RRB NTPC CBT 1 2021 Mathematics Questions with Solutions

Explanation

Using the trigonometric identities cosecθ = 1/sinθ and cotθ = cosθ/sinθ:

(cosecθ + cotθ) = (1 + cosθ) / sinθ

Multiplying this by (1 − cosθ):

(cosecθ + cotθ) × (1 − cosθ) = [(1 + cosθ) / sinθ] × (1 − cosθ) = (1 − cos²θ) / sinθ

Using the Pythagorean identity 1 − cos²θ = sin²θ:

= sin²θ / sinθ = sinθ

Explanation

For an even number of observations n = 8, the median is the arithmetic mean of the (n/2)th and (n/2 + 1)th terms, which are the 4th and 5th terms.

Median = (4th term + 5th term) / 2

Substituting the 4th term (x), 5th term (x + 1), and median (9.5):

(x + x + 1) / 2 = 9.5 2x + 1 = 19 2x = 18 x = 9

Explanation

Rearranging the given equation to group terms containing x on one side and constant terms on the other:

1/x - 1/(2x) = 3/4 - 2/3

Simplifying both sides:

(2 - 1)/(2x) = (9 - 8)/12

1/(2x) = 1/12

Equating the denominators yields:

2x = 12 x = 6

Explanation

Speed = Distance / Time.

Distance = 1000 m = 1 km Time = 10 min = 10/60 h = 1/6 h

Speed = 1 / (1/6) = 6 km/h.

Explanation

To convert 2.47 hours into hours, minutes, and seconds:

The whole number part gives 2 hours.

Multiply the remaining decimal portion by 60 to convert it into minutes: 0.47 × 60 = 28.2 minutes

The whole number part gives 28 minutes.

Multiply the remaining decimal portion of the minutes by 60 to convert it into seconds: 0.2 × 60 = 12 seconds

Combining these components yields 2 hours, 28 minutes, and 12 seconds.

Explanation

The total number of smaller cubes formed by dividing a larger cube is given by the formula N = (L / l)³, where L is the side length of the larger cube and l is the side length of each smaller cube.

Substituting L = 20 cm and l = 4 cm:

N = (20 / 4)³ = 5³ = 125

Explanation

The curved surface area of a right circular cylinder is 2πrh and its total surface area is 2πr(h + r), where h is the height and r is the radius.

Using the given ratio of curved surface area to total surface area: (2πrh) / [2πr(h + r)] = 2 / 5

Canceling the common terms 2πr: h / (h + r) = 2 / 5

Cross-multiplying to solve for the relation between h and r: 5h = 2(h + r) 5h = 2h + 2r 3h = 2r h / r = 2 / 3

The ratio of the height to the radius is h : r = 2 : 3.

Explanation

The empirical relationship between the measures of central tendency for a moderately skewed frequency distribution is:

Mode = 3 Median - 2 Mean

Rearranging this equation to solve for Mean:

2 Mean = 3 Median - Mode Mean = (3 Median - Mode) / 2

Explanation

To find the ratio a:d, multiply the successive ratios:

a/d = (a/b) × (b/c) × (c/d)

Substituting the given values:

a/d = (2/5) × (5/8) × (8/11)

Canceling out common terms in the numerator and denominator gives:

a/d = 2/11

Explanation

Let the two positive numbers be x and y.

From the given quotient, x ÷ y = 4/3, which gives x = (4/3)y.

Substituting x into the product equation x × y = 972: (4/3)y × y = 972 y² = 972 × (3/4) y² = 729 y = √729 = 27

Substituting y = 27 into x = (4/3)y gives: x = (4/3) × 27 = 36

Explanation

When the selling price of x items equals the cost price of y items, the gain percentage is given by the formula:

Gain % = [(y - x) / x] × 100

Here, x = 10 and y = 15.

Substituting the given values: Gain % = [(15 - 10) / 10] × 100 = (5 / 10) × 100 = 50%

Explanation

Let the breadth of the rectangular solar panel be b metres.

Length l = b + 5% of b = 1.05b = (21/20)b

Area = length × breadth 210 = (21/20)b × b 210 = (21/20)b² b² = (210 × 20) / 21 b² = 200 b = √200 = 10√2 m

Explanation

A rational number is any number that can be expressed in the form p/q, where p and q are integers and q ≠ 0. For any non-zero rational number p/q, its negative is -(p/q) = (-p)/q. Since integers are closed under negation, -p is also an integer. Therefore, (-p)/q is a ratio of two integers with a non-zero denominator, which satisfies the definition of a rational number.

