RRB NTPC CBT 1 2026 Mathematics Questions with Solutions

Explanation

Using the BODMAS rule for order of operations:

First, perform division and multiplication from left to right: (-56) ÷ (-14) = 4 (-4) × 6 = -24

Substitute these back into the expression: (-9) - 4 + (-24)

Next, perform subtraction and addition from left to right: -9 - 4 - 24 = -13 - 24 = -37

Explanation

The overall average spending is calculated using the weighted average formula:

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

Simplifying the ratio of members in groups A, B, and C gives: 42 : 14 : 21 = 6 : 2 : 3

Total effective weight = 6 + 2 + 3 = 11

Total weighted spending = (6 × 275) + (2 × 430) + (3 × 498) = 1650 + 860 + 1494 = 4004

Overall average per member = 4004 / 11 = ₹364

Explanation

Let the monthly incomes of A and B be 4x and 5x, and their monthly expenditures be 3y and 4y respectively.

Since Savings = Income - Expenditure: 4x - 3y = 10,000 5x - 4y = 12,000

Multiplying the first equation by 4 and the second equation by 3: 16x - 12y = 40,000 15x - 12y = 36,000

Subtracting the second equation from the first gives: x = 4,000

The monthly income of A is 4x = 4 × 4,000 = ₹16,000.

Explanation

Let the cost price of the item for P be x.

The final price paid by R after a 30% profit followed by a 52% loss is given by: x × (1 + 30/100) × (1 - 52/100) = 3900 x × 1.30 × 0.48 = 3900 0.624x = 3900 x = 3900 ÷ 0.624 = 6250

Explanation

The sum of all 10 observations is calculated using the mean formula: Sum = Mean × Total observations = 6 × 10 = 60.

Subtracting the sum of the first nine observations from the total sum gives the tenth observation: 60 - 49 = 11.

Explanation

Let the total work be the LCM of 40 and 72, which is 360 units.

Efficiency of (A + B + C) = 360 ÷ 40 = 9 units/day Efficiency of (B + C) = 360 ÷ 72 = 5 units/day

Efficiency of A = (A + B + C) - (B + C) = 9 - 5 = 4 units/day

Since B is 50% more efficient than A: Efficiency of B = 4 × 1.5 = 6 units/day

Work completed by A, B and C together in 16 days = 16 × 9 = 144 units Remaining work = 360 - 144 = 216 units

Time required by B alone to complete the remaining work = 216 ÷ 6 = 36 days

Explanation

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

Substituting a = 20 km/h and b = 45 km/h: Mean proportional = √(20 × 45) = √900 = 30 km/h.

Explanation

Let M be the maximum marks of the examination.

According to the given condition, 35% of maximum marks is equal to Girish's score plus 72 marks: 35% of M = 831 + 72 0.35 × M = 903 M = 903 / 0.35 = 2580

The percentage of marks scored by his friend is: Percentage = (774 / 2580) × 100 = 30%

Explanation

Length of an arc l = (θ / 360°) × 2πr

Given: r = 7 cm, θ = 240°, and π = 22/7.

Substituting the given values: l = (240° / 360°) × 2 × (22/7) × 7 l = (2 / 3) × 44 l = 88 / 3 l = 29.33 cm

Explanation

For a sum of money to become n times itself in T years at simple interest, the interest earned is (n - 1) times the principal.

Rate of interest R = [(n - 1) × 100] / T

Substituting n = 5 and T = 24: R = [(5 - 1) × 100] / 24 R = (4 × 100) / 24 = 400 / 24 = 50 / 3 = 16.666...% ≈ 16.67%

Explanation

Volume of a cylinder is V = πr²h.

Total volume of the reaction vessel: Vᵥessel = 3.14 × 15² × 40 = 3.14 × 225 × 40 = 28,260 cm³

Volume of three cooling tubes: Vₜubes = 3 × (3.14 × 4² × 40) = 3 × 3.14 × 16 × 40 = 6,028.8 cm³

Remaining volume in the vessel: Vᵣemaining = 28,260 - 6,028.8 = 22,231.2 cm³

Volume of the solution at 75% capacity: Vₛolution = 0.75 × 22,231.2 = 16,673.4 cm³

Converting cm³ to liters (1 liter = 1,000 cm³): Volume = 16,673.4 / 1,000 = 16.6734 liters ≈ 16.67 liters

Explanation

Let the Cost Price (CP) of the item be 100.

