RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

In compound interest, the amount at the end of the first year serves as the principal for the second year.

Interest earned during the second year = Amount after 2nd year - Amount after 1st year = ₹550 - ₹500 = ₹50

Rate of interest per annum = (Interest in 2nd year / Amount after 1st year) × 100 = (50 / 500) × 100 = 10% per annum.

Explanation

Sum of five consecutive terms of an arithmetic progression is given by 5 × middle term.

Here, the middle term is 373 and the number of terms is 5.

Sum = 5 × 373 = 1865

Since 5 is a prime factor of the product, the sum is divisible by 5.

Explanation

For successive percentage increases of A% and B%, the effective total percentage increase is given by the formula:

Net Percentage Increase = A + B + (A × B / 100)

Substituting A = 20% and B = 30%:

Net Percentage Increase = 20 + 30 + (20 × 30 / 100) = 50 + 6 = 56%

Alternatively, assuming an initial income of 100, the first increase makes it 120, and a further 30% increase raises it to 120 × 1.30 = 156, resulting in an overall increase of 56%.

Explanation

In a parallelogram, adjacent angles are supplementary, so their sum is equal to 180°.

Let the smallest angle be x. The adjacent angle is given as 2x - 39°.

x + (2x - 39°) = 180° 3x - 39° = 180° 3x = 219° x = 73°

Explanation

Let the number of ₹1, 50 paise, and 25 paise coins be x, 2x, and 3x respectively.

Converting the values to rupees: Value of ₹1 coins = 1 × x = x Value of 50 paise coins = 0.50 × 2x = x Value of 25 paise coins = 0.25 × 3x = 0.75x

Total value = x + x + 0.75x = 2.75x = ₹110

x = 110 / 2.75 = 40

Number of 50 paise coins = 2x = 2 × 40 = 80.

Explanation

According to the division algorithm: Dividend = (Divisor × Quotient) + Remainder

Substituting the given values into the formula: 15,971 = (Divisor × 55) + 21 Divisor × 55 = 15,971 - 21 Divisor × 55 = 15,950 Divisor = 15,950 ÷ 55 = 290

Explanation

Sum of all 6 numbers (a + b + c + d + e + f) = 36 × 6 = 216. Sum of 3 numbers (b + d + f) = 28 × 3 = 84.

Subtracting the sum of (b, d, f) from the total sum gives: a + c + e = 216 - 84 = 132.

Mean of a, c, and e = 132 / 3 = 44.

Explanation

For any two positive numbers x and y, the product of the numbers is equal to the product of their LCM and HCF: x × y = LCM × HCF = 72

The difference between the two numbers is given as 1, so y = x - 1. Substituting this into the product equation gives: x(x - 1) = 72 x² - x - 72 = 0 (x - 9)(x + 8) = 0

Since x must be a positive number, x = 9, which gives y = 9 - 1 = 8. Therefore, the two numbers are 8 and 9.

Explanation

First number = 3 × 75 = 225.

Using the formula for two numbers: First number × Second number = HCF × LCM

Substituting the given values: 225 × Second number = 75 × 450 Second number = (75 × 450) / 225 = 150

Explanation

To form the smallest four-digit number using the digits 3, 5, 0, 6, the thousands place must be filled by the smallest non-zero digit, which is 3. The remaining digits are then arranged in ascending order (0, 5, 6) in the hundreds, tens, and units places to minimize the overall value, giving 3056.

Explanation

Work done by Vicky in 8 days = 8/40 = 1/5 of the total work. Remaining work = 1 - 1/5 = 4/5 of the total work.

Gurpreet Singh completes 4/5 of the work in 32 days, so Gurpreet's total time to complete the work alone = 32 × (5/4) = 40 days.

Since both Vicky and Gurpreet Singh can complete the work individually in 40 days, their combined 1-day work = 1/40 + 1/40 = 2/40 = 1/20.

Time taken together = 1 / (1/20) = 20 days.

Explanation

The sum of the cubes of the first n natural numbers is given by the formula: Sum = [n(n + 1) / 2]²

Substituting n = 10 into the formula: Sum = [10 × 11 / 2]² = 55² = 3025

Explanation

When a person covers the same distance at two different speeds x and y, the average speed for the entire journey is given by the formula:

Average Speed = 2xy / (x + y)

Substituting x = 110 km/h and y = 50 km/h:

Average Speed = (2 × 110 × 50) / (110 + 50) = 11000 / 160 = 68.75 km/h

Explanation

Let x be the number added to each term. Setting up the linear equation from the given ratio:

(7 + x) / (9 + x) = 13 / 14

Cross-multiplying gives:

14 × (7 + x) = 13 × (9 + x) 98 + 14x = 117 + 13x 14x - 13x = 117 - 98 x = 19

Adding 19 yields 7 + 19 = 26 and 9 + 19 = 28, forming the ratio 26:28 = 13:14.

