RRB NTPC CBT 1 2026 Mathematics Questions with Solutions

Explanation

Actual distance = Map distance × Scale factor

Actual distance = 4 cm × 829,483 = 3,317,932 cm

To convert centimeters to kilometers, divide by 100,000 (since 1 km = 100,000 cm):

Actual distance = 3,317,932 / 100,000 = 33.17932 km

Rounding off to two decimal places gives 33.18 km.

Explanation

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

Mode = 3 × Median - 2 × Mean

Substituting the given values into the formula:

48 = 3 × Median - 2 × 72 48 = 3 × Median - 144 3 × Median = 48 + 144 3 × Median = 192 Median = 192 ÷ 3 = 64

Explanation

Let total work be 144 units (LCM of 36 and 48).

Work rate of A + B + C = 144 / 36 = 4 units/day Work rate of B + C = 144 / 48 = 3 units/day

Work rate of A = 4 - 3 = 1 unit/day

Since B is 75% more efficient than A: Work rate of B = 1 + (0.75 × 1) = 1.75 = 7/4 units/day

Work completed by A, B, and C in 15 days = 15 × 4 = 60 units Remaining work = 144 - 60 = 84 units

Days required by B to finish remaining work = 84 / (7/4) = 84 × 4 / 7 = 48 days

Explanation

Let a = 28x and b = 28y, where x and y are positive integers such that HCF(x, y) = 1.

Given a + b = 868, substituting the expressions for a and b yields: 28(x + y) = 868 x + y = 31

Since 31 is a prime number, every positive integer x in the range 1 ≤ x ≤ 30 is coprime to y = 31 - x, because HCF(x, 31 - x) = HCF(x, 31) = 1.

Thus, x can take any integral value from 1 to 30, yielding 30 distinct ordered pairs (x, y) and correspondingly 30 ordered pairs (a, b).

Explanation

The volume of a cylinder is given by the formula V = πr²h, where r is the radius and h is the height.

Substituting the given values: 1584 = (22/7) × r² × 14 1584 = 44 × r² r² = 36 r = 6 m

The diameter of the tank is twice the radius: Diameter = 2 × r = 2 × 6 = 12 m

Explanation

The weighted average formula for combined groups is:

Overall Average = (n₁a₁ + n₂a₂ + n₃a₃) ÷ (n₁ + n₂ + n₃)

Substitute the given values: Total expenditure = (46 × 496) + (15 × 106) + (11 × 406) = 22816 + 1590 + 4466 = ₹28872

Total members = 46 + 15 + 11 = 72

Overall average spent per member = 28872 ÷ 72 = 401

Explanation

Using the standard algebraic identity: x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - (xy + yz + zx))

From the given values: xyz = 150 / 5 = 30

Substituting the values into the identity: x³ + y³ + z³ - 3(30) = 10 × (38 - 31) x³ + y³ + z³ - 90 = 10 × 7 x³ + y³ + z³ - 90 = 70 x³ + y³ + z³ = 160

Therefore: 2(x³ + y³ + z³) = 2 × 160 = 320

Explanation

Let the total quantity of goods be 100 units.

First portion = 35 units at +22% profit = +7.70 units Remaining quantity = 100 - 35 = 65 units

Second portion = 40% of 65 = 26 units at +37% profit = +9.62 units

Third portion = 65 - 26 = 39 units at -38% loss = -14.82 units

Overall profit x = 7.70 + 9.62 - 14.82 = 2.5

Explanation

Under compound interest compounded annually, if an amount becomes k times in T years, it becomes kⁿ times itself in n × T years.

Given that the amount doubles (2¹ times) in T = 8 years: To become 8 times (2³ times) itself, n = 3.

Time required = n × T = 3 × 8 = 24 years.

