RRB NTPC CBT 1 2026 Mathematics Questions with Solutions

Explanation

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

Given: 48 × CP = 36 × SP SP / CP = 48 / 36 = 4 / 3

Profit percentage = [(SP - CP) / CP] × 100 = [(4 - 3) / 3] × 100 = (1 / 3) × 100 = 33.33%

Explanation

Butter required per waffle = 680 ÷ 4 = 170 g.

Using unitary method, the total butter required for 17 waffles = 170 × 17 = 2890 g.

Explanation

The time interval between simultaneous activations of the sprays is given by the Least Common Multiple (LCM) of their individual activation intervals.

LCM(12, 15, 20) = 60 minutes = 1 hour

Since the sprays activate together every 1 hour, the time of the 6th simultaneous activation (including the 1st one at the start) requires adding 5 interval periods of 1 hour to the starting time.

Time = 9:45 AM + (5 × 1 hour) = 2:45 PM

Explanation

The modal class is 70-80 because it has the highest frequency of 38.

Using the formula for the mode of grouped data: Mode = L + [(f₁ - f₀) / (2f₁ - f₀ - f₂)] × h

Here, lower limit of modal class L = 70, frequency of modal class f₁ = 38, frequency of preceding class f₀ = 18, frequency of succeeding class f₂ = 26, and class height h = 10.

Substituting the values: Mode = 70 + [(38 - 18) / (2 × 38 - 18 - 26)] × 10 Mode = 70 + [20 / (76 - 44)] × 10 Mode = 70 + (20 / 32) × 10 Mode = 70 + 6.25 = 76.25

Explanation

When a hemisphere of radius r is carved out from a face of a cube of edge length a, the net surface area of that face increases by πr² because the flat circular base area of πr² is removed while the curved surface area of 2πr² is added.

Total surface area of the remaining structure = 6a² + 6πr²

Given, a = 14 cm and r = 7 cm: Surface area = 6 × 14² + 6 × (22/7) × 7² = 6 × 196 + 6 × 22 × 7 = 1176 + 924 = 2100 cm²

Explanation

The perpendicular line drawn from the center of a circle to a chord bisects the chord.

Half length of the chord = 18 / 2 = 9 cm

Using the Pythagorean theorem in the right triangle formed by the radius (r), perpendicular distance from the center (12 cm), and half of the chord (9 cm):

r² = 9² + 12² = 81 + 144 = 225 r = √225 = 15 cm

Diameter = 2 × r = 2 × 15 = 30 cm

Explanation

Profit is distributed in proportion to the investments made by each partner.

Total investment = ₹1230 + ₹1710 + ₹1010 = ₹3950

Shanaya's share in profit = Total Profit × (Shanaya's Investment / Total Investment) = ₹1580 × (1010 / 3950) = 4 × 101 = ₹404

Explanation

Profit share in a partnership is proportional to the product of the investment amount and the time duration.

Let the investment amounts for Mansi, Neha, and Utsavi be 7 units, 9 units, and 9 units respectively. Mansi and Neha invested for 12 months, while Utsavi invested for 6 months.

Profit Ratio = (7 × 12) : (9 × 12) : (9 × 6)

Dividing all terms by 6: = (7 × 2) : (9 × 2) : (9 × 1) = 14 : 18 : 9

Explanation

Let the original price of fuel be 100 units.

Using the successive percentage change formula, the final price factor is: (1 - 0.30) × (1 - 0.20) × (1 - 0.50) × (1 + 0.40) = 0.70 × 0.80 × 0.50 × 1.40 = 0.392

Final price = 100 × 0.392 = 39.2 units

Percentage decrease = ((100 - 39.2) / 100) × 100 = 60.8%

Explanation

For a quadratic expression ax² + bx + c to be strictly positive for all real values of x, the coefficient of x² must be positive (a > 0) and its discriminant must be negative (D = b² - 4ac < 0).

Here, a = 12 > 0, b = -k, and c = 27.

Setting D < 0: (-k)² - 4(12)(27) < 0 k² - 1296 < 0 k² < 1296

Taking the square root on both sides gives -36 < k < 36.