Explanation

To determine the maximum number of 17 cm pieces, divide the total length of the rod by the length of each piece and take the integer portion:

Number of pieces = ⌊960 ÷ 17⌋ = 56

Multiplying 17 × 56 gives 952 cm, leaving a remainder of 8 cm, which is less than 17 cm and cannot form another complete piece.

Explanation

Cost Price (CP) = Selling Price × 100 / (100 + Profit%) CP = 230 × 100 / (100 + 15) = 230 × 100 / 115 = ₹200

When the book is sold for ₹210: Profit = New Selling Price - CP = 210 - 200 = ₹10 Profit% = (Profit / CP) × 100 = (10 / 200) × 100 = 5% profit

Explanation

Average weight = Σ(f × x) / Σf

Total weight of all passengers = (4 × 90) + (15 × 60) + (6 × 75) + (5 × 78) + (3 × 72) + (7 × 45) = 360 + 900 + 450 + 390 + 216 + 315 = 2631 kg

Total number of passengers = 40

Average weight = 2631 / 40 = 65.775 kg ≈ 65.77 kg

Explanation

Using the formula for time difference when distance is constant: Time difference = (Distance / Speed₁) − (Distance / Speed₂)

Let the distance be D km. Substituting the given speeds and time difference of half an hour (0.5 hours): D/3 − D/3.75 = 0.5 D/3 − 4D/15 = 0.5 (5D − 4D) / 15 = 0.5 D / 15 = 0.5 D = 15 × 0.5 = 7.5 km

Explanation

Let the cost price of the item be 100%.

The net percentage difference between a 7% profit and a 6% loss is: 7% + 6% = 13%

Given that this 13% difference corresponds to 104:

1% of Cost Price = 104 ÷ 13 = 8

Cost Price (100%) = 8 × 100 = 800

Explanation

Net profit percentage can be determined directly using the formula: Profit % = M - D - (M × D)/100, where M is the markup percentage and D is the discount percentage.

Substituting M = 50% and D = 25%: Profit % = 50 - 25 - (50 × 25)/100 = 25 - 12.5 = 12.50%

Alternatively, taking the cost price as ₹100: Marked Price = ₹100 + 50% of ₹100 = ₹150 Selling Price = ₹150 - 25% of ₹150 = 150 - 37.50 = ₹112.50 Actual Profit = ₹112.50 - ₹100 = ₹12.50, which equals a 12.50% profit.

Explanation

For equal annual installments under compound interest, the formula relating present debt P to annual installment x at interest rate r% over 2 years is:

P = x / (1 + r/100) + x / (1 + r/100)²

Substituting P = 1025 and r = 5:

1025 = x / (1 + 5/100) + x / (1 + 5/100)² 1025 = 20x / 21 + 400x / 441 1025 = (420x + 400x) / 441 1025 = 820x / 441 x = (1025 × 441) / 820 = 551.25

Explanation

Gain = Principal × Net Interest Rate Difference × Time / 100

Here, Principal = 8000, Rate difference = 7% - 5% = 2%, and Time = 3 years.

Gain = 8000 × 2 × 3 / 100 = 480.

Explanation

Using the rule of alligation, the ratio of the volume of the 80% drink to the volume of the 25% drink is:

(60 - 25) : (80 - 60) = 35 : 20 = 7 : 4

Since 4 parts of the ratio correspond to 36 litres of the 25% drink, 1 part equals 36 ÷ 4 = 9 litres.

The required volume of the 80% drink (7 parts) is 7 × 9 = 63 litres.

Explanation

Number of ice cubes = Volume of ice compartment / Volume of one ice cube

Volume of rectangular compartment = length × width × height = 8 cm × 5 cm × 4 cm = 160 cm³

Volume of one cubic ice cube = edge³ = (2 cm)³ = 8 cm³

Number of ice cubes = 160 / 8 = 20

Explanation

Subtracting 40 from a number reduces it to 60% of its original value, which means 40 represents 40% (100% − 60%) of the number.