Since the Marked Price (MP) is set 180% above the cost price: MP = 100 + 180 = 280

To achieve a gain of 152%, the Selling Price (SP) must be: SP = 100 + 152 = 252

Discount amount = MP - SP = 280 - 252 = 28

Discount percentage = (Discount / MP) × 100 = (28 / 280) × 100 = 10%

Explanation

The difference between compound interest and simple interest for a period of 2 years is given by the formula:

Difference = P × (r / 100)²

Substituting Difference = 47 and r = 4:

47 = P × (4 / 100)² 47 = P × (1 / 25)² 47 = P / 625 P = 47 × 625 = 29,375

Explanation

Since distance is constant, time is inversely proportional to speed.

When Lakshya walks at 4/7 of his normal speed, the ratio of his normal speed to new speed is 7 : 4.

Therefore, the ratio of his normal time to new time is 4 : 7.

The difference in time units is 7 - 4 = 3 units, which corresponds to 15 minutes late.

3 units = 15 minutes 1 unit = 5 minutes

Usual time = 4 units = 4 × 5 = 20 minutes.

Explanation

Volume of a cylinder = π × r² × h Volume of original cylinder = π × 12² × 25 = 3600π cm³

When a 45° sector is removed along the height, the remaining angle of the base circle is 360° - 45° = 315°.

Volume of the larger piece = (315° / 360°) × 3600π = (7 / 8) × 3600π = 3150π cm³

Explanation

Let the present age of the son be S years. Vanraj's present age = 3S Daughter's present age = S - 3

Sum of their present ages = 3S + S + (S - 3) = 5S - 3

Since there are 3 people, the sum of their ages 3 years ago was 3 × 3 = 9 years less than the sum of their present ages: (5S - 3) - 9 = 58 5S - 12 = 58 5S = 70 S = 14

Vanraj's present age = 3 × 14 = 42 years.

Explanation

Using the algebraic identity (a + b)³ = a³ + b³ + 3ab(a + b), set a = x and b = 1/(18x).

Given a + b = 3 and ab = x × 1/(18x) = 1/18.

Cubing both sides of x + 1/(18x) = 3: x³ + 1/(5832x³) + 3 × (1/18) × 3 = 27 x³ + 1/(5832x³) + ½ = 27 x³ + 1/(5832x³) = 27 - ½ = 53/2

Multiplying both sides by 18: 18x³ + 1/(324x³) = 18 × (53/2) = 9 × 53 = 477

Explanation

Total number of ratio parts = 17 + 11 = 28 parts.

Value of 1 part = 9,632 ÷ 28 = 344.

Number of boys = 17 × 344 = 5,848.

Explanation

Using the relation M₁ × D₁ = M₂ × D₂:

Total work or provision = 100 × 26 = 2600 man-days

Remaining men = 100 - 60 = 40 men

Number of days D₂ = 2600 / 40 = 65 days

Explanation

By the Alternate Segment Theorem, the angle between a tangent and a chord through the point of contact is equal to the angle subtended by the chord in the alternate segment of the circle.

For chord PY and tangent BP, the angle subtended in the alternate segment at vertex X is ∠PXY = ∠BPY = 60°.

For chord PX and tangent AP, the angle subtended in the alternate segment at vertex Y is ∠PYX = ∠APX = 50°.

Thus, ∠PXY = 60° and ∠PYX = 50°, respectively.

Explanation

34% of 50 = (34 × 50) / 100 = 17

∛512 = 8 and √36 = 6

Evaluating the product term: (4/6) × 8 × 6 = 4 × 8 = 32

Now, substituting these values into the expression for y: y = 17 - 32 = -15

Explanation

Let the time taken by pipe Q to fill the tank be x hours. Then, the time taken by pipe P is x + 15 hours, and the time taken by pipe R is x - 12 hours.

