RRB NTPC CBT 1 2026 Mathematics Questions with Solutions

Explanation

Average = Sum of values ÷ Number of values

Sum of marks = 72 + 68 + 74 + 70 + 76 = 360 Total number of subjects = 5

Average marks = 360 ÷ 5 = 72

Explanation

When two runners move in the same direction along a circular track, the number of distinct meeting points on the circumference is given by |a - b|, where a : b is the ratio of their speeds expressed in its simplest form.

Given: Time taken by Shahrukh = 85 seconds Time taken by Amir = 90 seconds

Speed of Shahrukh = 1530 / 85 = 18 m/s Speed of Amir = 1530 / 90 = 17 m/s

Speed ratio (Shahrukh : Amir) = 18 : 17

Since 18 and 17 are coprime, a = 18 and b = 17.

Number of distinct meeting points = |18 - 17| = 1

Explanation

The mode of a discrete frequency distribution is the value of the variable corresponding to the highest frequency.

In the given data: Height 14 cm has frequency 1 Height 27 cm has frequency 3 Height 23 cm has frequency 6 Height 21 cm has frequency 12 Height 19 cm has frequency 7 Height 17 cm has frequency 5

The maximum frequency is 12, which corresponds to the height of 21 cm.

Explanation

When the distance travelled in both directions is equal, average speed is given by the formula: Average Speed = 2 × v₁ × v₂ / (v₁ + v₂)

Let the morning speed be v km/h. Then the afternoon speed is (v - 2) km/h.

Substituting the values into the formula: 4.8 = 2 × v × (v - 2) / (v + v - 2) 4.8 = v(v - 2) / (v - 1) 24 / 5 = (v² - 2v) / (v - 1) 24v - 24 = 5v² - 10v 5v² - 34v + 24 = 0

Factoring the quadratic equation: (5v - 4)(v - 6) = 0

Since v must be greater than 2 km/h so that the afternoon speed remains positive, v = 6 km/h.

Morning speed = 6 km/h Afternoon speed = 4 km/h Sum of speeds = 6 + 4 = 10 km/h

Explanation

Subtracting the first equation from the second equation yields: (2X + 2Y + 3Z) - (X + 2Y + 4Z) = 10 - 12 X - Z = -2

Subtracting the second equation from the third equation yields: (3X + 2Y + Z) - (2X + 2Y + 3Z) = 8 - 10 X - 2Z = -2

Subtracting X - 2Z = -2 from X - Z = -2 gives: Z = 0

Substituting Z = 0 into X - Z = -2 gives X = -2. Substituting X = -2 and Z = 0 into the first equation X + 2Y + 4Z = 12 gives: -2 + 2Y + 0 = 12 2Y = 14 Y = 7

Explanation

According to the BODMAS rule, operations are performed in the order of Division, Multiplication, Addition, and Subtraction.

First, perform the division: (-60) ÷ (-6) = 10

Next, perform the multiplication: (-2) × 9 = -18

Now substitute these values back into the expression: (-9) - 10 + (-18) = -19 - 18 = -37

Explanation

Using the algebraic identity (a - b)² = (a + b)² - 4ab with a = 3x and b = 2y:

(3x - 2y)² = (3x + 2y)² - 4(3x)(2y)

Given that 3x + 2y = 10 and 2xy = 3: (3x - 2y)² = (10)² - 12(2xy) (3x - 2y)² = 100 - 12(3) (3x - 2y)² = 100 - 36 = 64

Taking the positive square root gives: 3x - 2y = √64 = 8

Explanation

Using the inverse proportion formula M₁ × D₁ = M₂ × D₂:

Remaining men M₂ = 100 - 46 = 54

100 × 27 = 54 × D₂

D₂ = (100 × 27) / 54 = 50 days

Explanation

If A's salary is r% more than that of B, then B's salary is less than that of A by [r / (100 + r)] × 100%.

Substituting r = 13: Percentage less = (13 / 113) × 100% = 1300 / 113 % ≈ 11.50%

Explanation

To combine the given ratios A:B = 2:3, B:C = 4:5, and C:D = 10:3 into a single compound ratio A:B:C:D:

First, combine A:B and B:C by making the term for B common (LCM of 3 and 4 is 12): A:B:C = (2 × 4) : (3 × 4) : (5 × 3) = 8 : 12 : 15

Next, combine A:B:C and C:D by making the term for C common (LCM of 15 and 10 is 30): A:B:C:D = (8 × 2) : (12 × 2) : (15 × 2) : (3 × 3) = 16 : 24 : 30 : 9

Sum of all ratio parts = 16 + 24 + 30 + 9 = 79 parts.

