RRB NTPC CBT 1 2026 Mathematics Questions with Solutions

Explanation

The ratio of the sides of triangle PQR is 4:5:6, so the sides are 4x, 5x, and 6x. Perimeter = 4x + 5x + 6x = 15x = 90 cm, giving x = 6 cm. The side lengths are a = 24 cm, b = 30 cm, and c = 36 cm.

The semi-perimeter s = (24 + 30 + 36) / 2 = 45 cm.

Using Heron's formula for the area of triangle PQR: Area(PQR) = √(s(s - a)(s - b)(s - c)) Area(PQR) = √(45 × (45 - 24) × (45 - 30) × (45 - 36)) Area(PQR) = √(45 × 21 × 15 × 9) = 135√7 cm²

Substituting √7 = 2.6: Area(PQR) = 135 × 2.6 = 351 cm²

The area of quadrilateral ABCD is twice the area of triangle PQR: Area(ABCD) = 2 × 351 = 702 cm²

Explanation

Let the number be x.

20% of 620 = 0.20 × 620 = 124

According to the given condition: 20% of x - 124 = 210 0.20x = 210 + 124 0.20x = 334 x = 334 / 0.20 = 1670

Explanation

Using the compound interest formula A = P(1 + r/100)ⁿ, set Anil's amount after 15 years equal to Ashok's amount after 16 years:

Anil × (1 + 10/100)¹⁵ = Ashok × (1 + 10/100)¹⁶

Dividing both sides by (1 + 10/100)¹⁵ gives: Anil / Ashok = 1 + 10/100 = 11/10

The ratio of Anil's share to Ashok's share is 11 : 10. Total parts = 11 + 10 = 21 parts.

Value of 1 part = 1407 / 21 = ₹67.

Ashok's share = 10 × ₹67 = ₹670.

Explanation

If every observation in a dataset is multiplied by a constant factor k, the new average is equal to the original average multiplied by k.

New Average = Original Average × k

Given that the original average of 13 numbers is 8 and each number is multiplied by 7:

New Average = 8 × 7 = 56

Explanation

Given y - 12 = 4√5, adding 9 to both sides gives: y - 3 = 9 + 4√5

We can rewrite 9 + 4√5 as a perfect square: y - 3 = (√5)² + (2)² + 2(2)(√5) = (√5 + 2)²

Taking the square root on both sides: √(y - 3) = √5 + 2

Rationalizing the reciprocal: 1/√(y - 3) = 1 / (√5 + 2) = (√5 - 2) / (5 - 4) = √5 - 2

Substituting these into the given expression: √(y - 3) - 1/√(y - 3) = (√5 + 2) - (√5 - 2) = 4

Explanation

Let M be the maximum marks in the examination.

According to the question: 28% of M − 30 = 572 0.28 × M = 572 + 30 = 602 M = 602 / 0.28 = 2150

Percentage of marks scored by his friend: (473 / 2150) × 100% = 22%

Explanation

Rate of the first tap = +1/4 cistern per hour Rate of the second tap = +1/38 cistern per hour Rate of the third tap = -1/38 cistern per hour

When all three taps are opened simultaneously, the net rate of filling per hour is: Net rate = 1/4 + 1/38 - 1/38 = 1/4 cistern per hour

Time required to fill 1/4 of the empty cistern = (Fraction to be filled) / (Net rate) Time = (1/4) / (1/4) = 1 hour

Explanation

The empirical formula relating mean, median, and mode is:

Mode = 3 × Median - 2 × Mean

Subtracting median from both sides rearranges the formula to:

Mode - Median = 2 × (Median - Mean)

Given that the median is 61.54 less than the mode, Mode - Median = 61.54. Substituting this value into the formula gives:

61.54 = 2 × (Median - Mean) Median - Mean = 61.54 ÷ 2 = 30.77

Explanation

When two trains move in opposite directions, their relative speed is the sum of their individual speeds: Relative Speed = 60 km/hr + 84 km/hr = 144 km/hr

Converting relative speed to meters per second: 144 × (5/18) = 40 m/s

The combined length of both trains equals the distance covered at this relative speed in 30 seconds: Combined Length = Relative Speed × Time = 40 m/s × 30 s = 1200 m

Explanation

Let the speed of the boat in still water be v km/h. The speed of the stream is 3 km/h.