Explanation

The third proportional c of two numbers a and b is given by the formula c = b² / a.

Substituting a = 10 and b = 20: c = (20 × 20) / 10 = 400 / 10 = 40.

Explanation

By the Triangle Inequality Theorem, three lengths can form a triangle if and only if the sum of the lengths of the two shorter sides is strictly greater than the length of the longest side (a + b > c).

For the side lengths 4, 5, and 7, the two shorter sides are 4 and 5, and the longest side is 7. Since 4 + 5 = 9 > 7, the triangle inequality condition is satisfied.

Explanation

When two objects move in opposite directions, their relative speed is the sum of their individual speeds: Relative speed = 64 + 8 = 72 km/h

Converting relative speed to m/s: 72 × (5/18) = 20 m/s

Time taken = Length of train / Relative speed = 100 / 20 = 5 sec.

Explanation

Area of a parallelogram = base × perpendicular height

Since both parallelograms have equal bases (b) and lie between the same pair of parallel lines, they share the same perpendicular distance/height (h) between those parallels.

Ratio of areas = (b × h) / (b × h) = 1 / 1 = 1:1

Explanation

The surface area of a sphere of radius r is given by A = 4πr². When the radius is tripled (r' = 3r), the new surface area becomes A' = 4π(3r)² = 36πr² = 9A.

The increase in surface area is A' - A = 9A - A = 8A.

Percentage increase = (Increase / Original Area) × 100 = (8A / A) × 100 = 800%.

Explanation

The largest four-digit number is 9999.

Dividing 9999 by 77 gives a quotient of 129 and a remainder of 66 (9999 = 77 × 129 + 66).

Subtracting the remainder from 9999 yields 9999 - 66 = 9933, which is the largest four-digit number exactly divisible by 77.

Explanation

The total work required to complete the task remains constant, following the inverse variation formula:

M₁ × D₁ = M₂ × D₂

Substituting the given values: 36 × 25 = M₂ × 60 900 = 60 × M₂ M₂ = 900 / 60 = 15 men

Explanation

Given that x% of y is 2000: (x / 100) × y = 2000 ⇒ x × y = 200000

Given that y% of z is 4000: (y / 100) × z = 4000 ⇒ y × z = 400000

Dividing the second equation by the first equation gives: (y × z) / (x × y) = 400000 / 200000 z / x = 2 z = 2x

Explanation

Following the BODMAS rule, evaluate the innermost bracket first: 2.3 × 1.02 = 2.346 3.1 - 2.346 = 0.754 0.5 × 0.754 = 0.377

Next, evaluate the square bracket: 3 + 0.377 = 3.377

Finally, subtract this result from 9.377: 9.377 - 3.377 = 6

Explanation

Let the speed of the steamer in still water be v km/h. The speed of the stream is given as u = 3 km/h.

Downstream speed = v + 3 Upstream speed = v - 3

Since Distance = Speed × Time and the distance covered both downstream and upstream is the same:

6 × (v + 3) = 8 × (v - 3) 6v + 18 = 8v - 24 2v = 42 v = 21 km/h

Explanation

First, evaluate the given radical expression: √64 = 8

Thus, the two numbers are 8 and 16.

For any two positive integers where one number completely divides the other, the smaller number is their Highest Common Factor (HCF) and the larger number is their Least Common Multiple (LCM).

Since 8 divides 16 completely: HCF(8, 16) = 8 LCM(8, 16) = 16

The required ratio of HCF to LCM is: HCF : LCM = 8 : 16 = 1 : 2

Explanation

Using the algebraic identity (x - y)/(x + y) - (x + y)/(x - y) = -4xy / (x² - y²):

Substituting x = √7 and y = 1 gives: [(√7 - 1)/(√7 + 1)] - [(√7 + 1)/(√7 - 1)] = -4(√7)(1) / ((√7)² - 1²)

Simplifying the expression: -4√7 / (7 - 1) = -4√7 / 6 = -2√7 / 3

Equating this result to a + b√7: 0 + (-2/3)√7 = a + b√7

Comparing corresponding rational and irrational terms gives a = 0 and b = -2/3.

