RRB NTPC CBT 1 2026 Mathematics Questions with Solutions

Explanation

From the first equation, v + w = 2 - u.

Substituting this into the third equation gives: 5u + 2(2 - u) = 13 3u + 4 = 13 u = 3

With u = 3, the first equation simplifies to v + w = -1, so v = -1 - w.

Substituting u = 3 and v = -1 - w into the second equation gives: 6(3) + 4(-1 - w) + 5w = 31 18 - 4 + w = 31 14 + w = 31 w = 17

Substituting w = 17 into v = -1 - w yields: v = -1 - 17 = -18

Explanation

Weighted Average = (n₁x₁ + n₂x₂ + n₃x₃) / (n₁ + n₂ + n₃)

Total members = 16 + 86 + 21 = 123

Total amount spent = (16 × 447) + (86 × 396) + (21 × 158) = 7152 + 34056 + 3318 = 44526

Overall average = 44526 / 123 = 362

Explanation

Using the standard order of operations (BODMAS/PEMDAS):

Perform division and multiplication first from left to right: (-56) ÷ (-14) = 4 (-2) × 6 = -12

Substitute these values back into the expression: (-9) - 4 + (-12)

Perform addition and subtraction from left to right: -9 - 4 - 12 = -25

Explanation

The empirical relationship between mode, median, and mean is given by the formula: Mode = 3 Median - 2 Mean

Subtracting mean from both sides gives: Mode - Mean = 3 (Median - Mean)

Since Mode - Mean = 33.6, we find: Median - Mean = 33.6 / 3 = 11.2

Now, subtracting median from both sides of the empirical formula gives: Mode - Median = 2 (Median - Mean)

Substituting the value of (Median - Mean): Mode - Median = 2 × 11.2 = 22.4

Explanation

First, calculate the square root of 3136: √3136 = 56

Substitute this value into the given equation: 18 × 7 × 560 ÷ 56 = y + 703

Simplify the division step: 560 ÷ 56 = 10

Evaluate the left side: 18 × 7 × 10 = 1260

Now, solve for y: 1260 = y + 703 y = 1260 - 703 = 557

Explanation

The length of a chord of a larger circle of radius R that is tangent to a concentric smaller circle of radius r is given by the formula L = 2√(R² - r²).

Substituting R = 17 cm and r = 15 cm: L = 2√(17² - 15²) L = 2√(289 - 225) L = 2√64 L = 2 × 8 = 16 cm

Explanation

Sum of three numbers = Average × 3 = 20 × 3 = 60.

Third number = Total sum - Sum of given numbers = 60 - (13 + 22) = 60 - 35 = 25.

Explanation

When three numbers a, b, and c are in continued proportion, their ratio satisfies a/b = b/c, which simplifies to b² = ac.

Substituting a = 8 and c = 18 into x² = 8 × 18 gives: x² = 144 x = √144 = 12

Explanation

At compound interest, if an investment becomes k times in t years, it becomes kⁿ times in (n × t) years.

Here, the money doubles (2 times) in 6 years. To become 16 times, which is 2⁴ times, the required time is 4 × 6 = 24 years.

Explanation

Using the BODMAS rule (Brackets, Orders, Division/Multiplication, Addition/Subtraction), operations are evaluated in order of priority:

First, perform division and multiplication: 20 ÷ 4 = 5 3 × 4 = 12

Substitute these back into the expression: 16 + 5 - 12

Finally, perform addition and subtraction from left to right: 16 + 5 = 21 21 - 12 = 9

Explanation

Let the three numbers be x, y, and z.

Given that x increased by 25% equals y: x × 1.25 = y y = 5x/4

Also, y is 20% less than z: y = z × 0.80 y = 4z/5

Equating the expressions for y: 5x/4 = 4z/5 x/z = (4 × 4) / (5 × 5) x/z = 16/25

Explanation

The formula for Simple Interest is SI = (P × R × T) / 100, where P is the Principal, R is the Rate of interest per annum, and T is the Time in years.

Substituting P = ₹7,200, SI = ₹1,080, and R = 6%:

1,080 = (7,200 × 6 × T) / 100 1,080 = 432 × T T = 1,080 / 432 = 2.5 years

Explanation

Let the speeds of Amit and Bhushan be S_A and S_B m/s, respectively.