Value per ratio part = 6320 / 79 = 80.

Share of A = 16 × 80 = 1280.

Explanation

Marked Price = ₹1,800 Selling Price = ₹234 Total Discount = 1800 − 234 = ₹1,566

The nominal value of the first discount of α% on marked price = (α / 100) × 1800 = 18α. Since the second discount has the same nominal rupee value as the first discount, the total discount is 18α + 18α = 36α.

Equating total discount: 36α = 1566 α = 1566 / 36 = 43.5

Explanation

The product of two numbers is equal to the product of their HCF and LCM:

First Number × Second Number = HCF × LCM

Substitute the given values into the formula: 208 × Second Number = 104 × 1040 Second Number = (104 × 1040) / 208 = 1040 / 2 = 520

The sum of the two numbers is: 208 + 520 = 728

Explanation

From the given proportion a : b :: b : c, the relation between the terms is b² = ac.

Given b = 6 and c = 4a, express a in terms of c as a = c/4. Substituting these values into the proportion equation:

6² = (c/4) × c 36 = c²/4 c² = 144 c = √144 = 12

Explanation

In rhombus ABCD, the diagonals AC and BD intersect at right angles at point O, so ∠BOC = 90°.

In right-angled triangle OBC: ∠CBO = 180° - (90° + ∠ACB) = 180° - (90° + 50°) = 40°

Since AD is parallel to BC, the alternate interior angles formed by transversal BD are equal, which gives ∠ADB = ∠CBO = 40°.

Explanation

Cost Price per dozen = ₹620 Selling Price per dozen = ₹93 × 12 = ₹1116

Profit per dozen = Selling Price - Cost Price = 1116 - 620 = ₹496

Profit Percentage = (Profit / Cost Price) × 100 = (496 / 620) × 100 = 80%

Since the quantity (33 dozens) remains constant for both purchasing and selling, calculating the profit percentage for one dozen gives the overall profit percentage directly.

Explanation

For a sum of money to become n times itself in T years on simple interest, the rate of interest R per year is given by the formula:

R = [(n - 1) × 100] / T

Given n = 5 and T = 20 years:

R = [(5 - 1) × 100] / 20 = (4 × 100) / 20 = 20%

Explanation

Using the BODMAS rule, division and multiplication are evaluated before addition and subtraction:

First, perform the division: (-56) ÷ (-8) = 7

Next, perform the multiplication: (-4) × 6 = -24

Substitute these values back into the expression: (-9) - 7 + (-24)

Finally, perform addition and subtraction from left to right: (-9) - 7 = -16 -16 + (-24) = -40

Explanation

Monthly savings = Monthly salary - (House rent + Bills) Monthly savings = 18000 - (5000 + 2500) = 18000 - 7500 = ₹10500

Since Rupesh spends his entire savings in his birthday month, he saves money for only 11 months in the year.

Total annual savings = 11 × 10500 = ₹115500

Explanation

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

For 4 : 12 :: 2.1 : x: 4 × x = 12 × 2.1 x = (12 × 2.1) / 4 = 6.3

For 9 : 36 :: 8 : y: 9 × y = 36 × 8 y = (36 × 8) / 9 = 32

The ratio of x to y is x : y = 6.3 : 32 = 63 / 320 = 63 : 320.

Explanation

For the given quadratic equation, the sum of the roots is 3/2 and the product of the roots is 5/2. Using the algebraic identity for the sum of squares of roots, alpha² + beta² = (alpha + beta)² - 2(alpha)(beta) = (3/2)² - 2(5/2) = 9/4 - 5 = -11/4.

Explanation

When two trains move in opposite directions, their relative speed is the sum of their individual speeds: Relative speed = 110 + 120 = 230 km/hr

Converting the relative speed to meters per second: Relative speed = 230 × (5/18) = 575/9 m/s

The total distance covered while crossing each other equals the sum of their lengths: Total distance = Relative speed × Time Length of first train + Length of second train = (575/9) × 12 = 2300/3 ≈ 766.67 m

Subtracting the length of the first train gives the length of the second train: Length of second train = 766.67 - 450 = 316.67 m

Explanation

To find the median for grouped data, first determine the cumulative frequency (CF) for each class interval:

0 - 10: frequency = 9, CF = 9 10 - 30: frequency = 19, CF = 28 30 - 60: frequency = 34, CF = 62 60 - 80: frequency = 12, CF = 74 80 - 90: frequency = 6, CF = 80

Total frequency N = 80, so N/2 = 40.