Explanation

Using the BODMAS rule (Brackets, Orders, Division, Multiplication, Addition, Subtraction), simplify the expression step by step:

First, perform division: 20 ÷ 5 = 4

Next, perform multiplication: 2 × 4 = 8

Now substitute these back into the expression: 16 + 4 - 8

Perform addition and subtraction from left to right: 16 + 4 = 20 20 - 8 = 12

Explanation

By definition, in a right-angled triangle, tanθ is the ratio of the length of the opposite side to the length of the adjacent side:

tanθ = Opposite side / Adjacent side

Substituting the given values:

tanθ = 7 / 21 = 1/3

Explanation

Using the standard algebraic identity: a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)

Let a = 2.33, b = 1.55, and c = 1.22.

Notice that: ab = 2.33 × 1.55 = 3.6115 ac = 2.33 × 1.22 = 2.8426 bc = 1.55 × 1.22 = 1.891

The given expression is in the form: (a³ + b³ + c³ - 3abc) / (a² + b² + c² - ab - ac - bc) = a + b + c

Substituting the values: a + b + c = 2.33 + 1.55 + 1.22 = 5.1

Explanation

Cost price of one dozen articles = ₹520 Selling price of one dozen articles = 12 × ₹52 = ₹624

Profit per dozen = ₹624 - ₹520 = ₹104

Profit percentage = (Profit / Cost price) × 100 = (104 / 520) × 100 = 20%

Since profit percentage depends only on the unit prices, the total quantity of 59 dozens does not alter the percentage profit.

Explanation

Let the cost price of 1000 gm of goods be ₹100. Since the shopkeeper professes to sell at a 35% loss, his selling price for the claimed 1000 gm is ₹65.

Let x gm be the actual weight used instead of 1000 gm. Cost price of x gm = ₹(x / 10)

Using the profit percentage formula: Profit % = [(Selling Price - Cost Price) / Cost Price] × 100 55 = [(65 - x/10) / (x/10)] × 100 0.55 = (65 - x/10) / (x/10) 0.55x / 10 = 65 - x / 10 1.55x = 650 x = 650 / 1.55 = 419.35... gm ≈ 419.4 gm

Explanation

In right-angled triangle ABO, the tangent AB is perpendicular to the radius OB, so ∠ABO = 90°.

Using trigonometry in ΔABO: cos(∠AOB) = OB / AO = 3 / 6 = ½ ∠AOB = 60°

By symmetry of tangents from an external point, OA bisects ∠BOC, giving: ∠BOC = 2 × ∠AOB = 2 × 60° = 120°

In triangle OBC, sides OB and OC are both radii of length 3 cm, so ΔOBC is an isosceles triangle with ∠OBC = ∠OCB.

Sum of angles in ΔOBC: ∠BOC + 2(∠OBC) = 180° 120° + 2(∠OBC) = 180° 2(∠OBC) = 60° ∠OBC = 30°

Explanation

From the first equation, x = 3z - 5.

Substituting x = 3z - 5 into the second equation gives: 2(3z - 5) - y + 2z = 16 8z - y = 26 y = 8z - 26

Substituting x and y into the third equation gives: 7(3z - 5) - 3(8z - 26) - 5z = 19 21z - 35 - 24z + 78 - 5z = 19 -8z + 43 = 19 -8z = -24 z = 3

Substituting z = 3 back into the expressions for x and y: x = 3(3) - 5 = 4 y = 8(3) - 26 = -2

Thus, the values are x = 4, y = -2, and z = 3.

Explanation

Using the work equivalence formula (M₁ × T₁) ÷ W₁ = (M₂ × T₂) ÷ W₂:

Given: M₁ = 9 students, T₁ = 3 hours, W₁ = 117 charts M₂ = required students, T₂ = 1 hour, W₂ = 208 charts

Substituting the values: (9 × 3) ÷ 117 = (M₂ × 1) ÷ 208 27 ÷ 117 = M₂ ÷ 208

Simplifying 27 ÷ 117 gives 3 ÷ 13: 3 ÷ 13 = M₂ ÷ 208 M₂ = (3 × 208) ÷ 13 M₂ = 3 × 16 = 48

Explanation

The mean proportional between two numbers a and b is given by √(a × b).

