RRB NTPC CBT 1 2026 Mathematics Questions with Solutions

Explanation

Let cost price be C and actual selling price be S.

Since selling at 4/11 of S results in a 20% loss: (4/11) × S = 0.80 × C (4/11) × S = (4/5) × C S = (11/5) × C = 2.2 × C

Selling at 71% of actual selling price gives: New Selling Price = 0.71 × S = 0.71 × 2.2 × C = 1.562 × C

Profit percentage = [(1.562 × C - C) / C] × 100 = 56.2%

Explanation

The time interval after which all three bells ring together is given by the Least Common Multiple (LCM) of their individual ringing intervals: LCM(120, 180, 240).

Prime factorizations: 120 = 2³ × 3 × 5 180 = 2² × 3² × 5 240 = 2⁴ × 3 × 5

LCM(120, 180, 240) = 2⁴ × 3² × 5 = 720 minutes.

Converting minutes into hours: 720 ÷ 60 = 12 hours.

Adding 12 hours to the initial time of 9:00 a.m. yields 9:00 p.m.

Explanation

Using the simple interest formula, SI = (P × R × T) / 100, where P = ₹7,200, R = 4%, and SI = ₹360.

Rearranging for time T: T = (SI × 100) / (P × R) T = (360 × 100) / (7200 × 4) T = 36000 / 28800 = 5 / 4 = 1.25 years

Explanation

From the given equality, subtracting pairwise yields: p - m = (m - n) / (mn) m - n = (n - p) / (np) n - p = (p - m) / (pm)

Multiplying these three equations together: (p - m)(m - n)(n - p) = [(p - m)(m - n)(n - p)] / (p m n)²

Since p, m, and n are all distinct (p ≠ m ≠ n), the product (p - m)(m - n)(n - p) is non-zero. Dividing both sides by this product gives: 1 = 1 / (p m n)² (p m n)² = 1 p × m × n = ±1

Explanation

Let the present age of the younger brother be x years.

Vanraj's present age = 2x Other brother's present age = x - 3

Sum of their present ages = 2x + x + (x - 3) = 4x - 3

Since there are 3 brothers, the sum of their ages 3 years ago was less by 3 × 3 = 9 years: (4x - 3) - 9 = 20 4x - 12 = 20 4x = 32 x = 8

Vanraj's present age = 2x = 2 × 8 = 16 years.

Explanation

To find the combined ratio A : B : C : D: A : B = 2 : 3 B : C = 4 : 5 C : D = 10 : 3

First, combining A : B and B : C gives A : B : C = (2 × 4) : (3 × 4) : (3 × 5) = 8 : 12 : 15.

Next, combining A : B : C with C : D = 10 : 3 (making the C-term equal to LCM(15, 10) = 30): A : B : C : D = (8 × 2) : (12 × 2) : (15 × 2) : (3 × 3) = 16 : 24 : 30 : 9.

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

Value of 1 part = 6241 / 79 = 79.

Share of A = 16 × 79 = 1264.

Explanation

Given equation: 3(sin²θ + cosec²θ) = 10

Let sin²θ = x. Then cosec²θ = 1/x. 3(x + 1/x) = 10 3x² - 10x + 3 = 0 (3x - 1)(x - 3) = 0

Since sin²θ ≤ 1, x = 1/3. Thus, sin²θ = 1/3 and cosec²θ = 3.

Using the identity cot²θ = cosec²θ - 1: cot²θ = 3 - 1 = 2

Now, evaluate the required expression: cot⁶θ + cosec⁶θ = (cot²θ)³ + (cosec²θ)³ = 2³ + 3³ = 8 + 27 = 35

Explanation

Let the present ages of A and B be 5x and 8x years respectively.

According to the given condition, after 6 years: (5x + 6) / (8x + 6) = 3 / 4

Cross-multiplying to solve for x: 4 × (5x + 6) = 3 × (8x + 6) 20x + 24 = 24x + 18 4x = 6 x = 1.5

A's present age = 5x = 5 × 1.5 = 7.5 years.

Explanation

The relationship between mean, median, and mode is given by the empirical formula:

Mode = 3 × Median - 2 × Mean

Substitute the given values into the formula: 24.7 = 3 × Median - 2 × 17.9 24.7 = 3 × Median - 35.8 3 × Median = 24.7 + 35.8 3 × Median = 60.5 Median = 60.5 ÷ 3 = 20.166... ≈ 20.2

Explanation

Relative speed between A and B moving in the same direction = 7 - 5 = 2 m/s.

Time taken for A and B to meet for the first time = Track length / Relative speed = 840 / 2 = 420 seconds.