Let the number be x. 0.40 × x = 40 x = 40 / 0.40 = 100

Explanation

Since radius OA is perpendicular to tangent PA at point A, ∠OAP = 90°.

Given ∠PAB = 55°, ∠OAB = ∠OAP − ∠PAB = 90° − 55° = 35°.

Since OA = OB (radii of the circle), ΔOAB is an isosceles triangle with ∠OBA = ∠OAB = 35°.

In ΔOAB, the sum of interior angles is 180°: ∠AOB = 180° − (∠OAB + ∠OBA) = 180° − (35° + 35°) = 110°.

Explanation

When a book is fully opened flat, the two opposing pages spread out completely along a single straight line in cross-section, forming a flat plane. The measure of the angle formed at the spine between the left and right pages is 180°, which is geometrically classified as a straight angle.

Explanation

The number 2 is an even prime number, so not every prime number is odd. Furthermore, the product of 2 and any other prime number, such as 2 × 3 = 6, results in an even number rather than an odd number.

Explanation

Expand the product using the distributive property:

((1/5)x - (1/6)y)(5x + 6y) = (1/5)x(5x + 6y) - (1/6)y(5x + 6y) = x² + (6/5)xy - (5/6)xy - y²

Combine the middle xy terms by finding a common denominator of 30:

(6/5 - 5/6)xy = ((36 - 25) / 30)xy = (11/30)xy

This simplifies the expression to x² + 11xy/30 - y².

Explanation

A 25% reduction on an expenditure of ₹600 saves 25% × 600 = ₹150. This saved amount buys 5 kg of additional wheat, so the reduced price of wheat per kg = 150 ÷ 5 = ₹30. Since the reduced price represents 75% (or 0.75) of the original price, the original price per kg = 30 ÷ 0.75 = ₹40.

Explanation

Let the three numbers be 1x, 2x, and 3x, where x is their highest common factor. Since the numbers 1, 2, and 3 are co-prime, the HCF of 1x, 2x, and 3x is x.

Given HCF = 12, we get x = 12.

Substituting x = 12 into the ratio: First number = 1 × 12 = 12 Second number = 2 × 12 = 24 Third number = 3 × 12 = 36

Explanation

Total mirrors in a dozen = 12.

For a ratio of broken to unbroken mirrors to be valid, the sum of the ratio terms must be a factor of the total number of mirrors, because individual mirrors cannot be fractional.

For the ratio 5:1, the sum of the ratio terms is 5 + 1 = 6. Since 6 divides 12 completely (12 ÷ 6 = 2), this gives 10 broken mirrors and 2 unbroken mirrors, which is a valid whole-number distribution.

Explanation

When a shopkeeper cheats by x% while buying and y% while selling, the overall gain percentage is calculated using the successive percentage change formula:

Net Gain % = x + y + (x × y) / 100

Substituting x = 10 and y = 10:

Net Gain % = 10 + 10 + (10 × 10) / 100 = 20 + 1 = 21%

Explanation

To arrange soldiers in equal rows and columns, the number of arranged soldiers must form a perfect square.

The largest integer whose square is less than or equal to 40125 is 200, because: 200² = 40000 201² = 40401

The maximum number of soldiers that can form a square grid is 40000 (200 rows of 200 soldiers each).

Number of soldiers left out = 40125 − 40000 = 125.

Explanation

Using the algebraic identity a² + b² - 2ab = (a - b)², let a = 0.2 and b = 0.02:

Numerator = (0.2)² + (0.02)² - 2 × 0.2 × 0.02 = (0.2 - 0.02)² = (0.18)² = 0.0324

Dividing by the denominator: 0.0324 / 0.36 = 0.09

Explanation

A rational number is defined as any number that can be expressed in the form p/q, where p and q are integers and q ≠ 0. Every integer z can be written as z/1, where both z and 1 are integers and the denominator is non-zero. This satisfies the definition of a rational number, which makes statement 2 true.

Explanation

By the inverse trigonometric identity, for any real number x satisfying |x| ≥ 1:

sec⁻¹x + cosec⁻¹x = π/2

Substituting this identity into the given expression yields:

cos(sec⁻¹x + cosec⁻¹x) = cos(π/2) = 0

Explanation

Let the number be x.

According to the given condition: (2/5) × (1/4) × (3/7) × x = 15

Simplifying the fraction: (3/70) × x = 15 x = (15 × 70) / 3 x = 350

Half of the number = 350 / 2 = 175

Explanation

Let the five-digit number be represented by digits in the positions of ten thousand's (T), thousand's (Th), hundred's (H), ten's (t), and unit's (u).