Given that the mean proportional between x and 36 is 18: √(x × 36) = 18

Squaring both sides: 36x = 18² 36x = 324 x = 324 / 36 = 9

Explanation

Let x be the present age of the younger brother. Then Sharad's present age is 2x, and the other brother's present age is x - 3.

The sum of their present ages is 2x + x + (x - 3) = 4x - 3.

Three years ago, each of the three brothers was 3 years younger, making the sum of their ages 9 years less than the present total: (4x - 3) - 9 = 28 4x - 12 = 28 4x = 40 x = 10

Sharad's present age is 2x = 2 × 10 = 20 years.

Explanation

First, evaluate √2601: Since 51² = 2601, √2601 = 51.

Substitute this back into the equation: (22 × 3 × 816) ÷ 51 = y + 794

Dividing 816 by 51 gives 16: 22 × 3 × 16 = y + 794

Multiply the left side: 66 × 16 = 1056

Solve for y: 1056 = y + 794 y = 1056 − 794 = 262

Explanation

Let the original price of fuel be 100.

Using the successive percentage change formula, the final price is: Final Price = 100 × (1 - 0.40) × (1 - 0.60) × (1 - 0.60) × (1 + 0.70) = 100 × 0.60 × 0.40 × 0.40 × 1.70 = 16.32

Percentage change = 16.32 - 100 = -83.68%

The negative value indicates a decrease of 83.68%.

Explanation

Profit share in a partnership is proportional to the product of investment and time period. Let the initial investments of Trisha and Riya be 8x and 5x respectively. Since Poorva invests an amount equal to Riya's, Poorva's investment is 5x.

Trisha's investment period = 12 months Riya's investment period = 12 months Poorva's investment period = 6 months

Ratio of profits = (8x × 12) : (5x × 12) : (5x × 6) Dividing throughout by 6x gives (8 × 2) : (5 × 2) : (5 × 1) = 16 : 10 : 5.

Explanation

Let the cost price of 1 gram of goods be ₹1. The cost price of 1000 grams is ₹1000.

Since the shopkeeper professes to sell at a 45% loss, the selling price for 1000 grams is: Selling Price = 1000 × (1 - 0.45) = ₹550

Let x be the actual weight used in grams instead of 1000 grams. The actual cost price to the shopkeeper for this weight is ₹x.

With an overall gain of 50%: Selling Price = Cost Price × (1 + Gain%) 550 = x × (1 + 0.50) 1.5 x = 550 x = 550 / 1.5 = 366.666... gm

Rounding off to one decimal place gives 366.7 gm.

Explanation

To find the ratio A : B : C when 3A = 4B = 8C, take the LCM of the coefficients 3, 4, and 8, which is 24.

Dividing the LCM by each coefficient gives the corresponding ratio terms: A = 24 / 3 = 8 B = 24 / 4 = 6 C = 24 / 8 = 3

This gives A : B : C = 8 : 6 : 3.

Explanation

The average of the first n even natural numbers is given by the formula (n + 1).

Substituting n = 62 into the formula: Average = 62 + 1 = 63

Explanation

Slant height l of each conical cup = √(r² + h²) = √(7² + 24²) = √625 = 25 cm.

Curved surface area of one cup = π × r × l = 3.14 × 7 × 25 = 549.5 cm².

Total paper required for 1000 cups = 549.5 × 1000 = 549500 cm².

Converting to square meters (since 1 m² = 10000 cm²): Required paper = 549500 / 10000 = 54.95 m².

Explanation

The total surface area of a solid hemisphere is given by the formula 3πr², where r is the radius.

3πr² = 200

Substituting π = 22/7: 3 × (22/7) × r² = 200 (66/7) × r² = 200 r² = (200 × 7) / 66 r² = 700 / 33 r = √(700/33) cm

Explanation

Overall average = Total amount spent / Total number of members

Total members = 69 + 65 + 88 = 222

Total amount spent = (69 × 468) + (65 × 120) + (88 × 276) = 32,292 + 7,800 + 24,288 = 64,380

Overall average = 64,380 / 222 = 290

Explanation

Relative speed when two objects move towards each other = Speed₁ + Speed₂ = 49 + 59 = 108 km/hr.

Converting relative speed to m/s: 108 × (5/18) = 30 m/s.