Total distance covered by A = Speed × Time = 7 × 420 = 2940 m.

Rounds run by A = 2940 / 840 = 3.5 rounds, which corresponds to 3 complete rounds plus an additional distance of 420 m from the starting point in the clockwise direction.

Explanation

Article 243R(2)(a) of the Indian Constitution, inserted by the 74th Constitutional Amendment Act, 1992, empowers the State Legislature to provide for the representation in a municipality of persons having special knowledge or experience in municipal administration. The proviso to this clause explicitly specifies that such nominated or co-opted members shall not have the right to vote in the meetings of the municipality.

Explanation

Globalisation refers to the process of increasing interconnectedness and integration among countries worldwide through the free flow of goods, services, capital, technology, and labor across international borders. In the contemporary period, it facilitates cross-border trade and investment, removes tariff and non-tariff barriers, and integrates national economies into a unified global market.

Explanation

Overall average amount spent per member = Total amount spent / Total number of members

Total number of members = 30 + 21 + 40 = 91

Total amount spent by each group: Group A = 30 × 378 = ₹11340 Group B = 21 × 355 = ₹7455 Group C = 40 × 224 = ₹8960

Total expenditure = 11340 + 7455 + 8960 = ₹27755

Overall average = 27755 / 91 = ₹305

Explanation

Let M be the maximum marks in the examination.

According to the question, 19% of M - 13 = 424 19% of M = 424 + 13 = 437

M = (437 × 100) / 19 = 2300

Percentage of marks scored by his friend = (Friend's marks / M) × 100 Percentage = (529 / 2300) × 100 = 529 / 23 = 23%

Explanation

In a cyclic quadrilateral, the exterior angle formed by producing any side is equal to the interior opposite angle.

Here, side PQ of the cyclic quadrilateral PQRS is extended to point T, making ∠RQT the exterior angle at vertex Q. The interior opposite angle to ∠RQT is ∠PSR.

Therefore, ∠PSR = ∠RQT = 58°.

Explanation

Translate the given percentage condition into an algebraic equation: 10% of A + 20% of B = ¾ (20% of A + 10% of B)

0.10A + 0.20B = ¾ (0.20A + 0.10B)

Multiply both sides by 100 to clear the decimals: 10A + 20B = ¾ (20A + 10B)

Multiply both sides by 4: 4(10A + 20B) = 3(20A + 10B) 40A + 80B = 60A + 30B

Simplify and group like terms: 80B - 30B = 60A - 40A 50B = 20A

Solving for the ratio of A to B: A / B = 50 / 20 = 5 / 2

Explanation

Total work = LCM(10, 15, 25) = 150 units.

Efficiencies per day: P = 150 / 10 = 15 units/day Q = 150 / 15 = 10 units/day R = 150 / 25 = 6 units/day

Work done on an odd day (P + Q) = 15 + 10 = 25 units Work done on an even day (P + R) = 15 + 6 = 21 units

In a 2-day cycle, work completed = 25 + 21 = 46 units.

In 3 cycles (6 days), work completed = 3 × 46 = 138 units. Remaining work = 150 - 138 = 12 units.

On Day 7 (odd day), P and Q work together with an efficiency of 25 units/day. Time required for remaining work = 12/25 days.

Total time = 6 + 12/25 = 6 12/25 days.

Explanation

According to the BODMAS rule, operations are evaluated in the order of Division and Multiplication (left to right), followed by Addition and Subtraction (left to right).

Evaluate division and multiplication first: 20 ÷ 4 = 5 2 × 3 = 6

Substitute these results into the expression: 16 + 5 - 6

Perform addition and subtraction: 16 + 5 = 21 21 - 6 = 15

Explanation

When two objects move in opposite directions, their relative speed is the sum of their individual speeds: Relative speed = 90 km/hr + 70 km/hr = 160 km/hr

Converting relative speed into m/s: 160 × 5/18 = 400/9 m/s

The total distance covered while crossing each other equals the sum of the lengths of both trains: Total distance = Relative speed × Time Length₁ + Length₂ = (400/9) × 27 = 1200 m

Given the length of one train is 480 m: Length of the other train = 1200 − 480 = 720 m

Explanation

Let the total capacity of the cistern be 1 unit.

Filling rate of the first tap = ½ unit per hour Filling rate of the second tap = 1/57 unit per hour Emptying rate of the third tap = 1/57 unit per hour

When all three taps are opened together, the net filling rate per hour is: Net rate = ½ + 1/57 - 1/57 = ½ unit per hour

To fill half of the cistern (½ unit): Time required = (Volume to fill) / (Net rate) = (½) / (½) = 1 hour

Explanation

Profit is shared in the ratio of investments.