Amit overtakes Bhushan 4 minutes (240 seconds) after Amit starts. By then, Bhushan has been running for 240 + 20 = 260 seconds. Equating the distances covered from Amit's starting line: 240 S_A = 200 + 260 S_B 12 S_A - 13 S_B = 10 S_B = (12 S_A - 10) / 13

For the full race, Amit runs 1200 m and Bhushan runs 1000 m (1200 - 200). Since Amit starts 20 seconds after Bhushan and finishes 10 seconds before him, Bhushan's total running time exceeds Amit's running time by 30 seconds: 1000 / S_B - 1200 / S_A = 30

Substituting S_B = (12 S_A - 10) / 13 into the time equation: 1300 / (12 S_A - 10) - 120 / S_A = 3

Solving for S_A gives S_A = 40 / 9 = 4 4/9 m/s.

Explanation

Total value of 12 kits received = 12 × ₹12,000 = ₹1,44,000.

Under the "Buy 10 kits, get 2 free" scheme, only 10 kits are billed: Billed amount = 10 × ₹12,000 = ₹1,20,000.

Applying the successive volume discount of 15% and institutional discount of 5% on the billed amount: Final amount paid = ₹1,20,000 × (1 - 0.15) × (1 - 0.05) = ₹1,20,000 × 0.85 × 0.95 = ₹96,900.

Total discount = ₹1,44,000 - ₹96,900 = ₹47,100.

Effective discount percentage = (₹47,100 / ₹1,44,000) × 100 = 32.7083% ≈ 32.71%.

Explanation

Let total registered voters be R.

Votes polled = 90% of R Valid votes = 80% of (90% of R) = 72% of R

In a two-candidate election, if the winner gets 51% of valid votes, the loser gets 49% of valid votes. Winning margin = 51% - 49% = 2% of valid votes

Margin = 2% of 72% of R = 0.0144 R

Given margin = 3870: 0.0144 R = 3870 R = 3870 ÷ 0.0144 = 268750

Explanation

When circles are placed sequentially between two common tangent lines such that each circle touches the adjacent ones, their radii form a geometric progression (GP).

For a sequence of five circles with radii r₁, r₂, r₃, r₄, r₅ in geometric progression, the radius of the smallest circle is r₁ = 4 units and the radius of the largest circle is r₅ = 9 units.

The radius of the middle circle is the third term r₃, which is the geometric mean of the first and fifth terms:

r₃ = √(r₁ × r₅)

r₃ = √(4 × 9) = √36 = 6 units

Explanation

Adding the first equation (x - y + z = 2) and the second equation (2x - y - z = -6) eliminates z: 3x - 2y = -4

Adding the second equation and the third equation (2x + 2y + z = -3) eliminates z: 4x + y = -9

Multiplying 4x + y = -9 by 2 gives 8x + 2y = -18. Adding this to 3x - 2y = -4 yields: 11x = -22 x = -2

Substituting x = -2 into 4x + y = -9: 4(-2) + y = -9 y = -1

Substituting x = -2 and y = -1 into x - y + z = 2: -2 - (-1) + z = 2 -1 + z = 2 z = 3

Explanation

A right-angled triangle with sides 3 cm, 4 cm, and 5 cm (where 3² + 4² = 5²) revolved about the 3 cm side forms a right circular cone with height h = 3 cm and base radius r = 4 cm.

Volume of cone = (1/3) × π × r² × h

Substituting r = 4 cm, h = 3 cm, and π = 3.14:

Volume = (1/3) × 3.14 × 4² × 3 Volume = 3.14 × 16 Volume = 50.24 cubic cm

Explanation

The profit shared among partners is directly proportional to their investments.

Ratio of investments of Siddharth, Lata, and Vivek = 1570 : 1760 : 1620 = 157 : 176 : 162

Sum of ratio terms = 157 + 176 + 162 = 495

Vivek's share in profit = (162 / 495) × 1540 = (18 / 11) × 308 = 18 × 28 = ₹504

Explanation

Let the common ratio multiplier for the number of articles be x.

The quantity of articles of types P, Q, and R are 3x, 2x, and 5x respectively.

Total cost = (3x × 280) + (2x × 180) + (5x × 120) Total cost = 840x + 360x + 600x = 1800x

Given total cost = ₹12,600

1800x = 12,600 x = 12,600 / 1800 = 7

Number of articles of type Q = 2x = 2 × 7 = 14

Explanation

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

Total time required to type 90 pages = 90 × 20 = 1800 minutes.