Explanation

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

According to the given condition: y + (30/100)x = (140/100)y

Subtracting y from both sides: (30/100)x = (140/100)y - y (30/100)x = (40/100)y 30x = 40y 3x = 4y

Rearranging to find the ratio of the first number to the second number: x / y = 4 / 3 = 4 : 3

Explanation

Using the algebraic identity: (x² + 1/x²)² = x⁴ + 1/x⁴ + 2

Substituting x⁴ + 1/x⁴ = 6239: (x² + 1/x²)² = 6239 + 2 = 6241 x² + 1/x² = √6241 = 79

Next, applying the identity for (x + 1/x): (x + 1/x)² = x² + 1/x² + 2 (x + 1/x)² = 79 + 2 = 81

Since x > 0, taking the positive square root gives: x + 1/x = √81 = 9

Explanation

Let the required number be x.

According to the given condition: (1/9) × (1/8) × (1/7) × (1/5) × (1/4) × (1/3) × (1/2) × x = 32

Multiplying the denominators: 9 × 8 × 7 × 5 × 4 × 3 × 2 = 60,480

So, x / 60,480 = 32

x = 32 × 60,480 = 1,935,360

Explanation

Rate of interest = 12½% = ⅛ per annum Time = 2 years

Ratio of Principal to Amount for 1 year = 8 : 9 Ratio of Principal to Amount for 2 years = 8² : 9² = 64 : 81

Compound Interest = 81 - 64 = 17 units

Given that 17 units = ₹6,800 1 unit = 6,800 ÷ 17 = ₹400

Principal = 64 units = 64 × 400 = ₹25,600

Explanation

Using the algebraic identity: a³ - b³ = (a - b)³ + 3ab(a - b)

Substituting the given values a - b = 5 and a³ - b³ = 5 into the identity: 5 = 5³ + 3ab(5) 5 = 125 + 15ab

Rearranging the equation to solve for ab: 15ab = 5 - 125 15ab = -120 ab = -120 / 15 ab = -8

Explanation

Work done by A and B in 1 day = 1/24 Work done by A and B in 18 days = 18/24 = 3/4

Remaining work = 1 - 3/4 = 1/4

Since C finishes 1/4 of the work in 10 days: Work done by C in 1 day = (1/4) ÷ 10 = 1/40

Combined 1-day work of A, B, and C = 1/24 + 1/40 = (5 + 3)/120 = 8/120 = 1/15

Time taken by A, B, and C working together = 1 ÷ (1/15) = 15 days.

Explanation

Total outcomes n(S) = 30.

By the Principle of Inclusion-Exclusion, the number of favorable outcomes divisible by 4 or 6 is given by: n(A ∪ B) = n(A) + n(B) - n(A ∩ B)

The count of numbers divisible by 4 is n(A) = ⌊30/4⌋ = 7. The count of numbers divisible by 6 is n(B) = ⌊30/6⌋ = 5. The count of numbers divisible by both 4 and 6 (multiples of LCM(4, 6) = 12) is n(A ∩ B) = ⌊30/12⌋ = 2.

Favorable outcomes = 7 + 5 - 2 = 10.

Probability = Favorable outcomes / Total outcomes = 10 / 30 = 1/3.

Explanation

Let the present age of the son be x years. Then, the present age of the father is 2x years.

Twenty years ago, their ages were (x - 20) and (2x - 20) respectively.

According to the given condition: 2x - 20 = 12 × (x - 20) 2x - 20 = 12x - 240 10x = 220 x = 22

The present age of the son is 22 years, and the present age of the father is 2 × 22 = 44 years.

The difference in their present ages is 44 - 22 = 22 years.

Explanation

Selling Price₂ = Selling Price₁ × (100 + Gain₂%) / (100 + Gain₁%)

Substituting the given values: Selling Price₂ = 21,600 × 120 / 108 = 200 × 120 = ₹24,000

Explanation

Evaluating the given mathematical expression using the order of operations (BODMAS):

Step 1: Evaluate the terms inside the innermost parentheses: (-5) × (-4) = 20 (3 - 5) = -2

Step 2: Solve the expression inside the square brackets: [20 - (-2)] = 20 + 2 = 22

Step 3: Subtract this result from the main expression: 2 - 22 = -20

Explanation

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19. Counting these numbers gives a total of 8 prime numbers.

Explanation

The difference between each divisor and its corresponding remainder is constant: 6 - 4 = 2 8 - 6 = 2 12 - 10 = 2

LCM(6, 8, 12) = 24.