Since the combined rate of pipes P and Q operating together equals the rate of pipe R alone: 1/(x + 15) + 1/x = 1/(x - 12)

Simplifying the left side: (2x + 15) / (x² + 15x) = 1 / (x - 12)

Cross-multiplying gives: (2x + 15)(x - 12) = x² + 15x 2x² - 9x - 180 = x² + 15x x² - 24x - 180 = 0

Factoring the quadratic equation: (x - 30)(x + 6) = 0

Since time cannot be negative, x = 30 hours.

The time required by pipe P to fill the tank is x + 15 = 30 + 15 = 45 hours.

Explanation

The fourth proportional of three numbers a, b, and c is given by x = (b × c) / a.

For 10, 80, and 30: First fourth proportional = (80 × 30) / 10 = 240

For 7, 28, and 19: Second fourth proportional = (28 × 19) / 7 = 4 × 19 = 76

Ratio = 240 : 76 = 60 : 19

Explanation

For a quadratic expression ax² + bx + c where a > 0, the minimum value occurs at x = -b/(2a) and is given by (4ac - b²)/(4a).

Given a = 7, b = -7, and c = 22:

Minimum value = (4 × 7 × 22 - (-7)²) / (4 × 7) = (616 - 49) / 28 = 567 / 28 = 81/4

Alternatively, substituting x = -b/(2a) = 7/14 = 1/2 directly into the expression: 7(1/2)² - 7(1/2) + 22 = 7/4 - 7/2 + 22 = (7 - 14 + 88)/4 = 81/4.

Explanation

The final population after successive percentage increases is given by: Final Population = Initial Population × (1 + r₁/100) × (1 + r₂/100)

Substituting Initial Population = 2,90,000, r₁ = 15%, and r₂ = 25%: Final Population = 2,90,000 × (115/100) × (125/100) = 2,90,000 × (23/20) × (5/4) = 3,625 × 23 × 5 = 4,16,875

Explanation

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

Overall Average = (N_A × A_A + N_B × A_B + N_C × A_C) / (N_A + N_B + N_C)

Substitute the given values into the formula:

Total expenditure = (75 × 493) + (47 × 397) + (15 × 465) = 36975 + 18659 + 6975 = 62609

Total members = 75 + 47 + 15 = 137

Overall Average = 62609 / 137 = 457

Explanation

Let M be the maximum marks of the examination.

33% of M − 79 = 416 33% of M = 416 + 79 = 495 M = (495 × 100) / 33 = 1500

Percentage of marks scored by his friend = (555 / 1500) × 100 = 37%

Explanation

Using the algebraic identity (x + 1/x)³ = x³ + 1/x³ + 3(x + 1/x), we can express x³ + 1/x³ as (x + 1/x)³ - 3(x + 1/x).

Substituting x + 1/x = -3 into the formula: x³ + 1/x³ = (-3)³ - 3(-3) = -27 + 9 = -18

Explanation

To find the time when both machines complete a cycle together again, calculate the Least Common Multiple (LCM) of their cycle times, 12 minutes and 18 minutes.

12 = 2² × 3 18 = 2 × 3²

LCM(12, 18) = 2² × 3² = 4 × 9 = 36 minutes.

Explanation

Area of a circle is πr² and its perimeter (circumference) is 2πr, where r is the radius. Equating the area to twice the perimeter gives:

πr² = 2 × (2πr) πr² = 4πr

Dividing both sides by πr gives r = 4 units.

The diameter of the circle is 2r = 2 × 4 = 8 units.

Explanation

Using the compound interest formula A = P(1 + r/100)ⁿ, where r = 10%:

Ishan's Amount after 11 years = Krish's Amount after 12 years I × (1 + 10/100)¹¹ = K × (1 + 10/100)¹² I / K = (1.1)¹² / (1.1)¹¹ = 1.1 = 11/10

Thus, the ratio of Ishan's share to Krish's share is 11 : 10.