Ratio of investment of A to B = 3 : 2 Total ratio parts = 3 + 2 = 5

B's share = (2 / 5) × 25,000 = ₹10,000

Explanation

For a cone inscribed in a sphere, the radius R of the circumscribing sphere is given by the formula: R = (r² + h²) / (2h)

Substituting r = 3 cm and h = 3√2 cm: R = (3² + (3√2)²) / (2 × 3√2) = (9 + 18) / (6√2) = 27 / (6√2) = 9 / (2√2) cm

Squaring R gives: R² = 81 / 8

Using the formula for the surface area of a sphere: Surface Area = 4πR² = 4 × (22 / 7) × (81 / 8) = (11 × 81) / 7 = 891 / 7 = 127 2/7 cm²

Explanation

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

Total time required to type 84 pages = 84 × 20 = 1680 minutes.

Explanation

By the Remainder Theorem, dividing f(x) by (x - r) leaves a remainder equal to f(r). Since the leading coefficient is 1, the polynomial is of the form f(x) = x² + bx + c.

Using the given conditions: f(2) = 2² + 2b + c = 5 → 2b + c = 1 f(4) = 4² + 4b + c = -3 → 4b + c = -19

Subtracting the first equation from the second: (4b + c) - (2b + c) = -19 - 1 → 2b = -20 → b = -10

Substituting b = -10 into 2b + c = 1: 2(-10) + c = 1 → c = 21

The resulting polynomial is f(x) = x² - 10x + 21, which sets the equation f(x) = 0 as x² - 10x + 21 = 0.

Explanation

Let N be the total number of registered voters.

Votes polled = 80% of N = 0.80N Valid votes = 97% of 0.80N = 0.776N

The winning candidate obtained 52% of the valid votes, while the losing candidate received 48% of the valid votes. The winning margin is the difference between their shares, which is 52% − 48% = 4% of the valid votes.

4% of 0.776N = 291 0.04 × 0.776N = 291 0.03104N = 291 N = 291 / 0.03104 = 9375

Explanation

Using the standard order of operations (BODMAS):

First, perform division and multiplication from left to right: 20 ÷ 4 = 5 2 × 4 = 8

Then, perform addition and subtraction from left to right: 16 + 5 - 8 = 21 - 8 = 13

Explanation

Profit is shared among partners in the ratio of their investments.

Total investment = 1050 + 1510 + 1280 = ₹3840

Pooja's share in profit = (Pooja's Investment ÷ Total Investment) × Total Profit = (1280 ÷ 3840) × 1650 = (1 ÷ 3) × 1650 = ₹550

Explanation

Comparing the given ratios to the benchmark value of ½ (0.5):

5/9 = 0.555... which is greater than ½ because the numerator 5 is greater than half of the denominator (4.5).

The remaining ratios—3/17 (≈ 0.176), 13/67 (≈ 0.194), and 10/37 (≈ 0.270)—all have numerators less than half of their respective denominators, making them strictly less than ½.

Explanation

In right-angled triangle ABC with ∠C = 90°, the side adjacent to ∠A is AC and the hypotenuse is AB.

Using the trigonometric ratio cos(A) = AC / AB: cos(45°) = 7 / AB

Since cos(45°) = 1 / √2: 1 / √2 = 7 / AB AB = 7√2 units

Explanation

Prime factorize the given GCD and LCM values: gcd(a, b) = 8 = 2³ gcd(b, c) = 12 = 2² × 3 gcd(c, a) = 4 = 2² LCM(a, b, c) = 240 = 2⁴ × 3¹ × 5¹

Analyzing the prime exponents for a, b, and c:

For prime 2: Since min(a, b) = 3, min(b, c) = 2, and min(c, a) = 2, the exponent of 2 in c must be 2. Since max(a, b, c) = 4, the exponents of 2 for (a, b, c) are either (4, 3, 2) or (3, 4, 2).

For prime 3: gcd(b, c) contains 3¹, whereas gcd(a, b) and gcd(c, a) contain 3⁰. Thus, the exponents of 3 for (a, b, c) are (0, 1, 1).

For prime 5: At least one integer must contain 5¹.