The median class is 30 - 60, as its cumulative frequency (62) is the first to exceed 40.

Using the grouped median formula: Median = L + [(N/2 - CFₚrev) / f] × h

Here, lower limit L = 30, cumulative frequency preceding median class CFₚrev = 28, frequency of median class f = 34, and class width h = 30.

Median = 30 + [(40 - 28) / 34] × 30 Median = 30 + (12 / 34) × 30 Median = 30 + 360 / 34 Median = 30 + 10.5882 = 40.5882...

Rounding to two decimal places gives 40.59.

Explanation

Downstream speed = Speed in still water + Speed of current = 8 + 3 = 11 km/hr Upstream speed = Speed in still water − Speed of current = 8 − 3 = 5 km/hr

Time taken downstream = Distance / Downstream speed = 11 / 11 = 1 hour Time taken upstream = Distance / Upstream speed = 15 / 5 = 3 hours

Total time = 1 + 3 = 4 hours

Explanation

Let length be L and breadth be B. Given B = (2/3)L and Area = 600 m².

Area = L × B = L × (2/3)L = (2/3)L² = 600 m² L² = (600 × 3) / 2 = 900 L = 30 m

Breadth B = (2/3) × 30 = 20 m

Perimeter = 2(L + B) = 2(30 + 20) = 100 m

Cost of fencing = Perimeter × Rate = 100 × ₹200 = ₹20,000

Explanation

Let total work be 36 units (LCM of 12 and 36).

Efficiency of Reema = 36 ÷ 12 = 3 units/day Efficiency of Juhi = 36 ÷ 36 = 1 unit/day

In one 2-day cycle (Reema on Day 1, Juhi on Day 2), total work completed = 3 + 1 = 4 units.

Number of 2-day cycles required = 36 ÷ 4 = 9 cycles.

Total time taken = 9 cycles × 2 days = 18 days.

Explanation

Area of a rectangle = length × width

Substituting the given values: Area = 10 × √5 sq. cm.

Since √5 ≈ 2.23606, Area = 10 × 2.23606 = 22.3606 ≈ 22.36 sq. cm.

Explanation

Total distance travelled in 10 hours is the sum of distances covered in each 1-hour interval, which corresponds to the sum of cubes of the first 10 natural numbers:

Total Distance = Σ i³ (for i = 1 to 10) = [n(n + 1) / 2]²

Substituting n = 10: Total Distance = [10 × 11 / 2]² = 55² = 3025 km

Average speed is defined as total distance divided by total time: Average Speed = Total Distance / Total Time = 3025 / 10 = 302.5 km/h

Explanation

By the linear property of arithmetic mean, if every observation in a dataset is multiplied by a constant k, the mean of the dataset is also multiplied by k.

New Average = Original Average × k

Substituting the given values: New Average = 40 × 2 = 80

Explanation

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

Total time required to type 95 pages = 95 × 20 = 1900 minutes.

Explanation

The ratio A : D can be calculated directly by multiplying the individual consecutive ratios:

A / D = (A / B) × (B / C) × (C / D)

Substituting the given ratios:

A / D = (3 / 5) × (2 / 3) × (3 / 4)

Canceling common factors in the numerator and denominator:

A / D = (3 × 2 × 3) / (5 × 3 × 4) = 3 / 10

Thus, the ratio of the ages of A and D is 3 : 10.

Explanation

The mode for grouped data is calculated using the formula:

Mode = L + [(f₁ - f₀) / (2f₁ - f₀ - f₂)] × h

Here, the modal class is 20-25 because it has the highest frequency of 72.

The values for the formula are: L (lower limit of modal class) = 20 f₁ (frequency of modal class) = 72 f₀ (frequency of preceding class) = 48 f₂ (frequency of succeeding class) = 36 h (class interval width) = 5

Substituting the values into the formula:

Mode = 20 + [(72 - 48) / (2 × 72 - 48 - 36)] × 5 Mode = 20 + [24 / (144 - 84)] × 5 Mode = 20 + (24 / 60) × 5 Mode = 20 + 2 = 22

Explanation

According to the order of operations, multiplication is evaluated before addition and subtraction.