Total distance to be covered to completely pass each other = Length₁ + Length₂ = 349 + 371 = 720 m.

Time taken = Total distance / Relative speed = 720 / 30 = 24 seconds.

Explanation

In the right-angled triangle formed by the ladder, the wall, and the ground:

sin(60°) = Height / Length of ladder

Height = Length of ladder × sin(60°) Height = 28 × (√3 / 2) Height = 14 × √3

Using the given value √3 = 1.74: Height = 14 × 1.74 = 24.36 m

Explanation

Under the Merchandise Exports from India Scheme (MEIS), introduced as part of the Foreign Trade Policy 2015-20, eligible exporters are granted Duty Credit Scrips as incentives calculated as a percentage of the Free on Board (FOB) value of exported goods. These Duty Credit Scrips are freely transferable, allowing exporters to either use them to pay basic customs duty and other specified taxes on imported goods or sell them in the open market to other importers.

Explanation

Under simple interest, when a sum of money P becomes 5 times itself, the total amount is 5P, which means the interest earned SI = 5P - P = 4P.

Using the rate formula R = (SI × 100) / (P × T): R = (4P × 100) / (P × 4) = 100% per annum.

Explanation

Using the standard algebraic identities, for a + 1/a = k, the cube is a³ + 1/a³ = k³ - 3k, and for aⁿ + 1/aⁿ = m, the square is a²ⁿ + 1/a²ⁿ = m² - 2.

Given a + 1/a = 3:

a³ + 1/a³ = 3³ - 3(3) = 27 - 9 = 18

Squaring this expression gives:

a⁶ + 1/a⁶ = (a³ + 1/a³)² - 2 = 18² - 2 = 324 - 2 = 322

Explanation

Using exponent rules, converting all terms to the base (9/28):

(28/9)¹¹ = (9/28)⁻¹¹ (28/9)¹⁷ = (9/28)⁻¹⁷

Multiplying the terms on the left side by adding their powers:

(9/28)⁻¹¹ × (9/28)¹⁶ × (9/28)⁻¹⁷ = (9/28)⁻¹¹ ⁺ ¹⁶ ⁻ ¹⁷ = (9/28)⁻¹²

Equating the exponents from both sides:

-12 = 2m + 8 2m = -20 m = -10

Explanation

An equation of the form ax² + bx + c = 0 represents a quadratic equation if and only if the coefficient of x² is non-zero (a ≠ 0).

In the given equation, the coefficient of x² is (2m - 3). For the equation to represent a quadratic equation, this coefficient must not equal zero:

2m - 3 ≠ 0 2m ≠ 3 m ≠ 3/2

Explanation

Using the compound interest formula A = P(1 + r/100)ᵗ, let Akshay's share be P₁ and Arjun's share be P₂.

Given that the amount accumulated by Akshay in 16 years equals the amount accumulated by Arjun in 17 years: P₁ × (1 + 5/100)¹⁶ = P₂ × (1 + 5/100)¹⁷ P₁ / P₂ = 1 + 5/100 = 21/20

Thus, the ratio of Akshay's share to Arjun's share is 21 : 20.

Total share = 21 + 20 = 41 units = ₹1394 1 unit = 1394 / 41 = 34

Arjun's share = 20 × 34 = ₹680

Explanation

Converting each ratio into its decimal value:

5/12 = 0.4167 14/31 = 0.4516 8/33 = 0.2424 17/39 = 0.4359

Comparing the decimal values: 0.4516 (14/31) > 0.4359 (17/39) > 0.4167 (5/12) > 0.2424 (8/33)

Explanation

Cost Price per article = ₹300 / 12 = ₹25

Selling Price per article = ₹90

Profit per article = ₹90 - ₹25 = ₹65

Profit percentage = (Profit / Cost Price) × 100 = (65 / 25) × 100 = 260%

Explanation

The mean of grouped data is given by the formula Mean = Σ(f × x) / Σf, where x is the class mark (midpoint) and f is the frequency of each class.

The class marks (x) for the intervals 10-20, 20-30, 30-40, 40-50, and 50-60 are 15, 25, 35, 45, and 55 respectively.