To minimize a + b + c: If exponents of 2 are (4, 3, 2) and factor 5 is assigned to c: a = 2⁴ = 16 b = 2³ × 3 = 24 c = 2² × 3 × 5 = 60 Sum = 16 + 24 + 60 = 100

If exponents of 2 are (3, 4, 2) and factor 5 is assigned to a: a = 2³ × 5 = 40 b = 2⁴ × 3 = 48 c = 2² × 3 = 12 Sum = 40 + 48 + 12 = 100

Explanation

To solve the system of linear equations: x + y + z = 2 --- (1) 6x - 4y + 5z = 31 --- (2) 5x + 2y + 2z = 13 --- (3)

Subtracting 2 × (1) from (3) eliminates y and z: (5x + 2y + 2z) - 2(x + y + z) = 13 - 2(2) 3x = 9 x = 3

Substituting x = 3 into (1) gives y + z = -1, or z = -1 - y.

Substituting x = 3 and z = -1 - y into (2): 6(3) - 4y + 5(-1 - y) = 31 18 - 4y - 5 - 5y = 31 13 - 9y = 31 -9y = 18 y = -2

Substituting y = -2 into z = -1 - y gives: z = -1 - (-2) = 1

This yields the solution (x, y, z) = (3, -2, 1).

Explanation

By the Area of Similar Triangles Theorem, the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

Area Ratio = (Side 1 / Side 2)² = (16 / 14)² = 16² : 14² = 256 : 196

Explanation

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

Weighted Average = (N₁A₁ + N₂A₂ + N₃A₃) / (N₁ + N₂ + N₃)

Total expenditure = (12 × 160) + (55 × 109) + (89 × 181) Total expenditure = 1920 + 5995 + 16109 = 24024

Total number of members = 12 + 55 + 89 = 156

Overall average = 24024 / 156 = 154

Explanation

Let CP be the cost price per piece and SP be the selling price per piece.

Given that the selling price of 48 pieces equals the cost price of x pieces: 48 × SP = x × CP SP / CP = x / 48

With a profit of 25%, the ratio of selling price to cost price is: SP / CP = (100 + Profit%) / 100 = 125 / 100 = 5 / 4

Equating the two expressions for SP / CP: x / 48 = 5 / 4 x = (5 × 48) / 4 x = 60

Explanation

For any two numbers, Product of two numbers = LCM × HCF.

Number 2 = (LCM × HCF) ÷ Number 1 Number 2 = (1116 × 186) ÷ 372

Since 372 ÷ 186 = 2, the calculation simplifies directly to: Number 2 = 1116 ÷ 2 = 558

Explanation

Let x + y = 6k, y + z = 7k, and z + x = 8k.

Adding these three equations together: 2(x + y + z) = 6k + 7k + 8k = 21k x + y + z = (21/2)k

Since x + y + z = 14: (21/2)k = 14 k = 14 × 2 / 21 = 4/3

The value of z is given by subtracting (x + y) from (x + y + z): z = (x + y + z) - (x + y) z = 14 - 6k z = 14 - 6 × (4/3) = 14 - 8 = 6

Explanation

Circumference of each coin = 2πr = 5.5 cm 2 × (22/7) × r = 5.5 r = (5.5 × 7) / (2 × 22) = 7/8 cm

Thickness of each coin h = 2 mm = 0.2 cm

Volume of one cylindrical coin = πr²h = (22/7) × (7/8)² × 0.2 = 154 / 320 cm³

Volume of the cuboid = length × breadth × height = 14 × 11 × 8 = 1232 cm³

Number of coins required = Volume of cuboid / Volume of one coin = 1232 / (154 / 320) = (1232 × 320) / 154 = 8 × 320 = 2560

Explanation

According to the algebraic identity, if a + b + c = 0, then a³ + b³ + c³ = 3abc.

Here, a = 55, b = -30, and c = -25. Since 55 + (-30) + (-25) = 0, the sum of their cubes is given by:

55³ + (-30)³ + (-25)³ = 3 × 55 × (-30) × (-25) = 3 × 55 × 750 = 165 × 750 = 123750

Explanation

Under compound interest, an amount multiplies by the same factor in equal intervals of time. Since the amount doubles (2¹) in 7 years, to become 8 times (2³) it requires 3 such time cycles.

Time required = 3 × 7 = 21 years.

Using the relation (1 + r)⁷ = 2, to find n such that (1 + r)ⁿ = 8: (1 + r)ⁿ = 2³ = ((1 + r)⁷)³ = (1 + r)²¹ Equating powers gives n = 21.

Explanation

For a sum of money P to become 4 times itself under simple interest, the total interest earned is SI = 4P - P = 3P.

Using the rate formula R = [(n - 1) / T] × 100, where n = 4 is the multiple and T = 25 years:

R = [(4 - 1) / 25] × 100 = (3 / 25) × 100 = 12% per annum.

Explanation

Profit is shared in the ratio of the product of capital and time period.

Ratio of initial capital invested by X, Y, and Z is 9 : 4 : 8.