First, simplify the multiplication term: (5/4) × 2.4 = 5 × 0.6 = 3

Convert the fraction 1/2 to decimal form: 1/2 = 0.5

Substitute these values back into the expression: 3.6 + 3 - 0.5 = 6.6 - 0.5 = 6.1

Explanation

The overall average amount spent per member is calculated using the weighted average formula:

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

Substitute the given values:

Total amount spent = (45 × 295) + (31 × 311) + (48 × 233) = 13275 + 9641 + 11184 = ₹34,100

Total number of members = 45 + 31 + 48 = 124

Overall average = 34100 / 124 = ₹275

Explanation

Let Arun's share be A and Mahesh's share be M.

Using the compound interest formula, Amount = P(1 + r/100)ᵗ: A × (1 + 8/100)¹⁷ = M × (1 + 8/100)¹⁸

Dividing both sides by (1 + 8/100)¹⁷: A = M × (1 + 8/100)¹ A = M × (27/25)

Therefore, the ratio of Arun's share to Mahesh's share is A : M = 27 : 25.

Total ratio parts = 27 + 25 = 52 parts

Value of 1 part = 1404 / 52 = ₹27

Mahesh's share = 25 × 27 = ₹675

Explanation

Slant height of the conical tent, l = √(r² + h²) = √(7² + 24²) = √625 = 25 m.

Curved Surface Area (CSA) = π × r × l = (22/7) × 7 × 25 = 550 m².

Total cost of cloth = CSA × Rate per m² = 550 × 40 = ₹22,000.

Explanation

For successive percentage changes of x% and y%, the net percentage change is given by the formula: Net % change = x + y + (x × y) / 100

Here, x = -5 (a loss of 5%) and y = 60 (a gain of 60%).

Substituting the values into the formula: Net % change = -5 + 60 + (-5 × 60) / 100 = 55 - 300 / 100 = 55 - 3 = 52%

Explanation

When two runners move in opposite directions along a circular track, their relative speed is the sum of their individual speeds: Relative speed = 6 m/s + 4 m/s = 10 m/s

The time taken for them to meet for the first time is given by: Time = Total Distance / Relative Speed = 400 / 10 = 40 seconds

Explanation

Since points P, M, N, and Q lie on the transversal line PQ in order, ∠PMB and ∠BMN form a linear pair at point M. ∠PMB + ∠BMN = 180° 5x + (3x + 2y) = 180° 8x + 2y = 180° 4x + y = 90° --- (Equation 1)

Since AB is parallel to CD, the corresponding angles formed by the transversal PQ are equal. Therefore, ∠BMN and ∠DNQ are equal. ∠BMN = ∠DNQ 3x + 2y = x + 3y 2x = y --- (Equation 2)

Substituting y = 2x into Equation 1: 4x + 2x = 90° 6x = 90° x = 15°

This gives y = 2(15°) = 30°.

Calculating the measure of ∠BMN: ∠BMN = 3x + 2y = 3(15°) + 2(30°) = 45° + 60° = 105°.

Explanation

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

When a sum becomes 5 times itself, the Amount = 5P, which means Interest (SI) = Amount - Principal = 5P - P = 4P.

Substituting SI = 4P and T = 11 into the formula: 4P = (P × R × 11) / 100 R = (4 × 100) / 11 R = 400 / 11 = 36 4/11 %

Explanation

Subtracting the second equation (x + 2y + z = 7) from the first equation (x + 2y + 4z = 14) eliminates x and y: 3z = 7, which gives z = 7/3.

Substituting z = 7/3 into the second equation yields: x + 2y + 7/3 = 7 x + 2y = 14/3

Substituting z = 7/3 into the third equation (2x + y + 2z = 2) yields: 2x + y + 14/3 = 2 2x + y = -8/3

Multiplying this by 2 gives: 4x + 2y = -16/3

Subtracting x + 2y = 14/3 from 4x + 2y = -16/3: 3x = -30/3 = -10 x = -10/3

Substituting x = -10/3 back into x + 2y = 14/3: -10/3 + 2y = 14/3 2y = 24/3 = 8 y = 4

The solution is x = -10/3, y = 4, and z = 7/3.