Explanation

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

Given Principal P = ₹8,400, Simple Interest SI = ₹840, and Rate R = 4% per annum:

840 = (8400 × 4 × T) / 100 840 = 336 × T T = 840 / 336 = 2.5 years

Explanation

Using exponent rules, convert all terms to the base (4/17):

(17/4)⁹ = (4/17)⁻⁹ (17/4)¹⁵ = (4/17)⁻¹⁵

Multiply the powers on the left-hand side by adding exponents:

(4/17)⁻⁹ × (4/17)⁸ × (4/17)⁻¹⁵ = (4/17)⁻⁹ ⁺ ⁸ ⁻ ¹⁵ = (4/17)⁻¹⁶

Equate the exponents on both sides of the equation:

5m + 15 = -16 5m = -31 m = 31/5

Explanation

The area of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) is given by the formula:

Area = ½ | x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂) |

Substituting the coordinates P(2, 5), Q(8, 1), and R(4, 9):

Area = ½ | 2(1 - 9) + 8(9 - 5) + 4(5 - 1) | Area = ½ | 2(-8) + 8(4) + 4(4) | Area = ½ | -16 + 32 + 16 | Area = ½ | 32 | = 16 square kilometers.

Explanation

In the right-angled triangle formed by the observer, the base of the flagpole, and the top of the flagpole, the relationship between the height of the flagpole (h), the distance from the observer (d), and the angle of elevation (θ) is given by the trigonometric ratio:

tan(θ) = height / distance

Substituting θ = 30° and d = 30 m into the formula:

tan(30°) = h / 30

Since tan(30°) = 1 / √3:

1 / √3 = h / 30

h = 30 / √3 = (30 × √3) / 3 = 10√3 m

Explanation

Average = (Sum of values) / (Total count) = (7 + 9 + 11) / 3 = 27 / 3 = 9.

Alternatively, since 7, 9, and 11 form an arithmetic progression with an odd number of terms, the average is directly given by the middle term, which is 9.

Explanation

First, use the total frequency to form the first equation: 6 + 19 + x + 5 + y = 59 30 + x + y = 59 x + y = 29

Next, calculate the sum of products of frequencies and class marks mᵢ (10, 30, 50, 70, 90): Σ(fᵢ × mᵢ) = (6 × 10) + (19 × 30) + (x × 50) + (5 × 70) + (y × 90) Σ(fᵢ × mᵢ) = 60 + 570 + 50x + 350 + 90y = 980 + 50x + 90y

Using the mean formula, Mean = Σ(fᵢ × mᵢ) / Total Frequency: 50 = (980 + 50x + 90y) / 59 2950 = 980 + 50x + 90y 50x + 90y = 1970 5x + 9y = 197

Subtracting 5 × (x + y = 29) from 5x + 9y = 197 gives: 4y = 197 - 145 4y = 52 y = 13

Substituting y = 13 into x + y = 29 gives x = 16.

Now, calculate the required expression: 7x + 6y = 7(16) + 6(13) = 112 + 78 = 190

Explanation

Let the initial salaries of A and B be 4x and 5x, respectively.

A's new salary = 4x × (1 + 20/100) = 4x × 1.2 = 4.8x B's new salary = 5x × (1 - 10/100) = 5x × 0.9 = 4.5x

New ratio of their salaries = 4.8x : 4.5x = 48 : 45 = 16 : 15.

Explanation

Downstream speed of the boat = Speed in still water + Speed of stream = 12 + 3 = 15 km/hr

Time taken = Distance / Downstream speed = 60 / 15 = 4 hours

Explanation

In a cyclic quadrilateral, opposite angles are supplementary, so ∠P + ∠R = 180° and ∠Q + ∠S = 180°. Since PS is parallel to QR, consecutive interior angles on transversal PQ sum to 180°, giving ∠P + ∠Q = 180°.

Using ∠P = 85°: ∠Q = 180° - 85° = 95° ∠R = 180° - 85° = 95° ∠S = 180° - 95° = 85°

Explanation

Let x be the initial speed of the stream in km/h.

The formula for the total time T taken for a round trip of distance d with boat speed v and stream speed x is: T = d / (v + x) + d / (v - x) = 2dv / (v² - x²)

Given d = 40 km and boat speed v = 7 km/h: T₁ = 2 × 40 × 7 / (7² - x²) = 560 / (49 - x²) hours

For x = 3 km/h: T₁ = 560 / (49 - 9) = 14 hours

When the stream speed increases by 2 km/h to 5 km/h, the downstream speed is 12 km/h and upstream speed is 2 km/h: T₂ = 40 / 12 + 40 / 2 = 10/3 + 20 = 70/3 hours

The time difference is: T₂ - T₁ = 70/3 - 14 = 28/3 hours = (28/3) × 60 = 560 minutes

This satisfies the given condition.