Σf = 15 + 21 + 43 + 41 + X = 120 + X

Σ(f × x) = (15 × 15) + (21 × 25) + (43 × 35) + (41 × 45) + (X × 55) = 225 + 525 + 1505 + 1845 + 55X = 4100 + 55X

Equating to the given mean of 35: 35 = (4100 + 55X) / (120 + X) 35 × (120 + X) = 4100 + 55X 4200 + 35X = 4100 + 55X 20X = 100 X = 5

Explanation

For any two positive integers, the product of the two numbers is equal to the product of their HCF and LCM:

First Number × Second Number = HCF × LCM

Substituting the given values:

84 × Second Number = 12 × 840

Second Number = (12 × 840) / 84 = 840 / 7 = 120

Explanation

Using the inverse proportion formula for total work or provisions, M₁ × D₁ = M₂ × D₂:

100 × 26 = (100 - 50) × D₂

100 × 26 = 50 × D₂

D₂ = (100 × 26) / 50 = 52 days

Explanation

Let B's salary be 100. Since A's salary is 113% more than that of B, A's salary = 100 + 113 = 213.

The percentage by which B's salary is less than A's salary is given by: Percentage less = [r / (100 + r)] × 100

Substituting r = 113: Percentage less = [113 / (100 + 113)] × 100 = (113 / 213) × 100 = 11300 / 213 = 53.0516% ≈ 53.05%

Explanation

The average of an arithmetic progression is given by the formula (First term + Last term) / 2, or directly for the first n multiples of a number k as k × (n + 1) / 2.

Here, k = 7 and n = 350.

Substituting the given values into the formula: Average = 7 × (350 + 1) / 2 Average = 7 × 351 / 2 Average = 2457 / 2 = 1228.5

Explanation

Let v be the speed of the boat in still water in km/h.

Downstream speed = v + 4 Upstream speed = v - 4

Since the distance covered downstream and upstream is equal: Distance = Speed × Time 6(v + 4) = 8(v - 4) 6v + 24 = 8v - 32 2v = 56 v = 28 km/h

Explanation

Let the monthly incomes of Vipul and Vijay be 6x and 7x respectively. Since Expenditure = Income - Savings, setting up the ratio of their expenditures gives:

(6x - 66,000) / (7x - 66,000) = 1 / 3

3 × (6x - 66,000) = 7x - 66,000 18x - 1,98,000 = 7x - 66,000 11x = 1,32,000 x = 12,000

Vipul's monthly income = 6x = 6 × 12,000 = ₹72,000.

Explanation

The mean is Σ(f × x) / Σf = (7 × 12 + 8 × 13 + 9 × 12 + 10 × 11) / 48 = 406 / 48 ≈ 8.485, matching i with 1.

The variance is [Σ(f × x²) / Σf] - (Mean)² = (3492 / 48) - (8.458)² = 72.75 - 71.543 = 1.207, matching iii with 2.

The standard deviation is √(Variance) = √1.207 ≈ 1.099, matching ii with 3.

Explanation

The sum of interior angles of an n-sided polygon is given by the formula (n - 2) × 180°. A trapezium is a four-sided polygon (quadrilateral), so n = 4. Substituting this into the formula gives (4 - 2) × 180° = 2 × 180° = 360°.

Explanation

Selling price of article X = 800 + 523 = ₹1323.

Since X is sold after two successive discounts of 45% and 73% on marked price M: 1323 = M × (1 - 0.45) × (1 - 0.73) 1323 = M × 0.55 × 0.27

Selling price of article Y after a single discount of 89%: Selling price = M × (1 - 0.89) = M × 0.11

Substituting M from the first equation: Selling price = (1323 × 0.11) / (0.55 × 0.27) Selling price = 1323 / (5 × 0.27) Selling price = 1323 / 1.35 = ₹980

Profit on Y = 980 - 800 = ₹180. Profit percentage on Y = (180 / 800) × 100 = 22.5%.

Explanation

Weighted Arithmetic Mean = (w₁x₁ + w₂x₂) / (w₁ + w₂)

Assignment contribution = 40% × 75 = 0.40 × 75 = 30 Exam contribution = 60% × 85 = 0.60 × 85 = 51

Weighted Mean = 30 + 51 = 81