Explanation

Let the total capacity of the cistern be 32 units (LCM of 4 and 32).

Filling rate of first tap = +8 units/hour Filling rate of second tap = +1 unit/hour Emptying rate of third tap = -8 units/hour

Combined filling rate = 8 + 1 - 8 = 1 unit/hour

Volume to be filled = ¼ × 32 = 8 units

Time taken = 8 units / 1 unit per hour = 8 hours

Explanation

When a three-dimensional object is melted and reshaped, its volume remains unchanged.

Volume of cylinder = Volume of cone πr²h_cylinder = ⅓πr²h_cone

Since both shapes share the same base radius r, dividing both sides by πr² gives: h_cone = 3 × h_cylinder h_cone = 3 × 10 = 30 cm

Explanation

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

Therefore, time required to type 82 pages = 82 × 20 = 1640 minutes.

Explanation

Overall Average = Total Amount Spent / Total Number of Members

Total number of members = 52 + 22 + 36 = 110

Total amount spent = (52 × 420) + (22 × 437) + (36 × 101) = 21,840 + 9,614 + 3,636 = ₹35,090

Overall average amount spent per member = 35,090 / 110 = ₹319

Explanation

Convert the given distances into kilometers: 20,000 m = 20 km, 44,000 m = 44 km, 15,000 m = 15 km, and 22,000 m = 22 km.

Let upstream speed be u km/h and downstream speed be d km/h.

Setting up the time equations from the problem: 20/u + 44/d = 8 15/u + 22/d = 5

Multiplying the second equation by 2 gives: 30/u + 44/d = 10

Subtracting the first equation from this result: 10/u = 2 ⇒ u = 5 km/h

Substituting u = 5 into the first equation: 20/5 + 44/d = 8 ⇒ 4 + 44/d = 8 ⇒ d = 11 km/h

Speed of boat in still water = (u + d) / 2 = (5 + 11) / 2 = 8 km/h

Converting 8 km/h into m/s: 8 × (5/18) = 40/18 = 20/9 = 2 2/9 m/s

Explanation

In a rhombus, the diagonals bisect each other at right angles, forming four congruent right-angled triangles with sides equal to half the diagonals and the hypotenuse equal to the side of the rhombus.

Given side a = 17 cm and diagonal d₁ = 30 cm, half of d₁ is 15 cm.

Using the Pythagorean theorem: (d₂/2) = √(a² - (d₁/2)²) = √(17² - 15²) = √(289 - 225) = √64 = 8 cm

Therefore, second diagonal d₂ = 2 × 8 = 16 cm.

Area of the rhombus = ½ × d₁ × d₂ = ½ × 30 × 16 = 240 cm²

Explanation

Let CP be the cost price per article and MP be the marked price per article.

The trader gives a total of 104 + 16 = 120 articles to the customer, so the total cost price for the trader is 120 × CP.

The customer pays for 104 articles at a 10% discount on the marked price. Total Selling Price = 104 × 0.90 × MP = 93.6 × MP

Given an overall profit of 17%: Total Selling Price = Total Cost Price × 1.17 93.6 × MP = 120 × CP × 1.17 93.6 × MP = 140.4 × CP MP / CP = 140.4 / 93.6 = 1.5

Markup percentage = (1.5 - 1) × 100 = 50%

Explanation

Sum of all 11 results = 11 × 61 = 671 Sum of first 6 results = 6 × 64 = 384 Sum of last 6 results = 6 × 61 = 366

Since the 6th result is included in both the first 6 and last 6 groups, it is counted twice: 6th result = (Sum of first 6 results) + (Sum of last 6 results) - (Sum of all 11 results) 6th result = 384 + 366 - 671 = 750 - 671 = 79

Using deviation method: Deviation of first 6 results = 6 × (64 - 61) = +18 Deviation of last 6 results = 6 × (61 - 61) = 0 6th result = Base average + Net deviation = 61 + 18 = 79

Explanation

Average = Sum of values / Total number of values

Substituting the given values into the formula: Average = 540 / 12 = 45

Explanation

The empirical relationship between mean, median, and mode for a moderately skewed frequency distribution is given by:

Mode = 3 × Median - 2 × Mean

Substituting the given values into the formula:

50 = 3 × Median - 2 × 55 50 = 3 × Median - 110 3 × Median = 50 + 110 3 × Median = 160 Median = 160 / 3 = 53.33