹

Since 64 and 128 do not contain factors of 5 or 7, A must contain both 5¹ and 7¹. The power of 2 in A can be any power from 2⁰ to 2⁷, as 128 already contributes 2⁷ to the least common multiple.

Thus, A = 2ᵏ × 5 × 7 = 35 × 2ᵏ, where 0 ≤ k ≤ 7.

Minimum possible value of A occurs when k = 0: A_min = 35 × 2⁰ = 35

Maximum possible value of A occurs when k = 7: A_max = 35 × 2⁷ = 4480

Difference between maximum and minimum values of A = 4480 - 35 = 4445

Explanation

For a quadratic equation ax² + bx + c = 0 to have identical roots, its discriminant D = b² - 4ac must be equal to zero.

Given the equation x² - (2 + h)x + (h² - 4h + 4) = 0, the coefficients are: a = 1 b = -(2 + h) c = h² - 4h + 4

Substituting these into D = 0: [-(2 + h)]² - 4(1)(h² - 4h + 4) = 0 (4 + 4h + h²) - (4h² - 16h + 16) = 0 -3h² + 20h - 12 = 0 3h² - 20h + 12 = 0

Factoring the equation: 3h² - 18h - 2h + 12 = 0 3h(h - 6) - 2(h - 6) = 0 (3h - 2)(h - 6) = 0

Thus, h = 2/3 or h = 6.

Explanation

The overall average amount spent per member is given by the weighted average formula:

Weighted Average = (n₁ × A₁ + n₂ × A₂ + n₃ × A₃) / (n₁ + n₂ + n₃)

Substituting the given values:

Total amount spent = (25 × 101) + (32 × 249) + (31 × 269) Total amount spent = 2525 + 7968 + 8339 = 18832

Total number of members = 25 + 32 + 31 = 88

Overall average = 18832 / 88 = 214

Explanation

Let CP and SP be the cost price and selling price of one book respectively.

25 × SP = 19 × CP SP / CP = 19 / 25

Since SP < CP, there is a loss.

Loss % = [(CP - SP) / CP] × 100 Loss % = [(25 - 19) / 25] × 100 = (6 / 25) × 100 = 24%

Explanation

The average of the first n multiples of a number k is given by the formula: Average = k × (n + 1) ÷ 2.

Here, k = 9 and n = 13.

Substituting the given values: Average = 9 × (13 + 1) ÷ 2 Average = 9 × 14 ÷ 2 Average = 9 × 7 = 63.

Alternatively, because the multiples of 9 form an arithmetic progression, the average is equal to the middle (7th) term: 9 × 7 = 63.

Explanation

When a sum becomes n times itself under simple interest, the rate of interest is given by the formula R = [(n - 1) × 100] / T, where n is the multiplier and T is the time in years.

Here, n = 5 and T = 16 years.

Substituting the values: R = [(5 - 1) × 100] / 16 R = (4 × 100) / 16 R = 400 / 16 = 25% per year.

Explanation

Mode for grouped data is calculated using the formula: Mode = L + [(f₁ - f₀) / (2f₁ - f₀ - f₂)] × h

From the given data, the class interval 50-60 has the highest frequency of 35, making it the modal class.

Here, lower limit of modal class (L) = 50, frequency of modal class (f₁) = 35, frequency of preceding class (f₀) = 26, frequency of succeeding class (f₂) = 20, and class width (h) = 10.

Substituting the values into the formula: Mode = 50 + [(35 - 26) / (2 × 35 - 26 - 20)] × 10 Mode = 50 + [9 / (70 - 46)] × 10 Mode = 50 + (9 / 24) × 10 Mode = 50 + 3.75 Mode = 53.75

Explanation

Downstream speed = 72 km / 8 hours = 9 km/hr Upstream speed = 72 km / 12 hours = 6 km/hr

Boat's rate in still water = (Downstream speed + Upstream speed) / 2 = (9 + 6) / 2 = 7.5 km/hr

Rate of current = (Downstream speed - Upstream speed) / 2 = (9 - 6) / 2 = 1.5 km/hr

Explanation

P takes 40 ÷ 5 = 8 minutes to complete 1 round, giving P a speed of ⅛ rounds per minute.

Time taken by P to complete 6 rounds = 6 × 8 = 48 minutes. Since R completes 7 rounds in 48 minutes, R's speed is 7/48 rounds per minute.

When moving in the same direction, relative speed = 7/48 - ⅛ = 1/48 rounds per minute.

Starting from opposite ends of a diameter, the distance between them is ½ round.

Time to meet = Distance ÷ Relative speed = ½ ÷ (1/48) = 24 minutes.

Explanation

Cost price of each article = ₹2300.

Selling price of X = Cost price + Profit = 2300 + 460 = ₹2760.

Let M be the marked price. The selling price of X after two successive discounts of 46% and 68% is given by: Selling price of X = M × (1 - 0.46) × (1 - 0.68) = M × 0.54 × 0.32 = 0.1728M

So, 0.1728M = 2760, which gives M = 2760 / 0.1728.