Sum of ratio terms = 11 + 10 = 21

Krish's share = (10 / 21) × 1407 = 10 × 67 = ₹670

Explanation

The volume of the cube with edge x is V_cube = x³. The cuboid on top has a square base of dimensions x × x and height (x - 4), so its volume is V_cuboid = x²(x - 4).

The total volume of the solid is Vₜotal = V_cube + V_cuboid = x³ + x²(x - 4).

Given that the total volume is 1.5 times the volume of the cube: x³ + x²(x - 4) = 1.5 x³

Since x > 0, dividing the entire equation by x² gives: x + (x - 4) = 1.5 x 2x - 4 = 1.5 x 0.5 x = 4 x = 8

Explanation

Average Speed = Total Distance ÷ Total Time

Total Distance = 810 + 492 + 988 = 2290 km

Total Time = (810 ÷ 30) + (492 ÷ 12) + (988 ÷ 38) = 27 + 41 + 26 = 94 hours

Average Speed = 2290 ÷ 94 = 1145 ÷ 47 = 24 17/47 km/hr

Explanation

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

Substituting the given values: 3 × (22/7) × r² = 272 (66/7) × r² = 272 r² = (272 × 7) / 66 r² = (136 × 7) / 33 r² = 952 / 33 r = √(952/33) cm

Explanation

Selling price of article X, SP_X = Cost Price + Profit = 500 + 636 = ₹1136.

After successive discounts of 20% and 45%, SP_X = Marked Price (MP) × (1 - 0.20) × (1 - 0.45) = MP × 0.44. Therefore, MP × 0.44 = 1136.

For article Y sold at a discount of 67%: Selling price of Y, SP_Y = MP × (1 - 0.67) = MP × 0.33.

Substituting the value from X: SP_Y = 1136 × (0.33 / 0.44) = 1136 × ¾ = ₹852.

Profit on article Y = SP_Y - Cost Price = 852 - 500 = ₹352. Profit percentage on Y = (352 / 500) × 100 = 70.4%.

Explanation

Cost price per article = 480 / 12 = ₹40 Selling price per article = ₹78

Profit percentage = [(Selling Price - Cost Price) / Cost Price] × 100 = [(78 - 40) / 40] × 100 = (38 / 40) × 100 = 95%

Explanation

When distance is constant, speed and time are inversely proportional.

Walking at 3/7 of his normal speed means the new time taken is 7/3 of the normal time T.

The delay is the difference between the new time and normal time: Delay = (7/3)T - T = (4/3)T

Given that the delay is 20 minutes: (4/3)T = 20 T = 20 × (3/4) = 15 minutes

Explanation

The mode for grouped frequency data is calculated using the formula:

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

The highest frequency is 25, so the modal class is 30-40. Lower limit of modal class (L) = 30 Frequency of modal class (f₁) = 25 Frequency of preceding class (f₀) = 18 Frequency of succeeding class (f₂) = 10 Class width (h) = 10

Substituting the values into the formula:

Mode = 30 + [(25 - 18) / (2 × 25 - 18 - 10)] × 10 Mode = 30 + [7 / (50 - 28)] × 10 Mode = 30 + (7 / 22) × 10 Mode = 30 + 3.1818 ≈ 33.18

Explanation

Using the exponent property (a/b)ⁿ = (b/a)⁻ⁿ, express all terms on the left-hand side with the common base (5/16):

(16/5)¹³ = (5/16)⁻¹³ (16/5)⁵ = (5/16)⁻⁵

Combining the exponents on the left-hand side using aˣ × aʸ = aˣ⁺ʸ:

(5/16)⁻¹³ × (5/16)¹⁶ × (5/16)⁻⁵ = (5/16)⁻¹³⁺¹⁶⁻⁵ = (5/16)⁻²

Now, equate the exponents on both sides of the equation:

-2 = 6m + 16 6m = -18 m = -3

Explanation

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

Selling at 2/5 (or 0.4) of S results in a 24% loss, which means the selling price is 76% of C: 0.4S = 0.76C S = 1.9C

When the article is sold at 84% of S: New Selling Price = 0.84 × S = 0.84 × 1.9C = 1.596C

Profit percentage = ((1.596C - C) / C) × 100 = 59.6%

Explanation

For a sum of money to become 5 times itself on simple interest, the interest earned is 5 - 1 = 4 times the principal sum.