Explanation

Simplifying each term by removing the decimal point:

1.2 ÷ 0.3 = 12 ÷ 3 = 4

2.4 ÷ 0.6 = 24 ÷ 6 = 4

Adding the two terms gives:

4 + 4 = 8

Explanation

Cost Price of 1 dozen articles = ₹400 Selling Price of 1 dozen articles = ₹58 × 12 = ₹696

Profit per dozen = ₹696 - ₹400 = ₹296

Profit percentage = (Profit / Cost Price) × 100 = (296 / 400) × 100 = 74%

Explanation

The profit is divided among A, B, and C in the ratio of their investments, which is 3 : 5 : 4.

Let the profit shares of A, B, and C be 3x, 5x, and 4x respectively. Total profit = 3x + 5x + 4x = 12x

When B gives ₹15,000 to A, A's new share becomes equal to C's share: 3x + 15,000 = 4x 4x - 3x = 15,000 x = 15,000

Total profit = 12 × 15,000 = ₹1,80,000

Explanation

Let Cost Price (CP) = 100. Since the Marked Price (MP) is set 100% above CP: MP = 100 + (100% of 100) = 200

To make a gain of 2%, the Selling Price (SP) must be: SP = CP + 2% of CP = 102

Discount required = MP − SP = 200 − 102 = 98

Discount Percentage = (Discount / MP) × 100 = (98 / 200) × 100 = 49%

Explanation

Let the ratio of trained to untrained teachers be 5:3 and their individual salaries be in the ratio 4:1.

Total salary spent on trained teachers = 5 × 4 = 20 units Total salary spent on untrained teachers = 3 × 1 = 3 units Total salary spent on all teachers = 20 + 3 = 23 units

Given that total monthly salary = ₹4,60,000: 1 unit = 4,60,000 / 23 = ₹20,000

Total salary paid to untrained teachers = 3 units = 3 × 20,000 = ₹60,000

Explanation

In any quadrilateral ABCD, the angle formed by the bisectors of two adjacent angles (A and B) is equal to half the sum of the remaining two angles (C and D):

∠AOB = ½ (∠C + ∠D)

Substituting the given value of ∠AOB = 41°:

41° = ½ (∠C + ∠D) ∠C + ∠D = 2 × 41° = 82°

Explanation

For a principal P to become 5 times itself (Amount A = 5P), the Simple Interest (SI) earned is A - P = 4P.

Using the Simple Interest formula: SI = (P × R × T) / 100

Substituting SI = 4P and T = 18: 4P = (P × R × 18) / 100 4 = (18 × R) / 100 R = 400 / 18 = 200 / 9 = 22.22%

Explanation

The ratio of effective investment (capital × time in months) for X, Y, and Z is:

X = (6 × 6) + (3 × 6) = 36 + 18 = 54 unit-months Y = 3 × 12 = 36 unit-months Z = 5 × 12 = 60 unit-months

Profit ratio X : Y : Z = 54 : 36 : 60 = 9 : 6 : 10

Total profit units = 9 + 6 + 10 = 25 units Difference between X's and Y's profit share = 9 - 6 = 3 units

Difference in profit = (3 / 25) × ₹69,000 = ₹8,280

Explanation

The average of the first n natural numbers is given by the formula (n + 1) / 2. For n = 64, substituting the value into the formula gives (64 + 1) / 2 = 65 / 2 = 32.5.

Explanation

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

Substituting a = 9 and b = 25: Mean proportional = √(9 × 25) = √225 = 15.

Explanation

For grouped data, the median is calculated using the formula:

Median = L + [((N / 2) - CF) / f] × h

Here, total frequency N = 100, so N / 2 = 50.

The cumulative frequencies for the intervals are: 0-10: 5 10-20: 14 20-30: 28 30-40: 50 40-50: 80 50-60: 100

Since the cumulative frequency reaches 50 at the class interval 30-40, the median class is 30-40.

