RRB NTPC CBT 1 2026 Mathematics Questions with Solutions

Explanation

Cost price of X = ₹1,000 Selling price of X = 1,000 + 337 = ₹1,337

After two successive discounts of 15% and 30%, the selling price of X in terms of marked price (MP) is: SP of X = MP × (1 − 0.15) × (1 − 0.30) = MP × 0.85 × 0.70 = MP × 0.595

After a single discount of 49%, the selling price of Y is: SP of Y = MP × (1 − 0.49) = MP × 0.51

Taking the ratio of SP of Y to SP of X: SP of Y / SP of X = 0.51 / 0.595 = 6 / 7

SP of Y = 1,337 × (6 / 7) = 191 × 6 = ₹1,146

With CP of Y = ₹1,000: Profit on Y = 1,146 − 1,000 = ₹146 Profit percentage on Y = (146 / 1,000) × 100 = 14.6%

Explanation

The product of two numbers is equal to the product of their HCF and LCM. ab = HCF × LCM = 3 × 1350 = 4050

Using the algebraic identity: (b - a)² = (a + b)² - 4ab (b - a)² = 129² - 4 × 4050 (b - a)² = 16641 - 16200 = 441

Since a < b, b - a = √441 = 21.

Now, calculate 1/a - 1/b: 1/a - 1/b = (b - a) / ab = 21 / 4050 = 7 / 1350

Explanation

Side of the square = √3969 = 63 cm.

Perimeter of the square = 4 × 63 = 252 cm.

Since the perimeter of the square equals the perimeter of the regular hexagon: Side of the regular hexagon = 252 / 6 = 42 cm.

Area of a regular hexagon = (3√3 / 2) × side² = (3√3 / 2) × 42² = (3√3 / 2) × 1764 = 3√3 × 882 = 2646√3 cm²

Explanation

Number of words per line = 36 / 4 = 9 words.

Total number of words in 93 lines = 93 × 9 = 837 words.

Explanation

When a journey covers equal distances at speeds v₁ and v₂, the average speed for the round trip is given by the harmonic mean formula:

v_avg = (2 × v₁ × v₂) / (v₁ + v₂)

Substituting v₁ = 144 km/h and v_avg = 126 km/h into the formula:

126 = (2 × 144 × v₂) / (144 + v₂)

Simplifying by dividing both sides by 18:

7 = (16 × v₂) / (144 + v₂)

7 × (144 + v₂) = 16 × v₂

1008 + 7 × v₂ = 16 × v₂

9 × v₂ = 1008

v₂ = 112 km/h

Explanation

Population after 2 years = Initial Population × (1 + r₁/100) × (1 + r₂/100)

Substituting the given values: Population = 2,90,000 × (105/100) × (130/100) = 2,90,000 × 1.05 × 1.3 = 2,90,000 × 1.365 = 3,95,850

Explanation

The mode of grouped frequency distribution is given by the formula:

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

The modal class is 20-30 because it has the highest frequency (36).

Lower limit of modal class (L) = 20 Frequency of modal class (f₁) = 36 Frequency of preceding class (f₀) = 16 Frequency of succeeding class (f₂) = 34 Class interval width (h) = 10

Substituting the values into the formula:

Mode = 20 + [(36 - 16) / (2 × 36 - 16 - 34)] × 10 Mode = 20 + [20 / (72 - 50)] × 10 Mode = 20 + (20 / 22) × 10 Mode = 20 + 9.0909 = 29.09

Explanation

Downstream speed = Distance / Time = 120 / 6 = 20 km/hr Upstream speed = Distance / Time = 120 / 10 = 12 km/hr

Speed of the boat in still water = ½ × (Downstream speed + Upstream speed) = ½ × (20 + 12) = 16 km/hr

Explanation

When two parallel lines are intersected by a transversal, alternate interior angles are equal in measure. Therefore, if one alternate interior angle measures 65°, the corresponding alternate interior angle also measures 65°.

Explanation

When two parallel lines are intersected by a transversal, alternate interior angles are equal. In this configuration, ∠AMN and ∠MND are alternate interior angles situated on opposite sides of the transversal PQ between the parallel lines AB and CD. Therefore, ∠AMN = ∠MND = 47°.

Explanation

Speed of Ankur = 5/3 m/s Speed of Mahima = 25/12 m/s Total length of track X and track Y = 0.325 km = 325 m

Since time = distance / speed, equating the time taken for 9 rounds of track X by Ankur and 5 rounds of track Y by Mahima: (9 × Lx) / (5/3) = (5 × Ly) / (25/12) 27 Lx / 5 = 12 Ly / 5 27 Lx = 12 Ly Lx / Ly = 4 / 9

Using Lx + Ly = 325 m: Lx = 325 × 4 / 13 = 100 m

Time taken by Mahima to complete one round of track X: Time = Lx / Speed of Mahima = 100 / (25/12) = 100 × 12 / 25 = 48 seconds.