Article Y is sold at a single discount of 73%, so its selling price is: Selling price of Y = M × (1 - 0.73) = 0.27M Selling price of Y = 0.27 × (2760 / 0.1728) = 2760 × (2700 / 1728) = 2760 × (25 / 16) = ₹4312.5

Profit on Y = Selling price - Cost price = 4312.5 - 2300 = ₹2012.5.

Profit percentage on Y = (2012.5 / 2300) × 100 = 87.5%

Explanation

Amount A = P × (1 + R/100)ⁿ

Given: Principal (P) = ₹28,450 Rate of interest (R) = 20% Time (n) = 2 years

Substituting the values into the formula: A = 28,450 × (1 + 20/100)² A = 28,450 × (6/5)² A = 28,450 × (36 / 25) A = 1,138 × 36 A = ₹40,968

Explanation

Let a = 732 and b = 268.

Using the identity a³ + b³ = (a + b)(a² - ab + b²), the numerator becomes: a³ + b³ = (732 + 268)(732² - 732 × 268 + 268²) = 1000 × (732² - 732 × 268 + 268²)

Since 73.2 = a / 10 and 26.8 = b / 10, the denominator simplifies to: (a / 10)² + (b / 10)² - (a / 10)(b / 10) = (a² - ab + b²) / 100

Dividing the numerator by the denominator: [1000 × (a² - ab + b²)] / [(a² - ab + b²) / 100] = 1000 × 100 = 100000

Explanation

In a right-angled triangle formed by the string, the height, and the ground:

height = length of string × sin(θ)

Substituting the given values: height = 24.4 × sin(30°) height = 24.4 × ½ = 12.2 m

Explanation

Convert mixed fractions and decimals into simple fractions: 5 1/6 = 31/6 6.2 = 31/5 0.75 = 3/4

First, evaluate the expression inside the parenthesis: (1/5 × 40) = 8

Now, carry out division and multiplication from left to right: 31/6 ÷ 31/5 × 3/4 ÷ 8 = (31/6 × 5/31) × 3/4 ÷ 8 = 5/6 × 3/4 ÷ 8 = 5/8 ÷ 8 = 5/8 × 1/8 = 5/64

Explanation

When two quadratic equations A₁x² + B₁x + C₁ = 0 and A₂x² + B₂x + C₂ = 0 have both roots in common, their corresponding coefficients are proportional:

A₁/A₂ = B₁/B₂ = C₁/C₂

Substituting the given values: (a² - 7a) / 240 = 1 / 8 = 12 / (a² - 4)

Solving (a² - 7a) / 240 = 1/8: a² - 7a = 30 a² - 7a - 30 = 0 (a - 10)(a + 3) = 0

Since a is a positive integer, a = 10.

Verifying with the constant term ratio: 12 / (10² - 4) = 12 / 96 = 1/8, which holds true.

Evaluating the given expression for a = 10: (a + 2)(a + 3) / (a - 4) = (10 + 2)(10 + 3) / (10 - 4) = (12 × 13) / 6 = 26

Explanation

Using the weighted average method for overall profit percentage:

Weighted Profit % = Σ (Weight × Profit or Loss %) ÷ Total Weight

Let the total quantity of goods be 100 units.

First batch = 50% of 100 = 50 units at +16% profit Second batch = 40% of remaining 50 units = 20 units at +35% profit Third batch = Remaining 100 - (50 + 20) = 30 units at -25% loss

Overall Profit % x = (50 × 16 + 20 × 35 + 30 × (-25)) ÷ 100 x = (800 + 700 - 750) ÷ 100 x = 750 ÷ 100 = 7.5

Explanation

Let the cost price be 100.

Since the marked price is fixed 180% above the cost price: Marked Price = 100 + 180 = 280

To gain 12%, the selling price must be: Selling Price = 100 + 12 = 112

Discount amount = Marked Price - Selling Price = 280 - 112 = 168

Discount percentage = (Discount / Marked Price) × 100 Discount percentage = (168 / 280) × 100 = 60%

Explanation

Cost Price (CP) per dozen = ₹100

Since 1 dozen = 12 articles, Selling Price (SP) per dozen = 12 × 75 = ₹900

Profit per dozen = SP - CP = 900 - 100 = ₹800

Profit percentage = (Profit / CP) × 100 = (800 / 100) × 100 = 800%

Explanation

Average = Sum of values / Total number of values

Sum = 312 + 126 + 100 + 99 = 637 Total count = 4

Average = 637 / 4 = 159.25

Explanation

The overall average is calculated using the weighted average formula:

Weighted Average = (n₁A₁ + n₂A₂ + n₃A₃) / (n₁ + n₂ + n₃)

The ratio of members in groups A, B, and C is 72 : 12 : 87 = 24 : 4 : 29.

Overall Average = (24 × 404 + 4 × 316 + 29 × 424) / (24 + 4 + 29) = (9696 + 1264 + 12296) / 57 = 23256 / 57 = 408

Explanation

Expressing the given percentage relations mathematically:

160% of 75% of A = (1.6) × (0.75) × A = 1.2 A

Two times 150% of B = 2 × 1.5 × B = 3 B

Equating the two values: 1.2 A = 3 B A / B = 3 / 1.2 = 2.5

To find how much percentage A is more than B: Percentage increase = ((A - B) / B) × 100 = ((2.5 - 1) / 1) × 100 = 150%

Explanation

The hydrostatic pressure at the base of a liquid column is given by the formula P = ρgh, where ρ is the density of the liquid, g is the acceleration due to gravity, and h is the height of the column.