Given: Hundred's digit (H) = 2 Unit's digit (u) = 2 × H = 2 × 2 = 4 Thousand's digit (Th) = 0 Ten thousand's digit (T) = H + u = 2 + 4 = 6 Ten's digit (t) = T - 1 = 6 - 1 = 5

Placing these digits together gives T Th H t u = 60254.

Explanation

Let the daily efficiency of B be 2 units/day. Since A is 50% as efficient as B, the efficiency of A is 1 unit/day. The combined daily work of A and B is 1 + 2 = 3 units/day.

Since C completes half of the work done by A and B together in a day, the efficiency of C is 3 / 2 = 1.5 units/day.

Total work = Efficiency of C × Time taken by C = 1.5 × 40 = 60 units.

Combined daily efficiency of A, B, and C = 1 + 2 + 1.5 = 4.5 units/day.

Time taken by A, B, and C together = Total work / Combined efficiency = 60 / 4.5 = 40/3 = 13⅓ days.

Explanation

Time = Distance / Speed

Given Distance = 250 km and Speed = 75 km/h: Time = 250 / 75 = 10 / 3 hours = 3⅓ hours

Converting ⅓ hour into minutes: ⅓ × 60 = 20 minutes

Therefore, the total time required is 3 hours 20 minutes.

Explanation

The principal value branch of sin⁻¹(x) is [-π/2, π/2].

Since sin(π/4) = 1/√2 and π/4 lies in the interval [-π/2, π/2], we have sin⁻¹(1/√2) = π/4.

Explanation

The minute hand completes one full revolution of 360° in 1 hour (60 minutes), giving it a speed of 6° per minute. The hour hand completes one full revolution of 360° in 12 hours (720 minutes), giving it a speed of 0.5° per minute.

The ratio of the velocity of the hour hand to that of the minute hand is 0.5 : 6 = 1 : 12.

Explanation

Complementary angles sum to 90°, so the complement of 15° is 90° − 15° = 75°.

Supplementary angles sum to 180°, so the supplement of 125° is 180° − 125° = 55°.

The required sum is 75° + 55° = 130°.

Explanation

Prime factorization of 294 gives: 294 = 2¹ × 3¹ × 7²

For a number to be a perfect square, the exponent of each prime factor in its factorization must be an even number. The prime factors 2 and 3 both have an odd exponent of 1.

To make all exponents even, 294 must be multiplied by 2 × 3 = 6.

Explanation

To find the least number to be added to 1300 to make it a perfect square, identify the smallest perfect square greater than 1300.

Evaluating adjacent squares around 1300: 36² = 1296 37² = 1369

Since 1296 < 1300 < 1369, the smallest perfect square greater than 1300 is 37² = 1369.

Subtracting 1300 from 1369 gives the required least number: 1369 - 1300 = 69

Explanation

Substituting B = 2 and A = 5 into the given expression: BA = 25 B3 = 23

Multiplying these two numbers gives: 25 × 23 = 575

This matches 57A where the units digit A is 5.

Explanation

Let the measure of angle C be x. Then, angle A = x + 12° and angle B = 4x.

Using the angle sum property of a triangle: Angle A + Angle B + Angle C = 180° (x + 12°) + 4x + x = 180° 6x + 12° = 180° 6x = 168° x = 28°

Substituting x = 28° gives: Angle A = 28° + 12° = 40° Angle B = 4 × 28° = 112° Angle C = 28°

Explanation

Mean = Sum of observations / Number of observations = 52 / 13 = 4.

Arranging the 13 observations in ascending order: 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, 8.

The mode is the value with the highest frequency, which is 3 (appearing 3 times).

Since the mean is 4 and the mode is 3, Mean ≠ Mode.

Explanation

When two articles are purchased at the same cost price, the overall profit percentage is simply the arithmetic mean of the individual profit percentages.

Overall Profit Percentage = (Profit₁% + Profit₂%) / 2 = (10% + 5%) / 2 = 15% / 2 = 7.5%

Alternatively, total cost price = ₹3,000 + ₹3,000 = ₹6,000. Total profit = (10% of ₹3,000) + (5% of ₹3,000) = ₹300 + ₹150 = ₹450. Total Profit Percentage = (450 / 6000) × 100 = 7.5%