Explanation

The highest frequency is 40, which lies in the class interval 65–75. Thus, the modal class is 65–75.

Using the mode formula for grouped data: Mode = L + [(f₁ - f₀) / (2f₁ - f₀ - f₂)] × h

Where: L = lower limit of the modal class = 65 f₁ = frequency of the modal class = 40 f₀ = frequency of the class preceding the modal class = 30 f₂ = frequency of the class succeeding the modal class = 34 h = class width = 10

Mode = 65 + [(40 - 30) / (2 × 40 - 30 - 34)] × 10 Mode = 65 + [10 / (80 - 64)] × 10 Mode = 65 + (10 / 16) × 10 Mode = 65 + 6.25 = 71.25

Explanation

Let x, y, and z be the number of ₹5, ₹2, and ₹1 coins respectively.

From the given conditions: x + y + z = 112 5x + 2y + z = 212 y = x + 50

Substituting y = x + 50 into the total coins and total value equations: 2x + z = 62 7x + z = 112

Subtracting the first equation from the second equation: 5x = 50 x = 10

Substituting x = 10 into 2x + z = 62: 2(10) + z = 62 z = 42

Explanation

Let the total capacity of the tank be 24 units (LCM of 4, 24, and 6).

Filling rate of first pipe = 24 ÷ 4 = 6 units/hour Filling rate of second pipe = 24 ÷ 24 = 1 unit/hour Emptying rate of third pipe = 24 ÷ 6 = 4 units/hour

Net rate with all three pipes open = 6 + 1 - 4 = 3 units/hour

Half capacity of the tank = 24 ÷ 2 = 12 units

Time required to fill half the tank = 12 ÷ 3 = 4 hours.

Explanation

Since the slant height equals the base diameter, the vertical cross-section of the cone is an equilateral triangle with side length equal to the diameter, a = 2 × 42√3 = 84√3 cm.

The radius of the largest sphere inscribed in such a cone equals the inradius of this equilateral triangle: Inradius rₛ = a / (2√3) = (84√3) / (2√3) = 42 cm.

Volume of the inscribed sphere = (4/3) × π × (rₛ)³ = (4/3) × (22/7) × 42 × 42 × 42 = 4 × 22 × 2 × 42 × 42 = 310464 cm³

Explanation

The average of consecutive numbers in an arithmetic progression is given by the formula (First term + Last term) / 2. For natural numbers from 1 to 60, the first term is 1 and the last term is 60. Substituting these values into the formula gives Average = (1 + 60) / 2 = 61 / 2 = 30.5. Alternatively, using the formula for the average of the first n natural numbers, Average = (n + 1) / 2 = (60 + 1) / 2 = 30.5.

Explanation

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

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

Given: n₁ = 39, A₁ = ₹377 n₂ = 33, A₂ = ₹401 n₃ = 52, A₃ = ₹233

Total expenditure = (39 × 377) + (33 × 401) + (52 × 233) = 14,703 + 13,233 + 12,116 = ₹40,052

Total number of members = 39 + 33 + 52 = 124

Overall average spent per member = 40,052 / 124 = ₹323

Explanation

Convert all terms on the left-hand side to powers of 5: (125)¹.3 = (5³)¹.3 = 5³.9 (25)¹.8 = (5²)¹.8 = 5³.6

Substitute these into the expression: 5³.9 × 5³.6 ÷ 5².5 = 5ˣ

Applying the laws of exponents: 5⁽3.9 + 3.6 - 2.5) = 5ˣ 5⁵ = 5ˣ

Equating exponents gives x = 5.

Now, calculate the value of 18x - 5: 18(5) - 5 = 90 - 5 = 85

Explanation

To combine the ratios a : b = 1 : 3, b : c = 5 : 7, and c : d = 9 : 5, equalize the common terms:

First, equalize b between a : b and b : c by multiplying (1 : 3) by 5 and (5 : 7) by 3: a : b : c = 5 : 15 : 21

Next, equalize c between a : b : c and c : d by finding the LCM of 21 and 9, which is 63. Multiply (5 : 15 : 21) by 3 and (9 : 5) by 7: a : b = 15 : 45 b : c = 45 : 63 c : d = 63 : 35

Combining these gives a : b : c : d = 15 : 45 : 63 : 35.