Explanation

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

Overall Average = (n₁x₁ + n₂x₂ + n₃x₃) ÷ (n₁ + n₂ + n₃)

Given data: Group A: n₁ = 25, x₁ = 137 → Total = 25 × 137 = 3425 Group B: n₂ = 26, x₂ = 443 → Total = 26 × 443 = 11518 Group C: n₃ = 21, x₃ = 461 → Total = 21 × 461 = 9681

Total amount spent = 3425 + 11518 + 9681 = 24624 Total number of members = 25 + 26 + 21 = 72

Overall Average = 24624 ÷ 72 = 342

Explanation

Let radius r = 20x and slant height l = 29x. Using height h = √(l² - r²), we get h = √((29x)² - (20x)²) = 21x.

Volume of a cone = (1/3) π r² h (1/3) π (20x)² (21x) = (24192 / 5) π 2800 x³ = 24192 / 5 x³ = 24192 / 14000 = 1.728 x = 1.2

Thus, radius r = 20 × 1.2 = 24 cm and slant height l = 29 × 1.2 = 34.8 cm.

Curved Surface Area = π r l = π × 24 × 34.8 = 835.2π cm²

Explanation

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

For n = 63, the average = 63 + 1 = 64.

Explanation

When two trains move in opposite directions, their relative speed is the sum of their individual speeds: Relative speed = 130 + 70 = 200 km/hr = 200 × (5 / 18) = 500 / 9 m/s

The total distance covered while crossing each other equals the sum of the lengths of both trains: Total distance = L₁ + L₂ = Relative speed × Time

130 + L₂ = (500 / 9) × 14 130 + L₂ = 7000 / 9 = 777.78 m L₂ = 777.78 − 130 = 647.78 m

Explanation

Using the successive transaction formula: Final Price = Initial Price × (1 + Profit%/100) × (1 - Loss%/100)

Substituting the given values: 6895 = P × (1 + 0.40) × (1 - 0.50) 6895 = P × 1.4 × 0.5 6895 = 0.7 × P P = 6895 / 0.7 = 9850

Explanation

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

The midpoints (x) for the class intervals 30-40, 40-50, 50-60, 60-70, and 70-80 are 35, 45, 55, 65, and 75 respectively.

Total frequency Σf = 49 + 33 + 24 + 21 + X = 127 + X

Sum of products Σ(f × x) = (49 × 35) + (33 × 45) + (24 × 55) + (21 × 65) + (X × 75) = 1715 + 1485 + 1320 + 1365 + 75X = 5885 + 75X

Substituting Mean = 47 into the formula: 47 = (5885 + 75X) / (127 + X) 47 × (127 + X) = 5885 + 75X 5969 + 47X = 5885 + 75X 28X = 84 X = 3

Explanation

Since the equation x² - 3px + q = 0 has equal roots, its discriminant is zero: (-3p)² - 4(1)(q) = 0 9p² - 4q = 0 q = 9p²/4

The repeated root of this equation is x = 3p/2.

Since both equations share a common root, x = 3p/2 must satisfy x² - 2kx + t = 0: (3p/2)² - 2k(3p/2) + t = 0 9p²/4 - 3pk + t = 0 t = 3pk - 9p²/4

Adding t and q: t + q = (3pk - 9p²/4) + 9p²/4 = 3pk

Squaring both sides gives: (t + q)² = (3pk)² = 9p²k²

Explanation

Percentage Increase = (Required Increase / Current Base) × 100%

Let original production = 100 units. New production after a 52% decrease = 100 - 52 = 48 units.

To restore production to 100 units, an increase of 52 units is required from the current base of 48 units.

Required percentage increase = (52 / 48) × 100% = (13 / 12) × 100% = 108⅓%

Explanation

Let the cost price be CP and the actual selling price be SP.

Selling at 5/12 of SP results in a 20% loss: (5/12) × SP = 0.8 × CP SP = (0.8 × 12) / 5 × CP SP = 1.92 × CP

Selling at 80% of SP gives a new selling price of: New SP = 0.8 × SP = 0.8 × 1.92 × CP = 1.536 × CP

Profit percentage = ((1.536 × CP - CP) / CP) × 100 = 0.536 × 100 = 53.6%

Explanation

Since cosec A is the reciprocal of sin A, cosec A = 5/3.