Converting the density of water to SI units: ρ = 1 kg/litre = 1000 kg/m³

Substituting the given values into the formula: P = 1000 kg/m³ × 9.8 m/s² × 20 m = 1,96,000 Pa

Explanation

The internal surface area of a cuboidal box open at the top is equal to the area of the base plus the area of the four side walls:

Surface Area = lb + 2h(l + b)

Given length l = 8 cm, breadth b = 17 cm, and height h = 22 cm:

Base area = 8 × 17 = 136 cm² Area of 4 walls = 2 × 22 × (8 + 17) = 44 × 25 = 1100 cm²

Total internal surface area = 136 + 1100 = 1236 cm²

Explanation

Express the given LCM values in prime factorization form: lcm(A, B) = 24 = 2³ × 3¹ lcm(B, C) = 36 = 2² × 3² lcm(C, A) = 72 = 2³ × 3²

Let A = 2ᵃ × 3ˣ, B = 2ᵇ × 3ʸ, and C = 2ᶜ × 3ᶻ.

For the powers of 2: max(a, b) = 3, max(b, c) = 2, and max(c, a) = 3. Since max(b, c) = 2, neither b nor c can exceed 2. Therefore, a must equal 3. To satisfy max(b, c) = 2, at least one of b or c must equal 2.

For the powers of 3: max(x, y) = 1, max(y, z) = 2, and max(z, x) = 2. Since max(x, y) = 1, neither x nor y can exceed 1. Therefore, z must equal 2. To satisfy max(x, y) = 1, at least one of x or y must equal 1.

To minimize the sum A + B + C: Set x = 0, which makes y = 1 and gives A = 2³ × 3⁰ = 8. Set c = 0, which makes b = 2 and gives C = 2⁰ × 3² = 9. With b = 2 and y = 1, B = 2² × 3¹ = 12.

The minimal values are A = 8, B = 12, and C = 9. A + B + C = 8 + 12 + 9 = 29

Explanation

The formula for the median of grouped data is Median = L + [(N/2 - cf) / f] × h.

Total frequency N = 11 + 18 + 28 + 15 + 22 + 31 + 13 = 138, giving N/2 = 69.

The cumulative frequencies for the intervals are 11 (for 20-30), 29 (30-40), 57 (40-50), 72 (50-60), 94 (60-70), 125 (70-80), and 138 (80-90).

The median class is 50-60, as its cumulative frequency (72) is the first to exceed 69.

Here, lower limit L = 50, preceding cumulative frequency cf = 57, frequency of median class f = 15, and class width h = 10.

Median = 50 + [(69 - 57) / 15] × 10 = 50 + (12 / 15) × 10 = 50 + 8 = 58

Explanation

Let F and S be the present ages of the father and son respectively.

Six years hence: F + 6 = 3(S + 6) F - 3S = 12

Six years ago: F - 6 = 7(S - 6) F - 7S = -36

Subtracting the second equation from the first gives: 4S = 48 S = 12

Substituting S = 12 into F - 3S = 12 gives F = 48.

Explanation

Simplify each part of the expression:

30% of 80 = (30 / 100) × 80 = 24

∛729 = 9

√49 = 7

Substitute these values into the given equation: y = 24 - (4 / 7) × 9 × 7

Simplify the multiplication: y = 24 - 4 × 9 y = 24 - 36 y = -12

Explanation

Time taken to type 1 page = 40 ÷ 2 = 20 minutes.

Total time required to type 91 pages = 91 × 20 = 1820 minutes.

Explanation

In the right-angled triangle formed by the wall, the ground, and the ladder, cos(θ) = base / hypotenuse, where the base is the distance of the foot of the ladder from the wall and the hypotenuse is the length of the ladder (L).

cos(60°) = 11.5 / L ½ = 11.5 / L L = 11.5 × 2 = 23 m

Explanation

Evaluating the given expression step by step:

45% of 20 = (45 ÷ 100) × 20 = 9

∛729 = 9

√36 = 6

Substituting these values into the second term: (4 / 6) × ∛729 × √36 = (4 / 6) × 9 × 6 = 36

Subtracting the second term from the first: y = 9 - 36 = -27

Explanation

For any non-zero real number a, the algebraic expansion for a⁵ + 1/a⁵ is given by: a⁵ + 1/a⁵ = (a² + 1/a²)(a³ + 1/a³) - (a + 1/a)

Given a + 1/a = 4: a² + 1/a² = (a + 1/a)² - 2 = 4² - 2 = 14 a³ + 1/a³ = (a + 1/a)³ - 3(a + 1/a) = 4³ - 3(4) = 52

Substituting these values into the main identity: a⁵ + 1/a⁵ = 14 × 52 - 4 = 728 - 4 = 724