Explanation

The volume of a sphere is given by the formula V = (4/3) × π × r³, where r is the radius.

Given the diameter d = 37.8 cm, the radius r = 37.8 / 2 = 18.9 cm.

Substituting the values into the formula with π = 22/7: V = (4/3) × (22/7) × (18.9)³ V = (4 / 21) × 22 × 6751.269 V = (88 / 21) × 6751.269 V = 88 × 321.489 V = 28291.032 cm³

Rounding to two places of decimal gives 28291.03 cm³.

Explanation

Let the cost price (CP) be ₹100.

Marked price (MP) = 100 + 150% of 100 = ₹250.

To achieve a profit of 60%, selling price (SP) = 100 + 60% of 100 = ₹160.

Discount allowed = MP - SP = 250 - 160 = ₹90.

Discount percentage = (Discount / MP) × 100 = (90 / 250) × 100 = 36%.

Explanation

Let f₁(p) = p² - 2p + 4 = (p - 1)² + 3 and f₂(p) = -p² + 2p - 4 = -((p - 1)² + 3).

For any p in the interval 0 ≤ p ≤ 1, (p - 1)² ≥ 0, which means f₁(p) ≥ 3 > 0, while f₂(p) ≤ -3 < 0.

Since f₁(p) is strictly greater than f₂(p) over the entire range, the maximum function simplifies to: max {p² - 2p + 4, -p² + 2p - 4} = p² - 2p + 4 = (p - 1)² + 3

To find the minimum value of (p - 1)² + 3 for 0 ≤ p ≤ 1, the term (p - 1)² is minimized at p = 1: Minimum value = (1 - 1)² + 3 = 3

Explanation

For a principal P to become n times itself on simple interest in T years, the interest earned is (n - 1) × P.

Using the rate formula: R = [(n - 1) × 100] / T

Given n = 5 and T = 12 years: R = [(5 - 1) × 100] / 12 = (4 × 100) / 12 = 400 / 12 = 33.33%

Explanation

Chetan's normal monthly savings = ₹18000 - ₹5000 - ₹2000 = ₹11000.

Since he spends his entire monthly savings during his birthday month, he saves money for only 11 months in the year.

Total annual savings = 11 × ₹11000 = ₹121000.

Explanation

To find the least common multiple (LCM) of algebraic monomials, take the highest power of each variable present across the given expressions.

1st term = x³.⁵ × y¹.⁵ 2nd term = y².⁵ / x⁻⁴.⁵ = x⁴.⁵ × y².⁵ 3rd term = x / y⁻³ = x × y³

Highest exponent of x = max(3.5, 4.5, 1) = 4.5 Highest exponent of y = max(1.5, 2.5, 3) = 3

LCM = x⁴.⁵ × y³ = x³ × x¹⁵ × y³ = (xy)³ × x³⁾² = (xy)³ √(x³)

Explanation

Let the initial costs of compounds A, B, and C be 3, 5, and 7 units respectively. The total initial cost is 3 + 5 + 7 = 15 units.

Adding compound D increases the total cost by 40%, which means the cost of compound D is 40% of 15 = 6 units.

Since the relative ratio of A, B, and C remains 3:5:7, the value of x representing D's share in the combined ratio 3:5:7:x is 6.

Explanation

The proportion of boys in a class with a boys-to-girls ratio of B:G is given by B / (B + G).

First class (4:5): 4 / (4 + 5) = 4/9 ≈ 44.44% Second class (3:2): 3 / (3 + 2) = 3/5 = 60% Third class (6:5): 6 / (6 + 5) = 6/11 ≈ 54.55% Fourth class (5:6): 5 / (5 + 6) = 5/11 ≈ 45.45%

Comparing the fractions, 3/5 (60%) is greater than 6/11 (54.55%), 5/11 (45.45%), and 4/9 (44.44%).

Explanation

Evaluating the roots: ∛2197 = 13, since 13³ = 2197 ⁴√10000 = 10, since 10⁴ = 10000

Substituting these into the expression: (13 / 10) × (5 / 39) × 48

Grouping and simplifying the fractions: = (13 / 39) × (5 / 10) × 48 = (1 / 3) × (1 / 2) × 48 = 48 / 6 = 8

Explanation

Since all calculators have the same cost price, the overall profit or loss percentage can be calculated directly using the weighted average method based on the ratio of the quantities sold.

Ratio of calculators sold at a profit to those sold at a loss = 30 : 15 = 2 : 1

Overall Percentage = [2 × (+25%) + 1 × (-20%)] / (2 + 1) = (50% - 20%) / 3 = 30% / 3 = +10%

The positive result indicates an overall profit of 10%.