Substituting sin A = 3/5 and cosec A = 5/3 into the given expression:

Numerator = 11 + 3 cosec A = 11 + 3 × (5/3) = 11 + 5 = 16

Denominator = 4 - 5 sin A = 4 - 5 × (3/5) = 4 - 3 = 1

Expression = √(16 / 1) = √16 = 4

Explanation

Simplify the given quadratic expression: 10(x - 1) - 5x² + 4(2 - x) + 8 = 10x - 10 - 5x² + 8 - 4x + 8 = -5x² + 6x + 6

For a quadratic expression ax² + bx + c where a < 0, the maximum value is given by c - b² / (4a).

Here, a = -5, b = 6, and c = 6.

Maximum value = 6 - (6)² / (4 × -5) = 6 + 36 / 20 = 6 + 1.8 = 7.8.

Explanation

An operating system is the essential system software that acts as an interface between computer hardware and user applications. It directly controls and manages hardware resources, including CPU process scheduling, RAM memory allocation, storage management, and input/output peripheral devices.

Explanation

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

Initial man-days = 100 × 27 = 2700

When 40 men leave, the remaining number of men M₂ = 100 - 40 = 60.

Number of days D₂ = (100 × 27) / 60 = 2700 / 60 = 45 days.

Explanation

Since points A, C, D, and B lie on the transversal line AB in order, ∠ACQ and ∠QCD form a linear pair at point C: ∠ACQ + ∠QCD = 180° 6x + 5x + y = 180° 11x + y = 180°

Since line PQ is parallel to line RS, corresponding angles at intersections C and D are equal: ∠QCD = ∠BDS 5x + y = 2x + 3y 3x = 2y y = 1.5x

Substituting y = 1.5x into the linear pair equation: 11x + 1.5x = 180° 12.5x = 180° x = 14.4°

Thus, y = 1.5 × 14.4° = 21.6°.

The measure of ∠QCD is: ∠QCD = 5(14.4°) + 21.6° = 72° + 21.6° = 93.60°

Explanation

As part of the 1991 economic reforms based on the recommendations of the Raja Chelliah Committee, India initiated comprehensive tax structure rationalization. Direct tax rates on personal income and corporate profits were progressively lowered to encourage voluntary compliance and broaden the tax base. Simultaneously, high indirect tax rates, including peak customs duties and central excise rates, were substantially reduced and simplified to enhance domestic industrial efficiency and integrate the Indian economy with global markets.

Explanation

The given expression can be factored using the algebraic identity: a⁴ + b⁴ - 2a²b² = (a² - b²)² = [(a + b)(a - b)]²

Calculating the sum and difference of a and b: a + b = (x + 2/x) + (x - 2/x) = 2x a - b = (x + 2/x) - (x - 2/x) = 4/x

Multiplying the two terms: a² - b² = (a + b)(a - b) = (2x) × (4/x) = 8

Squaring the result gives: (a² - b²)² = 8² = 64

Explanation

The Ranji Trophy was first held during the 1934–35 season. The inaugural match of India's premier domestic first-class cricket tournament took place on 4 November 1934 between Madras and Mysore at the Chepauk Ground in Madras. The championship was named in honour of Maharaja Kumar Shri Ranjitsinhji, the first Indian to play international cricket, and Bombay emerged as the champions of this first edition by defeating Northern Punjab in the final.

Explanation

The mean x̄ of a frequency distribution is given by the formula x̄ = ∑(fᵢ × xᵢ) / ∑ fᵢ, where xᵢ represents the marks and fᵢ represents the number of students (frequency).

Sum of frequencies: ∑ fᵢ = 24 + 38 + 26 + 32 + 13 = 133

Sum of products: ∑(fᵢ × xᵢ) = (16 × 24) + (21 × 38) + (47 × 26) + (80 × 32) + (99 × 13) = 384 + 798 + 1222 + 2560 + 1287 = 6251

Calculating the mean: x̄ = 6251 / 133 = 47

Explanation

Let the initial investments of A and B be 5x and 7x respectively.

C's investment = 3.5x active for 8 months. A's investment = 5x for 8 months + 3x for 4 months (after withdrawing 40% of capital). B's investment = 7x for 12 months.