Downstream speed = (v + 3) km/h Upstream speed = (v - 3) km/h

Since time = distance / speed, and the time taken for both journeys is equal: 36 / (v + 3) = 24 / (v - 3)

Dividing the numerators by 12: 3 / (v + 3) = 2 / (v - 3)

Cross-multiplying: 3(v - 3) = 2(v + 3) 3v - 9 = 2v + 6 v = 15 km/h

Explanation

In the right-angled triangle formed by the tower, the ground, and the line of sight: tan(θ) = Height / Distance

Given θ = 45° and Distance = 28.6 m: tan(45°) = Height / 28.6

Since tan(45°) = 1: 1 = Height / 28.6 Height = 28.6 m

Explanation

The mean mark (x̄) of a discrete frequency distribution is calculated using the formula:

x̄ = Σ(f × x) / Σf

Sum of the products of marks and number of students: Σ(f × x) = (81 × 15) + (35 × 4) + (73 × 3) + (56 × 16) Σ(f × x) = 1215 + 140 + 219 + 896 = 2470

Total number of students: Σf = 15 + 4 + 3 + 16 = 38

Calculating the mean: x̄ = 2470 / 38 = 65

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 816A0408, the last two digits are 08. Since 8 is divisible by 4, the entire number is divisible by 4 regardless of the value of digit A. As A can be any single digit from 0 to 9, its maximum possible value is 9.

Explanation

Using the standard order of operations (BODMAS/PEMDAS), evaluate division and multiplication from left to right first, followed by addition and subtraction:

First, calculate the division and multiplication terms: (-56) ÷ (-14) = 4 (-4) × 7 = -28

Now, substitute these back into the expression: (-9) - 4 + (-28) = -9 - 4 - 28 = -41

Explanation

Profit is shared in the ratio of the product of capital and time for each partner.

Let the initial monthly investments of X, Y, and Z be 4x, 2x, and 5x respectively.

Effective investment for 12 months: X = (4x × 6) + (2x × 6) = 36x Y = 2x × 12 = 24x Z = 5x × 12 = 60x

Profit sharing ratio (X : Y : Z) = 36 : 24 : 60 = 3 : 2 : 5

Total profit ratio parts = 3 + 2 + 5 = 10 parts Difference between X's and Y's profit share = 3 - 2 = 1 part

Difference in profit = (1 / 10) × ₹37,000 = ₹3,700

Explanation

Let the son's present age be x years. Then, Sharad's present age is 3x years and his daughter's present age is (x - 3) years.

The sum of their present ages is 3x + x + (x - 3) = 5x - 3.

Since 3 years ago each person was 3 years younger, the sum of their ages reduced by 3 × 3 = 9 years: (5x - 3) - 9 = 63 5x - 12 = 63 5x = 75 x = 15

Sharad's present age = 3x = 3 × 15 = 45 years.

Explanation

Area of a regular hexagon = (3√3 / 2) × a², where a is the side length.

Given a = 14√3 cm: a² = (14√3)² = 196 × 3 = 588 cm²

Area = (3√3 / 2) × 588 = 3√3 × 294 = 882√3 cm²

Explanation

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

Selling the article at 6/15 (or 2/5) of its actual selling price results in a 28% loss: (2/5) × S = (100 - 28)% of C 0.4 × S = 0.72 × C S = (0.72 / 0.4) × C = 1.8 C

When sold at 89% of its actual selling price, the new selling price is: New SP = 0.89 × S = 0.89 × 1.8 C = 1.602 C

Profit percentage = [(1.602 C - C) / C] × 100 = 0.602 × 100 = 60.2%

Explanation

When two quadratic equations a₁x² + b₁x + c₁ = 0 and a₂x² + b₂x + c₂ = 0 have both roots in common, the ratio of their corresponding coefficients is equal: a₁/a₂ = b₁/b₂ = c₁/c₂.

Rearranging the given equations into standard form: (8A + 4)x² + Bx + (4A - 8) = 0 (6A + 3)x² + Kx + (3A - 6) = 0

Applying the common roots condition: (8A + 4)/(6A + 3) = B/K = (4A - 8)/(3A - 6)

Simplifying the constant ratios: 4(2A + 1) / 3(2A + 1) = 4/3 4(A - 2) / 3(A - 2) = 4/3

Thus: B/K = 4/3 3B = 4K 3B - 4K = 0

Substituting this into the expression: 9B² - 16K² = (3B - 4K)(3B + 4K) = 0 × (3B + 4K) = 0

Explanation

Given a:b = 2:3, substitute a = 2 and b = 3 into the ratio:

7a³ - 2b³ = 7(2)³ - 2(3)³ = 7(8) - 2(27) = 56 - 54 = 2

b³ - a³ = 3³ - 2³ = 27 - 8 = 19

The ratio (7a³ - 2b³) : (b³ - a³) evaluates to 2 : 19.