Explanation

In a right-angled triangle ABC with ∠B = 90°, angles A and C are complementary, which means A + C = 90°, or C = 90° - A.

By complementary angle trigonometric identities, cot C = cot(90° - A) = tan A.

Therefore, tan A - cot C = tan A - tan A = 0.

Explanation

Let the cost price of 1 gram of goods be ₹1, making the cost price of 1 kg (1000 grams) equal to ₹1000.

Since the shopkeeper professes to sell at a 20% loss, the selling price for 1 kg is: Selling Price = ₹1000 × (1 - 0.20) = ₹800

Let x be the actual weight in grams supplied to the customer. The cost price of x grams is ₹x. Since he earns an overall profit of 40%: Selling Price = Cost Price × (1 + Gain%) 800 = x × 1.40 x = 800 / 1.40 = 4000 / 7 ≈ 571.4 gm

Explanation

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

Sum of digits of 445A0208 = 4 + 4 + 5 + A + 0 + 2 + 0 + 8 = 23 + A

For (23 + A) to be divisible by 3, the single-digit values A can take are 1, 4, and 7 (since 24, 27, and 30 are divisible by 3).

The maximum possible value for A is 7.

Explanation

Let the total capacity of the cistern be C units.

Filling rate of the first tap = C/4 units/hour Filling rate of the second tap = C/32 units/hour Emptying rate of the third tap = C/32 units/hour

Net filling rate when all three taps are opened together: Net rate = C/4 + C/32 - C/32 = C/4 units/hour

Time required to fill 1/4 of the cistern: Time = (C/4) / (C/4) = 1 hour

Explanation

For a period of 2 years, the difference between compound interest and simple interest is given by the formula D = P × (R / 100)², where D is the interest difference, P is the principal amount invested, and R is the rate of interest per annum.

Given D = ₹53 and R = 4%: 53 = P × (4 / 100)² 53 = P × (1 / 25)² 53 = P / 625 P = 53 × 625 = ₹33,125

Explanation

Let the cost price (CP) be ₹100.

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

To gain 88%, the selling price (SP) must be: SP = 100 + 88 = ₹188

Discount allowed = MP - SP = 200 - 188 = ₹12

Discount percentage = (Discount / MP) × 100 = (12 / 200) × 100 = 6%

Explanation

For 7 consecutive integers with mean 0, the set of numbers is centered at zero: -3, -2, -1, 0, 1, 2, 3.

The sum of squared deviations from the mean is (-3)² + (-2)² + (-1)² + 0² + 1² + 2² + 3² = 9 + 4 + 1 + 0 + 1 + 4 + 9 = 28.

Variance = 28 / 7 = 4.

Standard deviation = √4 = 2.

Explanation

To convert a mixed recurring decimal 0.26̄ into a vulgar fraction, use the formula:

Fraction = (Total number formed by decimal digits - Non-recurring part) / (9 for each recurring digit followed by 0 for each non-recurring digit)

Numerator = 26 - 2 = 24 Denominator = 90

Simplifying 24/90 by dividing both numerator and denominator by 6 gives 4/15.

Explanation

In right-angled triangle ABC with ∠B = 90° and ∠ACB = 30°:

sin 30° = AB / AC ½ = 5 / AC AC = 10 cm

tan 30° = AB / BC 1 / √3 = 5 / BC BC = 5√3 cm

AC + BC = 10 + 5√3 cm

Explanation

When equal time is spent during each stage of a journey, the overall average speed is the arithmetic mean of the speeds in all stages.

Speed in stage 1, v₁ = x km/hr Speed in stage 2, v₂ = x + 0.5x = 1.5x km/hr Speed in stage 3, v₃ = 1.5x × 0.8 = 1.2x km/hr

Average speed = (v₁ + v₂ + v₃) / 3 48 = (x + 1.5x + 1.2x) / 3 48 = 3.7x / 3 3.7x = 144 x = 1440 / 37 ≈ 38.9 km/hr

Explanation

Let B's salary be 100. Since A's salary is 130% more than B's, A's salary = 100 + 130 = 230.

The percentage by which B's salary is less than A's salary is given by the formula: Percentage = [(A - B) / A] × 100

Substituting the values: Percentage = [(230 - 100) / 230] × 100 = (130 / 230) × 100 = 1300 / 23 = 56.52%

Explanation

Gopal Krishna Gokhale founded the Servants of India Society on June 12, 1905, in Pune, Maharashtra. The primary objective of the society was to unite and train Indians to work for the service of the nation, promote secular education, improve sanitation and health care, and advance social and economic reforms across the country.