Lower limit of median class (L) = 30 Preceding cumulative frequency (CF) = 28 Frequency of median class (f) = 22 Class width (h) = 10

Substituting the values into the formula: Median = 30 + [(50 - 28) / 22] × 10 Median = 30 + (22 / 22) × 10 Median = 30 + 10 = 40.0

Explanation

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

Given values: n₁ = 32, A₁ = ₹266 n₂ = 52, A₂ = ₹470 n₃ = 73, A₃ = ₹168

Total expenditure = (32 × 266) + (52 × 470) + (73 × 168) = 8512 + 24440 + 12264 = ₹45,216

Total members = 32 + 52 + 73 = 157

Overall average amount spent per member = 45216 / 157 = ₹288

Explanation

The mean of grouped data is calculated using the formula Mean = Σ(f × x) / Σf, where f represents frequency and x represents the class midpoint.

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

Σf = 42 + 19 + 23 + 26 + X = 110 + X

Σ(f × x) = (42 × 15) + (19 × 25) + (23 × 35) + (26 × 45) + (X × 55) Σ(f × x) = 630 + 475 + 805 + 1170 + 55X = 3080 + 55X

Substituting the given mean of 40: 40 = (3080 + 55X) / (110 + X) 40 × (110 + X) = 3080 + 55X 4400 + 40X = 3080 + 55X 15X = 1320 X = 88

Explanation

Expressing cot θ as cos θ / sin θ:

Numerator: cot θ - cos θ = (cos θ / sin θ) - cos θ = cos θ × (1 / sin θ - 1) = cos θ × (csc θ - 1)

Denominator: cot θ + cos θ = (cos θ / sin θ) + cos θ = cos θ × (1 / sin θ + 1) = cos θ × (csc θ + 1)

Dividing the numerator by the denominator and cancelling the common factor cos θ: (cot θ - cos θ) / (cot θ + cos θ) = [cos θ × (csc θ - 1)] / [cos θ × (csc θ + 1)] = (csc θ - 1) / (csc θ + 1)

Explanation

Let the required number be x.

According to the question: 20% of x − 20% of 620 = 190

20% of 620 = (20 / 100) × 620 = 124

0.20 × x − 124 = 190 0.20 × x = 190 + 124 0.20 × x = 314 x = 314 / 0.20 = 1570

Explanation

According to the order of operations (BODMAS):

Perform division and multiplication first: (-56) ÷ (-14) = 4 (-3) × 6 = -18

Now perform subtraction and addition from left to right: (-9) - 4 + (-18) = -13 - 18 = -31

Explanation

Average speed = Total distance / Total time

Total distance = (20 × 90) + (70 × 60) = 1800 + 4200 = 6000 m Total time = 90 + 60 = 150 min

Average speed = 6000 / 150 = 40 m/min

Explanation

Cost price per article = 150 / 12 = ₹12.5

Selling price per article = ₹78

Profit per article = 78 - 12.5 = ₹65.5

Profit percentage = (Profit / Cost Price) × 100 = (65.5 / 12.5) × 100 = 524%

Explanation

Let the son's present age be x years. Then Vanraj's present age is 3x years, and the daughter's present age is x - 3 years.

Sum of their present ages = 3x + x + (x - 3) = 5x - 3 years.

Since there are 3 people, the sum of their ages 3 years ago was 9 years less than the sum of their present ages: (5x - 3) - 9 = 93 5x - 12 = 93 5x = 105 x = 21

Vanraj's present age = 3 × 21 = 63 years.

Explanation

Circumference of the base, 2πr = 66 cm r = (66 × 7) / (2 × 22) = 10.5 cm

The maximum possible height of the cone corresponds to the breadth of the sheet, h = 30 cm.

Total volume of the cone, V = (1/3)πr²h V = (1/3) × (22/7) × (10.5)² × 30 = 3465 cm³

The volume of the small cone formed at the stapled tip (height = 2 cm from vertex) is proportional to the cube of the height ratio: v = (hₛtaple / h)³ × V v = (2 / 30)³ × 3465 = (1 / 15)³ × 3465 = 3465 / 3375 ≈ 1.03 cm³

Usable volume of the packet = 3465 − 1.03 = 3463.97 ≈ 3464 cm³

Explanation

A quadrilateral ABCD is a parallelogram if and only if any of these standard geometric conditions hold. Satisfying opposite sides being equal, each diagonal dividing the figure into two congruent triangles, each pair of opposite angles being equal, or diagonals bisecting each other directly implies that both pairs of opposite sides are parallel. Each of these four conditions serves as a defining characteristic of a parallelogram.