Explanation

For any triangle, the angle subtended by a side at the in-centre O is related to the opposite vertex angle by the formula: ∠QOR = 90° + ½ ∠QPR

Substituting the given value ∠QOR = 130°: 130° = 90° + ½ ∠QPR ½ ∠QPR = 40° ∠QPR = 80°

Explanation

Let total work to print one lakh books be LCM(8, 16, 24) = 48 units.

Work done per hour: P = 48 / 8 = 6 units Q = 48 / 16 = 3 units R = 48 / 24 = 2 units

In one complete 3-hour cycle (P, Q, R), work completed = 6 + 3 + 2 = 11 units.

In the first 9 hours (3 full cycles), total work done = 3 × 11 = 33 units. Remaining work = 48 - 33 = 15 units.

After P stops, Q and R work alternately. In a 2-hour cycle of Q and R, work done = 3 + 2 = 5 units.

Time required to complete the remaining 15 units = (15 / 5) × 2 = 6 hours.

Explanation

Number of cylinders cut along length = 8 / 1 = 8 Number of cylinders cut along width = 8 / 1 = 8 Number of cylinders cut along height = 8 / 4 = 2 Total number of cylinders cut = 8 × 8 × 2 = 128

Total surface area of one cylinder = 2πr(h + r) Given radius r = 0.5 m and height h = 4 m: Surface area = 2 × (22/7) × 0.5 × (4 + 0.5) = 99/7 m²

Cost of painting one cylinder = (99/7) × 14 = ₹198

Total cost to paint all 128 cylinders = 128 × 198 = ₹25,344

Explanation

Let the two numbers be x and y.

Given: Sum: x + y = 56 Difference: x - y = 56 × (1/7) = 8

Solving for x and y: x = (56 + 8) / 2 = 32 y = (56 - 8) / 2 = 24

To find the LCM of 32 and 24: Prime factorizations: 32 = 2⁵ and 24 = 2³ × 3 LCM(32, 24) = 2⁵ × 3 = 96

Explanation

For a sum of money to double itself under simple interest, the interest earned equals the principal sum, which means the total percentage interest earned is 100%.

Using the formula: Rate (R) × Time (T) = 100

Substituting T = 16 years: R = 100 / 16 = 6.25% per year

Explanation

The overall average spent per member is given by the weighted average formula:

Overall Average = Total Expenditure ÷ Total Members Total Expenditure = (78 × 136) + (22 × 383) + (34 × 457) = 10608 + 8426 + 15538 = ₹34572 Total Members = 78 + 22 + 34 = 134

Overall Average = 34572 ÷ 134 = ₹258

Explanation

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

Using the successive percentage formula: Final price paid by R = x × (1 + Profit%) × (1 - Loss%)

3234 = x × (1 + 0.5) × (1 - 0.6) 3234 = x × 1.5 × 0.4 3234 = 0.6x x = 3234 / 0.6 = 5390

Explanation

Cost price of X (CP) = ₹500 Selling price of X (SP_X) = CP + Profit = 500 + 958 = ₹1458

Two successive discounts of 50% and 28% result in a selling price equal to (1 − 0.50) × (1 − 0.28) = 0.36 or 36% of the Marked Price (MP). 36% of MP = ₹1458 MP = 1458 / 0.36 = ₹4050

For article Y: Single discount = 77% Selling price of Y (SP_Y) = 4050 × (1 − 0.77) = 4050 × 0.23 = ₹931.5

Profit on Y = 931.5 − 500 = ₹431.5 Profit percentage on Y = (431.5 / 500) × 100 = 86.3%

Explanation

The third proportional x to two numbers a and b satisfies the proportion a : b = b : x, giving the formula x = b² / a.

Substituting a = 16 and b = 24: x = 24² / 16 = 576 / 16 = 36

Explanation

Let the marked price (MP) of the article be 100.

The first selling price (SP₁) = 3/10 × 100 = 30. Since a loss of 25% is incurred at SP₁, the cost price (CP) = SP₁ / (1 - 0.25) = 30 / 0.75 = 40.

The second selling price (SP₂) = 69% of MP = 69.

Profit = SP₂ - CP = 69 - 40 = 29.

Profit percentage = (Profit / CP) × 100 = (29 / 40) × 100 = 72.5%.

Explanation

The profit-sharing ratio is calculated by multiplying each partner's capital by the duration for which it was invested.

Let the initial capitals of X, Y, and Z be 9, 5, and 2 units respectively.