Using the simple interest formula SI = (P × R × T) / 100: 4P = (P × R × 22) / 100 R = 400 / 22 = 18.1818... ≈ 18.18%

Explanation

The empirical relation connecting mean, median, and mode is given by:

Mode = 3 × Median - 2 × Mean

Subtracting Median from both sides gives:

Mode - Median = 2 × Median - 2 × Mean Mode - Median = 2 × (Median - Mean)

It is given that the median is 77.74 less than the mode, which means Mode - Median = 77.74. Substituting this value into the equation:

77.74 = 2 × (Median - Mean) Median - Mean = 77.74 / 2 = 38.87

Thus, the median exceeds the mean by 38.87.

Explanation

For a system of linear equations AX = B, the number of solutions is determined by the determinant of the coefficient matrix A and the ranks of A and the augmented matrix [A|B].

Calculating the determinant of the coefficient matrix: | -3 2 -5 | | 2 -3 4 | = -3(-3 - 4) - 2(2 - 4) - 5(2 + 3) = 21 + 4 - 25 = 0 | 1 1 1 |

Summing all three equations: (-3 + 2 + 1)α + (2 - 3 + 1)β + (-5 + 4 + 1)γ = -14 + 10 + 4 0α + 0β + 0γ = 0

Since det(A) = 0 and rank(A) = rank([A|B]) = 2, which is less than the number of variables (3), the system is consistent with dependent equations, giving infinitely many solutions.

Explanation

Squaring and adding the expressions for x and y:

x² = a² sin² θ y² = a² cos² θ

Adding the two equations:

x² + y² = a² sin² θ + a² cos² θ x² + y² = a²(sin² θ + cos² θ)

Since sin² θ + cos² θ = 1, substituting this identity yields:

x² + y² = a²

Explanation

Present sum of ages of Amit, Bharat, and Chitra = (36 × 3) + (3 × 4) = 108 + 12 = 120 years.

Present sum of ages of Amit and Bharat = (30 × 2) + (2 × 6) = 60 + 12 = 72 years.

Present age of Chitra = 120 - 72 = 48 years.

Chitra's age three years ago = 48 - 3 = 45 years.

Explanation

A number is divisible by 4 if the number formed by its last two digits is divisible by 4.

In the 8-digit number 588A9792, the last two digits are 92. Since 92 ÷ 4 = 23, the number is divisible by 4 regardless of the value of A. Therefore, any single-digit integer from 0 to 9 can be assigned to A, making the maximum possible value for A equal to 9.

Explanation

Using the work-time relation formula:

(M₁ × T₁) / W₁ = (M₂ × T₂) / W₂

Where M is the number of machines, T is the time in hours, and W is the work done (number of components manufactured).

Substituting the given values into the formula:

(6 × 8) / 480 = (10 × 12) / W₂

48 / 480 = 120 / W₂

1 / 10 = 120 / W₂

W₂ = 120 × 10 = 1200

Explanation

Daily work done by A and B together = 1/21 Daily work done by A alone = 1/28

Daily work done by B alone = 1/21 - 1/28 = (4 - 3)/84 = 1/84

Therefore, B alone takes 84 days to complete the full work.

Time required for B alone to complete ¾ of the work = ¾ × 84 = 63 days.

Explanation

Let M be the maximum marks of the examination.

According to the given condition, 30% of M - 75 = 441. 0.30 × M = 441 + 75 = 516 M = 516 / 0.30 = 1720

To find the percentage of marks scored by his friend who obtained 688 marks: Percentage = (688 / 1720) × 100 = 40%

Explanation

Factorizing the first equation: x² - 21x + 110 = 0 (x - 11)(x - 10) = 0 x = 11 or x = 10

Factorizing the second equation: 110 + x - x² = 0 x² - x - 110 = 0 (x - 11)(x + 10) = 0 x = 11 or x = -10

The root present in both equations is 11.