Explanation

For a number to be completely divisible by 6, it must be divisible by both 2 and 3.

Since the unit digit of 56339A8 is 8 (an even number), it is always divisible by 2 regardless of the value of A.

For divisibility by 3, the sum of all digits must be a multiple of 3. Sum of digits = 5 + 6 + 3 + 3 + 9 + A + 8 = 34 + A

The smallest non-negative single-digit value for A that makes (34 + A) divisible by 3 is A = 2, giving a sum of 36.

Explanation

Expanding the left-hand side gives (x² + 1)² - x⁴ = (x⁴ + 2x² + 1) - x⁴ = 2x² + 1. Equating this to the right-hand side yields 2x² + 1 = 2x + 1, which simplifies to 2x² - 2x = 0. Since the highest degree of the variable x in this simplified equation is 2, it represents a standard quadratic equation.

Explanation

Let the cost price of 1 gm be ₹1.

At a professed loss of 60%, the selling price charged for 1000 gm is 1000 × (1 - 0.60) = ₹400.

If the shopkeeper actually dispenses x gm instead of 1000 gm, his actual cost price is ₹x. Since his overall gain is 70%, the relationship between cost price and selling price is given by:

Selling Price = Cost Price × (1 + Gain%) 400 = x × (1 + 0.70) 1.7 x = 400 x = 4000 / 17 ≈ 235.29 gm

Rounding off to one decimal place gives 235.3 gm.

Explanation

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

Total time required to type 87 pages = 87 × 20 = 1740 minutes.

Explanation

Let N be the total number of registered voters.

Votes polled = 80% of N = 0.80N Valid votes = 85% of votes polled = 0.85 × 0.80N = 0.68N

The winning candidate received 75% of valid votes, which means the losing candidate received 25% of valid votes. The winning margin is therefore 75% - 25% = 50% of valid votes.

Margin = 0.50 × 0.68N = 0.34N

Given that the winning margin is 289 votes: 0.34N = 289 N = 289 / 0.34 = 850

Explanation

Area of a sector = (θ / 360°) × πr²

Substituting the given values, Area = 385 cm², r = 21 cm, and π = 22 / 7:

385 = (θ / 360°) × (22 / 7) × 21 × 21 385 = (θ / 360°) × 22 × 3 × 21 385 = (θ / 360°) × 1386

θ = (385 × 360°) / 1386 θ = (5 / 18) × 360° θ = 100°

Explanation

Under compound interest, if an amount doubles in T years, it becomes 2ⁿ times itself in n × T years.

Here, the doubling period T = 7 years. Since 16 = 2⁴, the value of n is 4.

Time required = 4 × 7 = 28 years.

Explanation

Cubing both x and y gives: x³ = 31 + 14√3 y³ = 31 - 14√3

Sum of x³ and y³: x³ + y³ = (31 + 14√3) + (31 - 14√3) = 62

Product of x³ and y³ using the identity (a + b)(a - b) = a² - b²: x³ y³ = (31 + 14√3)(31 - 14√3) = 31² - (14√3)² = 961 - 588 = 373

Substituting these values into the given expression: x³ + y³ + 3x³y³ = 62 + 3 × 373 = 62 + 1119 = 1181

Explanation

For inversely proportional variables, the product of x and y remains constant: x₁y₁ = x₂y₂.

Substituting the given values: 28 × 30 = x × 20 840 = 20x x = 840 / 20 = 42

Explanation

Using the difference of cubes identity, a³ - b³ = (a - b)(a² + ab + b²):

8x³ - 27y³ = (2x)³ - (3y)³ = (2x - 3y)(4x² + 9y² + 6xy)

Comparing this with (Ax + By)(Cx² - Dy² + Exy): A = 2 B = -3 C = 4 -D = 9, which gives D = -9 E = 6

Calculating the required sum: A + B + C + D + E = 2 + (-3) + 4 + (-9) + 6 = 0