X's effective investment for 12 months = (9 × 6) + (4.5 × 6) = 54 + 27 = 81 units Y's effective investment for 12 months = 5 × 12 = 60 units Z's effective investment for 12 months = 2 × 12 = 24 units

Profit ratio X : Y : Z = 81 : 60 : 24 = 27 : 20 : 8

Total profit units = 27 + 20 + 8 = 55 units

Value of 1 unit = 88,000 ÷ 55 = ₹1,600

Difference between X's and Y's profit = (27 - 20) × 1,600 = 7 × 1,600 = ₹11,200

Explanation

The algebraic identity a³ + b³ + c³ = 3abc holds when a + b + c = 0.

Rewrite the given equation as: (5x - 1)³ + (x + 3)³ + (6x - 8)³ = 3(5x - 1)(x + 3)(6x - 8)

Here, a = 5x - 1, b = x + 3, and c = 2(3x - 4) = 6x - 8.

Setting a + b + c = 0 gives: (5x - 1) + (x + 3) + (6x - 8) = 0 12x - 6 = 0 12x = 6 x = ½

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 = 281 (66/7) × r² = 281 r² = (281 × 7) / 66 r² = 1967 / 66 r = √(1967 / 66) cm

Explanation

Time required by pipe P to fill the full tank = 15 × (9/5) = 27 hours. Time required by pipe Q to fill the full tank = 21 × (9/7) = 27 hours.

Fraction of tank filled by P and Q together in 3 hours = 3 × (1/27 + 1/27) = 6/27 = 2/9.

Pipe R empties this 2/9 part of the tank in 4 hours, so the emptying rate of R = (2/9) / 4 = 1/18 tank per hour.

When P, Q, and R operate together, net filling rate = 1/27 + 1/27 - 1/18 = 2/27 - 1/18 = 4/54 - 3/54 = 1/54 tank per hour.

Time taken by P, Q, and R together to fill the empty tank = 1 / (1/54) = 54 hours.

Explanation

Rate of work of B = Rate of (A + B) - Rate of A = 1/20 - 1/45 = (9 - 4)/180 = 5/180 = 1/36 per day.

Since B alone completes the total work in 36 days, the time required for B alone to complete three-fourth (¾) of the work is ¾ × 36 = 27 days.

Explanation

Using the successive percentage change formula, net percentage increase = A + B + (A × B)/100 = 15 + 20 + (15 × 20)/100 = 38%.

Population after two years = 2,90,000 × (1 + 38/100) = 2,90,000 × 1.38 = 4,00,200.

Explanation

The formula for round-trip distance when total time t, speed in still water u, and stream speed v are given is:

Distance = t × (u² - v²) / (2 × u)

Given: u = 8 km/hr v = 2 km/hr Total time t = 3 hours 12 minutes = 3 + 12/60 = 16/5 hours

Substituting the values: Distance = (16/5) × (8² - 2²) / (2 × 8) Distance = (16/5) × 60 / 16 Distance = 60 / 5 = 12 km

Explanation

Combining the fractions over a common denominator gives:

[cos² θ + (1 + sin θ)²] / [cos θ (1 + sin θ)]

Expanding the numerator using (1 + sin θ)² = 1 + 2 sin θ + sin² θ:

[cos² θ + sin² θ + 1 + 2 sin θ] / [cos θ (1 + sin θ)]

Using the Pythagorean identity cos² θ + sin² θ = 1:

[1 + 1 + 2 sin θ] / [cos θ (1 + sin θ)] = [2(1 + sin θ)] / [cos θ (1 + sin θ)]

Canceling the common factor (1 + sin θ) yields 2 / cos θ = 2 sec θ.

Explanation

When a sum of money doubles itself on simple interest, the interest earned (SI) is equal to the principal (P).

Using the simple interest formula: SI = (P × R × T) / 100

Substituting SI = P and time T = 20 years: P = (P × R × 20) / 100 1 = 20R / 100 R = 100 / 20 = 5%

Alternatively, using the direct shortcut for a sum doubling itself: Rate (R) = 100 / T = 100 / 20 = 5% per annum.

Explanation

Let the first part be x and the second part be y.

According to the given condition, one-fifth of the first part and one-eighth of the second part are in the ratio 3:4:

(x / 5) / (y / 8) = 3 / 4

(x / 5) × (8 / y) = 3 / 4

8x / 5y = 3 / 4

x / y = (3 × 5) / (4 × 8) = 15 / 32

This gives the ratio x : y = 15 : 32.