Calculating the ratio of equivalent monthly investments (A : B : C): A = (5x × 8) + (3x × 4) = 52x B = 7x × 12 = 84x C = 3.5x × 8 = 28x

A : B : C = 52 : 84 : 28 = 13 : 21 : 7

Total ratio units = 13 + 21 + 7 = 41

Value per unit = 27,675 / 41 = ₹675

B's share of profit = 21 × 675 = ₹14,175

Explanation

According to the order of operations (BODMAS/PEMDAS), perform division and multiplication before addition and subtraction:

Division: (-56) ÷ (-14) = 4 Multiplication: (-3) × 7 = -21

Substitute these back into the expression: (-9) - 4 + (-21)

Now, evaluate from left to right: (-9) - 4 = -13 (-13) + (-21) = -34

Explanation

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

Since the shopkeeper professes to sell at a 60% loss, the selling price for 1000 grams is 1000 × (1 - 0.60) = ₹400.

Let x be the actual weight used for 1 kg. His actual cost price for x grams is ₹x.

Given that his actual gain is 90%: Gain % = [(Selling Price - Cost Price) / Cost Price] × 100 90 = [(400 - x) / x] × 100 0.9 × x = 400 - x 1.9 × x = 400 x = 4000 / 19 = 210.526... gm ≈ 210.5 gm

Explanation

Weight of teacher = New Average + (Initial number of students × Increase in average)

Given: Initial students = 10 Initial average = 50 kg New average = 50 + 1 = 51 kg Increase in average = 1 kg

Teacher's weight = 51 + (10 × 1) = 61 kg

Alternatively: Total weight of 10 students = 10 × 50 = 500 kg Total weight of 11 individuals (students + teacher) = 11 × 51 = 561 kg Teacher's weight = 561 − 500 = 61 kg

Explanation

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

Sum of digits of 54572A0 = 5 + 4 + 5 + 7 + 2 + A + 0 = 23 + A

For 23 + A to be a multiple of 3, the smallest non-negative single-digit value for A is 1, giving a digit sum of 24, which is divisible by 3.

Explanation

Let the cost price (CP) be ₹100.

Since the marked price (MP) is fixed 100% above the cost price: MP = 100 + 100 = ₹200

To achieve a gain of 26%, the selling price (SP) must be: SP = 100 + 26 = ₹126

Discount required = MP - SP = 200 - 126 = ₹74

Discount percentage = (Discount / MP) × 100 = (74 / 200) × 100 = 37%

Explanation

Since AB is a tangent to the inner circle C₂ at point P, the radius OP is perpendicular to AB (OP ⊥ AB). O is also the center of the outer circle C₁, making AB a chord of C₁. The perpendicular drawn from the center of a circle to a chord bisects the chord. Therefore, OP bisects AB at point P, which gives AP = PB.

Explanation

The surface area of a sphere is given by the formula 4πr², where r is the radius of the sphere.

4πr² = 154 4 × (22/7) × r² = 154 r² = (154 × 7) / (4 × 22) r² = (7 × 7) / 4 r² = 49 / 4 r = √(49 / 4) = 7 / 2 = 3.5 cm

Explanation

Solving the second quadratic equation x² + 2x - 3 = 0 by factoring gives (x + 3)(x - 1) = 0, which yields roots x = 1 and x = -3.

If the common root is x = 1, substituting x = 1 into x² + x + p = 0 gives: 1² + 1 + p = 0 2 + p = 0 p = -2

If the common root is x = -3, substituting x = -3 into x² + x + p = 0 gives: (-3)² + (-3) + p = 0 9 - 3 + p = 0 p = -6

Explanation

15% of 80 = (15 / 100) × 80 = 12

Evaluating the cube root and square root terms: ∛216 = 6 √36 = 6

Now calculating the multiplication term: (3 / 6) × ∛216 × √36 = ½ × 6 × 6 = 18

Subtracting the terms gives: y = 12 - 18 = -6

Explanation

The weighted mean of the first n natural numbers, where each weight equals the corresponding number, is given by the formula:

Weighted Mean = Σi² / Σi = [n(n + 1)(2n + 1) / 6] / [n(n + 1) / 2] = (2n + 1) / 3

For the first 5 natural numbers (n = 5):

Weighted Mean = (2 × 5 + 1) / 3 = 11 / 3