Explanation

From the third equation, x + y - z = 0, expressing z in terms of x and y gives z = x + y.

Substituting z = x + y into the second equation, 2x - y + z = 3, yields 2x - y + (x + y) = 3, which simplifies to 3x = 3, giving x = 1.

Substituting z = x + y into the first equation, x + 4y + 2z = 12, gives x + 4y + 2(x + y) = 12, which simplifies to 3x + 6y = 12.

Substituting x = 1 into 3x + 6y = 12 gives 3(1) + 6y = 12, so 6y = 9, which simplifies to y = 3/2.

Finally, using z = x + y with x = 1 and y = 3/2 gives z = 1 + 3/2 = 5/2.

Explanation

Arranging the seven boxes from top to bottom based on the given conditions: Since only A is kept above E, A is at the 1st position and E is at the 2nd position. Since only S is kept between E and T, S is at the 3rd position and T is at the 4th position. Since only K is kept below R, R is at the 6th position and K is at the 7th position. This places the remaining box, M, at the 5th position.

The final arrangement from top to bottom is A, E, S, T, M, R, K. Between box M (5th position) and box K (7th position), there is only 1 box (R).

Explanation

The mode of a grouped frequency distribution is given by:

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

Since the mode is 36, the modal class is 30-40.

Lower limit of modal class (L) = 30 Class width (h) = 10 Frequency of modal class (f₁) = 9 Frequency of preceding class 20-30 (f₀) = x Frequency of succeeding class 40-50 (f₂) = 5

Substituting these values into the formula:

36 = 30 + [(9 - x) / (18 - x - 5)] × 10 6 = [(9 - x) / (13 - x)] × 10 6(13 - x) = 10(9 - x) 78 - 6x = 90 - 10x 4x = 12 x = 3

Explanation

Article 226 of the Constitution of India empowers High Courts to issue writs, directions, or orders to any person, authority, or government not only for the enforcement of Fundamental Rights but also "for any other purpose". The phrase "for any other purpose" extends the High Court's writ jurisdiction to the enforcement of ordinary legal and statutory rights, giving High Courts a broader writ jurisdiction than the Supreme Court in terms of subject matter.

Explanation

The angle subtended by the minor arc AB at the centre O is ∠AOB = 120°. The reflex angle subtended by the major arc AB at the centre is 360° - 120° = 240°.

Since S is a point on the minor arc AB, the angle ∠ASB is subtended by the major arc AB at the circumference. By the inscribed angle theorem, the angle subtended by an arc at any point on the circumference is half the central angle subtended by the same arc.

∠ASB = ½ × reflex ∠AOB = ½ × 240° = 120°.

Alternatively, choosing any point P on the major arc gives ∠APB = ½ × 120° = 60°. Since ASBP forms a cyclic quadrilateral, opposite angles sum to 180°, yielding ∠ASB = 180° - 60° = 120°.

Explanation

Influenza is an acute respiratory infection caused by influenza viruses, characterized by the sudden onset of systemic and upper respiratory symptoms. The classic clinical presentation includes fever, cough, and severe body aches or myalgia, accompanied by headaches, fatigue, and sore throat. High fever and widespread body aches stem from the host's inflammatory cytokine response to viral infection, while cough results from viral irritation and inflammation of the respiratory tract lining.

Explanation

Net filling rate per minute = 1/24 + 1/36 - 1/18

Taking the LCM of 24, 36, and 18, which is 72:

Net filling rate = (3 + 2 - 4) / 72 = 1/72 tank per minute

Total time taken to fill the cistern = 1 / (1/72) = 72 minutes.

Explanation

Let the present ages of P and Q be 3x and 4x years respectively.

After 6 years, their ages will be (3x + 6) and (4x + 6) years.

According to the question: (3x + 6) / (4x + 6) = 4 / 5

Cross-multiplying gives: 5 × (3x + 6) = 4 × (4x + 6) 15x + 30 = 16x + 24 x = 6

Present age of P = 3x = 3 × 6 = 18 years.

Shortcut method: The ratio changes from 3 : 4 to 4 : 5, representing an increase of 1 ratio unit for both P and Q in 6 years. 1 unit = 6 years Present age of P = 3 units = 3 × 6 = 18 years.

Explanation

Let √m = x (where x ≥ 0). Then m = x², and the given equation becomes: 2x² = 5x - 2 2x² - 5x + 2 = 0

Solving this quadratic equation: (2x - 1)(x - 2) = 0 x = ½ or x = 2

Since m = x²: m₁ = (½)² = ¼ m₂ = 2² = 4

The difference between the two values of m is: 4 - ¼ = 15/4