Explanation

In a 150-meter race, when R covers 150 m, S covers 150 - 6 = 144 m. Ratio of speeds, v_R : v_S = 150 : 144 = 25 : 24, giving v_R = (25/24)v_S.

In a 250-meter race, R beats S by 2 seconds: (250 / v_S) - (250 / v_R) = 2 (250 / v_S) - (240 / v_S) = 2 10 / v_S = 2 v_S = 5 m/s

Speed of R, v_R = (25/24) × 5 = 125/24 m/s.

In 30 seconds: Distance covered by R = (125/24) × 30 = 156.25 m Distance covered by S = 5 × 30 = 150 m

Distance by which R beats S = 156.25 - 150 = 6.25 meters.

Explanation

Since the population increases at a rate of 25% (or ¼) per year, the population in any given year is 5/4 times the population of the preceding year. Working backward from the current population:

Population 2 years ago = 450000 × (4/5)² = 450000 × (16/25) = 288000 Population 3 years ago = 450000 × (4/5)³ = 450000 × (64/125) = 230400

The positive difference between the population 2 years ago and 3 years ago is: 288000 - 230400 = 57600

Alternatively, the increase from 3 years ago to 2 years ago is 25% (¼) of the population 3 years ago: 230400 ÷ 4 = 57600

Explanation

Let the maximum marks be M.

According to the given condition: 92% of M − 60 = 768 92% of M = 828 M = (828 × 100) / 92 = 900

Percentage scored by friend = (738 / 900) × 100 = 82%

Explanation

For a quadratic equation ax² + bx + c = 0, the nature of roots is determined by the discriminant D = b² - 4ac.

Rearranging x - 1/x = 17 gives x² - 17x - 1 = 0, where a = 1, b = -17, and c = -1.

Calculating the discriminant: D = (-17)² - 4(1)(-1) = 289 + 4 = 293

Since D > 0 and 293 is not a perfect square, the roots given by x = (17 ± √293) / 2 are real, distinct, and irrational.

Explanation

Supplementary angles add up to 180°. Let x be the measure of the smaller angle in degrees. The larger angle is given as x + 32°.

x + (x + 32) = 180 2x + 32 = 180 2x = 148 x = 74°

Explanation

For similar cones, the ratio of their surface areas equals the square of the ratio of their corresponding base radii.

Given that the smaller upper cone has half the curved surface area of the original cone: (r'r)² = ½ r'r = 1/√2 r = r' × √2

where r' is the radius of the smaller cone and r is the radius of the original cone.

Since the diameter of the smaller cone is 6√2 cm, its radius r' is (6√2) / 2 = 3√2 cm.

Substituting r' into the relation: r = 3√2 × √2 = 6 cm.

Explanation

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

Total number of articles given away by the trader = 14 + 2 = 16 articles. Total Cost Price for the trader = 16 × CP

The customer pays for 14 articles at a 5% discount on the marked price: Discounted price per article paid = 0.95 × MP Total Revenue = 14 × 0.95 × MP = 13.3 × MP

Since the trader gains 33% profit in the entire transaction: Total Revenue = 16 × CP × 1.33 = 21.28 × CP

Equating total revenue: 13.3 × MP = 21.28 × CP MP / CP = 21.28 / 13.3 = 1.6

Thus, the marked price is 60% more than the cost price.

Explanation

Cost Price (CP) per article = ₹102 ÷ 12 = ₹8.50 Selling Price (SP) per article = ₹68

Profit per article = SP - CP = ₹68 - ₹8.50 = ₹59.50

Profit Percentage = (Profit ÷ CP) × 100 = (59.50 ÷ 8.50) × 100 = 7 × 100 = 700%

Explanation

For an odd number of observations n, the median is given by the ((n + 1) / 2)th observation when the data is arranged in ascending order.

Here, n = 11 students, so the median is the ((11 + 1) / 2)th = 6th value.

Arranging the given marks in ascending order: 45, 49, 54, 57, 58, 60, 62, 63, 66, 68, 71

The 6th observation in this sorted list is 60.

Explanation

First expression = (x².5 / y⁰.5) × √(xy) = (x².5 / y⁰.5) × x⁰.5 × y⁰.5 = x⁽2.5 + 0.5) = x³

Second expression = (y¹.5 / x⁰.5) × √(xy) = (y¹.5 / x⁰.5) × x⁰.5 × y⁰.5 = y⁽1.5 + 0.5) = y²

Since x and y are co-prime integers greater than 1, x³ and y² have no common prime factors.

LCM(x³, y²) = x³ × y² = x × x² × y² = x(xy)²

HCF(x³, y²) = 1