The required number must be of the form N = 24k - 2, where k is a positive integer.

Testing integer values for k: For k = 1: N = 24(1) - 2 = 22 (not a multiple of 14) For k = 2: N = 24(2) - 2 = 46 (not a multiple of 14) For k = 3: N = 24(3) - 2 = 70, which is a multiple of 14 (14 × 5 = 70).

Explanation

The change in age over 12 years corresponds to an increase of (13 - 9) = 4 ratio units for Keshav and (14 - 10) = 4 ratio units for Vipul.

4 ratio units = 12 years 1 ratio unit = 3 years

Keshav's present age = 9 × 3 = 27 years.

Explanation

An equivalent fraction is obtained by multiplying or dividing both the numerator and the denominator of a fraction by the same non-zero integer. Since there are infinitely many non-zero integers (such as 1, 2, 3, 4, ...), multiplying the numerator and denominator by each distinct non-zero integer generates a unique equivalent fraction each time. Therefore, infinitely many equivalent fractions can be formed for any given fraction.

Explanation

To find the Highest Common Factor (HCF) of 15, 25, and 42, find the prime factorizations of each number: 15 = 3 × 5 25 = 5² 42 = 2 × 3 × 7

Since no prime factor is common to all three numbers, the only common factor is 1. Thus, the HCF is 1.

Explanation

Let total work be LCM(18, 15) = 90 units. Efficiency of A = 90 / 18 = 5 units/day Efficiency of B = 90 / 15 = 6 units/day

Since B left 7 days before completion, A worked alone for the final 7 days. Work done by A in the last 7 days = 7 × 5 = 35 units. Remaining work done by A and B together = 90 - 35 = 55 units.

Number of days A and B worked together = 55 / (5 + 6) = 55 / 11 = 5 days.

Explanation

For a quadratic polynomial x² + bx + c with roots α and β, the sum of roots is α + β = -b and the product of roots is αβ = c.

Here, α + β = -8 and αβ = 15k.

Using the identity α² + β² = (α + β)² - 2αβ: 34 = (-8)² - 2(15k) 34 = 64 - 30k 30k = 30 k = 1

Explanation

Using the direct mathematical translation of "of" as multiplication:

Value = ⅓ × ¼ × ²⁄₇ × 12

Simplifying the expression: Value = (1 × 1 × 2 × 12) / (3 × 4 × 7) = 24 / 84 = ²⁄₇

Explanation

The total number of outcomes is 20.

Using the Inclusion-Exclusion Principle, the number of balls divisible by 3 or 5 is given by: n(3 or 5) = ⌊20/3⌋ + ⌊20/5⌋ - ⌊20/15⌋ = 6 + 4 - 1 = 9

Probability = Favorable outcomes / Total outcomes = 9/20

Explanation

Summing the given numbers by aligning their decimal places:

17.60000

• 1.76000 = 19.36000
• 0.17600 = 19.53600
• 0.01760 = 19.55360
• 0.00176 = 19.55536

Explanation

Speed of train in m/s = 84 × (5/18) = 70/3 m/s

The train takes 9 seconds to pass a post, which corresponds to covering its own length. To cross the platform, it takes 30 seconds.

Time taken to cover the length of the platform alone = 30 - 9 = 21 seconds

Length of the platform = Speed × Time = (70/3) × 21 = 490 m

Explanation

Since total money remains constant at ₹42, quantity is inversely proportional to price. A gain of 25% means CP : SP = 100 : 125 = 4 : 5.

Therefore, Quantity Bought : Quantity Sold = 5 : 4.

Given Quantity Sold = 16, Quantity Bought = (5 / 4) × 16 = 20 oranges.

Explanation

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

Given: 65% of x = 130 (65/100) × x = 130 x = (130 × 100) / 65 = 200

40% of y = 100 (40/100) × y = 100 y = (100 × 100) / 40 = 250

The ratio of the first number to the second number is x : y = 200 : 250 = 4 : 5.

Explanation

LCM of fractions = (LCM of numerators) ÷ (HCF of denominators)

For the fractions 1/6, 1/4, and 1/2: LCM of numerators (1, 1, 1) = 1 HCF of denominators (6, 4, 2) = 2

LCM = 1/2

Explanation

Using the formula for Simple Interest, SI = (P × R × T) / 100:

12,600 = (70,000 × 12 × T) / 100 12,600 = 8,400 × T T = 12,600 / 8,400 = 3/2 years