X's total capital-months = (9 × 6) + (4.5 × 6) = 81 Y's total capital-months = 4 × 12 = 48 Z's total capital-months = 8 × 12 = 96

Profit sharing ratio X : Y : Z = 81 : 48 : 96 = 27 : 16 : 32

Total profit units = 27 + 16 + 32 = 75 units

Value per unit = ₹33,000 ÷ 75 = ₹440

Difference between X's and Y's share = 27 - 16 = 11 units

Difference in profit = 11 × 440 = ₹4,840

Explanation

Sum of interior angles of an n-sided regular polygon = (n - 2) × 180° = 1620° n - 2 = 1620 / 180 = 9 n = 11

Each interior angle = 1620° / 11 Each exterior angle = 360° / 11

Difference between interior and exterior angle = (1620° - 360°) / 11 = 1260° / 11 = 114 6/11°

Explanation

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

When sold at 3/11 of MP, the loss is 15%, which means the selling price is 85% of CP: (3/11) × MP = 0.85 × CP MP = (0.85 × 11 / 3) × CP = (9.35 / 3) × CP

When sold at 54% of MP, the new selling price is: New Selling Price = 0.54 × MP New Selling Price = 0.54 × (9.35 / 3) × CP = 0.18 × 9.35 × CP = 1.683 × CP

Profit percentage = (1.683 - 1) × 100 = 68.3%

Explanation

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

The modal class is 60-70, as it has the maximum frequency of 39.

Lower limit of modal class (L) = 60 Frequency of modal class (f1) = 39 Frequency of preceding class (f0) = 28 Frequency of succeeding class (f2) = 10 Class width (h) = 10

Substituting the values into the formula: Mode = 60 + [(39 - 28) / (2 × 39 - 28 - 10)] × 10 Mode = 60 + (11 / 40) × 10 Mode = 60 + 2.75 = 62.75

Explanation

Using the algebraic identity (a - b)³ = a³ - b³ - 3ab(a - b):

Let a = 7⁽2/3) and b = 7⁽-2/3). Here, ab = 7⁽2/3) × 7⁽-2/3) = 7⁰ = 1, and (a - b) = x.

Substituting these values into the identity: x³ = (7⁽2/3))³ - (7⁽-2/3))³ - 3(1)(x) x³ = 7² - 7⁻² - 3x x³ = 49 - 1/49 - 3x x³ + 3x = 49 - 1/49 x³ + 3x = (2401 - 1)/49 = 2400/49

Explanation

A number is divisible by 3 if the sum of its digits is divisible by 3.

Sum of digits of 230A7807 = 2 + 3 + 0 + A + 7 + 8 + 0 + 7 = 27 + A.

For 27 + A to be divisible by 3, A must be a single-digit multiple of 3, since 27 is already divisible by 3. The possible values for the single-digit variable A are 0, 3, 6, and 9.

To find the maximum value that can be assigned to A, we select the largest value from these possibilities, which is 9.

Explanation

Let r be the radius and h be the height of the original cylinder. Cutting it into 5 identical pieces parallel to its base produces 5 smaller cylinders, each with radius r and height h/5.

The total curved surface area of all 5 pieces is 2πrh, and the total flat surface area of all 5 pieces is 5 × 2πr² = 10πr². Equating the two gives 2πrh = 10πr², which simplifies to h = 5r.

For a single piece, its total surface area S is the sum of its curved area and two circular bases: S = 2πr(h/5) + 2πr² = 2πr(r) + 2πr² = 4πr².

The total surface area of the initial cylinder was: 2πrh + 2πr² = 2πr(5r) + 2πr² = 12πr² = 3 × 4πr² = 3S.

Explanation

Let the cost price (CP) be ₹100.

Marked Price (MP) = 100 + 136 = ₹236 Selling Price (SP) to gain 18% = 100 + 18 = ₹118

Discount required = MP - SP = 236 - 118 = ₹118

Discount Percentage = (Discount / MP) × 100 = (118 / 236) × 100 = 50%

Explanation

The ratio of the angles is 2:3:4, giving a total of 2 + 3 + 4 = 9 ratio parts.

The largest angle represents 4 parts out of the 9 parts.

Percentage = (4 / 9) × 100 = 44.44%

Rounding 44.44% to the nearest integer gives 44%.

Explanation

Net rate of filling per hour when all three taps are opened together = 1/4 + 1/30 - 1/30 = 1/4 of the cistern per hour.

Since the three taps together fill 1/4 of the cistern in one hour, the time required to fill one-fourth of the empty cistern is (1/4) ÷ (1/4) = 1 hour.